Research Article A New Fast Nonsingular Terminal Sliding Mode Control...

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Research Article A New Fast Nonsingular Terminal Sliding Mode Control for a Class of Second-Order Uncertain Systems Linjie Xin, Qinglin Wang, and Yuan Li School of Automation, Beijing Institute of Technology, Zhongguancun Street, Haidian District, Beijing, China Correspondence should be addressed to Yuan Li; [email protected] Received 12 July 2016; Revised 30 October 2016; Accepted 15 November 2016 Academic Editor: Asier Ibeas Copyright © 2016 Linjie Xin et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper considers the robust and adaptive nonsingular terminal sliding mode (NTSM) control for a class of second-order uncertain systems. First, a new fast NTSM was proposed which had global fast convergence rate in the sliding phase. en, a new form of robust NTSM controller was designed to handle a wider class of second-order uncertain systems. Moreover, an exponential-decline switching gain was introduced for chattering suppression. Aſter that, a double sliding surfaces control scheme was constructed to combine the NTSM control with the adaptive technique. e benefit is that a strict demonstration can be given for the stagnation problem in the stability analysis of NTSM. Finally, a case study for tracking control of a variable-length pendulum was performed to verify the proposed controllers. 1. Introduction It is well known that the sliding mode control (SMC) has invariant property to the matched uncertainties. It has become a popular method to control nonlinear uncertain systems [1, 2]. In the earlier studies of SMC, the research was focused on the linear sliding mode (LSM). In [3, 4], the concept of terminal attractor was proposed and used to design the sliding mode which was known as the terminal sliding mode (TSM). e TSM had advantage of finite time convergence, and it has been widely used in many applications, such as robots, spacecraſt, and DC-DC buck converters [5–7]. Compared to the LSM, the TSM has lower convergence speed when the states are far from the origin. Considering this problem, Yu et al. designed a fast terminal sliding mode (FTSM) which had global fast convergence in the sliding phase [8]. e FTSM technique has been developed for controlling nonlinear second-order systems with uncertain terms in [9–11]. Moreover, the concept of terminal attractor was also applied to the reaching law by Yu and Man [12]. As the negative fractional power exists in the TSM control signals, it may result in singularity of the control input. is is hard to accept in real implementations. So, the control input was switched to a general sliding mode control in order to avoid the singularity when the states converged into a small vicinity of origin in [13]. However, this switching method is a suboptimal solution which lost the advantage of TSM. In [14, 15], a new form of TSM known as nonsingular TSM (NTSM) was proposed and applied to robotic manipulators and piezoelectric actuators. It also has a lower convergence speed when the states are far from the origin. Although the NTSM control has solved the singularity problem, it needs an extra demonstration to show that the motion of sliding surface will not stagnate at some nonzero points in the reaching phase. According to the existing literatures, most of the TSM control methods were designed for the second-order systems. In [16–18], the recursive TSM control was developed for higher order nonlinear systems. For the robust TSM controllers, the boundary informa- tion of system uncertainties is usually required to be known in advance. However, it is difficult to obtain in many practical implementation processes. erefore, the adaptive technique was employed to cooperate with the TSM and FTSM control in [19–21]. Moreover, the adaptive NTSM control method was developed for uncertain systems with input nonlinearity in [22, 23]. Note that a strict proof for the stagnation problem of NTSM can be given in the robust NTSM control, for example, Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2016, Article ID 1743861, 12 pages http://dx.doi.org/10.1155/2016/1743861

Transcript of Research Article A New Fast Nonsingular Terminal Sliding Mode Control...

Research ArticleA New Fast Nonsingular Terminal Sliding Mode Control fora Class of Second-Order Uncertain Systems

Linjie Xin Qinglin Wang and Yuan Li

School of Automation Beijing Institute of Technology Zhongguancun Street Haidian District Beijing China

Correspondence should be addressed to Yuan Li liyuanbiteducn

Received 12 July 2016 Revised 30 October 2016 Accepted 15 November 2016

Academic Editor Asier Ibeas

Copyright copy 2016 Linjie Xin et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper considers the robust and adaptive nonsingular terminal sliding mode (NTSM) control for a class of second-orderuncertain systems First a new fast NTSM was proposed which had global fast convergence rate in the sliding phase Then anew form of robust NTSM controller was designed to handle a wider class of second-order uncertain systems Moreover anexponential-decline switching gain was introduced for chattering suppression After that a double sliding surfaces control schemewas constructed to combine the NTSM control with the adaptive technique The benefit is that a strict demonstration can be givenfor the stagnation problem in the stability analysis of NTSM Finally a case study for tracking control of a variable-length pendulumwas performed to verify the proposed controllers

1 Introduction

It is well known that the sliding mode control (SMC)has invariant property to the matched uncertainties It hasbecome a popular method to control nonlinear uncertainsystems [1 2] In the earlier studies of SMC the researchwas focused on the linear sliding mode (LSM) In [3 4]the concept of terminal attractor was proposed and used todesign the sliding mode which was known as the terminalsliding mode (TSM) The TSM had advantage of finitetime convergence and it has been widely used in manyapplications such as robots spacecraft and DC-DC buckconverters [5ndash7]

Compared to the LSM the TSM has lower convergencespeed when the states are far from the origin Consideringthis problem Yu et al designed a fast terminal sliding mode(FTSM) which had global fast convergence in the slidingphase [8] The FTSM technique has been developed forcontrolling nonlinear second-order systems with uncertainterms in [9ndash11] Moreover the concept of terminal attractorwas also applied to the reaching law by Yu and Man [12]As the negative fractional power exists in the TSM controlsignals it may result in singularity of the control inputThis ishard to accept in real implementations So the control input

was switched to a general sliding mode control in order toavoid the singularity when the states converged into a smallvicinity of origin in [13] However this switching methodis a suboptimal solution which lost the advantage of TSMIn [14 15] a new form of TSM known as nonsingular TSM(NTSM) was proposed and applied to robotic manipulatorsand piezoelectric actuators It also has a lower convergencespeed when the states are far from the origin Althoughthe NTSM control has solved the singularity problem itneeds an extra demonstration to show that the motion ofsliding surface will not stagnate at some nonzero points in thereaching phase According to the existing literatures most ofthe TSMcontrolmethodswere designed for the second-ordersystems In [16ndash18] the recursive TSM control was developedfor higher order nonlinear systems

For the robust TSM controllers the boundary informa-tion of system uncertainties is usually required to be knownin advance However it is difficult to obtain inmany practicalimplementation processes Therefore the adaptive techniquewas employed to cooperate with the TSM and FTSM controlin [19ndash21]Moreover the adaptiveNTSMcontrolmethodwasdeveloped for uncertain systems with input nonlinearity in[22 23] Note that a strict proof for the stagnation problem ofNTSM can be given in the robust NTSM control for example

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 1743861 12 pageshttpdxdoiorg10115520161743861

2 Mathematical Problems in Engineering

[14 15] However it was difficult to obtain for the case ofadaptive NTSM control in [22 23]

Previous studies have developed the TSM control andthus have great significance On the other hand there are stillsome points valuable for further research Most of the NTSMcontrollers were designed for a class of uncertain systems inwhich the coefficient of control input was a known function(or with an uncertain item) In the case that this coefficient istotally unknown except its boundary few NTSM controllerswere reported So the purpose of this paper is to develop theNTSM control for the latter case

The main contribution of this paper can be summarizedas follows

(1) A fast NTSM (FNTSM) is designed to combine theadvantages of LSM and NTSM together Thus theconvergence rate of the NTSM is enhanced when theinitial states are far away from the origin

(2) A new robust NTSM controller is proposed for thecase that the coefficient of control input is unknownIt is also applicable to the case that the coefficient isknown Moreover an exponential-decline switchinggain is designed to attenuate the chattering phe-nomenon

(3) An adaptive NTSM control scheme with doublesliding surfaces is proposed to give a strict prooffor the stagnation problem of NTSM The systemuncertainty is firstly compensated on an integralsliding surface after that the system trajectory isforced to the designed fast NTSM

2 Robust NTSM Controller Design

Consider a second-order nonlinear system as

1 = 11990922 = 120593 (119909) + 120574 (119909) 119906 (1)

where 120593(119909) and 120574(119909) denote bounded uncertain functions120574(119909) = 0Assumption 1 The uncertain functions 120593(119909) and 120574(119909) satisfy

1003816100381610038161003816120593 (119909)1003816100381610038161003816 le 120593 = 1205930 + 1205931 100381610038161003816100381611990911003816100381610038161003816 + 1205932 100381610038161003816100381611990921003816100381610038161003816 0 lt 119870119898 le 120574 (119909) le 119870119872

(2)

where 1205930 1205931 1205932 119870119898 and 119870119872 are positive constants 119870119898and 119870119872 are assumed to be known Robust and adaptivecontrollers are designed for the two cases that 120593 is known ornot respectively In the rest of the paper 120593(119909) and 120574(119909) aresimply denoted as 120593 and 12057421 FNTSMDesign In the study of [14] a kind of NTSMwasdesigned as

119904 = 1199091 + 120573 100381610038161003816100381611990921003816100381610038161003816120594 sign (1199092) 1199092 = 1 0 lt 120573 1 lt 120594 lt 2 (3)

NTSM

LSM

0 2minus2

x1

minus2

0

2

x2

Figure 1 The proposed sliding surface on the phase plane

It is equivalent to the following form which can beconverted to a conventional TSM

119904 =

1199091 + 1205731199092120594 1199092 gt 01199091 1199092 = 01199091 minus 120573 (minus1199092)120594 1199092 lt 0

(4)

As illustrated in [8] the TSM has slower convergence ratethan the LSM when the initial states are far away from theorigin It can be explained by its eigenvalue For TSM 1 +120573minus112059411990911205941 = 0 it is easy to obtain the eigenvalue as

12059711205971199091 = minus120573minus1120594 11205941199091120594minus11 (5)

As 1120594 minus 1 lt 0 the eigenvalue tends to be negativeinfinity at the origin This fact implies that the convergencerate is infinitely large This is the major merit of TSMHowever when the initial state is far away from the originthe convergence rate is smaller than the case 120594 = 1 that is aLSM

In order to enhance the convergence rate of NTSM (3) anew FNTSM is designed as

119904 = 1199091 + 120573 100381610038161003816100381611990921003816100381610038161003816119902 sign (1199092) 119902 = 120575 + (1 minus 120575) sign (100381610038161003816100381611990911003816100381610038161003816 minus 120576) 1 le 120576 1 lt 120575 lt 15 (6)

The FNTSM (6) is a switching sliding surface whichtransfers the system dynamic from a LSM (119902 = 1) to a NTSM(1 lt 119902 lt 2) (Figure 1 eg 120576 = 1) Thus it has global fastconvergence rate and reserves the advantage of NTSM

Here a simple example is given to compare the NTSM (3)and the FNTSM (6)

1 = 11990922 = 119906 (7)

Mathematical Problems in Engineering 3

FNTSMNTSM

1 2 3 4 5 6 7 80Time (s)

minus1

1

3

5

7

9

11

13

15O

utpu

t

4 5 6 730

1

2

(a) System response

FNTSMNTSM

0

10

20

30

40

50

60

Con

trol i

nput

1 2 3 4 5 6 7 80Time (s)

0

1

2

3 3525

Y 5711X 072

Y 5744X 0

(b) Control input

Figure 2 Comparison between the proposed FNTSM and the NTSM

Considering system (7) the control input was taken as theNTSM controller form of [14]

119906 = minus11199021003816100381610038161003816119909210038161003816100381610038162minus119902 sign (1199092) minus 2119904 minus |119904|79 sign (119904) (8)

With the same reaching law the parameters for the slidingsurfaces were chosen as

NTSM 119902 = 119902119873 = 117

FNTSM 119902 = 119902119865 = 97 + (1 minus 9

7) sign (100381610038161003816100381611990911003816100381610038161003816 minus 1) (9)

Note that the FNTSM became equivalent to the NTSMwithin the region |1199091| le 1 because of 119902119865 = 117 insidethis region As shown in Figure 2 the convergence rate ofthe proposed FNTSM controller was faster than the NTSMcontrollerrsquos while the control input of FNTSM controllerwas much smaller In particular for the FNTSM (6) theswitching of sliding surface is not continuousWhen the state1199091 converged into the prescribed region the sliding surfaceand control signal had slight discontinuous changes As aresult the control signal switched to a NTSM control froma LSM control As shown in Figure 2(b) there was a smallsaltation of the control signal (from 0967 to 0582) at 119905 =286 s because of the switching of sliding surface22 Controller Design Most of the NTSM controllers weredesigned for system (1) with a known 120574 (or with an uncertainitem) However these controllers are not theoretically appli-cable to totally unknown 120574 In this subsection a new form ofNTSMcontroller is designed for system (1) in the case that120593 isknown A proof is given to show that the stagnation problemof sliding surface will not occur Moreover the new NTSMcontroller is also applicable to the case that 120574 is known

Generally the NTSM control design contains the slidingsurface and the reaching law A discontinuous reaching

law with switching function is widely adopted to drive thesystem trajectory onto the sliding surface which leads to thechattering In the following theorem an exponential-declineswitching gain (EDSG) is designed to attenuate the chatteringphenomenon

The EDSG is formulated as

119860 (119905) 1205721 = (1198871 + 1198872119890minus119886119905) 1205721 (10)

where 1205721 gt 120593 119886 gt 0 1198871 ge 1 and 1198872 gt 0The EDSG acceleratesthe convergence rate of sliding surface at the beginning ofreaching phase and attenuates the amplitude of chatteringwithout deteriorating the system robustness

Theorem 2 For system (1) the following controller forces thesystem dynamic to the sliding surface (6) in finite time

119906 = minus 1119870119898 (1205722119904 + 119860 (119905) 1205721 sign (119904)

+ 1120573119902

1003816100381610038161003816119909210038161003816100381610038162minus119902 sign (119904)) (11)

where 1205721 = 120593 + 120578 1205722 ge 0 and 120578 is a small positive constant

Proof Consider the following Lyapunov function for thesliding surface

1198811 = 121199042 (12)

Differentiating (12) with respect to time yields

1 = 119904 119904 = 119904 (1199092 + 120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 2)= 119904 (1199092 + 120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 (120593 + 120574119906))

(13)

4 Mathematical Problems in Engineering

Substituting the controller (11) into (13) yields

1 = 119904 (1199092 + 120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 (120593 minus 120574119870119898 (1205722119904

+ 119860 (119905) 1205721 sign (119904) + 1120573119902

1003816100381610038161003816119909210038161003816100381610038162minus119902 sign (119904)))) = 1199041199092

+ 120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 (120593119904 minus 1205741198701198981205722119904

2 minus 120574119870119898119860 (119905) 1205721 |119904| minus 120574

119870119898sdot 1120573119902

1003816100381610038161003816119909210038161003816100381610038162minus119902 |119904|) = minus120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 1205741198701198981205722119904

2 + 1199041199092

minus 120574119870119898 |119904| 100381610038161003816100381611990921003816100381610038161003816 + 120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 (120593119904 minus 120574

119870119898119860 (119905) 1205721 |119904|)

(14)

With 120574 ge 119870119898 and |1199092|119902minus1 gt 0 (14) turns to1 le minus120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 12057221199042 + 1199041199092 minus |119904| 100381610038161003816100381611990921003816100381610038161003816

+ 120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 (10038161003816100381610038161205931003816100381610038161003816 |119904| minus 119860 (119905) 1205721 |119904|) (15)

As 119860(119905)1205721 ge 120593 + 120578 it is obtained as

1 le minus120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 12057221199042 minus 120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 120578 |119904|= minus120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 (12057221199042 + 120578 |119904|)= minus120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 (212057221198811 + 120578212119881121 )

(16)

Considering the last expression of (16) if the motion ofsliding surface does not stagnate at the points 1199091 = 0 1199092 =0 in the reaching phase it would converge to zero in finitetime according to the extended Lyapunov description of finitetime stability in Remark 2 of [14]

Here a demonstration is given to show that themotion ofsliding surface does not stagnate at the points 1199091 = 0 1199092 =0

For system (1) with the controller (11) taking 1199092 = 0 weobtain

2 = 120593 minus 120574119870119898 (1205722119904 + 119860 (119905) 1205721 sign (119904)

+ 1120573119902

1003816100381610038161003816119909210038161003816100381610038162minus119902 sign (119904)) = minus1205722 120574119870119898 119904 + 120593 minus 119860 (119905) 1205721

sdot 120574119870119898 sign (119904)

(17)

where 119904 = 1199091In the case of 1199091 gt 0 the following inequality is satisfied

for 22 = minus1205722 120574

119870119898 119904 + 120593 minus 119860 (119905) 1205721 120574119870119898

= minus1205722 120574119870119898 119904 + 120593 minus 120574

119870119898 (1198871 + 1198872119890minus119886119905) (120593 + 120578)le minus1205722119904⏟⏟⏟⏟⏟⏟⏟lt0

+ 120593 minus 120593 minus 120578⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟lt0

(18)

In the other case of 1199091 lt 0 it follows that2 = minus1205722 120574

119870119898 119904 + 120593 + 119860 (119905) 1205721 120574119870119898

= minus1205722 120574119870119898 119904 + 120593 + 120574

119870119898 (1198871 + 1198872119890minus119886119905) (120593 + 120578)ge minus1205722119904⏟⏟⏟⏟⏟⏟⏟gt0

+ 120593 + 120593 + 120578⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟gt0

(19)

It is concluded that the points 1199091 = 0 1199092 = 0 arenot attractors Thus the motion of sliding surface will notstagnate at these points and it converges to zero in finite time

This completes the proof

Remark 3 Note that the term |1199092|2minus119902sign(119904) of controller(11) is constructed for the stability analysis It is usuallytaken as |1199092|2minus119902sign(1199092) in the existing NTSM controllers forexample the controller (8) However it is not applicable tothe stability analysis of the discussed system (1)

Remark 4 In the case that 120574 is known the proposed con-troller (11) is also applicable by taking 119870119898 = 1205743 Adaptive NTSM Control Scheme Design

In practical situations it is usually difficult to obtain the priorknowledge of 120593 Therefore an adaptive controller is designedfor system (1) in this section In the studies of adaptive NTSMcontrol [22 23] the switching gain was directly adjusted bythe adaptive technique However it was difficult to provethat the points 1199091 = 0 1199092 = 0 were not attractors asin Theorem 2 So the adaptive switching gain was modifiedby the constant control gain in the extreme case that thestagnation problem may happen

In this section a double sliding surfaces control schemeis designed to combine the NTSM control with the adaptivetechnique The first layer is a common integral slidingmode (ISM) in which the impact of uncertain function 120593is compensated The second layer is a NTSM in which thesystem trajectory converges to the origin The purpose ofusing double sliding surfaces is to provide a strict proof forthe stagnation problem of NTSM

In this section the control law is designed as

119906 = 119906119886 + 119906119899 (20)

where 119906119886 denotes an adaptive controller which forces thesystemmotion to the first sliding surface After that a NTSMcontroller 119906119899 drives the system motion to the second slidingsurface

The first sliding surface ISM is designed as

1199041 = 1199092 + 119911 minus 119890minus120579119905 (1199092 (0) + 119911 (0)) = minus119906119899

(21)

where 120579 gt 0 and 1199092(0) 119911(0) are the initial values of 1199092 119911The exponential-decline term is introduced to eliminate thereaching phase [24] Then the system dynamic is on the firstsliding surface at the initial time

Mathematical Problems in Engineering 5

Differentiating (21) with respect to time yields

1199041 = 2 + + Λ = 120593 + 120574119906 minus 119906119899 + Λ= 120593 + Λ + (120574 minus 1) 119906119899 + 120574119906119886

(22)

where Λ = 120579119890minus120579119905(1199092(0) + 119911(0)) Once the first sliding surfaceis established the original system (1) will approach to thesystem motion of an integral chain system as 2 = minus119906119899

Here the NTSM controller 119906119899 for system 2 = minus119906119899 isdesigned as

119906119899 = minus 1120573119902

1003816100381610038161003816119909210038161003816100381610038162minus119902 sign (1199092) minus 12057221199042minus 1205723 1003816100381610038161003816119904210038161003816100381610038161199011 sign (1199042)

(23)

where 1205723 is a positive constant 1199011 is a constant satisfying 0 lt1199011 lt 1 and 1199042 denotes the FNTSM (6)

The continuous controller (23) had been verified in [14]and it also had been proved that the points 1199091 = 0 1199092 =0 were not attractors For an integral chain system withoutuncertainty (eg 2 = minus119906119899) the discontinuous control is notnecessary Although the proposed NTSM controller (11) inTheorem 2 can be used to design 119906119899 it is not employed as it isdiscontinuous

With the following assumption an adaptive controlleris designed in Theorem 6 for system (22) by combining theTSM reaching law with the familiar first-order SMC adaptivecontrol (eg [25ndash27])

Assumption 5 Assume 120593+(120574minus1)119906119899 is unknown and satisfiesthe following inequality

1003816100381610038161003816120593 + (120574 minus 1) 1199061198991003816100381610038161003816 le 1198880 + 1198881 100381610038161003816100381611990911003816100381610038161003816 + 1198882 100381610038161003816100381611990921003816100381610038161003816 (24)

where 1198880 1198881 and 1198882 are unknown positive constants

Theorem6 For system (22) the sliding surface 1199041 converges tozero asymptotically with the following adaptive controller

119906119886 = minus 1119870119898 (11989631199041 + 1198964 1003816100381610038161003816119904110038161003816100381610038161199012 sign (1199041) + |Λ| sign (1199041)

+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sign (1199041)) 1198880 = 1198960 100381610038161003816100381611990411003816100381610038161003816 1198881 = 1198961 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 1198882 = 1198962 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816

(25)

where 1198960 sim 1198964 ge 0 1 ge 1199012 gt 0 and 119894(0) gt 0 (119894 = 0 1 2)Proof Define a new Lyapunov function for 1199041 as

1198812 = 1211990421 + 1

21198960 20 + 1

21198961 21 + 1

21198962 22 (26)

where 119894 = 119888119894 minus 119894 Its derivative is2 = 1199041 1199041 minus 1

1198960 01198880 minus 1

1198961 11198881 minus 1

1198962 21198882 (27)

Substituting controller (25) into (27) yields

2 = 1199041 (120593 + Λ + (120574 minus 1) 119906119899 + 120574119906119886) minus ( 11198960 0

1198880

+ 11198961 1

1198881 + 11198962 2

1198882) = 1199041 (120593 + Λ + (120574 minus 1) 119906119899)+ 1205741199041119906119886 minus 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) le 100381610038161003816100381611990411003816100381610038161003816 1003816100381610038161003816120593+ (120574 minus 1) 1199061198991003816100381610038161003816 + 100381610038161003816100381611990411003816100381610038161003816 |Λ| + 1205741199041119906119886 minus 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816+ 2 100381610038161003816100381611990921003816100381610038161003816) le 100381610038161003816100381611990411003816100381610038161003816 (1198880 + 1198881 100381610038161003816100381611990911003816100381610038161003816 + 1198882 100381610038161003816100381611990921003816100381610038161003816) + 100381610038161003816100381611990411003816100381610038161003816 |Λ|minus 120574

119870119898 (119896311990421 + 1198964 1003816100381610038161003816119904110038161003816100381610038161199012+1 + 100381610038161003816100381611990411003816100381610038161003816 |Λ|

+ 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816)) minus 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816+ 2 100381610038161003816100381611990921003816100381610038161003816)

(28)

With the adaptive laws in (25) and 119894(0) gt 0 it ensuresthat 119894 gt 0 Since 120574 ge 119870119898 it follows that

2 le minus 120574119870119898 (119896311990421 + 1198964 1003816100381610038161003816119904110038161003816100381610038161199012+1)

+ 100381610038161003816100381611990411003816100381610038161003816 (1198880 + 1198881 100381610038161003816100381611990911003816100381610038161003816 + 1198882 100381610038161003816100381611990921003816100381610038161003816)minus 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816)minus 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816)

= minus 120574119870119898 (119896311990421 + 1198964 1003816100381610038161003816119904110038161003816100381610038161199012+1)

(29)

Applying the Barbalat lemma [28] the sliding surface 1199041converges to zero asymptotically

This completes the proof

The boundary layer approach is a common method forchattering reduction It is well known that the chatteringreduction is achieved with a small cost of control precisionMore specifically the sliding surface is driven to a vicinity ofzero In [29] a concept of real sliding surface was introducedwhich was similar to the boundary layer It also means thatthe sliding surface is forced to a small region rather than zeroBased on this concept a modified adaptive law was proposedto avoid the overestimation of adaptive gain in [29] Theadaptive gain slowly decreased to the necessary value oncethe real sliding surface was established

In the case that the boundary layer approach is adoptedfor the proposed adaptive controller (25) the chatteringamplitude can be further attenuated if the adaptive gains0 1 2 decrease to a low level Thus the above two meth-ods are combined to design a modified adaptive controller as

119906119886 = minus 1119870119898 (11989631199041 + 1198964 1003816100381610038161003816119904110038161003816100381610038161199012 sign (1199041) + |Λ| sat (1199041)

+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sat (1199041)) (30)

6 Mathematical Problems in Engineering

o

u

x

m

R(t)

Figure 3 Variable-length pendulum model

where the sat(sdot) function and the adaptive laws are taken as

sat (1199041) = sign (1199041) 100381610038161003816100381611990411003816100381610038161003816 gt 1198741199041119874

100381610038161003816100381611990411003816100381610038161003816 le 119874

1198880 = 1198960 100381610038161003816100381611990411003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 119874) 0 gt 12057211988801198960 0 le 1205721198880

1198881 = 1198961 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 119874) 1 gt 12057211988811198961 1 le 1205721198881

1198882 = 1198962 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 119874) 2 gt 12057211988821198962 2 le 1205721198882

(31)

where a positive constant 119874 denotes the boundary lay 1198960 sim1198962 and 1205721198880 sim 1205721198882 are positive constantsAccording toTheorem 6 the sliding surface 1199041 converges

to zero asymptotically Thus it converges into |1199041| le 119874 infinite time Then the adaptive gains begin to decrease and2 becomes sign indefinite As soon as the sliding surfaceescapes from the region |1199041| le 119874 the adaptive gains increaseagain so that the sliding surface is driven back into the region|1199041| le 119874 Finally the sliding surface 1199041 remains in a largerregion than |1199041| le 119874which is the so-called real sliding surfaceThe second expressions of the modified adaptive laws ensurethe positive values of the adaptive gains and they are valid ina short time

Remark 7 For the tracking control issues it is necessary tomodify the adaptive laws of 1 and 2 to reduce the chatteringamplitude However it is not necessary when the states1199091 1199092 converge to zero4 Simulation

In this section the designed robust and adaptive NTSMcontrollers were tested for the tracking control of a variable-length pendulum [25] (Figure 3) formulated as

= minus2 (119905)119877 (119905) minus 119892

119877 (119905) sin (119909) + 11198981198772 (119905)119906 (32)

The angular coordinate 119909 is impelled to track a referencesignal 119909119888 by the torque 119906 A known mass 119898 moves alonethe rod without friction and the distance 119877(119905) between thecenter 119900 and the mass 119898 is unmeasured It is assumed thatthe states [119909 ]119879 can be measured for the controller designThe parameters of (32) and the reference signal are taken as

119877 (119905) = 1 + 025 sin (4119905) + 05 cos (119905)119898 = 1 kg119892 = 981ms2

119909 (0) = 12119909119888 = 05 sin (05119905)

(33)

Defining the tracking error as 1198901 = 119909 minus 119909119888 the trackingcontrol is converted to the stabilization of the following errorsystem

1198901 = 11989021198902 = minus2 (119905)

119877 (119905) minus 119892119877 (119905) sin (119909) minus 119888 + 1

1198981198772 (119905)119906(34)

41 Robust Control In this subsection three robust control-lers (A B and C) were evaluated According to Theorem 2controllers A and B (without EDSG) were designed using theFNTSM of [11] and the proposed FNTSM (6) respectivelyThe controller C was the proposed FNTSM-EDSG controllerin Theorem 2

Controller A (based on FNTSM of [11])

119906 = minus16 (021199041198651 + 1205721 sign (1199041198651)

+ 790119890572 sign (1199041198651) sign (1198902)

+ 49450119890251 119890572 sign (1199041198651) sign (119890251 1198902))

1199041198651 = 1198901 + 119890751 + 119890972 1205721 = 04 + 04 100381610038161003816100381611990911003816100381610038161003816 + 04 100381610038161003816100381611990921003816100381610038161003816

(35)

Controller B (based on FNTSM (6))

119906 = minus16(021199041198652 + 1205721 sign (1199041198652) + 11199021003816100381610038161003816119890210038161003816100381610038162minus119902 sign (1199041198652))

1199041198652 = 1198901 + 100381610038161003816100381611989021003816100381610038161003816119902 sign (1198902) 119902 = 97 + (1 minus 9

7) sign (100381610038161003816100381611990911003816100381610038161003816 minus 1) (36)

Controller C (based on FNTSM (6) + EDSG)

119906 = minus16(021199041198652 + 119860 (119905) 1205721 sign (1199041198652)

+ 11199021003816100381610038161003816119890210038161003816100381610038162minus119902 sign (1199041198652)) 119860 (119905) = 1 + 119890minus2119905 1205721 = 02 + 02 100381610038161003816100381611990911003816100381610038161003816 + 02 100381610038161003816100381611990921003816100381610038161003816

(37)

The simulation results are given in Figure 4 As shownin Figures 4(a) and 4(b) the angular coordinate converged

Mathematical Problems in Engineering 7

Controller B Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad) Controller A Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad)

Controller C Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad)

(a) Angular coordinate

2 3 4

0

002

Controller A

1 2 3 4 5 6 7 8 9 100Time (s)

0

06

12

Trac

king

erro

r (ra

d)

Controller B

2 3 4

0

002

0

06

12

Trac

king

erro

r (ra

d)

1 2 3 4 5 6 7 8 9 100Time (s)

2 3 4

0

002

Controller C

0

06

12Tr

acki

ng er

ror (

rad)

1 2 3 4 5 6 7 8 9 100Time (s)

(b) Tracking error

1 2 3 4 5 6 7 8 9 100Time (s)

minus10123

Slid

ing

surfa

ces

Controller A

Controller B

minus10123

Slid

ing

surfa

ces

1 2 3 4 5 6 7 8 9 100Time (s)

Controller C

minus10123

Slid

ing

surfa

ces

1 2 3 4 5 6 7 8 9 100Time (s)

(c) Sliding surfaces

1 2 3 4 5 6 7 8 9 100Time (s)

Controller A

minus30

minus10

10

30

Con

trol t

orqu

es (N

)

Controller B

minus30

minus10

10

30

Con

trol t

orqu

es (N

)

1 2 3 4 5 6 7 8 9 100Time (s)

Controller C

minus30

minus10

10

30

Con

trol t

orqu

es (N

)

1 2 3 4 5 6 7 8 9 100Time (s)

(d) Control torque

Figure 4 Simulation result of the robust control

8 Mathematical Problems in Engineering

to the reference signal within 119905 = 4 s The sliding surfacesconverged to zero within 119905 = 15 s (Figure 4(c)) It is clear thatcontroller B cost less convergence time than the controllerA while the control inputs of these controllers had slightdifference (Figures 4(b) and 4(d)) This fact implies thatthe controller designed by the proposed FNTSM was moreefficient As expected the chattering amplitude of controllerC (plusmn5) was half of controller B (plusmn10) (Figure 4(d)) Howeverthe performance of controllerChad small differencewith thatof controller BThe result confirms that the EDSG is effectivefor chattering suppression without deteriorating the systemrobustness

42 Adaptive Control In this subsection the proposed adap-tive NTSM control scheme ((23) and (25)) was verified forthe tracking control As a comparison the adaptive methodin [25] was adopted to design the adaptive controller 119906119886 Thecontrollers were designed as follows

Controller D (Theorem 6)

119906119899 = minus11199021003816100381610038161003816119890210038161003816100381610038162minus119902 sign (1198902) minus 251199042 minus 10038161003816100381610038161199042100381610038161003816100381679 sign (1199042)

119906119886 = minus16 (051199041 + 10038161003816100381610038161199041100381610038161003816100381679 sign (1199041)+ 100381610038161003816100381610038165119890minus5119905 (1198902 (0) + 119911 (0))10038161003816100381610038161003816 sign (1199041)+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sign (1199041))

1199041 = 1198902 + 119911 minus 119890minus5119905 (1198902 (0) + 119911 (0)) = minus119906119899119911 (0) = 01198880 = 100381610038161003816100381611990411003816100381610038161003816 1198881 = 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 1198882 = 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 0 (0) = 1 (0) = 2 (0) = 001

(38)

Controller E (method in [25])

119906119886 = minus5119890minus5119905 (1198902 (0) + 119911 (0)) minus 0 sign (1199041) 1198880 = 5 100381610038161003816100381611990411003816100381610038161003816

(39)

ControllersD and E shared the sameNTSMcontroller 119906119899and the sliding surface 1199042 was taken as 1199041198652 of controller B

The simulation results are given in Figures 5 and 6 It isclear that both controllers D and E achieved fast tracking ofthe reference signal Controller D had shorter convergencetime and better transient performance than controller Eeven though the maximum control torque and the chatteringamplitude of controller D were smaller Meanwhile thesliding surfaces in Figure 6 converged to zero within 119905 =3 s In Figure 6(a) the motions of ISM started from zeroand there was a short time that ISM did not remain zerodue to the affection of uncertainty It is observed that the

FNTSM converged to zero after the ISM and the stagnationof FNTSM never happened It verifies that the designedadaptive NTSM control scheme is effective Since the twocontrollers shared the same NTSM controller 119906119899 the simula-tion results confirm that the proposed adaptive method hadbetter transient performance and was more efficient than themethod in [25]

Here themodified adaptive controller ((30) and (31)) waschecked for chattering reduction In the following controllerF the boundary layer approach and the modified adaptivelaws were both employed As a comparison controller Gonly employed the boundary layer approach In order toavoid the unbounded growing of adaptive gains inside theboundary layer the switching adaptive laws were adoptedfor controller G In addition controllers F and G used thesame boundary layer and NTSM controller 119906119899 of controllerD Since it is assumed that the states can be measured forthe controller design themeasurement noisemay deterioratethe performance of controller in real implementations Toillustrate the robustness of the controller random noise wastaken for the states [119909 ]119879 (mean value 0 and standarddeviation 001)

Controller F

119906119886 = minus16 (051199041 + 10038161003816100381610038161199041100381610038161003816100381679 sign (1199041)+ 100381610038161003816100381610038165119890minus5119905 (1198902 (0) + 119911 (0))10038161003816100381610038161003816 sat (1199041)+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sat (1199041))

sat (1199041) = sign (1199041) 100381610038161003816100381611990411003816100381610038161003816 gt 0041199041

004100381610038161003816100381611990411003816100381610038161003816 le 004

1198880 = 2 100381610038161003816100381611990411003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 004) 0 gt 0011 0 le 001

1198881 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 005) 1 gt 0011 1 le 001

1198882 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 004) 2 gt 0011 2 le 001

(40)

Controller G

119906119886 = minus16 (051199041 + 10038161003816100381610038161199041100381610038161003816100381679 sign (1199041)+ 100381610038161003816100381610038165119890minus5119905 (1198902 (0) + 119911 (0))10038161003816100381610038161003816 sat (1199041)+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sat (1199041))

1198880 = 2 100381610038161003816100381611990411003816100381610038161003816 1199041 gt 0040 1199041 le 004

Mathematical Problems in Engineering 9

Ang

ular

coor

dina

te (r

ad)

Controller DController E

Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus08

minus06

minus04

minus02

0

02

04

06

08

1

(a) Angular coordinate

Trac

king

erro

r (ra

d)

Controller DController E

minus02

0

02

04

06

08

1

1 2 3 4 5 6 7 8 9 100Time (s)

(b) Tracking error

Adaptive gain 1

Adaptive gain 1Adaptive gain 2

Adaptive gain 3

Controller D

Controller E

1 2 3 4 5 6 7 8 9 100Time (s)

0

01

02

Adap

tive g

ains

0

5

10

15

Adap

tive g

ains

1 2 3 4 5 6 7 8 9 100Time (s)

(c) Adaptive gains

0

Controller E

Controller D

minus20

20

Con

trol t

orqu

es (N

)

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus20

0

20C

ontro

l tor

ques

(N)

(d) Control torque

Figure 5 Simulation result of the adaptive control

1198881 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 1199041 gt 0040 1199041 le 004

1198882 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 1199041 gt 0040 1199041 le 004

(41)

The simulation result of controller F is given in Figure 7It is shown that the angular coordinate tracked the referencesignal with a high precision (Figure 7(a)) Due to the adop-tion of the boundary layer and themodified adaptive laws thesliding surface 1199041 converged to a small domain as |1199041| le 01

(Figure 7(b))Moreover the adaptive gains decreased to a lowlevel with approximate reduction ratios as 5108 8479and 1615 respectively (Figure 7(c)) As a result there was amarked reduction in the chattering of controller F comparedto that of controller G in Figure 7(d) Thus the effectivenessof the modified adaptive controller is verified

5 Conclusions

This paper studies the NTSM control for a class of second-order systems in which the coefficient of control input isunknown In this paper a new FNTSM was designed tointegrate the advantages of LSM and NTSM Based onthis FNTSM new forms of robust controller and adaptive

10 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 10minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

Time (s)

ISM

Controller DController E

(a) ISM

0 1 2 3 4 5 6 7 8 9 10minus2

minus15

minus1

minus05

0

05

1

15

2

25

3

Time (s)

FNTS

M

Controller DController E

(b) FNTSM

Figure 6 The response of sliding surfaces

Controller FReference signal

10 20 30 40 500Time (s)

minus1

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad)

(a) Angular coordinate

minus02

0

02

04

06

08

0 10 20 30 40 50Time (s)

Slid

ing

surfa

ces

ISMFNTSM

(b) Sliding surfaces

0

005

01

015

02

025

Adap

tive g

ains

Adaptive gain 1Adaptive gain 2

Adaptive gain 3

10 20 30 40 500Time (s)

(c) Adaptive gains

0 10 20 30 40 50Time (s)

Controller G

Controller F

minus10

0

10

Con

trol t

orqu

es (N

)

minus10

0

10

Con

trol t

orqu

es (N

)

10 20 30 40 500Time (s)

(d) Control torques

Figure 7 Simulation result of the modified adaptive control

Mathematical Problems in Engineering 11

controller were proposed for the discussed uncertain systemThe simulation verified the effectiveness of the proposed con-trollersThe proposed FNTSM shows faster convergence ratethan the NTSM and chattering reduction was achieved bythe EDSG The adaptive controller presented better transientperformance and more efficiency than the adaptive methodin [25] Moreover the modified adaptive controller achievedeffective chattering reduction

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this article

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (no 61375100 no 61472037 and no61433003)

References

[1] J M Zhang C Y Sun R M Zhang and C Qian ldquoAdaptivesliding mode control for re-entry attitude of near space hyper-sonic vehicle based on backstepping designrdquo IEEECAA Journalof Automatica Sinica vol 2 no 1 pp 94ndash101 2015

[2] J Liu Y Zhao B Geng and B Xiao ldquoAdaptive second ordersliding mode control of a fuel cell hybrid system for electricvehicle applicationsrdquo Mathematical Problems in Engineeringvol 2015 Article ID 370424 14 pages 2015

[3] M Zak ldquoTerminal attractors in neural networksrdquo NeuralNetworks vol 2 no 4 pp 259ndash274 1989

[4] S T Venkataraman and S Gulati ldquoTerminal sliding modes anew approach to nonlinear control synthesisrdquo in Proceedings ofthe 5th International Conference onAdvanced Robotics vol 1 pp443ndash448 Pisa Italy June 1991

[5] D Y Zhao S Y Li and F Gao ldquoA new terminal sliding modecontrol for robotic manipulatorsrdquo International Journal of Con-trol vol 82 no 10 pp 1804ndash1813 2009

[6] Z K Song H X Li and K B Sun ldquoFinite-time control fornonlinear spacecraft attitude based on terminal sliding modetechniquerdquo ISA Transactions vol 53 no 1 pp 117ndash124 2014

[7] H Komurcugil ldquoNon-singular terminal sliding-mode controlof DC-DC buck convertersrdquo Control Engineering Practice vol21 no 3 pp 321ndash332 2013

[8] X Yu M Zhihong and Y Wu ldquoTerminal sliding modes withfast transient performancerdquo in Proceedings of the 1997 36th IEEEConference on Decision and Control Part 1 (of 5) vol 2 pp 962ndash963 San Diego Calif USA December 1997

[9] S Mobayen ldquoFast terminal sliding mode controller design fornonlinear second-order systems with time-varying uncertain-tiesrdquo Complexity vol 21 no 2 pp 239ndash244 2015

[10] Z He C Liu Y Zhan H Li X Huang and Z ZhangldquoNonsingular fast terminal sliding mode control with extendedstate observer and tracking differentiator for uncertain nonlin-ear systemsrdquo Mathematical Problems in Engineering vol 2014Article ID 639707 16 pages 2014

[11] L Liu Q M Zhu L Cheng et al ldquoApplied methods and tech-niques for mechatronic systems modelling identification and

controlrdquo Lecture Notes in Control and Information Sciences vol452 pp 79ndash97 2014

[12] X H Yu and Z H Man ldquoOn finite time mechanism terminalsliding modesrdquo in Proceedings of the IEEE International Work-shop on Variable Structure Systems pp 164ndash167 Tokyo Japan1996

[13] L Y Wang T Y Chai and L F Zhai ldquoNeural-network-basedterminal sliding-mode control of robotic manipulators includ-ing actuator dynamicsrdquo IEEE Transactions on Industrial Elec-tronics vol 56 no 9 pp 3296ndash3304 2009

[14] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[15] A Al-Ghanimi J Zheng and Z Man ldquoRobust and fast non-singular terminal sliding mode control for piezoelectric actua-torsrdquo IET ControlTheory ampApplications vol 9 no 18 pp 2678ndash2687 2015

[16] S Mobayen ldquoFinite-time tracking control of chained-formnonholonomic systems with external disturbances based onrecursive terminal sliding mode methodrdquo Nonlinear Dynamicsvol 80 no 1-2 pp 669ndash683 2015

[17] S Mobayen ldquoFast terminal sliding mode tracking of non-holonomic systems with exponential decay raterdquo IET ControlTheory amp Applications vol 9 no 8 pp 1294ndash1301 2015

[18] Y Wu and J Wang ldquoContinuous recursive sliding modecontrol for hypersonic flight vehicle with extended disturbanceobserverrdquo Mathematical Problems in Engineering vol 2015Article ID 506906 26 pages 2015

[19] S Mobayen ldquoAn adaptive fast terminal sliding mode controlcombined with global slidingmode scheme for tracking controlof uncertain nonlinear third-order systemsrdquoNonlinear Dynam-ics vol 82 no 1-2 pp 599ndash610 2015

[20] N Boonsatit and C Pukdeboon ldquoAdaptive fast terminal slidingmode control of magnetic levitation systemrdquo Journal of ControlAutomation and Electrical Systems vol 27 no 4 pp 359ndash3672016

[21] L Y Fang T S Li Z F Li and R Li ldquoAdaptive terminal slidingmode control for anti-synchronization of uncertain chaoticsystemsrdquoNonlinear Dynamics vol 74 no 4 pp 991ndash1002 2013

[22] C-C Yang ldquoSynchronization of second-order chaotic systemsvia adaptive terminal sliding mode control with input nonlin-earityrdquo Journal of the Franklin Institute Engineering and AppliedMathematics vol 349 no 6 pp 2019ndash2032 2012

[23] C-C Yang and C-J Ou ldquoAdaptive terminal sliding mode con-trol subject to input nonlinearity for synchronization of chaoticgyrosrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 3 pp 682ndash691 2013

[24] J-J Kim J-J Lee K-B Park and M-J Youn ldquoDesign ofnew time-varying sliding surface for robot manipulator usingvariable structure controllerrdquo Electronics Letters vol 29 no 2pp 195ndash196 1993

[25] P Li and Z-Q Zheng ldquoRobust adaptive second-order sliding-mode control with fast transient performancerdquo IET ControlTheory amp Applications vol 6 no 2 pp 305ndash312 2012

[26] M Zhihong M OrsquoDay and X Yu ldquoA robust adaptive terminalsliding mode control for rigid robotic manipulatorsrdquo Journal ofIntelligent and Robotic SystemsTheory and Applications vol 24no 1 pp 23ndash41 1999

[27] Z Zhu Y Xia and M Fu ldquoAdaptive sliding mode control forattitude stabilization with actuator saturationrdquo IEEE Transac-tions on Industrial Electronics vol 58 no 10 pp 4898ndash49072011

12 Mathematical Problems in Engineering

[28] H K Khalil Nonlinear Systems Prentice Hall EnglewoodCliffs NJ USA 3rd edition 2002

[29] Y Shtessel M Taleb and F Plestan ldquoA novel adaptive-gainsupertwisting sliding mode controller methodology and appli-cationrdquo Automatica vol 48 no 5 pp 759ndash769 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

[14 15] However it was difficult to obtain for the case ofadaptive NTSM control in [22 23]

Previous studies have developed the TSM control andthus have great significance On the other hand there are stillsome points valuable for further research Most of the NTSMcontrollers were designed for a class of uncertain systems inwhich the coefficient of control input was a known function(or with an uncertain item) In the case that this coefficient istotally unknown except its boundary few NTSM controllerswere reported So the purpose of this paper is to develop theNTSM control for the latter case

The main contribution of this paper can be summarizedas follows

(1) A fast NTSM (FNTSM) is designed to combine theadvantages of LSM and NTSM together Thus theconvergence rate of the NTSM is enhanced when theinitial states are far away from the origin

(2) A new robust NTSM controller is proposed for thecase that the coefficient of control input is unknownIt is also applicable to the case that the coefficient isknown Moreover an exponential-decline switchinggain is designed to attenuate the chattering phe-nomenon

(3) An adaptive NTSM control scheme with doublesliding surfaces is proposed to give a strict prooffor the stagnation problem of NTSM The systemuncertainty is firstly compensated on an integralsliding surface after that the system trajectory isforced to the designed fast NTSM

2 Robust NTSM Controller Design

Consider a second-order nonlinear system as

1 = 11990922 = 120593 (119909) + 120574 (119909) 119906 (1)

where 120593(119909) and 120574(119909) denote bounded uncertain functions120574(119909) = 0Assumption 1 The uncertain functions 120593(119909) and 120574(119909) satisfy

1003816100381610038161003816120593 (119909)1003816100381610038161003816 le 120593 = 1205930 + 1205931 100381610038161003816100381611990911003816100381610038161003816 + 1205932 100381610038161003816100381611990921003816100381610038161003816 0 lt 119870119898 le 120574 (119909) le 119870119872

(2)

where 1205930 1205931 1205932 119870119898 and 119870119872 are positive constants 119870119898and 119870119872 are assumed to be known Robust and adaptivecontrollers are designed for the two cases that 120593 is known ornot respectively In the rest of the paper 120593(119909) and 120574(119909) aresimply denoted as 120593 and 12057421 FNTSMDesign In the study of [14] a kind of NTSMwasdesigned as

119904 = 1199091 + 120573 100381610038161003816100381611990921003816100381610038161003816120594 sign (1199092) 1199092 = 1 0 lt 120573 1 lt 120594 lt 2 (3)

NTSM

LSM

0 2minus2

x1

minus2

0

2

x2

Figure 1 The proposed sliding surface on the phase plane

It is equivalent to the following form which can beconverted to a conventional TSM

119904 =

1199091 + 1205731199092120594 1199092 gt 01199091 1199092 = 01199091 minus 120573 (minus1199092)120594 1199092 lt 0

(4)

As illustrated in [8] the TSM has slower convergence ratethan the LSM when the initial states are far away from theorigin It can be explained by its eigenvalue For TSM 1 +120573minus112059411990911205941 = 0 it is easy to obtain the eigenvalue as

12059711205971199091 = minus120573minus1120594 11205941199091120594minus11 (5)

As 1120594 minus 1 lt 0 the eigenvalue tends to be negativeinfinity at the origin This fact implies that the convergencerate is infinitely large This is the major merit of TSMHowever when the initial state is far away from the originthe convergence rate is smaller than the case 120594 = 1 that is aLSM

In order to enhance the convergence rate of NTSM (3) anew FNTSM is designed as

119904 = 1199091 + 120573 100381610038161003816100381611990921003816100381610038161003816119902 sign (1199092) 119902 = 120575 + (1 minus 120575) sign (100381610038161003816100381611990911003816100381610038161003816 minus 120576) 1 le 120576 1 lt 120575 lt 15 (6)

The FNTSM (6) is a switching sliding surface whichtransfers the system dynamic from a LSM (119902 = 1) to a NTSM(1 lt 119902 lt 2) (Figure 1 eg 120576 = 1) Thus it has global fastconvergence rate and reserves the advantage of NTSM

Here a simple example is given to compare the NTSM (3)and the FNTSM (6)

1 = 11990922 = 119906 (7)

Mathematical Problems in Engineering 3

FNTSMNTSM

1 2 3 4 5 6 7 80Time (s)

minus1

1

3

5

7

9

11

13

15O

utpu

t

4 5 6 730

1

2

(a) System response

FNTSMNTSM

0

10

20

30

40

50

60

Con

trol i

nput

1 2 3 4 5 6 7 80Time (s)

0

1

2

3 3525

Y 5711X 072

Y 5744X 0

(b) Control input

Figure 2 Comparison between the proposed FNTSM and the NTSM

Considering system (7) the control input was taken as theNTSM controller form of [14]

119906 = minus11199021003816100381610038161003816119909210038161003816100381610038162minus119902 sign (1199092) minus 2119904 minus |119904|79 sign (119904) (8)

With the same reaching law the parameters for the slidingsurfaces were chosen as

NTSM 119902 = 119902119873 = 117

FNTSM 119902 = 119902119865 = 97 + (1 minus 9

7) sign (100381610038161003816100381611990911003816100381610038161003816 minus 1) (9)

Note that the FNTSM became equivalent to the NTSMwithin the region |1199091| le 1 because of 119902119865 = 117 insidethis region As shown in Figure 2 the convergence rate ofthe proposed FNTSM controller was faster than the NTSMcontrollerrsquos while the control input of FNTSM controllerwas much smaller In particular for the FNTSM (6) theswitching of sliding surface is not continuousWhen the state1199091 converged into the prescribed region the sliding surfaceand control signal had slight discontinuous changes As aresult the control signal switched to a NTSM control froma LSM control As shown in Figure 2(b) there was a smallsaltation of the control signal (from 0967 to 0582) at 119905 =286 s because of the switching of sliding surface22 Controller Design Most of the NTSM controllers weredesigned for system (1) with a known 120574 (or with an uncertainitem) However these controllers are not theoretically appli-cable to totally unknown 120574 In this subsection a new form ofNTSMcontroller is designed for system (1) in the case that120593 isknown A proof is given to show that the stagnation problemof sliding surface will not occur Moreover the new NTSMcontroller is also applicable to the case that 120574 is known

Generally the NTSM control design contains the slidingsurface and the reaching law A discontinuous reaching

law with switching function is widely adopted to drive thesystem trajectory onto the sliding surface which leads to thechattering In the following theorem an exponential-declineswitching gain (EDSG) is designed to attenuate the chatteringphenomenon

The EDSG is formulated as

119860 (119905) 1205721 = (1198871 + 1198872119890minus119886119905) 1205721 (10)

where 1205721 gt 120593 119886 gt 0 1198871 ge 1 and 1198872 gt 0The EDSG acceleratesthe convergence rate of sliding surface at the beginning ofreaching phase and attenuates the amplitude of chatteringwithout deteriorating the system robustness

Theorem 2 For system (1) the following controller forces thesystem dynamic to the sliding surface (6) in finite time

119906 = minus 1119870119898 (1205722119904 + 119860 (119905) 1205721 sign (119904)

+ 1120573119902

1003816100381610038161003816119909210038161003816100381610038162minus119902 sign (119904)) (11)

where 1205721 = 120593 + 120578 1205722 ge 0 and 120578 is a small positive constant

Proof Consider the following Lyapunov function for thesliding surface

1198811 = 121199042 (12)

Differentiating (12) with respect to time yields

1 = 119904 119904 = 119904 (1199092 + 120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 2)= 119904 (1199092 + 120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 (120593 + 120574119906))

(13)

4 Mathematical Problems in Engineering

Substituting the controller (11) into (13) yields

1 = 119904 (1199092 + 120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 (120593 minus 120574119870119898 (1205722119904

+ 119860 (119905) 1205721 sign (119904) + 1120573119902

1003816100381610038161003816119909210038161003816100381610038162minus119902 sign (119904)))) = 1199041199092

+ 120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 (120593119904 minus 1205741198701198981205722119904

2 minus 120574119870119898119860 (119905) 1205721 |119904| minus 120574

119870119898sdot 1120573119902

1003816100381610038161003816119909210038161003816100381610038162minus119902 |119904|) = minus120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 1205741198701198981205722119904

2 + 1199041199092

minus 120574119870119898 |119904| 100381610038161003816100381611990921003816100381610038161003816 + 120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 (120593119904 minus 120574

119870119898119860 (119905) 1205721 |119904|)

(14)

With 120574 ge 119870119898 and |1199092|119902minus1 gt 0 (14) turns to1 le minus120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 12057221199042 + 1199041199092 minus |119904| 100381610038161003816100381611990921003816100381610038161003816

+ 120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 (10038161003816100381610038161205931003816100381610038161003816 |119904| minus 119860 (119905) 1205721 |119904|) (15)

As 119860(119905)1205721 ge 120593 + 120578 it is obtained as

1 le minus120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 12057221199042 minus 120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 120578 |119904|= minus120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 (12057221199042 + 120578 |119904|)= minus120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 (212057221198811 + 120578212119881121 )

(16)

Considering the last expression of (16) if the motion ofsliding surface does not stagnate at the points 1199091 = 0 1199092 =0 in the reaching phase it would converge to zero in finitetime according to the extended Lyapunov description of finitetime stability in Remark 2 of [14]

Here a demonstration is given to show that themotion ofsliding surface does not stagnate at the points 1199091 = 0 1199092 =0

For system (1) with the controller (11) taking 1199092 = 0 weobtain

2 = 120593 minus 120574119870119898 (1205722119904 + 119860 (119905) 1205721 sign (119904)

+ 1120573119902

1003816100381610038161003816119909210038161003816100381610038162minus119902 sign (119904)) = minus1205722 120574119870119898 119904 + 120593 minus 119860 (119905) 1205721

sdot 120574119870119898 sign (119904)

(17)

where 119904 = 1199091In the case of 1199091 gt 0 the following inequality is satisfied

for 22 = minus1205722 120574

119870119898 119904 + 120593 minus 119860 (119905) 1205721 120574119870119898

= minus1205722 120574119870119898 119904 + 120593 minus 120574

119870119898 (1198871 + 1198872119890minus119886119905) (120593 + 120578)le minus1205722119904⏟⏟⏟⏟⏟⏟⏟lt0

+ 120593 minus 120593 minus 120578⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟lt0

(18)

In the other case of 1199091 lt 0 it follows that2 = minus1205722 120574

119870119898 119904 + 120593 + 119860 (119905) 1205721 120574119870119898

= minus1205722 120574119870119898 119904 + 120593 + 120574

119870119898 (1198871 + 1198872119890minus119886119905) (120593 + 120578)ge minus1205722119904⏟⏟⏟⏟⏟⏟⏟gt0

+ 120593 + 120593 + 120578⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟gt0

(19)

It is concluded that the points 1199091 = 0 1199092 = 0 arenot attractors Thus the motion of sliding surface will notstagnate at these points and it converges to zero in finite time

This completes the proof

Remark 3 Note that the term |1199092|2minus119902sign(119904) of controller(11) is constructed for the stability analysis It is usuallytaken as |1199092|2minus119902sign(1199092) in the existing NTSM controllers forexample the controller (8) However it is not applicable tothe stability analysis of the discussed system (1)

Remark 4 In the case that 120574 is known the proposed con-troller (11) is also applicable by taking 119870119898 = 1205743 Adaptive NTSM Control Scheme Design

In practical situations it is usually difficult to obtain the priorknowledge of 120593 Therefore an adaptive controller is designedfor system (1) in this section In the studies of adaptive NTSMcontrol [22 23] the switching gain was directly adjusted bythe adaptive technique However it was difficult to provethat the points 1199091 = 0 1199092 = 0 were not attractors asin Theorem 2 So the adaptive switching gain was modifiedby the constant control gain in the extreme case that thestagnation problem may happen

In this section a double sliding surfaces control schemeis designed to combine the NTSM control with the adaptivetechnique The first layer is a common integral slidingmode (ISM) in which the impact of uncertain function 120593is compensated The second layer is a NTSM in which thesystem trajectory converges to the origin The purpose ofusing double sliding surfaces is to provide a strict proof forthe stagnation problem of NTSM

In this section the control law is designed as

119906 = 119906119886 + 119906119899 (20)

where 119906119886 denotes an adaptive controller which forces thesystemmotion to the first sliding surface After that a NTSMcontroller 119906119899 drives the system motion to the second slidingsurface

The first sliding surface ISM is designed as

1199041 = 1199092 + 119911 minus 119890minus120579119905 (1199092 (0) + 119911 (0)) = minus119906119899

(21)

where 120579 gt 0 and 1199092(0) 119911(0) are the initial values of 1199092 119911The exponential-decline term is introduced to eliminate thereaching phase [24] Then the system dynamic is on the firstsliding surface at the initial time

Mathematical Problems in Engineering 5

Differentiating (21) with respect to time yields

1199041 = 2 + + Λ = 120593 + 120574119906 minus 119906119899 + Λ= 120593 + Λ + (120574 minus 1) 119906119899 + 120574119906119886

(22)

where Λ = 120579119890minus120579119905(1199092(0) + 119911(0)) Once the first sliding surfaceis established the original system (1) will approach to thesystem motion of an integral chain system as 2 = minus119906119899

Here the NTSM controller 119906119899 for system 2 = minus119906119899 isdesigned as

119906119899 = minus 1120573119902

1003816100381610038161003816119909210038161003816100381610038162minus119902 sign (1199092) minus 12057221199042minus 1205723 1003816100381610038161003816119904210038161003816100381610038161199011 sign (1199042)

(23)

where 1205723 is a positive constant 1199011 is a constant satisfying 0 lt1199011 lt 1 and 1199042 denotes the FNTSM (6)

The continuous controller (23) had been verified in [14]and it also had been proved that the points 1199091 = 0 1199092 =0 were not attractors For an integral chain system withoutuncertainty (eg 2 = minus119906119899) the discontinuous control is notnecessary Although the proposed NTSM controller (11) inTheorem 2 can be used to design 119906119899 it is not employed as it isdiscontinuous

With the following assumption an adaptive controlleris designed in Theorem 6 for system (22) by combining theTSM reaching law with the familiar first-order SMC adaptivecontrol (eg [25ndash27])

Assumption 5 Assume 120593+(120574minus1)119906119899 is unknown and satisfiesthe following inequality

1003816100381610038161003816120593 + (120574 minus 1) 1199061198991003816100381610038161003816 le 1198880 + 1198881 100381610038161003816100381611990911003816100381610038161003816 + 1198882 100381610038161003816100381611990921003816100381610038161003816 (24)

where 1198880 1198881 and 1198882 are unknown positive constants

Theorem6 For system (22) the sliding surface 1199041 converges tozero asymptotically with the following adaptive controller

119906119886 = minus 1119870119898 (11989631199041 + 1198964 1003816100381610038161003816119904110038161003816100381610038161199012 sign (1199041) + |Λ| sign (1199041)

+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sign (1199041)) 1198880 = 1198960 100381610038161003816100381611990411003816100381610038161003816 1198881 = 1198961 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 1198882 = 1198962 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816

(25)

where 1198960 sim 1198964 ge 0 1 ge 1199012 gt 0 and 119894(0) gt 0 (119894 = 0 1 2)Proof Define a new Lyapunov function for 1199041 as

1198812 = 1211990421 + 1

21198960 20 + 1

21198961 21 + 1

21198962 22 (26)

where 119894 = 119888119894 minus 119894 Its derivative is2 = 1199041 1199041 minus 1

1198960 01198880 minus 1

1198961 11198881 minus 1

1198962 21198882 (27)

Substituting controller (25) into (27) yields

2 = 1199041 (120593 + Λ + (120574 minus 1) 119906119899 + 120574119906119886) minus ( 11198960 0

1198880

+ 11198961 1

1198881 + 11198962 2

1198882) = 1199041 (120593 + Λ + (120574 minus 1) 119906119899)+ 1205741199041119906119886 minus 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) le 100381610038161003816100381611990411003816100381610038161003816 1003816100381610038161003816120593+ (120574 minus 1) 1199061198991003816100381610038161003816 + 100381610038161003816100381611990411003816100381610038161003816 |Λ| + 1205741199041119906119886 minus 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816+ 2 100381610038161003816100381611990921003816100381610038161003816) le 100381610038161003816100381611990411003816100381610038161003816 (1198880 + 1198881 100381610038161003816100381611990911003816100381610038161003816 + 1198882 100381610038161003816100381611990921003816100381610038161003816) + 100381610038161003816100381611990411003816100381610038161003816 |Λ|minus 120574

119870119898 (119896311990421 + 1198964 1003816100381610038161003816119904110038161003816100381610038161199012+1 + 100381610038161003816100381611990411003816100381610038161003816 |Λ|

+ 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816)) minus 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816+ 2 100381610038161003816100381611990921003816100381610038161003816)

(28)

With the adaptive laws in (25) and 119894(0) gt 0 it ensuresthat 119894 gt 0 Since 120574 ge 119870119898 it follows that

2 le minus 120574119870119898 (119896311990421 + 1198964 1003816100381610038161003816119904110038161003816100381610038161199012+1)

+ 100381610038161003816100381611990411003816100381610038161003816 (1198880 + 1198881 100381610038161003816100381611990911003816100381610038161003816 + 1198882 100381610038161003816100381611990921003816100381610038161003816)minus 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816)minus 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816)

= minus 120574119870119898 (119896311990421 + 1198964 1003816100381610038161003816119904110038161003816100381610038161199012+1)

(29)

Applying the Barbalat lemma [28] the sliding surface 1199041converges to zero asymptotically

This completes the proof

The boundary layer approach is a common method forchattering reduction It is well known that the chatteringreduction is achieved with a small cost of control precisionMore specifically the sliding surface is driven to a vicinity ofzero In [29] a concept of real sliding surface was introducedwhich was similar to the boundary layer It also means thatthe sliding surface is forced to a small region rather than zeroBased on this concept a modified adaptive law was proposedto avoid the overestimation of adaptive gain in [29] Theadaptive gain slowly decreased to the necessary value oncethe real sliding surface was established

In the case that the boundary layer approach is adoptedfor the proposed adaptive controller (25) the chatteringamplitude can be further attenuated if the adaptive gains0 1 2 decrease to a low level Thus the above two meth-ods are combined to design a modified adaptive controller as

119906119886 = minus 1119870119898 (11989631199041 + 1198964 1003816100381610038161003816119904110038161003816100381610038161199012 sign (1199041) + |Λ| sat (1199041)

+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sat (1199041)) (30)

6 Mathematical Problems in Engineering

o

u

x

m

R(t)

Figure 3 Variable-length pendulum model

where the sat(sdot) function and the adaptive laws are taken as

sat (1199041) = sign (1199041) 100381610038161003816100381611990411003816100381610038161003816 gt 1198741199041119874

100381610038161003816100381611990411003816100381610038161003816 le 119874

1198880 = 1198960 100381610038161003816100381611990411003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 119874) 0 gt 12057211988801198960 0 le 1205721198880

1198881 = 1198961 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 119874) 1 gt 12057211988811198961 1 le 1205721198881

1198882 = 1198962 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 119874) 2 gt 12057211988821198962 2 le 1205721198882

(31)

where a positive constant 119874 denotes the boundary lay 1198960 sim1198962 and 1205721198880 sim 1205721198882 are positive constantsAccording toTheorem 6 the sliding surface 1199041 converges

to zero asymptotically Thus it converges into |1199041| le 119874 infinite time Then the adaptive gains begin to decrease and2 becomes sign indefinite As soon as the sliding surfaceescapes from the region |1199041| le 119874 the adaptive gains increaseagain so that the sliding surface is driven back into the region|1199041| le 119874 Finally the sliding surface 1199041 remains in a largerregion than |1199041| le 119874which is the so-called real sliding surfaceThe second expressions of the modified adaptive laws ensurethe positive values of the adaptive gains and they are valid ina short time

Remark 7 For the tracking control issues it is necessary tomodify the adaptive laws of 1 and 2 to reduce the chatteringamplitude However it is not necessary when the states1199091 1199092 converge to zero4 Simulation

In this section the designed robust and adaptive NTSMcontrollers were tested for the tracking control of a variable-length pendulum [25] (Figure 3) formulated as

= minus2 (119905)119877 (119905) minus 119892

119877 (119905) sin (119909) + 11198981198772 (119905)119906 (32)

The angular coordinate 119909 is impelled to track a referencesignal 119909119888 by the torque 119906 A known mass 119898 moves alonethe rod without friction and the distance 119877(119905) between thecenter 119900 and the mass 119898 is unmeasured It is assumed thatthe states [119909 ]119879 can be measured for the controller designThe parameters of (32) and the reference signal are taken as

119877 (119905) = 1 + 025 sin (4119905) + 05 cos (119905)119898 = 1 kg119892 = 981ms2

119909 (0) = 12119909119888 = 05 sin (05119905)

(33)

Defining the tracking error as 1198901 = 119909 minus 119909119888 the trackingcontrol is converted to the stabilization of the following errorsystem

1198901 = 11989021198902 = minus2 (119905)

119877 (119905) minus 119892119877 (119905) sin (119909) minus 119888 + 1

1198981198772 (119905)119906(34)

41 Robust Control In this subsection three robust control-lers (A B and C) were evaluated According to Theorem 2controllers A and B (without EDSG) were designed using theFNTSM of [11] and the proposed FNTSM (6) respectivelyThe controller C was the proposed FNTSM-EDSG controllerin Theorem 2

Controller A (based on FNTSM of [11])

119906 = minus16 (021199041198651 + 1205721 sign (1199041198651)

+ 790119890572 sign (1199041198651) sign (1198902)

+ 49450119890251 119890572 sign (1199041198651) sign (119890251 1198902))

1199041198651 = 1198901 + 119890751 + 119890972 1205721 = 04 + 04 100381610038161003816100381611990911003816100381610038161003816 + 04 100381610038161003816100381611990921003816100381610038161003816

(35)

Controller B (based on FNTSM (6))

119906 = minus16(021199041198652 + 1205721 sign (1199041198652) + 11199021003816100381610038161003816119890210038161003816100381610038162minus119902 sign (1199041198652))

1199041198652 = 1198901 + 100381610038161003816100381611989021003816100381610038161003816119902 sign (1198902) 119902 = 97 + (1 minus 9

7) sign (100381610038161003816100381611990911003816100381610038161003816 minus 1) (36)

Controller C (based on FNTSM (6) + EDSG)

119906 = minus16(021199041198652 + 119860 (119905) 1205721 sign (1199041198652)

+ 11199021003816100381610038161003816119890210038161003816100381610038162minus119902 sign (1199041198652)) 119860 (119905) = 1 + 119890minus2119905 1205721 = 02 + 02 100381610038161003816100381611990911003816100381610038161003816 + 02 100381610038161003816100381611990921003816100381610038161003816

(37)

The simulation results are given in Figure 4 As shownin Figures 4(a) and 4(b) the angular coordinate converged

Mathematical Problems in Engineering 7

Controller B Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad) Controller A Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad)

Controller C Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad)

(a) Angular coordinate

2 3 4

0

002

Controller A

1 2 3 4 5 6 7 8 9 100Time (s)

0

06

12

Trac

king

erro

r (ra

d)

Controller B

2 3 4

0

002

0

06

12

Trac

king

erro

r (ra

d)

1 2 3 4 5 6 7 8 9 100Time (s)

2 3 4

0

002

Controller C

0

06

12Tr

acki

ng er

ror (

rad)

1 2 3 4 5 6 7 8 9 100Time (s)

(b) Tracking error

1 2 3 4 5 6 7 8 9 100Time (s)

minus10123

Slid

ing

surfa

ces

Controller A

Controller B

minus10123

Slid

ing

surfa

ces

1 2 3 4 5 6 7 8 9 100Time (s)

Controller C

minus10123

Slid

ing

surfa

ces

1 2 3 4 5 6 7 8 9 100Time (s)

(c) Sliding surfaces

1 2 3 4 5 6 7 8 9 100Time (s)

Controller A

minus30

minus10

10

30

Con

trol t

orqu

es (N

)

Controller B

minus30

minus10

10

30

Con

trol t

orqu

es (N

)

1 2 3 4 5 6 7 8 9 100Time (s)

Controller C

minus30

minus10

10

30

Con

trol t

orqu

es (N

)

1 2 3 4 5 6 7 8 9 100Time (s)

(d) Control torque

Figure 4 Simulation result of the robust control

8 Mathematical Problems in Engineering

to the reference signal within 119905 = 4 s The sliding surfacesconverged to zero within 119905 = 15 s (Figure 4(c)) It is clear thatcontroller B cost less convergence time than the controllerA while the control inputs of these controllers had slightdifference (Figures 4(b) and 4(d)) This fact implies thatthe controller designed by the proposed FNTSM was moreefficient As expected the chattering amplitude of controllerC (plusmn5) was half of controller B (plusmn10) (Figure 4(d)) Howeverthe performance of controllerChad small differencewith thatof controller BThe result confirms that the EDSG is effectivefor chattering suppression without deteriorating the systemrobustness

42 Adaptive Control In this subsection the proposed adap-tive NTSM control scheme ((23) and (25)) was verified forthe tracking control As a comparison the adaptive methodin [25] was adopted to design the adaptive controller 119906119886 Thecontrollers were designed as follows

Controller D (Theorem 6)

119906119899 = minus11199021003816100381610038161003816119890210038161003816100381610038162minus119902 sign (1198902) minus 251199042 minus 10038161003816100381610038161199042100381610038161003816100381679 sign (1199042)

119906119886 = minus16 (051199041 + 10038161003816100381610038161199041100381610038161003816100381679 sign (1199041)+ 100381610038161003816100381610038165119890minus5119905 (1198902 (0) + 119911 (0))10038161003816100381610038161003816 sign (1199041)+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sign (1199041))

1199041 = 1198902 + 119911 minus 119890minus5119905 (1198902 (0) + 119911 (0)) = minus119906119899119911 (0) = 01198880 = 100381610038161003816100381611990411003816100381610038161003816 1198881 = 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 1198882 = 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 0 (0) = 1 (0) = 2 (0) = 001

(38)

Controller E (method in [25])

119906119886 = minus5119890minus5119905 (1198902 (0) + 119911 (0)) minus 0 sign (1199041) 1198880 = 5 100381610038161003816100381611990411003816100381610038161003816

(39)

ControllersD and E shared the sameNTSMcontroller 119906119899and the sliding surface 1199042 was taken as 1199041198652 of controller B

The simulation results are given in Figures 5 and 6 It isclear that both controllers D and E achieved fast tracking ofthe reference signal Controller D had shorter convergencetime and better transient performance than controller Eeven though the maximum control torque and the chatteringamplitude of controller D were smaller Meanwhile thesliding surfaces in Figure 6 converged to zero within 119905 =3 s In Figure 6(a) the motions of ISM started from zeroand there was a short time that ISM did not remain zerodue to the affection of uncertainty It is observed that the

FNTSM converged to zero after the ISM and the stagnationof FNTSM never happened It verifies that the designedadaptive NTSM control scheme is effective Since the twocontrollers shared the same NTSM controller 119906119899 the simula-tion results confirm that the proposed adaptive method hadbetter transient performance and was more efficient than themethod in [25]

Here themodified adaptive controller ((30) and (31)) waschecked for chattering reduction In the following controllerF the boundary layer approach and the modified adaptivelaws were both employed As a comparison controller Gonly employed the boundary layer approach In order toavoid the unbounded growing of adaptive gains inside theboundary layer the switching adaptive laws were adoptedfor controller G In addition controllers F and G used thesame boundary layer and NTSM controller 119906119899 of controllerD Since it is assumed that the states can be measured forthe controller design themeasurement noisemay deterioratethe performance of controller in real implementations Toillustrate the robustness of the controller random noise wastaken for the states [119909 ]119879 (mean value 0 and standarddeviation 001)

Controller F

119906119886 = minus16 (051199041 + 10038161003816100381610038161199041100381610038161003816100381679 sign (1199041)+ 100381610038161003816100381610038165119890minus5119905 (1198902 (0) + 119911 (0))10038161003816100381610038161003816 sat (1199041)+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sat (1199041))

sat (1199041) = sign (1199041) 100381610038161003816100381611990411003816100381610038161003816 gt 0041199041

004100381610038161003816100381611990411003816100381610038161003816 le 004

1198880 = 2 100381610038161003816100381611990411003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 004) 0 gt 0011 0 le 001

1198881 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 005) 1 gt 0011 1 le 001

1198882 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 004) 2 gt 0011 2 le 001

(40)

Controller G

119906119886 = minus16 (051199041 + 10038161003816100381610038161199041100381610038161003816100381679 sign (1199041)+ 100381610038161003816100381610038165119890minus5119905 (1198902 (0) + 119911 (0))10038161003816100381610038161003816 sat (1199041)+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sat (1199041))

1198880 = 2 100381610038161003816100381611990411003816100381610038161003816 1199041 gt 0040 1199041 le 004

Mathematical Problems in Engineering 9

Ang

ular

coor

dina

te (r

ad)

Controller DController E

Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus08

minus06

minus04

minus02

0

02

04

06

08

1

(a) Angular coordinate

Trac

king

erro

r (ra

d)

Controller DController E

minus02

0

02

04

06

08

1

1 2 3 4 5 6 7 8 9 100Time (s)

(b) Tracking error

Adaptive gain 1

Adaptive gain 1Adaptive gain 2

Adaptive gain 3

Controller D

Controller E

1 2 3 4 5 6 7 8 9 100Time (s)

0

01

02

Adap

tive g

ains

0

5

10

15

Adap

tive g

ains

1 2 3 4 5 6 7 8 9 100Time (s)

(c) Adaptive gains

0

Controller E

Controller D

minus20

20

Con

trol t

orqu

es (N

)

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus20

0

20C

ontro

l tor

ques

(N)

(d) Control torque

Figure 5 Simulation result of the adaptive control

1198881 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 1199041 gt 0040 1199041 le 004

1198882 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 1199041 gt 0040 1199041 le 004

(41)

The simulation result of controller F is given in Figure 7It is shown that the angular coordinate tracked the referencesignal with a high precision (Figure 7(a)) Due to the adop-tion of the boundary layer and themodified adaptive laws thesliding surface 1199041 converged to a small domain as |1199041| le 01

(Figure 7(b))Moreover the adaptive gains decreased to a lowlevel with approximate reduction ratios as 5108 8479and 1615 respectively (Figure 7(c)) As a result there was amarked reduction in the chattering of controller F comparedto that of controller G in Figure 7(d) Thus the effectivenessof the modified adaptive controller is verified

5 Conclusions

This paper studies the NTSM control for a class of second-order systems in which the coefficient of control input isunknown In this paper a new FNTSM was designed tointegrate the advantages of LSM and NTSM Based onthis FNTSM new forms of robust controller and adaptive

10 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 10minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

Time (s)

ISM

Controller DController E

(a) ISM

0 1 2 3 4 5 6 7 8 9 10minus2

minus15

minus1

minus05

0

05

1

15

2

25

3

Time (s)

FNTS

M

Controller DController E

(b) FNTSM

Figure 6 The response of sliding surfaces

Controller FReference signal

10 20 30 40 500Time (s)

minus1

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad)

(a) Angular coordinate

minus02

0

02

04

06

08

0 10 20 30 40 50Time (s)

Slid

ing

surfa

ces

ISMFNTSM

(b) Sliding surfaces

0

005

01

015

02

025

Adap

tive g

ains

Adaptive gain 1Adaptive gain 2

Adaptive gain 3

10 20 30 40 500Time (s)

(c) Adaptive gains

0 10 20 30 40 50Time (s)

Controller G

Controller F

minus10

0

10

Con

trol t

orqu

es (N

)

minus10

0

10

Con

trol t

orqu

es (N

)

10 20 30 40 500Time (s)

(d) Control torques

Figure 7 Simulation result of the modified adaptive control

Mathematical Problems in Engineering 11

controller were proposed for the discussed uncertain systemThe simulation verified the effectiveness of the proposed con-trollersThe proposed FNTSM shows faster convergence ratethan the NTSM and chattering reduction was achieved bythe EDSG The adaptive controller presented better transientperformance and more efficiency than the adaptive methodin [25] Moreover the modified adaptive controller achievedeffective chattering reduction

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this article

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (no 61375100 no 61472037 and no61433003)

References

[1] J M Zhang C Y Sun R M Zhang and C Qian ldquoAdaptivesliding mode control for re-entry attitude of near space hyper-sonic vehicle based on backstepping designrdquo IEEECAA Journalof Automatica Sinica vol 2 no 1 pp 94ndash101 2015

[2] J Liu Y Zhao B Geng and B Xiao ldquoAdaptive second ordersliding mode control of a fuel cell hybrid system for electricvehicle applicationsrdquo Mathematical Problems in Engineeringvol 2015 Article ID 370424 14 pages 2015

[3] M Zak ldquoTerminal attractors in neural networksrdquo NeuralNetworks vol 2 no 4 pp 259ndash274 1989

[4] S T Venkataraman and S Gulati ldquoTerminal sliding modes anew approach to nonlinear control synthesisrdquo in Proceedings ofthe 5th International Conference onAdvanced Robotics vol 1 pp443ndash448 Pisa Italy June 1991

[5] D Y Zhao S Y Li and F Gao ldquoA new terminal sliding modecontrol for robotic manipulatorsrdquo International Journal of Con-trol vol 82 no 10 pp 1804ndash1813 2009

[6] Z K Song H X Li and K B Sun ldquoFinite-time control fornonlinear spacecraft attitude based on terminal sliding modetechniquerdquo ISA Transactions vol 53 no 1 pp 117ndash124 2014

[7] H Komurcugil ldquoNon-singular terminal sliding-mode controlof DC-DC buck convertersrdquo Control Engineering Practice vol21 no 3 pp 321ndash332 2013

[8] X Yu M Zhihong and Y Wu ldquoTerminal sliding modes withfast transient performancerdquo in Proceedings of the 1997 36th IEEEConference on Decision and Control Part 1 (of 5) vol 2 pp 962ndash963 San Diego Calif USA December 1997

[9] S Mobayen ldquoFast terminal sliding mode controller design fornonlinear second-order systems with time-varying uncertain-tiesrdquo Complexity vol 21 no 2 pp 239ndash244 2015

[10] Z He C Liu Y Zhan H Li X Huang and Z ZhangldquoNonsingular fast terminal sliding mode control with extendedstate observer and tracking differentiator for uncertain nonlin-ear systemsrdquo Mathematical Problems in Engineering vol 2014Article ID 639707 16 pages 2014

[11] L Liu Q M Zhu L Cheng et al ldquoApplied methods and tech-niques for mechatronic systems modelling identification and

controlrdquo Lecture Notes in Control and Information Sciences vol452 pp 79ndash97 2014

[12] X H Yu and Z H Man ldquoOn finite time mechanism terminalsliding modesrdquo in Proceedings of the IEEE International Work-shop on Variable Structure Systems pp 164ndash167 Tokyo Japan1996

[13] L Y Wang T Y Chai and L F Zhai ldquoNeural-network-basedterminal sliding-mode control of robotic manipulators includ-ing actuator dynamicsrdquo IEEE Transactions on Industrial Elec-tronics vol 56 no 9 pp 3296ndash3304 2009

[14] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[15] A Al-Ghanimi J Zheng and Z Man ldquoRobust and fast non-singular terminal sliding mode control for piezoelectric actua-torsrdquo IET ControlTheory ampApplications vol 9 no 18 pp 2678ndash2687 2015

[16] S Mobayen ldquoFinite-time tracking control of chained-formnonholonomic systems with external disturbances based onrecursive terminal sliding mode methodrdquo Nonlinear Dynamicsvol 80 no 1-2 pp 669ndash683 2015

[17] S Mobayen ldquoFast terminal sliding mode tracking of non-holonomic systems with exponential decay raterdquo IET ControlTheory amp Applications vol 9 no 8 pp 1294ndash1301 2015

[18] Y Wu and J Wang ldquoContinuous recursive sliding modecontrol for hypersonic flight vehicle with extended disturbanceobserverrdquo Mathematical Problems in Engineering vol 2015Article ID 506906 26 pages 2015

[19] S Mobayen ldquoAn adaptive fast terminal sliding mode controlcombined with global slidingmode scheme for tracking controlof uncertain nonlinear third-order systemsrdquoNonlinear Dynam-ics vol 82 no 1-2 pp 599ndash610 2015

[20] N Boonsatit and C Pukdeboon ldquoAdaptive fast terminal slidingmode control of magnetic levitation systemrdquo Journal of ControlAutomation and Electrical Systems vol 27 no 4 pp 359ndash3672016

[21] L Y Fang T S Li Z F Li and R Li ldquoAdaptive terminal slidingmode control for anti-synchronization of uncertain chaoticsystemsrdquoNonlinear Dynamics vol 74 no 4 pp 991ndash1002 2013

[22] C-C Yang ldquoSynchronization of second-order chaotic systemsvia adaptive terminal sliding mode control with input nonlin-earityrdquo Journal of the Franklin Institute Engineering and AppliedMathematics vol 349 no 6 pp 2019ndash2032 2012

[23] C-C Yang and C-J Ou ldquoAdaptive terminal sliding mode con-trol subject to input nonlinearity for synchronization of chaoticgyrosrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 3 pp 682ndash691 2013

[24] J-J Kim J-J Lee K-B Park and M-J Youn ldquoDesign ofnew time-varying sliding surface for robot manipulator usingvariable structure controllerrdquo Electronics Letters vol 29 no 2pp 195ndash196 1993

[25] P Li and Z-Q Zheng ldquoRobust adaptive second-order sliding-mode control with fast transient performancerdquo IET ControlTheory amp Applications vol 6 no 2 pp 305ndash312 2012

[26] M Zhihong M OrsquoDay and X Yu ldquoA robust adaptive terminalsliding mode control for rigid robotic manipulatorsrdquo Journal ofIntelligent and Robotic SystemsTheory and Applications vol 24no 1 pp 23ndash41 1999

[27] Z Zhu Y Xia and M Fu ldquoAdaptive sliding mode control forattitude stabilization with actuator saturationrdquo IEEE Transac-tions on Industrial Electronics vol 58 no 10 pp 4898ndash49072011

12 Mathematical Problems in Engineering

[28] H K Khalil Nonlinear Systems Prentice Hall EnglewoodCliffs NJ USA 3rd edition 2002

[29] Y Shtessel M Taleb and F Plestan ldquoA novel adaptive-gainsupertwisting sliding mode controller methodology and appli-cationrdquo Automatica vol 48 no 5 pp 759ndash769 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

FNTSMNTSM

1 2 3 4 5 6 7 80Time (s)

minus1

1

3

5

7

9

11

13

15O

utpu

t

4 5 6 730

1

2

(a) System response

FNTSMNTSM

0

10

20

30

40

50

60

Con

trol i

nput

1 2 3 4 5 6 7 80Time (s)

0

1

2

3 3525

Y 5711X 072

Y 5744X 0

(b) Control input

Figure 2 Comparison between the proposed FNTSM and the NTSM

Considering system (7) the control input was taken as theNTSM controller form of [14]

119906 = minus11199021003816100381610038161003816119909210038161003816100381610038162minus119902 sign (1199092) minus 2119904 minus |119904|79 sign (119904) (8)

With the same reaching law the parameters for the slidingsurfaces were chosen as

NTSM 119902 = 119902119873 = 117

FNTSM 119902 = 119902119865 = 97 + (1 minus 9

7) sign (100381610038161003816100381611990911003816100381610038161003816 minus 1) (9)

Note that the FNTSM became equivalent to the NTSMwithin the region |1199091| le 1 because of 119902119865 = 117 insidethis region As shown in Figure 2 the convergence rate ofthe proposed FNTSM controller was faster than the NTSMcontrollerrsquos while the control input of FNTSM controllerwas much smaller In particular for the FNTSM (6) theswitching of sliding surface is not continuousWhen the state1199091 converged into the prescribed region the sliding surfaceand control signal had slight discontinuous changes As aresult the control signal switched to a NTSM control froma LSM control As shown in Figure 2(b) there was a smallsaltation of the control signal (from 0967 to 0582) at 119905 =286 s because of the switching of sliding surface22 Controller Design Most of the NTSM controllers weredesigned for system (1) with a known 120574 (or with an uncertainitem) However these controllers are not theoretically appli-cable to totally unknown 120574 In this subsection a new form ofNTSMcontroller is designed for system (1) in the case that120593 isknown A proof is given to show that the stagnation problemof sliding surface will not occur Moreover the new NTSMcontroller is also applicable to the case that 120574 is known

Generally the NTSM control design contains the slidingsurface and the reaching law A discontinuous reaching

law with switching function is widely adopted to drive thesystem trajectory onto the sliding surface which leads to thechattering In the following theorem an exponential-declineswitching gain (EDSG) is designed to attenuate the chatteringphenomenon

The EDSG is formulated as

119860 (119905) 1205721 = (1198871 + 1198872119890minus119886119905) 1205721 (10)

where 1205721 gt 120593 119886 gt 0 1198871 ge 1 and 1198872 gt 0The EDSG acceleratesthe convergence rate of sliding surface at the beginning ofreaching phase and attenuates the amplitude of chatteringwithout deteriorating the system robustness

Theorem 2 For system (1) the following controller forces thesystem dynamic to the sliding surface (6) in finite time

119906 = minus 1119870119898 (1205722119904 + 119860 (119905) 1205721 sign (119904)

+ 1120573119902

1003816100381610038161003816119909210038161003816100381610038162minus119902 sign (119904)) (11)

where 1205721 = 120593 + 120578 1205722 ge 0 and 120578 is a small positive constant

Proof Consider the following Lyapunov function for thesliding surface

1198811 = 121199042 (12)

Differentiating (12) with respect to time yields

1 = 119904 119904 = 119904 (1199092 + 120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 2)= 119904 (1199092 + 120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 (120593 + 120574119906))

(13)

4 Mathematical Problems in Engineering

Substituting the controller (11) into (13) yields

1 = 119904 (1199092 + 120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 (120593 minus 120574119870119898 (1205722119904

+ 119860 (119905) 1205721 sign (119904) + 1120573119902

1003816100381610038161003816119909210038161003816100381610038162minus119902 sign (119904)))) = 1199041199092

+ 120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 (120593119904 minus 1205741198701198981205722119904

2 minus 120574119870119898119860 (119905) 1205721 |119904| minus 120574

119870119898sdot 1120573119902

1003816100381610038161003816119909210038161003816100381610038162minus119902 |119904|) = minus120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 1205741198701198981205722119904

2 + 1199041199092

minus 120574119870119898 |119904| 100381610038161003816100381611990921003816100381610038161003816 + 120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 (120593119904 minus 120574

119870119898119860 (119905) 1205721 |119904|)

(14)

With 120574 ge 119870119898 and |1199092|119902minus1 gt 0 (14) turns to1 le minus120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 12057221199042 + 1199041199092 minus |119904| 100381610038161003816100381611990921003816100381610038161003816

+ 120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 (10038161003816100381610038161205931003816100381610038161003816 |119904| minus 119860 (119905) 1205721 |119904|) (15)

As 119860(119905)1205721 ge 120593 + 120578 it is obtained as

1 le minus120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 12057221199042 minus 120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 120578 |119904|= minus120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 (12057221199042 + 120578 |119904|)= minus120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 (212057221198811 + 120578212119881121 )

(16)

Considering the last expression of (16) if the motion ofsliding surface does not stagnate at the points 1199091 = 0 1199092 =0 in the reaching phase it would converge to zero in finitetime according to the extended Lyapunov description of finitetime stability in Remark 2 of [14]

Here a demonstration is given to show that themotion ofsliding surface does not stagnate at the points 1199091 = 0 1199092 =0

For system (1) with the controller (11) taking 1199092 = 0 weobtain

2 = 120593 minus 120574119870119898 (1205722119904 + 119860 (119905) 1205721 sign (119904)

+ 1120573119902

1003816100381610038161003816119909210038161003816100381610038162minus119902 sign (119904)) = minus1205722 120574119870119898 119904 + 120593 minus 119860 (119905) 1205721

sdot 120574119870119898 sign (119904)

(17)

where 119904 = 1199091In the case of 1199091 gt 0 the following inequality is satisfied

for 22 = minus1205722 120574

119870119898 119904 + 120593 minus 119860 (119905) 1205721 120574119870119898

= minus1205722 120574119870119898 119904 + 120593 minus 120574

119870119898 (1198871 + 1198872119890minus119886119905) (120593 + 120578)le minus1205722119904⏟⏟⏟⏟⏟⏟⏟lt0

+ 120593 minus 120593 minus 120578⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟lt0

(18)

In the other case of 1199091 lt 0 it follows that2 = minus1205722 120574

119870119898 119904 + 120593 + 119860 (119905) 1205721 120574119870119898

= minus1205722 120574119870119898 119904 + 120593 + 120574

119870119898 (1198871 + 1198872119890minus119886119905) (120593 + 120578)ge minus1205722119904⏟⏟⏟⏟⏟⏟⏟gt0

+ 120593 + 120593 + 120578⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟gt0

(19)

It is concluded that the points 1199091 = 0 1199092 = 0 arenot attractors Thus the motion of sliding surface will notstagnate at these points and it converges to zero in finite time

This completes the proof

Remark 3 Note that the term |1199092|2minus119902sign(119904) of controller(11) is constructed for the stability analysis It is usuallytaken as |1199092|2minus119902sign(1199092) in the existing NTSM controllers forexample the controller (8) However it is not applicable tothe stability analysis of the discussed system (1)

Remark 4 In the case that 120574 is known the proposed con-troller (11) is also applicable by taking 119870119898 = 1205743 Adaptive NTSM Control Scheme Design

In practical situations it is usually difficult to obtain the priorknowledge of 120593 Therefore an adaptive controller is designedfor system (1) in this section In the studies of adaptive NTSMcontrol [22 23] the switching gain was directly adjusted bythe adaptive technique However it was difficult to provethat the points 1199091 = 0 1199092 = 0 were not attractors asin Theorem 2 So the adaptive switching gain was modifiedby the constant control gain in the extreme case that thestagnation problem may happen

In this section a double sliding surfaces control schemeis designed to combine the NTSM control with the adaptivetechnique The first layer is a common integral slidingmode (ISM) in which the impact of uncertain function 120593is compensated The second layer is a NTSM in which thesystem trajectory converges to the origin The purpose ofusing double sliding surfaces is to provide a strict proof forthe stagnation problem of NTSM

In this section the control law is designed as

119906 = 119906119886 + 119906119899 (20)

where 119906119886 denotes an adaptive controller which forces thesystemmotion to the first sliding surface After that a NTSMcontroller 119906119899 drives the system motion to the second slidingsurface

The first sliding surface ISM is designed as

1199041 = 1199092 + 119911 minus 119890minus120579119905 (1199092 (0) + 119911 (0)) = minus119906119899

(21)

where 120579 gt 0 and 1199092(0) 119911(0) are the initial values of 1199092 119911The exponential-decline term is introduced to eliminate thereaching phase [24] Then the system dynamic is on the firstsliding surface at the initial time

Mathematical Problems in Engineering 5

Differentiating (21) with respect to time yields

1199041 = 2 + + Λ = 120593 + 120574119906 minus 119906119899 + Λ= 120593 + Λ + (120574 minus 1) 119906119899 + 120574119906119886

(22)

where Λ = 120579119890minus120579119905(1199092(0) + 119911(0)) Once the first sliding surfaceis established the original system (1) will approach to thesystem motion of an integral chain system as 2 = minus119906119899

Here the NTSM controller 119906119899 for system 2 = minus119906119899 isdesigned as

119906119899 = minus 1120573119902

1003816100381610038161003816119909210038161003816100381610038162minus119902 sign (1199092) minus 12057221199042minus 1205723 1003816100381610038161003816119904210038161003816100381610038161199011 sign (1199042)

(23)

where 1205723 is a positive constant 1199011 is a constant satisfying 0 lt1199011 lt 1 and 1199042 denotes the FNTSM (6)

The continuous controller (23) had been verified in [14]and it also had been proved that the points 1199091 = 0 1199092 =0 were not attractors For an integral chain system withoutuncertainty (eg 2 = minus119906119899) the discontinuous control is notnecessary Although the proposed NTSM controller (11) inTheorem 2 can be used to design 119906119899 it is not employed as it isdiscontinuous

With the following assumption an adaptive controlleris designed in Theorem 6 for system (22) by combining theTSM reaching law with the familiar first-order SMC adaptivecontrol (eg [25ndash27])

Assumption 5 Assume 120593+(120574minus1)119906119899 is unknown and satisfiesthe following inequality

1003816100381610038161003816120593 + (120574 minus 1) 1199061198991003816100381610038161003816 le 1198880 + 1198881 100381610038161003816100381611990911003816100381610038161003816 + 1198882 100381610038161003816100381611990921003816100381610038161003816 (24)

where 1198880 1198881 and 1198882 are unknown positive constants

Theorem6 For system (22) the sliding surface 1199041 converges tozero asymptotically with the following adaptive controller

119906119886 = minus 1119870119898 (11989631199041 + 1198964 1003816100381610038161003816119904110038161003816100381610038161199012 sign (1199041) + |Λ| sign (1199041)

+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sign (1199041)) 1198880 = 1198960 100381610038161003816100381611990411003816100381610038161003816 1198881 = 1198961 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 1198882 = 1198962 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816

(25)

where 1198960 sim 1198964 ge 0 1 ge 1199012 gt 0 and 119894(0) gt 0 (119894 = 0 1 2)Proof Define a new Lyapunov function for 1199041 as

1198812 = 1211990421 + 1

21198960 20 + 1

21198961 21 + 1

21198962 22 (26)

where 119894 = 119888119894 minus 119894 Its derivative is2 = 1199041 1199041 minus 1

1198960 01198880 minus 1

1198961 11198881 minus 1

1198962 21198882 (27)

Substituting controller (25) into (27) yields

2 = 1199041 (120593 + Λ + (120574 minus 1) 119906119899 + 120574119906119886) minus ( 11198960 0

1198880

+ 11198961 1

1198881 + 11198962 2

1198882) = 1199041 (120593 + Λ + (120574 minus 1) 119906119899)+ 1205741199041119906119886 minus 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) le 100381610038161003816100381611990411003816100381610038161003816 1003816100381610038161003816120593+ (120574 minus 1) 1199061198991003816100381610038161003816 + 100381610038161003816100381611990411003816100381610038161003816 |Λ| + 1205741199041119906119886 minus 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816+ 2 100381610038161003816100381611990921003816100381610038161003816) le 100381610038161003816100381611990411003816100381610038161003816 (1198880 + 1198881 100381610038161003816100381611990911003816100381610038161003816 + 1198882 100381610038161003816100381611990921003816100381610038161003816) + 100381610038161003816100381611990411003816100381610038161003816 |Λ|minus 120574

119870119898 (119896311990421 + 1198964 1003816100381610038161003816119904110038161003816100381610038161199012+1 + 100381610038161003816100381611990411003816100381610038161003816 |Λ|

+ 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816)) minus 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816+ 2 100381610038161003816100381611990921003816100381610038161003816)

(28)

With the adaptive laws in (25) and 119894(0) gt 0 it ensuresthat 119894 gt 0 Since 120574 ge 119870119898 it follows that

2 le minus 120574119870119898 (119896311990421 + 1198964 1003816100381610038161003816119904110038161003816100381610038161199012+1)

+ 100381610038161003816100381611990411003816100381610038161003816 (1198880 + 1198881 100381610038161003816100381611990911003816100381610038161003816 + 1198882 100381610038161003816100381611990921003816100381610038161003816)minus 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816)minus 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816)

= minus 120574119870119898 (119896311990421 + 1198964 1003816100381610038161003816119904110038161003816100381610038161199012+1)

(29)

Applying the Barbalat lemma [28] the sliding surface 1199041converges to zero asymptotically

This completes the proof

The boundary layer approach is a common method forchattering reduction It is well known that the chatteringreduction is achieved with a small cost of control precisionMore specifically the sliding surface is driven to a vicinity ofzero In [29] a concept of real sliding surface was introducedwhich was similar to the boundary layer It also means thatthe sliding surface is forced to a small region rather than zeroBased on this concept a modified adaptive law was proposedto avoid the overestimation of adaptive gain in [29] Theadaptive gain slowly decreased to the necessary value oncethe real sliding surface was established

In the case that the boundary layer approach is adoptedfor the proposed adaptive controller (25) the chatteringamplitude can be further attenuated if the adaptive gains0 1 2 decrease to a low level Thus the above two meth-ods are combined to design a modified adaptive controller as

119906119886 = minus 1119870119898 (11989631199041 + 1198964 1003816100381610038161003816119904110038161003816100381610038161199012 sign (1199041) + |Λ| sat (1199041)

+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sat (1199041)) (30)

6 Mathematical Problems in Engineering

o

u

x

m

R(t)

Figure 3 Variable-length pendulum model

where the sat(sdot) function and the adaptive laws are taken as

sat (1199041) = sign (1199041) 100381610038161003816100381611990411003816100381610038161003816 gt 1198741199041119874

100381610038161003816100381611990411003816100381610038161003816 le 119874

1198880 = 1198960 100381610038161003816100381611990411003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 119874) 0 gt 12057211988801198960 0 le 1205721198880

1198881 = 1198961 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 119874) 1 gt 12057211988811198961 1 le 1205721198881

1198882 = 1198962 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 119874) 2 gt 12057211988821198962 2 le 1205721198882

(31)

where a positive constant 119874 denotes the boundary lay 1198960 sim1198962 and 1205721198880 sim 1205721198882 are positive constantsAccording toTheorem 6 the sliding surface 1199041 converges

to zero asymptotically Thus it converges into |1199041| le 119874 infinite time Then the adaptive gains begin to decrease and2 becomes sign indefinite As soon as the sliding surfaceescapes from the region |1199041| le 119874 the adaptive gains increaseagain so that the sliding surface is driven back into the region|1199041| le 119874 Finally the sliding surface 1199041 remains in a largerregion than |1199041| le 119874which is the so-called real sliding surfaceThe second expressions of the modified adaptive laws ensurethe positive values of the adaptive gains and they are valid ina short time

Remark 7 For the tracking control issues it is necessary tomodify the adaptive laws of 1 and 2 to reduce the chatteringamplitude However it is not necessary when the states1199091 1199092 converge to zero4 Simulation

In this section the designed robust and adaptive NTSMcontrollers were tested for the tracking control of a variable-length pendulum [25] (Figure 3) formulated as

= minus2 (119905)119877 (119905) minus 119892

119877 (119905) sin (119909) + 11198981198772 (119905)119906 (32)

The angular coordinate 119909 is impelled to track a referencesignal 119909119888 by the torque 119906 A known mass 119898 moves alonethe rod without friction and the distance 119877(119905) between thecenter 119900 and the mass 119898 is unmeasured It is assumed thatthe states [119909 ]119879 can be measured for the controller designThe parameters of (32) and the reference signal are taken as

119877 (119905) = 1 + 025 sin (4119905) + 05 cos (119905)119898 = 1 kg119892 = 981ms2

119909 (0) = 12119909119888 = 05 sin (05119905)

(33)

Defining the tracking error as 1198901 = 119909 minus 119909119888 the trackingcontrol is converted to the stabilization of the following errorsystem

1198901 = 11989021198902 = minus2 (119905)

119877 (119905) minus 119892119877 (119905) sin (119909) minus 119888 + 1

1198981198772 (119905)119906(34)

41 Robust Control In this subsection three robust control-lers (A B and C) were evaluated According to Theorem 2controllers A and B (without EDSG) were designed using theFNTSM of [11] and the proposed FNTSM (6) respectivelyThe controller C was the proposed FNTSM-EDSG controllerin Theorem 2

Controller A (based on FNTSM of [11])

119906 = minus16 (021199041198651 + 1205721 sign (1199041198651)

+ 790119890572 sign (1199041198651) sign (1198902)

+ 49450119890251 119890572 sign (1199041198651) sign (119890251 1198902))

1199041198651 = 1198901 + 119890751 + 119890972 1205721 = 04 + 04 100381610038161003816100381611990911003816100381610038161003816 + 04 100381610038161003816100381611990921003816100381610038161003816

(35)

Controller B (based on FNTSM (6))

119906 = minus16(021199041198652 + 1205721 sign (1199041198652) + 11199021003816100381610038161003816119890210038161003816100381610038162minus119902 sign (1199041198652))

1199041198652 = 1198901 + 100381610038161003816100381611989021003816100381610038161003816119902 sign (1198902) 119902 = 97 + (1 minus 9

7) sign (100381610038161003816100381611990911003816100381610038161003816 minus 1) (36)

Controller C (based on FNTSM (6) + EDSG)

119906 = minus16(021199041198652 + 119860 (119905) 1205721 sign (1199041198652)

+ 11199021003816100381610038161003816119890210038161003816100381610038162minus119902 sign (1199041198652)) 119860 (119905) = 1 + 119890minus2119905 1205721 = 02 + 02 100381610038161003816100381611990911003816100381610038161003816 + 02 100381610038161003816100381611990921003816100381610038161003816

(37)

The simulation results are given in Figure 4 As shownin Figures 4(a) and 4(b) the angular coordinate converged

Mathematical Problems in Engineering 7

Controller B Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad) Controller A Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad)

Controller C Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad)

(a) Angular coordinate

2 3 4

0

002

Controller A

1 2 3 4 5 6 7 8 9 100Time (s)

0

06

12

Trac

king

erro

r (ra

d)

Controller B

2 3 4

0

002

0

06

12

Trac

king

erro

r (ra

d)

1 2 3 4 5 6 7 8 9 100Time (s)

2 3 4

0

002

Controller C

0

06

12Tr

acki

ng er

ror (

rad)

1 2 3 4 5 6 7 8 9 100Time (s)

(b) Tracking error

1 2 3 4 5 6 7 8 9 100Time (s)

minus10123

Slid

ing

surfa

ces

Controller A

Controller B

minus10123

Slid

ing

surfa

ces

1 2 3 4 5 6 7 8 9 100Time (s)

Controller C

minus10123

Slid

ing

surfa

ces

1 2 3 4 5 6 7 8 9 100Time (s)

(c) Sliding surfaces

1 2 3 4 5 6 7 8 9 100Time (s)

Controller A

minus30

minus10

10

30

Con

trol t

orqu

es (N

)

Controller B

minus30

minus10

10

30

Con

trol t

orqu

es (N

)

1 2 3 4 5 6 7 8 9 100Time (s)

Controller C

minus30

minus10

10

30

Con

trol t

orqu

es (N

)

1 2 3 4 5 6 7 8 9 100Time (s)

(d) Control torque

Figure 4 Simulation result of the robust control

8 Mathematical Problems in Engineering

to the reference signal within 119905 = 4 s The sliding surfacesconverged to zero within 119905 = 15 s (Figure 4(c)) It is clear thatcontroller B cost less convergence time than the controllerA while the control inputs of these controllers had slightdifference (Figures 4(b) and 4(d)) This fact implies thatthe controller designed by the proposed FNTSM was moreefficient As expected the chattering amplitude of controllerC (plusmn5) was half of controller B (plusmn10) (Figure 4(d)) Howeverthe performance of controllerChad small differencewith thatof controller BThe result confirms that the EDSG is effectivefor chattering suppression without deteriorating the systemrobustness

42 Adaptive Control In this subsection the proposed adap-tive NTSM control scheme ((23) and (25)) was verified forthe tracking control As a comparison the adaptive methodin [25] was adopted to design the adaptive controller 119906119886 Thecontrollers were designed as follows

Controller D (Theorem 6)

119906119899 = minus11199021003816100381610038161003816119890210038161003816100381610038162minus119902 sign (1198902) minus 251199042 minus 10038161003816100381610038161199042100381610038161003816100381679 sign (1199042)

119906119886 = minus16 (051199041 + 10038161003816100381610038161199041100381610038161003816100381679 sign (1199041)+ 100381610038161003816100381610038165119890minus5119905 (1198902 (0) + 119911 (0))10038161003816100381610038161003816 sign (1199041)+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sign (1199041))

1199041 = 1198902 + 119911 minus 119890minus5119905 (1198902 (0) + 119911 (0)) = minus119906119899119911 (0) = 01198880 = 100381610038161003816100381611990411003816100381610038161003816 1198881 = 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 1198882 = 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 0 (0) = 1 (0) = 2 (0) = 001

(38)

Controller E (method in [25])

119906119886 = minus5119890minus5119905 (1198902 (0) + 119911 (0)) minus 0 sign (1199041) 1198880 = 5 100381610038161003816100381611990411003816100381610038161003816

(39)

ControllersD and E shared the sameNTSMcontroller 119906119899and the sliding surface 1199042 was taken as 1199041198652 of controller B

The simulation results are given in Figures 5 and 6 It isclear that both controllers D and E achieved fast tracking ofthe reference signal Controller D had shorter convergencetime and better transient performance than controller Eeven though the maximum control torque and the chatteringamplitude of controller D were smaller Meanwhile thesliding surfaces in Figure 6 converged to zero within 119905 =3 s In Figure 6(a) the motions of ISM started from zeroand there was a short time that ISM did not remain zerodue to the affection of uncertainty It is observed that the

FNTSM converged to zero after the ISM and the stagnationof FNTSM never happened It verifies that the designedadaptive NTSM control scheme is effective Since the twocontrollers shared the same NTSM controller 119906119899 the simula-tion results confirm that the proposed adaptive method hadbetter transient performance and was more efficient than themethod in [25]

Here themodified adaptive controller ((30) and (31)) waschecked for chattering reduction In the following controllerF the boundary layer approach and the modified adaptivelaws were both employed As a comparison controller Gonly employed the boundary layer approach In order toavoid the unbounded growing of adaptive gains inside theboundary layer the switching adaptive laws were adoptedfor controller G In addition controllers F and G used thesame boundary layer and NTSM controller 119906119899 of controllerD Since it is assumed that the states can be measured forthe controller design themeasurement noisemay deterioratethe performance of controller in real implementations Toillustrate the robustness of the controller random noise wastaken for the states [119909 ]119879 (mean value 0 and standarddeviation 001)

Controller F

119906119886 = minus16 (051199041 + 10038161003816100381610038161199041100381610038161003816100381679 sign (1199041)+ 100381610038161003816100381610038165119890minus5119905 (1198902 (0) + 119911 (0))10038161003816100381610038161003816 sat (1199041)+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sat (1199041))

sat (1199041) = sign (1199041) 100381610038161003816100381611990411003816100381610038161003816 gt 0041199041

004100381610038161003816100381611990411003816100381610038161003816 le 004

1198880 = 2 100381610038161003816100381611990411003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 004) 0 gt 0011 0 le 001

1198881 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 005) 1 gt 0011 1 le 001

1198882 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 004) 2 gt 0011 2 le 001

(40)

Controller G

119906119886 = minus16 (051199041 + 10038161003816100381610038161199041100381610038161003816100381679 sign (1199041)+ 100381610038161003816100381610038165119890minus5119905 (1198902 (0) + 119911 (0))10038161003816100381610038161003816 sat (1199041)+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sat (1199041))

1198880 = 2 100381610038161003816100381611990411003816100381610038161003816 1199041 gt 0040 1199041 le 004

Mathematical Problems in Engineering 9

Ang

ular

coor

dina

te (r

ad)

Controller DController E

Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus08

minus06

minus04

minus02

0

02

04

06

08

1

(a) Angular coordinate

Trac

king

erro

r (ra

d)

Controller DController E

minus02

0

02

04

06

08

1

1 2 3 4 5 6 7 8 9 100Time (s)

(b) Tracking error

Adaptive gain 1

Adaptive gain 1Adaptive gain 2

Adaptive gain 3

Controller D

Controller E

1 2 3 4 5 6 7 8 9 100Time (s)

0

01

02

Adap

tive g

ains

0

5

10

15

Adap

tive g

ains

1 2 3 4 5 6 7 8 9 100Time (s)

(c) Adaptive gains

0

Controller E

Controller D

minus20

20

Con

trol t

orqu

es (N

)

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus20

0

20C

ontro

l tor

ques

(N)

(d) Control torque

Figure 5 Simulation result of the adaptive control

1198881 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 1199041 gt 0040 1199041 le 004

1198882 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 1199041 gt 0040 1199041 le 004

(41)

The simulation result of controller F is given in Figure 7It is shown that the angular coordinate tracked the referencesignal with a high precision (Figure 7(a)) Due to the adop-tion of the boundary layer and themodified adaptive laws thesliding surface 1199041 converged to a small domain as |1199041| le 01

(Figure 7(b))Moreover the adaptive gains decreased to a lowlevel with approximate reduction ratios as 5108 8479and 1615 respectively (Figure 7(c)) As a result there was amarked reduction in the chattering of controller F comparedto that of controller G in Figure 7(d) Thus the effectivenessof the modified adaptive controller is verified

5 Conclusions

This paper studies the NTSM control for a class of second-order systems in which the coefficient of control input isunknown In this paper a new FNTSM was designed tointegrate the advantages of LSM and NTSM Based onthis FNTSM new forms of robust controller and adaptive

10 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 10minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

Time (s)

ISM

Controller DController E

(a) ISM

0 1 2 3 4 5 6 7 8 9 10minus2

minus15

minus1

minus05

0

05

1

15

2

25

3

Time (s)

FNTS

M

Controller DController E

(b) FNTSM

Figure 6 The response of sliding surfaces

Controller FReference signal

10 20 30 40 500Time (s)

minus1

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad)

(a) Angular coordinate

minus02

0

02

04

06

08

0 10 20 30 40 50Time (s)

Slid

ing

surfa

ces

ISMFNTSM

(b) Sliding surfaces

0

005

01

015

02

025

Adap

tive g

ains

Adaptive gain 1Adaptive gain 2

Adaptive gain 3

10 20 30 40 500Time (s)

(c) Adaptive gains

0 10 20 30 40 50Time (s)

Controller G

Controller F

minus10

0

10

Con

trol t

orqu

es (N

)

minus10

0

10

Con

trol t

orqu

es (N

)

10 20 30 40 500Time (s)

(d) Control torques

Figure 7 Simulation result of the modified adaptive control

Mathematical Problems in Engineering 11

controller were proposed for the discussed uncertain systemThe simulation verified the effectiveness of the proposed con-trollersThe proposed FNTSM shows faster convergence ratethan the NTSM and chattering reduction was achieved bythe EDSG The adaptive controller presented better transientperformance and more efficiency than the adaptive methodin [25] Moreover the modified adaptive controller achievedeffective chattering reduction

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this article

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (no 61375100 no 61472037 and no61433003)

References

[1] J M Zhang C Y Sun R M Zhang and C Qian ldquoAdaptivesliding mode control for re-entry attitude of near space hyper-sonic vehicle based on backstepping designrdquo IEEECAA Journalof Automatica Sinica vol 2 no 1 pp 94ndash101 2015

[2] J Liu Y Zhao B Geng and B Xiao ldquoAdaptive second ordersliding mode control of a fuel cell hybrid system for electricvehicle applicationsrdquo Mathematical Problems in Engineeringvol 2015 Article ID 370424 14 pages 2015

[3] M Zak ldquoTerminal attractors in neural networksrdquo NeuralNetworks vol 2 no 4 pp 259ndash274 1989

[4] S T Venkataraman and S Gulati ldquoTerminal sliding modes anew approach to nonlinear control synthesisrdquo in Proceedings ofthe 5th International Conference onAdvanced Robotics vol 1 pp443ndash448 Pisa Italy June 1991

[5] D Y Zhao S Y Li and F Gao ldquoA new terminal sliding modecontrol for robotic manipulatorsrdquo International Journal of Con-trol vol 82 no 10 pp 1804ndash1813 2009

[6] Z K Song H X Li and K B Sun ldquoFinite-time control fornonlinear spacecraft attitude based on terminal sliding modetechniquerdquo ISA Transactions vol 53 no 1 pp 117ndash124 2014

[7] H Komurcugil ldquoNon-singular terminal sliding-mode controlof DC-DC buck convertersrdquo Control Engineering Practice vol21 no 3 pp 321ndash332 2013

[8] X Yu M Zhihong and Y Wu ldquoTerminal sliding modes withfast transient performancerdquo in Proceedings of the 1997 36th IEEEConference on Decision and Control Part 1 (of 5) vol 2 pp 962ndash963 San Diego Calif USA December 1997

[9] S Mobayen ldquoFast terminal sliding mode controller design fornonlinear second-order systems with time-varying uncertain-tiesrdquo Complexity vol 21 no 2 pp 239ndash244 2015

[10] Z He C Liu Y Zhan H Li X Huang and Z ZhangldquoNonsingular fast terminal sliding mode control with extendedstate observer and tracking differentiator for uncertain nonlin-ear systemsrdquo Mathematical Problems in Engineering vol 2014Article ID 639707 16 pages 2014

[11] L Liu Q M Zhu L Cheng et al ldquoApplied methods and tech-niques for mechatronic systems modelling identification and

controlrdquo Lecture Notes in Control and Information Sciences vol452 pp 79ndash97 2014

[12] X H Yu and Z H Man ldquoOn finite time mechanism terminalsliding modesrdquo in Proceedings of the IEEE International Work-shop on Variable Structure Systems pp 164ndash167 Tokyo Japan1996

[13] L Y Wang T Y Chai and L F Zhai ldquoNeural-network-basedterminal sliding-mode control of robotic manipulators includ-ing actuator dynamicsrdquo IEEE Transactions on Industrial Elec-tronics vol 56 no 9 pp 3296ndash3304 2009

[14] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[15] A Al-Ghanimi J Zheng and Z Man ldquoRobust and fast non-singular terminal sliding mode control for piezoelectric actua-torsrdquo IET ControlTheory ampApplications vol 9 no 18 pp 2678ndash2687 2015

[16] S Mobayen ldquoFinite-time tracking control of chained-formnonholonomic systems with external disturbances based onrecursive terminal sliding mode methodrdquo Nonlinear Dynamicsvol 80 no 1-2 pp 669ndash683 2015

[17] S Mobayen ldquoFast terminal sliding mode tracking of non-holonomic systems with exponential decay raterdquo IET ControlTheory amp Applications vol 9 no 8 pp 1294ndash1301 2015

[18] Y Wu and J Wang ldquoContinuous recursive sliding modecontrol for hypersonic flight vehicle with extended disturbanceobserverrdquo Mathematical Problems in Engineering vol 2015Article ID 506906 26 pages 2015

[19] S Mobayen ldquoAn adaptive fast terminal sliding mode controlcombined with global slidingmode scheme for tracking controlof uncertain nonlinear third-order systemsrdquoNonlinear Dynam-ics vol 82 no 1-2 pp 599ndash610 2015

[20] N Boonsatit and C Pukdeboon ldquoAdaptive fast terminal slidingmode control of magnetic levitation systemrdquo Journal of ControlAutomation and Electrical Systems vol 27 no 4 pp 359ndash3672016

[21] L Y Fang T S Li Z F Li and R Li ldquoAdaptive terminal slidingmode control for anti-synchronization of uncertain chaoticsystemsrdquoNonlinear Dynamics vol 74 no 4 pp 991ndash1002 2013

[22] C-C Yang ldquoSynchronization of second-order chaotic systemsvia adaptive terminal sliding mode control with input nonlin-earityrdquo Journal of the Franklin Institute Engineering and AppliedMathematics vol 349 no 6 pp 2019ndash2032 2012

[23] C-C Yang and C-J Ou ldquoAdaptive terminal sliding mode con-trol subject to input nonlinearity for synchronization of chaoticgyrosrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 3 pp 682ndash691 2013

[24] J-J Kim J-J Lee K-B Park and M-J Youn ldquoDesign ofnew time-varying sliding surface for robot manipulator usingvariable structure controllerrdquo Electronics Letters vol 29 no 2pp 195ndash196 1993

[25] P Li and Z-Q Zheng ldquoRobust adaptive second-order sliding-mode control with fast transient performancerdquo IET ControlTheory amp Applications vol 6 no 2 pp 305ndash312 2012

[26] M Zhihong M OrsquoDay and X Yu ldquoA robust adaptive terminalsliding mode control for rigid robotic manipulatorsrdquo Journal ofIntelligent and Robotic SystemsTheory and Applications vol 24no 1 pp 23ndash41 1999

[27] Z Zhu Y Xia and M Fu ldquoAdaptive sliding mode control forattitude stabilization with actuator saturationrdquo IEEE Transac-tions on Industrial Electronics vol 58 no 10 pp 4898ndash49072011

12 Mathematical Problems in Engineering

[28] H K Khalil Nonlinear Systems Prentice Hall EnglewoodCliffs NJ USA 3rd edition 2002

[29] Y Shtessel M Taleb and F Plestan ldquoA novel adaptive-gainsupertwisting sliding mode controller methodology and appli-cationrdquo Automatica vol 48 no 5 pp 759ndash769 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Substituting the controller (11) into (13) yields

1 = 119904 (1199092 + 120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 (120593 minus 120574119870119898 (1205722119904

+ 119860 (119905) 1205721 sign (119904) + 1120573119902

1003816100381610038161003816119909210038161003816100381610038162minus119902 sign (119904)))) = 1199041199092

+ 120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 (120593119904 minus 1205741198701198981205722119904

2 minus 120574119870119898119860 (119905) 1205721 |119904| minus 120574

119870119898sdot 1120573119902

1003816100381610038161003816119909210038161003816100381610038162minus119902 |119904|) = minus120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 1205741198701198981205722119904

2 + 1199041199092

minus 120574119870119898 |119904| 100381610038161003816100381611990921003816100381610038161003816 + 120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 (120593119904 minus 120574

119870119898119860 (119905) 1205721 |119904|)

(14)

With 120574 ge 119870119898 and |1199092|119902minus1 gt 0 (14) turns to1 le minus120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 12057221199042 + 1199041199092 minus |119904| 100381610038161003816100381611990921003816100381610038161003816

+ 120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 (10038161003816100381610038161205931003816100381610038161003816 |119904| minus 119860 (119905) 1205721 |119904|) (15)

As 119860(119905)1205721 ge 120593 + 120578 it is obtained as

1 le minus120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 12057221199042 minus 120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 120578 |119904|= minus120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 (12057221199042 + 120578 |119904|)= minus120573119902 100381610038161003816100381611990921003816100381610038161003816119902minus1 (212057221198811 + 120578212119881121 )

(16)

Considering the last expression of (16) if the motion ofsliding surface does not stagnate at the points 1199091 = 0 1199092 =0 in the reaching phase it would converge to zero in finitetime according to the extended Lyapunov description of finitetime stability in Remark 2 of [14]

Here a demonstration is given to show that themotion ofsliding surface does not stagnate at the points 1199091 = 0 1199092 =0

For system (1) with the controller (11) taking 1199092 = 0 weobtain

2 = 120593 minus 120574119870119898 (1205722119904 + 119860 (119905) 1205721 sign (119904)

+ 1120573119902

1003816100381610038161003816119909210038161003816100381610038162minus119902 sign (119904)) = minus1205722 120574119870119898 119904 + 120593 minus 119860 (119905) 1205721

sdot 120574119870119898 sign (119904)

(17)

where 119904 = 1199091In the case of 1199091 gt 0 the following inequality is satisfied

for 22 = minus1205722 120574

119870119898 119904 + 120593 minus 119860 (119905) 1205721 120574119870119898

= minus1205722 120574119870119898 119904 + 120593 minus 120574

119870119898 (1198871 + 1198872119890minus119886119905) (120593 + 120578)le minus1205722119904⏟⏟⏟⏟⏟⏟⏟lt0

+ 120593 minus 120593 minus 120578⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟lt0

(18)

In the other case of 1199091 lt 0 it follows that2 = minus1205722 120574

119870119898 119904 + 120593 + 119860 (119905) 1205721 120574119870119898

= minus1205722 120574119870119898 119904 + 120593 + 120574

119870119898 (1198871 + 1198872119890minus119886119905) (120593 + 120578)ge minus1205722119904⏟⏟⏟⏟⏟⏟⏟gt0

+ 120593 + 120593 + 120578⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟gt0

(19)

It is concluded that the points 1199091 = 0 1199092 = 0 arenot attractors Thus the motion of sliding surface will notstagnate at these points and it converges to zero in finite time

This completes the proof

Remark 3 Note that the term |1199092|2minus119902sign(119904) of controller(11) is constructed for the stability analysis It is usuallytaken as |1199092|2minus119902sign(1199092) in the existing NTSM controllers forexample the controller (8) However it is not applicable tothe stability analysis of the discussed system (1)

Remark 4 In the case that 120574 is known the proposed con-troller (11) is also applicable by taking 119870119898 = 1205743 Adaptive NTSM Control Scheme Design

In practical situations it is usually difficult to obtain the priorknowledge of 120593 Therefore an adaptive controller is designedfor system (1) in this section In the studies of adaptive NTSMcontrol [22 23] the switching gain was directly adjusted bythe adaptive technique However it was difficult to provethat the points 1199091 = 0 1199092 = 0 were not attractors asin Theorem 2 So the adaptive switching gain was modifiedby the constant control gain in the extreme case that thestagnation problem may happen

In this section a double sliding surfaces control schemeis designed to combine the NTSM control with the adaptivetechnique The first layer is a common integral slidingmode (ISM) in which the impact of uncertain function 120593is compensated The second layer is a NTSM in which thesystem trajectory converges to the origin The purpose ofusing double sliding surfaces is to provide a strict proof forthe stagnation problem of NTSM

In this section the control law is designed as

119906 = 119906119886 + 119906119899 (20)

where 119906119886 denotes an adaptive controller which forces thesystemmotion to the first sliding surface After that a NTSMcontroller 119906119899 drives the system motion to the second slidingsurface

The first sliding surface ISM is designed as

1199041 = 1199092 + 119911 minus 119890minus120579119905 (1199092 (0) + 119911 (0)) = minus119906119899

(21)

where 120579 gt 0 and 1199092(0) 119911(0) are the initial values of 1199092 119911The exponential-decline term is introduced to eliminate thereaching phase [24] Then the system dynamic is on the firstsliding surface at the initial time

Mathematical Problems in Engineering 5

Differentiating (21) with respect to time yields

1199041 = 2 + + Λ = 120593 + 120574119906 minus 119906119899 + Λ= 120593 + Λ + (120574 minus 1) 119906119899 + 120574119906119886

(22)

where Λ = 120579119890minus120579119905(1199092(0) + 119911(0)) Once the first sliding surfaceis established the original system (1) will approach to thesystem motion of an integral chain system as 2 = minus119906119899

Here the NTSM controller 119906119899 for system 2 = minus119906119899 isdesigned as

119906119899 = minus 1120573119902

1003816100381610038161003816119909210038161003816100381610038162minus119902 sign (1199092) minus 12057221199042minus 1205723 1003816100381610038161003816119904210038161003816100381610038161199011 sign (1199042)

(23)

where 1205723 is a positive constant 1199011 is a constant satisfying 0 lt1199011 lt 1 and 1199042 denotes the FNTSM (6)

The continuous controller (23) had been verified in [14]and it also had been proved that the points 1199091 = 0 1199092 =0 were not attractors For an integral chain system withoutuncertainty (eg 2 = minus119906119899) the discontinuous control is notnecessary Although the proposed NTSM controller (11) inTheorem 2 can be used to design 119906119899 it is not employed as it isdiscontinuous

With the following assumption an adaptive controlleris designed in Theorem 6 for system (22) by combining theTSM reaching law with the familiar first-order SMC adaptivecontrol (eg [25ndash27])

Assumption 5 Assume 120593+(120574minus1)119906119899 is unknown and satisfiesthe following inequality

1003816100381610038161003816120593 + (120574 minus 1) 1199061198991003816100381610038161003816 le 1198880 + 1198881 100381610038161003816100381611990911003816100381610038161003816 + 1198882 100381610038161003816100381611990921003816100381610038161003816 (24)

where 1198880 1198881 and 1198882 are unknown positive constants

Theorem6 For system (22) the sliding surface 1199041 converges tozero asymptotically with the following adaptive controller

119906119886 = minus 1119870119898 (11989631199041 + 1198964 1003816100381610038161003816119904110038161003816100381610038161199012 sign (1199041) + |Λ| sign (1199041)

+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sign (1199041)) 1198880 = 1198960 100381610038161003816100381611990411003816100381610038161003816 1198881 = 1198961 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 1198882 = 1198962 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816

(25)

where 1198960 sim 1198964 ge 0 1 ge 1199012 gt 0 and 119894(0) gt 0 (119894 = 0 1 2)Proof Define a new Lyapunov function for 1199041 as

1198812 = 1211990421 + 1

21198960 20 + 1

21198961 21 + 1

21198962 22 (26)

where 119894 = 119888119894 minus 119894 Its derivative is2 = 1199041 1199041 minus 1

1198960 01198880 minus 1

1198961 11198881 minus 1

1198962 21198882 (27)

Substituting controller (25) into (27) yields

2 = 1199041 (120593 + Λ + (120574 minus 1) 119906119899 + 120574119906119886) minus ( 11198960 0

1198880

+ 11198961 1

1198881 + 11198962 2

1198882) = 1199041 (120593 + Λ + (120574 minus 1) 119906119899)+ 1205741199041119906119886 minus 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) le 100381610038161003816100381611990411003816100381610038161003816 1003816100381610038161003816120593+ (120574 minus 1) 1199061198991003816100381610038161003816 + 100381610038161003816100381611990411003816100381610038161003816 |Λ| + 1205741199041119906119886 minus 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816+ 2 100381610038161003816100381611990921003816100381610038161003816) le 100381610038161003816100381611990411003816100381610038161003816 (1198880 + 1198881 100381610038161003816100381611990911003816100381610038161003816 + 1198882 100381610038161003816100381611990921003816100381610038161003816) + 100381610038161003816100381611990411003816100381610038161003816 |Λ|minus 120574

119870119898 (119896311990421 + 1198964 1003816100381610038161003816119904110038161003816100381610038161199012+1 + 100381610038161003816100381611990411003816100381610038161003816 |Λ|

+ 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816)) minus 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816+ 2 100381610038161003816100381611990921003816100381610038161003816)

(28)

With the adaptive laws in (25) and 119894(0) gt 0 it ensuresthat 119894 gt 0 Since 120574 ge 119870119898 it follows that

2 le minus 120574119870119898 (119896311990421 + 1198964 1003816100381610038161003816119904110038161003816100381610038161199012+1)

+ 100381610038161003816100381611990411003816100381610038161003816 (1198880 + 1198881 100381610038161003816100381611990911003816100381610038161003816 + 1198882 100381610038161003816100381611990921003816100381610038161003816)minus 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816)minus 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816)

= minus 120574119870119898 (119896311990421 + 1198964 1003816100381610038161003816119904110038161003816100381610038161199012+1)

(29)

Applying the Barbalat lemma [28] the sliding surface 1199041converges to zero asymptotically

This completes the proof

The boundary layer approach is a common method forchattering reduction It is well known that the chatteringreduction is achieved with a small cost of control precisionMore specifically the sliding surface is driven to a vicinity ofzero In [29] a concept of real sliding surface was introducedwhich was similar to the boundary layer It also means thatthe sliding surface is forced to a small region rather than zeroBased on this concept a modified adaptive law was proposedto avoid the overestimation of adaptive gain in [29] Theadaptive gain slowly decreased to the necessary value oncethe real sliding surface was established

In the case that the boundary layer approach is adoptedfor the proposed adaptive controller (25) the chatteringamplitude can be further attenuated if the adaptive gains0 1 2 decrease to a low level Thus the above two meth-ods are combined to design a modified adaptive controller as

119906119886 = minus 1119870119898 (11989631199041 + 1198964 1003816100381610038161003816119904110038161003816100381610038161199012 sign (1199041) + |Λ| sat (1199041)

+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sat (1199041)) (30)

6 Mathematical Problems in Engineering

o

u

x

m

R(t)

Figure 3 Variable-length pendulum model

where the sat(sdot) function and the adaptive laws are taken as

sat (1199041) = sign (1199041) 100381610038161003816100381611990411003816100381610038161003816 gt 1198741199041119874

100381610038161003816100381611990411003816100381610038161003816 le 119874

1198880 = 1198960 100381610038161003816100381611990411003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 119874) 0 gt 12057211988801198960 0 le 1205721198880

1198881 = 1198961 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 119874) 1 gt 12057211988811198961 1 le 1205721198881

1198882 = 1198962 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 119874) 2 gt 12057211988821198962 2 le 1205721198882

(31)

where a positive constant 119874 denotes the boundary lay 1198960 sim1198962 and 1205721198880 sim 1205721198882 are positive constantsAccording toTheorem 6 the sliding surface 1199041 converges

to zero asymptotically Thus it converges into |1199041| le 119874 infinite time Then the adaptive gains begin to decrease and2 becomes sign indefinite As soon as the sliding surfaceescapes from the region |1199041| le 119874 the adaptive gains increaseagain so that the sliding surface is driven back into the region|1199041| le 119874 Finally the sliding surface 1199041 remains in a largerregion than |1199041| le 119874which is the so-called real sliding surfaceThe second expressions of the modified adaptive laws ensurethe positive values of the adaptive gains and they are valid ina short time

Remark 7 For the tracking control issues it is necessary tomodify the adaptive laws of 1 and 2 to reduce the chatteringamplitude However it is not necessary when the states1199091 1199092 converge to zero4 Simulation

In this section the designed robust and adaptive NTSMcontrollers were tested for the tracking control of a variable-length pendulum [25] (Figure 3) formulated as

= minus2 (119905)119877 (119905) minus 119892

119877 (119905) sin (119909) + 11198981198772 (119905)119906 (32)

The angular coordinate 119909 is impelled to track a referencesignal 119909119888 by the torque 119906 A known mass 119898 moves alonethe rod without friction and the distance 119877(119905) between thecenter 119900 and the mass 119898 is unmeasured It is assumed thatthe states [119909 ]119879 can be measured for the controller designThe parameters of (32) and the reference signal are taken as

119877 (119905) = 1 + 025 sin (4119905) + 05 cos (119905)119898 = 1 kg119892 = 981ms2

119909 (0) = 12119909119888 = 05 sin (05119905)

(33)

Defining the tracking error as 1198901 = 119909 minus 119909119888 the trackingcontrol is converted to the stabilization of the following errorsystem

1198901 = 11989021198902 = minus2 (119905)

119877 (119905) minus 119892119877 (119905) sin (119909) minus 119888 + 1

1198981198772 (119905)119906(34)

41 Robust Control In this subsection three robust control-lers (A B and C) were evaluated According to Theorem 2controllers A and B (without EDSG) were designed using theFNTSM of [11] and the proposed FNTSM (6) respectivelyThe controller C was the proposed FNTSM-EDSG controllerin Theorem 2

Controller A (based on FNTSM of [11])

119906 = minus16 (021199041198651 + 1205721 sign (1199041198651)

+ 790119890572 sign (1199041198651) sign (1198902)

+ 49450119890251 119890572 sign (1199041198651) sign (119890251 1198902))

1199041198651 = 1198901 + 119890751 + 119890972 1205721 = 04 + 04 100381610038161003816100381611990911003816100381610038161003816 + 04 100381610038161003816100381611990921003816100381610038161003816

(35)

Controller B (based on FNTSM (6))

119906 = minus16(021199041198652 + 1205721 sign (1199041198652) + 11199021003816100381610038161003816119890210038161003816100381610038162minus119902 sign (1199041198652))

1199041198652 = 1198901 + 100381610038161003816100381611989021003816100381610038161003816119902 sign (1198902) 119902 = 97 + (1 minus 9

7) sign (100381610038161003816100381611990911003816100381610038161003816 minus 1) (36)

Controller C (based on FNTSM (6) + EDSG)

119906 = minus16(021199041198652 + 119860 (119905) 1205721 sign (1199041198652)

+ 11199021003816100381610038161003816119890210038161003816100381610038162minus119902 sign (1199041198652)) 119860 (119905) = 1 + 119890minus2119905 1205721 = 02 + 02 100381610038161003816100381611990911003816100381610038161003816 + 02 100381610038161003816100381611990921003816100381610038161003816

(37)

The simulation results are given in Figure 4 As shownin Figures 4(a) and 4(b) the angular coordinate converged

Mathematical Problems in Engineering 7

Controller B Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad) Controller A Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad)

Controller C Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad)

(a) Angular coordinate

2 3 4

0

002

Controller A

1 2 3 4 5 6 7 8 9 100Time (s)

0

06

12

Trac

king

erro

r (ra

d)

Controller B

2 3 4

0

002

0

06

12

Trac

king

erro

r (ra

d)

1 2 3 4 5 6 7 8 9 100Time (s)

2 3 4

0

002

Controller C

0

06

12Tr

acki

ng er

ror (

rad)

1 2 3 4 5 6 7 8 9 100Time (s)

(b) Tracking error

1 2 3 4 5 6 7 8 9 100Time (s)

minus10123

Slid

ing

surfa

ces

Controller A

Controller B

minus10123

Slid

ing

surfa

ces

1 2 3 4 5 6 7 8 9 100Time (s)

Controller C

minus10123

Slid

ing

surfa

ces

1 2 3 4 5 6 7 8 9 100Time (s)

(c) Sliding surfaces

1 2 3 4 5 6 7 8 9 100Time (s)

Controller A

minus30

minus10

10

30

Con

trol t

orqu

es (N

)

Controller B

minus30

minus10

10

30

Con

trol t

orqu

es (N

)

1 2 3 4 5 6 7 8 9 100Time (s)

Controller C

minus30

minus10

10

30

Con

trol t

orqu

es (N

)

1 2 3 4 5 6 7 8 9 100Time (s)

(d) Control torque

Figure 4 Simulation result of the robust control

8 Mathematical Problems in Engineering

to the reference signal within 119905 = 4 s The sliding surfacesconverged to zero within 119905 = 15 s (Figure 4(c)) It is clear thatcontroller B cost less convergence time than the controllerA while the control inputs of these controllers had slightdifference (Figures 4(b) and 4(d)) This fact implies thatthe controller designed by the proposed FNTSM was moreefficient As expected the chattering amplitude of controllerC (plusmn5) was half of controller B (plusmn10) (Figure 4(d)) Howeverthe performance of controllerChad small differencewith thatof controller BThe result confirms that the EDSG is effectivefor chattering suppression without deteriorating the systemrobustness

42 Adaptive Control In this subsection the proposed adap-tive NTSM control scheme ((23) and (25)) was verified forthe tracking control As a comparison the adaptive methodin [25] was adopted to design the adaptive controller 119906119886 Thecontrollers were designed as follows

Controller D (Theorem 6)

119906119899 = minus11199021003816100381610038161003816119890210038161003816100381610038162minus119902 sign (1198902) minus 251199042 minus 10038161003816100381610038161199042100381610038161003816100381679 sign (1199042)

119906119886 = minus16 (051199041 + 10038161003816100381610038161199041100381610038161003816100381679 sign (1199041)+ 100381610038161003816100381610038165119890minus5119905 (1198902 (0) + 119911 (0))10038161003816100381610038161003816 sign (1199041)+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sign (1199041))

1199041 = 1198902 + 119911 minus 119890minus5119905 (1198902 (0) + 119911 (0)) = minus119906119899119911 (0) = 01198880 = 100381610038161003816100381611990411003816100381610038161003816 1198881 = 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 1198882 = 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 0 (0) = 1 (0) = 2 (0) = 001

(38)

Controller E (method in [25])

119906119886 = minus5119890minus5119905 (1198902 (0) + 119911 (0)) minus 0 sign (1199041) 1198880 = 5 100381610038161003816100381611990411003816100381610038161003816

(39)

ControllersD and E shared the sameNTSMcontroller 119906119899and the sliding surface 1199042 was taken as 1199041198652 of controller B

The simulation results are given in Figures 5 and 6 It isclear that both controllers D and E achieved fast tracking ofthe reference signal Controller D had shorter convergencetime and better transient performance than controller Eeven though the maximum control torque and the chatteringamplitude of controller D were smaller Meanwhile thesliding surfaces in Figure 6 converged to zero within 119905 =3 s In Figure 6(a) the motions of ISM started from zeroand there was a short time that ISM did not remain zerodue to the affection of uncertainty It is observed that the

FNTSM converged to zero after the ISM and the stagnationof FNTSM never happened It verifies that the designedadaptive NTSM control scheme is effective Since the twocontrollers shared the same NTSM controller 119906119899 the simula-tion results confirm that the proposed adaptive method hadbetter transient performance and was more efficient than themethod in [25]

Here themodified adaptive controller ((30) and (31)) waschecked for chattering reduction In the following controllerF the boundary layer approach and the modified adaptivelaws were both employed As a comparison controller Gonly employed the boundary layer approach In order toavoid the unbounded growing of adaptive gains inside theboundary layer the switching adaptive laws were adoptedfor controller G In addition controllers F and G used thesame boundary layer and NTSM controller 119906119899 of controllerD Since it is assumed that the states can be measured forthe controller design themeasurement noisemay deterioratethe performance of controller in real implementations Toillustrate the robustness of the controller random noise wastaken for the states [119909 ]119879 (mean value 0 and standarddeviation 001)

Controller F

119906119886 = minus16 (051199041 + 10038161003816100381610038161199041100381610038161003816100381679 sign (1199041)+ 100381610038161003816100381610038165119890minus5119905 (1198902 (0) + 119911 (0))10038161003816100381610038161003816 sat (1199041)+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sat (1199041))

sat (1199041) = sign (1199041) 100381610038161003816100381611990411003816100381610038161003816 gt 0041199041

004100381610038161003816100381611990411003816100381610038161003816 le 004

1198880 = 2 100381610038161003816100381611990411003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 004) 0 gt 0011 0 le 001

1198881 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 005) 1 gt 0011 1 le 001

1198882 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 004) 2 gt 0011 2 le 001

(40)

Controller G

119906119886 = minus16 (051199041 + 10038161003816100381610038161199041100381610038161003816100381679 sign (1199041)+ 100381610038161003816100381610038165119890minus5119905 (1198902 (0) + 119911 (0))10038161003816100381610038161003816 sat (1199041)+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sat (1199041))

1198880 = 2 100381610038161003816100381611990411003816100381610038161003816 1199041 gt 0040 1199041 le 004

Mathematical Problems in Engineering 9

Ang

ular

coor

dina

te (r

ad)

Controller DController E

Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus08

minus06

minus04

minus02

0

02

04

06

08

1

(a) Angular coordinate

Trac

king

erro

r (ra

d)

Controller DController E

minus02

0

02

04

06

08

1

1 2 3 4 5 6 7 8 9 100Time (s)

(b) Tracking error

Adaptive gain 1

Adaptive gain 1Adaptive gain 2

Adaptive gain 3

Controller D

Controller E

1 2 3 4 5 6 7 8 9 100Time (s)

0

01

02

Adap

tive g

ains

0

5

10

15

Adap

tive g

ains

1 2 3 4 5 6 7 8 9 100Time (s)

(c) Adaptive gains

0

Controller E

Controller D

minus20

20

Con

trol t

orqu

es (N

)

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus20

0

20C

ontro

l tor

ques

(N)

(d) Control torque

Figure 5 Simulation result of the adaptive control

1198881 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 1199041 gt 0040 1199041 le 004

1198882 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 1199041 gt 0040 1199041 le 004

(41)

The simulation result of controller F is given in Figure 7It is shown that the angular coordinate tracked the referencesignal with a high precision (Figure 7(a)) Due to the adop-tion of the boundary layer and themodified adaptive laws thesliding surface 1199041 converged to a small domain as |1199041| le 01

(Figure 7(b))Moreover the adaptive gains decreased to a lowlevel with approximate reduction ratios as 5108 8479and 1615 respectively (Figure 7(c)) As a result there was amarked reduction in the chattering of controller F comparedto that of controller G in Figure 7(d) Thus the effectivenessof the modified adaptive controller is verified

5 Conclusions

This paper studies the NTSM control for a class of second-order systems in which the coefficient of control input isunknown In this paper a new FNTSM was designed tointegrate the advantages of LSM and NTSM Based onthis FNTSM new forms of robust controller and adaptive

10 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 10minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

Time (s)

ISM

Controller DController E

(a) ISM

0 1 2 3 4 5 6 7 8 9 10minus2

minus15

minus1

minus05

0

05

1

15

2

25

3

Time (s)

FNTS

M

Controller DController E

(b) FNTSM

Figure 6 The response of sliding surfaces

Controller FReference signal

10 20 30 40 500Time (s)

minus1

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad)

(a) Angular coordinate

minus02

0

02

04

06

08

0 10 20 30 40 50Time (s)

Slid

ing

surfa

ces

ISMFNTSM

(b) Sliding surfaces

0

005

01

015

02

025

Adap

tive g

ains

Adaptive gain 1Adaptive gain 2

Adaptive gain 3

10 20 30 40 500Time (s)

(c) Adaptive gains

0 10 20 30 40 50Time (s)

Controller G

Controller F

minus10

0

10

Con

trol t

orqu

es (N

)

minus10

0

10

Con

trol t

orqu

es (N

)

10 20 30 40 500Time (s)

(d) Control torques

Figure 7 Simulation result of the modified adaptive control

Mathematical Problems in Engineering 11

controller were proposed for the discussed uncertain systemThe simulation verified the effectiveness of the proposed con-trollersThe proposed FNTSM shows faster convergence ratethan the NTSM and chattering reduction was achieved bythe EDSG The adaptive controller presented better transientperformance and more efficiency than the adaptive methodin [25] Moreover the modified adaptive controller achievedeffective chattering reduction

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this article

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (no 61375100 no 61472037 and no61433003)

References

[1] J M Zhang C Y Sun R M Zhang and C Qian ldquoAdaptivesliding mode control for re-entry attitude of near space hyper-sonic vehicle based on backstepping designrdquo IEEECAA Journalof Automatica Sinica vol 2 no 1 pp 94ndash101 2015

[2] J Liu Y Zhao B Geng and B Xiao ldquoAdaptive second ordersliding mode control of a fuel cell hybrid system for electricvehicle applicationsrdquo Mathematical Problems in Engineeringvol 2015 Article ID 370424 14 pages 2015

[3] M Zak ldquoTerminal attractors in neural networksrdquo NeuralNetworks vol 2 no 4 pp 259ndash274 1989

[4] S T Venkataraman and S Gulati ldquoTerminal sliding modes anew approach to nonlinear control synthesisrdquo in Proceedings ofthe 5th International Conference onAdvanced Robotics vol 1 pp443ndash448 Pisa Italy June 1991

[5] D Y Zhao S Y Li and F Gao ldquoA new terminal sliding modecontrol for robotic manipulatorsrdquo International Journal of Con-trol vol 82 no 10 pp 1804ndash1813 2009

[6] Z K Song H X Li and K B Sun ldquoFinite-time control fornonlinear spacecraft attitude based on terminal sliding modetechniquerdquo ISA Transactions vol 53 no 1 pp 117ndash124 2014

[7] H Komurcugil ldquoNon-singular terminal sliding-mode controlof DC-DC buck convertersrdquo Control Engineering Practice vol21 no 3 pp 321ndash332 2013

[8] X Yu M Zhihong and Y Wu ldquoTerminal sliding modes withfast transient performancerdquo in Proceedings of the 1997 36th IEEEConference on Decision and Control Part 1 (of 5) vol 2 pp 962ndash963 San Diego Calif USA December 1997

[9] S Mobayen ldquoFast terminal sliding mode controller design fornonlinear second-order systems with time-varying uncertain-tiesrdquo Complexity vol 21 no 2 pp 239ndash244 2015

[10] Z He C Liu Y Zhan H Li X Huang and Z ZhangldquoNonsingular fast terminal sliding mode control with extendedstate observer and tracking differentiator for uncertain nonlin-ear systemsrdquo Mathematical Problems in Engineering vol 2014Article ID 639707 16 pages 2014

[11] L Liu Q M Zhu L Cheng et al ldquoApplied methods and tech-niques for mechatronic systems modelling identification and

controlrdquo Lecture Notes in Control and Information Sciences vol452 pp 79ndash97 2014

[12] X H Yu and Z H Man ldquoOn finite time mechanism terminalsliding modesrdquo in Proceedings of the IEEE International Work-shop on Variable Structure Systems pp 164ndash167 Tokyo Japan1996

[13] L Y Wang T Y Chai and L F Zhai ldquoNeural-network-basedterminal sliding-mode control of robotic manipulators includ-ing actuator dynamicsrdquo IEEE Transactions on Industrial Elec-tronics vol 56 no 9 pp 3296ndash3304 2009

[14] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[15] A Al-Ghanimi J Zheng and Z Man ldquoRobust and fast non-singular terminal sliding mode control for piezoelectric actua-torsrdquo IET ControlTheory ampApplications vol 9 no 18 pp 2678ndash2687 2015

[16] S Mobayen ldquoFinite-time tracking control of chained-formnonholonomic systems with external disturbances based onrecursive terminal sliding mode methodrdquo Nonlinear Dynamicsvol 80 no 1-2 pp 669ndash683 2015

[17] S Mobayen ldquoFast terminal sliding mode tracking of non-holonomic systems with exponential decay raterdquo IET ControlTheory amp Applications vol 9 no 8 pp 1294ndash1301 2015

[18] Y Wu and J Wang ldquoContinuous recursive sliding modecontrol for hypersonic flight vehicle with extended disturbanceobserverrdquo Mathematical Problems in Engineering vol 2015Article ID 506906 26 pages 2015

[19] S Mobayen ldquoAn adaptive fast terminal sliding mode controlcombined with global slidingmode scheme for tracking controlof uncertain nonlinear third-order systemsrdquoNonlinear Dynam-ics vol 82 no 1-2 pp 599ndash610 2015

[20] N Boonsatit and C Pukdeboon ldquoAdaptive fast terminal slidingmode control of magnetic levitation systemrdquo Journal of ControlAutomation and Electrical Systems vol 27 no 4 pp 359ndash3672016

[21] L Y Fang T S Li Z F Li and R Li ldquoAdaptive terminal slidingmode control for anti-synchronization of uncertain chaoticsystemsrdquoNonlinear Dynamics vol 74 no 4 pp 991ndash1002 2013

[22] C-C Yang ldquoSynchronization of second-order chaotic systemsvia adaptive terminal sliding mode control with input nonlin-earityrdquo Journal of the Franklin Institute Engineering and AppliedMathematics vol 349 no 6 pp 2019ndash2032 2012

[23] C-C Yang and C-J Ou ldquoAdaptive terminal sliding mode con-trol subject to input nonlinearity for synchronization of chaoticgyrosrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 3 pp 682ndash691 2013

[24] J-J Kim J-J Lee K-B Park and M-J Youn ldquoDesign ofnew time-varying sliding surface for robot manipulator usingvariable structure controllerrdquo Electronics Letters vol 29 no 2pp 195ndash196 1993

[25] P Li and Z-Q Zheng ldquoRobust adaptive second-order sliding-mode control with fast transient performancerdquo IET ControlTheory amp Applications vol 6 no 2 pp 305ndash312 2012

[26] M Zhihong M OrsquoDay and X Yu ldquoA robust adaptive terminalsliding mode control for rigid robotic manipulatorsrdquo Journal ofIntelligent and Robotic SystemsTheory and Applications vol 24no 1 pp 23ndash41 1999

[27] Z Zhu Y Xia and M Fu ldquoAdaptive sliding mode control forattitude stabilization with actuator saturationrdquo IEEE Transac-tions on Industrial Electronics vol 58 no 10 pp 4898ndash49072011

12 Mathematical Problems in Engineering

[28] H K Khalil Nonlinear Systems Prentice Hall EnglewoodCliffs NJ USA 3rd edition 2002

[29] Y Shtessel M Taleb and F Plestan ldquoA novel adaptive-gainsupertwisting sliding mode controller methodology and appli-cationrdquo Automatica vol 48 no 5 pp 759ndash769 2012

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Differentiating (21) with respect to time yields

1199041 = 2 + + Λ = 120593 + 120574119906 minus 119906119899 + Λ= 120593 + Λ + (120574 minus 1) 119906119899 + 120574119906119886

(22)

where Λ = 120579119890minus120579119905(1199092(0) + 119911(0)) Once the first sliding surfaceis established the original system (1) will approach to thesystem motion of an integral chain system as 2 = minus119906119899

Here the NTSM controller 119906119899 for system 2 = minus119906119899 isdesigned as

119906119899 = minus 1120573119902

1003816100381610038161003816119909210038161003816100381610038162minus119902 sign (1199092) minus 12057221199042minus 1205723 1003816100381610038161003816119904210038161003816100381610038161199011 sign (1199042)

(23)

where 1205723 is a positive constant 1199011 is a constant satisfying 0 lt1199011 lt 1 and 1199042 denotes the FNTSM (6)

The continuous controller (23) had been verified in [14]and it also had been proved that the points 1199091 = 0 1199092 =0 were not attractors For an integral chain system withoutuncertainty (eg 2 = minus119906119899) the discontinuous control is notnecessary Although the proposed NTSM controller (11) inTheorem 2 can be used to design 119906119899 it is not employed as it isdiscontinuous

With the following assumption an adaptive controlleris designed in Theorem 6 for system (22) by combining theTSM reaching law with the familiar first-order SMC adaptivecontrol (eg [25ndash27])

Assumption 5 Assume 120593+(120574minus1)119906119899 is unknown and satisfiesthe following inequality

1003816100381610038161003816120593 + (120574 minus 1) 1199061198991003816100381610038161003816 le 1198880 + 1198881 100381610038161003816100381611990911003816100381610038161003816 + 1198882 100381610038161003816100381611990921003816100381610038161003816 (24)

where 1198880 1198881 and 1198882 are unknown positive constants

Theorem6 For system (22) the sliding surface 1199041 converges tozero asymptotically with the following adaptive controller

119906119886 = minus 1119870119898 (11989631199041 + 1198964 1003816100381610038161003816119904110038161003816100381610038161199012 sign (1199041) + |Λ| sign (1199041)

+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sign (1199041)) 1198880 = 1198960 100381610038161003816100381611990411003816100381610038161003816 1198881 = 1198961 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 1198882 = 1198962 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816

(25)

where 1198960 sim 1198964 ge 0 1 ge 1199012 gt 0 and 119894(0) gt 0 (119894 = 0 1 2)Proof Define a new Lyapunov function for 1199041 as

1198812 = 1211990421 + 1

21198960 20 + 1

21198961 21 + 1

21198962 22 (26)

where 119894 = 119888119894 minus 119894 Its derivative is2 = 1199041 1199041 minus 1

1198960 01198880 minus 1

1198961 11198881 minus 1

1198962 21198882 (27)

Substituting controller (25) into (27) yields

2 = 1199041 (120593 + Λ + (120574 minus 1) 119906119899 + 120574119906119886) minus ( 11198960 0

1198880

+ 11198961 1

1198881 + 11198962 2

1198882) = 1199041 (120593 + Λ + (120574 minus 1) 119906119899)+ 1205741199041119906119886 minus 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) le 100381610038161003816100381611990411003816100381610038161003816 1003816100381610038161003816120593+ (120574 minus 1) 1199061198991003816100381610038161003816 + 100381610038161003816100381611990411003816100381610038161003816 |Λ| + 1205741199041119906119886 minus 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816+ 2 100381610038161003816100381611990921003816100381610038161003816) le 100381610038161003816100381611990411003816100381610038161003816 (1198880 + 1198881 100381610038161003816100381611990911003816100381610038161003816 + 1198882 100381610038161003816100381611990921003816100381610038161003816) + 100381610038161003816100381611990411003816100381610038161003816 |Λ|minus 120574

119870119898 (119896311990421 + 1198964 1003816100381610038161003816119904110038161003816100381610038161199012+1 + 100381610038161003816100381611990411003816100381610038161003816 |Λ|

+ 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816)) minus 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816+ 2 100381610038161003816100381611990921003816100381610038161003816)

(28)

With the adaptive laws in (25) and 119894(0) gt 0 it ensuresthat 119894 gt 0 Since 120574 ge 119870119898 it follows that

2 le minus 120574119870119898 (119896311990421 + 1198964 1003816100381610038161003816119904110038161003816100381610038161199012+1)

+ 100381610038161003816100381611990411003816100381610038161003816 (1198880 + 1198881 100381610038161003816100381611990911003816100381610038161003816 + 1198882 100381610038161003816100381611990921003816100381610038161003816)minus 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816)minus 100381610038161003816100381611990411003816100381610038161003816 (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816)

= minus 120574119870119898 (119896311990421 + 1198964 1003816100381610038161003816119904110038161003816100381610038161199012+1)

(29)

Applying the Barbalat lemma [28] the sliding surface 1199041converges to zero asymptotically

This completes the proof

The boundary layer approach is a common method forchattering reduction It is well known that the chatteringreduction is achieved with a small cost of control precisionMore specifically the sliding surface is driven to a vicinity ofzero In [29] a concept of real sliding surface was introducedwhich was similar to the boundary layer It also means thatthe sliding surface is forced to a small region rather than zeroBased on this concept a modified adaptive law was proposedto avoid the overestimation of adaptive gain in [29] Theadaptive gain slowly decreased to the necessary value oncethe real sliding surface was established

In the case that the boundary layer approach is adoptedfor the proposed adaptive controller (25) the chatteringamplitude can be further attenuated if the adaptive gains0 1 2 decrease to a low level Thus the above two meth-ods are combined to design a modified adaptive controller as

119906119886 = minus 1119870119898 (11989631199041 + 1198964 1003816100381610038161003816119904110038161003816100381610038161199012 sign (1199041) + |Λ| sat (1199041)

+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sat (1199041)) (30)

6 Mathematical Problems in Engineering

o

u

x

m

R(t)

Figure 3 Variable-length pendulum model

where the sat(sdot) function and the adaptive laws are taken as

sat (1199041) = sign (1199041) 100381610038161003816100381611990411003816100381610038161003816 gt 1198741199041119874

100381610038161003816100381611990411003816100381610038161003816 le 119874

1198880 = 1198960 100381610038161003816100381611990411003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 119874) 0 gt 12057211988801198960 0 le 1205721198880

1198881 = 1198961 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 119874) 1 gt 12057211988811198961 1 le 1205721198881

1198882 = 1198962 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 119874) 2 gt 12057211988821198962 2 le 1205721198882

(31)

where a positive constant 119874 denotes the boundary lay 1198960 sim1198962 and 1205721198880 sim 1205721198882 are positive constantsAccording toTheorem 6 the sliding surface 1199041 converges

to zero asymptotically Thus it converges into |1199041| le 119874 infinite time Then the adaptive gains begin to decrease and2 becomes sign indefinite As soon as the sliding surfaceescapes from the region |1199041| le 119874 the adaptive gains increaseagain so that the sliding surface is driven back into the region|1199041| le 119874 Finally the sliding surface 1199041 remains in a largerregion than |1199041| le 119874which is the so-called real sliding surfaceThe second expressions of the modified adaptive laws ensurethe positive values of the adaptive gains and they are valid ina short time

Remark 7 For the tracking control issues it is necessary tomodify the adaptive laws of 1 and 2 to reduce the chatteringamplitude However it is not necessary when the states1199091 1199092 converge to zero4 Simulation

In this section the designed robust and adaptive NTSMcontrollers were tested for the tracking control of a variable-length pendulum [25] (Figure 3) formulated as

= minus2 (119905)119877 (119905) minus 119892

119877 (119905) sin (119909) + 11198981198772 (119905)119906 (32)

The angular coordinate 119909 is impelled to track a referencesignal 119909119888 by the torque 119906 A known mass 119898 moves alonethe rod without friction and the distance 119877(119905) between thecenter 119900 and the mass 119898 is unmeasured It is assumed thatthe states [119909 ]119879 can be measured for the controller designThe parameters of (32) and the reference signal are taken as

119877 (119905) = 1 + 025 sin (4119905) + 05 cos (119905)119898 = 1 kg119892 = 981ms2

119909 (0) = 12119909119888 = 05 sin (05119905)

(33)

Defining the tracking error as 1198901 = 119909 minus 119909119888 the trackingcontrol is converted to the stabilization of the following errorsystem

1198901 = 11989021198902 = minus2 (119905)

119877 (119905) minus 119892119877 (119905) sin (119909) minus 119888 + 1

1198981198772 (119905)119906(34)

41 Robust Control In this subsection three robust control-lers (A B and C) were evaluated According to Theorem 2controllers A and B (without EDSG) were designed using theFNTSM of [11] and the proposed FNTSM (6) respectivelyThe controller C was the proposed FNTSM-EDSG controllerin Theorem 2

Controller A (based on FNTSM of [11])

119906 = minus16 (021199041198651 + 1205721 sign (1199041198651)

+ 790119890572 sign (1199041198651) sign (1198902)

+ 49450119890251 119890572 sign (1199041198651) sign (119890251 1198902))

1199041198651 = 1198901 + 119890751 + 119890972 1205721 = 04 + 04 100381610038161003816100381611990911003816100381610038161003816 + 04 100381610038161003816100381611990921003816100381610038161003816

(35)

Controller B (based on FNTSM (6))

119906 = minus16(021199041198652 + 1205721 sign (1199041198652) + 11199021003816100381610038161003816119890210038161003816100381610038162minus119902 sign (1199041198652))

1199041198652 = 1198901 + 100381610038161003816100381611989021003816100381610038161003816119902 sign (1198902) 119902 = 97 + (1 minus 9

7) sign (100381610038161003816100381611990911003816100381610038161003816 minus 1) (36)

Controller C (based on FNTSM (6) + EDSG)

119906 = minus16(021199041198652 + 119860 (119905) 1205721 sign (1199041198652)

+ 11199021003816100381610038161003816119890210038161003816100381610038162minus119902 sign (1199041198652)) 119860 (119905) = 1 + 119890minus2119905 1205721 = 02 + 02 100381610038161003816100381611990911003816100381610038161003816 + 02 100381610038161003816100381611990921003816100381610038161003816

(37)

The simulation results are given in Figure 4 As shownin Figures 4(a) and 4(b) the angular coordinate converged

Mathematical Problems in Engineering 7

Controller B Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad) Controller A Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad)

Controller C Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad)

(a) Angular coordinate

2 3 4

0

002

Controller A

1 2 3 4 5 6 7 8 9 100Time (s)

0

06

12

Trac

king

erro

r (ra

d)

Controller B

2 3 4

0

002

0

06

12

Trac

king

erro

r (ra

d)

1 2 3 4 5 6 7 8 9 100Time (s)

2 3 4

0

002

Controller C

0

06

12Tr

acki

ng er

ror (

rad)

1 2 3 4 5 6 7 8 9 100Time (s)

(b) Tracking error

1 2 3 4 5 6 7 8 9 100Time (s)

minus10123

Slid

ing

surfa

ces

Controller A

Controller B

minus10123

Slid

ing

surfa

ces

1 2 3 4 5 6 7 8 9 100Time (s)

Controller C

minus10123

Slid

ing

surfa

ces

1 2 3 4 5 6 7 8 9 100Time (s)

(c) Sliding surfaces

1 2 3 4 5 6 7 8 9 100Time (s)

Controller A

minus30

minus10

10

30

Con

trol t

orqu

es (N

)

Controller B

minus30

minus10

10

30

Con

trol t

orqu

es (N

)

1 2 3 4 5 6 7 8 9 100Time (s)

Controller C

minus30

minus10

10

30

Con

trol t

orqu

es (N

)

1 2 3 4 5 6 7 8 9 100Time (s)

(d) Control torque

Figure 4 Simulation result of the robust control

8 Mathematical Problems in Engineering

to the reference signal within 119905 = 4 s The sliding surfacesconverged to zero within 119905 = 15 s (Figure 4(c)) It is clear thatcontroller B cost less convergence time than the controllerA while the control inputs of these controllers had slightdifference (Figures 4(b) and 4(d)) This fact implies thatthe controller designed by the proposed FNTSM was moreefficient As expected the chattering amplitude of controllerC (plusmn5) was half of controller B (plusmn10) (Figure 4(d)) Howeverthe performance of controllerChad small differencewith thatof controller BThe result confirms that the EDSG is effectivefor chattering suppression without deteriorating the systemrobustness

42 Adaptive Control In this subsection the proposed adap-tive NTSM control scheme ((23) and (25)) was verified forthe tracking control As a comparison the adaptive methodin [25] was adopted to design the adaptive controller 119906119886 Thecontrollers were designed as follows

Controller D (Theorem 6)

119906119899 = minus11199021003816100381610038161003816119890210038161003816100381610038162minus119902 sign (1198902) minus 251199042 minus 10038161003816100381610038161199042100381610038161003816100381679 sign (1199042)

119906119886 = minus16 (051199041 + 10038161003816100381610038161199041100381610038161003816100381679 sign (1199041)+ 100381610038161003816100381610038165119890minus5119905 (1198902 (0) + 119911 (0))10038161003816100381610038161003816 sign (1199041)+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sign (1199041))

1199041 = 1198902 + 119911 minus 119890minus5119905 (1198902 (0) + 119911 (0)) = minus119906119899119911 (0) = 01198880 = 100381610038161003816100381611990411003816100381610038161003816 1198881 = 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 1198882 = 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 0 (0) = 1 (0) = 2 (0) = 001

(38)

Controller E (method in [25])

119906119886 = minus5119890minus5119905 (1198902 (0) + 119911 (0)) minus 0 sign (1199041) 1198880 = 5 100381610038161003816100381611990411003816100381610038161003816

(39)

ControllersD and E shared the sameNTSMcontroller 119906119899and the sliding surface 1199042 was taken as 1199041198652 of controller B

The simulation results are given in Figures 5 and 6 It isclear that both controllers D and E achieved fast tracking ofthe reference signal Controller D had shorter convergencetime and better transient performance than controller Eeven though the maximum control torque and the chatteringamplitude of controller D were smaller Meanwhile thesliding surfaces in Figure 6 converged to zero within 119905 =3 s In Figure 6(a) the motions of ISM started from zeroand there was a short time that ISM did not remain zerodue to the affection of uncertainty It is observed that the

FNTSM converged to zero after the ISM and the stagnationof FNTSM never happened It verifies that the designedadaptive NTSM control scheme is effective Since the twocontrollers shared the same NTSM controller 119906119899 the simula-tion results confirm that the proposed adaptive method hadbetter transient performance and was more efficient than themethod in [25]

Here themodified adaptive controller ((30) and (31)) waschecked for chattering reduction In the following controllerF the boundary layer approach and the modified adaptivelaws were both employed As a comparison controller Gonly employed the boundary layer approach In order toavoid the unbounded growing of adaptive gains inside theboundary layer the switching adaptive laws were adoptedfor controller G In addition controllers F and G used thesame boundary layer and NTSM controller 119906119899 of controllerD Since it is assumed that the states can be measured forthe controller design themeasurement noisemay deterioratethe performance of controller in real implementations Toillustrate the robustness of the controller random noise wastaken for the states [119909 ]119879 (mean value 0 and standarddeviation 001)

Controller F

119906119886 = minus16 (051199041 + 10038161003816100381610038161199041100381610038161003816100381679 sign (1199041)+ 100381610038161003816100381610038165119890minus5119905 (1198902 (0) + 119911 (0))10038161003816100381610038161003816 sat (1199041)+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sat (1199041))

sat (1199041) = sign (1199041) 100381610038161003816100381611990411003816100381610038161003816 gt 0041199041

004100381610038161003816100381611990411003816100381610038161003816 le 004

1198880 = 2 100381610038161003816100381611990411003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 004) 0 gt 0011 0 le 001

1198881 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 005) 1 gt 0011 1 le 001

1198882 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 004) 2 gt 0011 2 le 001

(40)

Controller G

119906119886 = minus16 (051199041 + 10038161003816100381610038161199041100381610038161003816100381679 sign (1199041)+ 100381610038161003816100381610038165119890minus5119905 (1198902 (0) + 119911 (0))10038161003816100381610038161003816 sat (1199041)+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sat (1199041))

1198880 = 2 100381610038161003816100381611990411003816100381610038161003816 1199041 gt 0040 1199041 le 004

Mathematical Problems in Engineering 9

Ang

ular

coor

dina

te (r

ad)

Controller DController E

Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus08

minus06

minus04

minus02

0

02

04

06

08

1

(a) Angular coordinate

Trac

king

erro

r (ra

d)

Controller DController E

minus02

0

02

04

06

08

1

1 2 3 4 5 6 7 8 9 100Time (s)

(b) Tracking error

Adaptive gain 1

Adaptive gain 1Adaptive gain 2

Adaptive gain 3

Controller D

Controller E

1 2 3 4 5 6 7 8 9 100Time (s)

0

01

02

Adap

tive g

ains

0

5

10

15

Adap

tive g

ains

1 2 3 4 5 6 7 8 9 100Time (s)

(c) Adaptive gains

0

Controller E

Controller D

minus20

20

Con

trol t

orqu

es (N

)

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus20

0

20C

ontro

l tor

ques

(N)

(d) Control torque

Figure 5 Simulation result of the adaptive control

1198881 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 1199041 gt 0040 1199041 le 004

1198882 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 1199041 gt 0040 1199041 le 004

(41)

The simulation result of controller F is given in Figure 7It is shown that the angular coordinate tracked the referencesignal with a high precision (Figure 7(a)) Due to the adop-tion of the boundary layer and themodified adaptive laws thesliding surface 1199041 converged to a small domain as |1199041| le 01

(Figure 7(b))Moreover the adaptive gains decreased to a lowlevel with approximate reduction ratios as 5108 8479and 1615 respectively (Figure 7(c)) As a result there was amarked reduction in the chattering of controller F comparedto that of controller G in Figure 7(d) Thus the effectivenessof the modified adaptive controller is verified

5 Conclusions

This paper studies the NTSM control for a class of second-order systems in which the coefficient of control input isunknown In this paper a new FNTSM was designed tointegrate the advantages of LSM and NTSM Based onthis FNTSM new forms of robust controller and adaptive

10 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 10minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

Time (s)

ISM

Controller DController E

(a) ISM

0 1 2 3 4 5 6 7 8 9 10minus2

minus15

minus1

minus05

0

05

1

15

2

25

3

Time (s)

FNTS

M

Controller DController E

(b) FNTSM

Figure 6 The response of sliding surfaces

Controller FReference signal

10 20 30 40 500Time (s)

minus1

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad)

(a) Angular coordinate

minus02

0

02

04

06

08

0 10 20 30 40 50Time (s)

Slid

ing

surfa

ces

ISMFNTSM

(b) Sliding surfaces

0

005

01

015

02

025

Adap

tive g

ains

Adaptive gain 1Adaptive gain 2

Adaptive gain 3

10 20 30 40 500Time (s)

(c) Adaptive gains

0 10 20 30 40 50Time (s)

Controller G

Controller F

minus10

0

10

Con

trol t

orqu

es (N

)

minus10

0

10

Con

trol t

orqu

es (N

)

10 20 30 40 500Time (s)

(d) Control torques

Figure 7 Simulation result of the modified adaptive control

Mathematical Problems in Engineering 11

controller were proposed for the discussed uncertain systemThe simulation verified the effectiveness of the proposed con-trollersThe proposed FNTSM shows faster convergence ratethan the NTSM and chattering reduction was achieved bythe EDSG The adaptive controller presented better transientperformance and more efficiency than the adaptive methodin [25] Moreover the modified adaptive controller achievedeffective chattering reduction

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this article

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (no 61375100 no 61472037 and no61433003)

References

[1] J M Zhang C Y Sun R M Zhang and C Qian ldquoAdaptivesliding mode control for re-entry attitude of near space hyper-sonic vehicle based on backstepping designrdquo IEEECAA Journalof Automatica Sinica vol 2 no 1 pp 94ndash101 2015

[2] J Liu Y Zhao B Geng and B Xiao ldquoAdaptive second ordersliding mode control of a fuel cell hybrid system for electricvehicle applicationsrdquo Mathematical Problems in Engineeringvol 2015 Article ID 370424 14 pages 2015

[3] M Zak ldquoTerminal attractors in neural networksrdquo NeuralNetworks vol 2 no 4 pp 259ndash274 1989

[4] S T Venkataraman and S Gulati ldquoTerminal sliding modes anew approach to nonlinear control synthesisrdquo in Proceedings ofthe 5th International Conference onAdvanced Robotics vol 1 pp443ndash448 Pisa Italy June 1991

[5] D Y Zhao S Y Li and F Gao ldquoA new terminal sliding modecontrol for robotic manipulatorsrdquo International Journal of Con-trol vol 82 no 10 pp 1804ndash1813 2009

[6] Z K Song H X Li and K B Sun ldquoFinite-time control fornonlinear spacecraft attitude based on terminal sliding modetechniquerdquo ISA Transactions vol 53 no 1 pp 117ndash124 2014

[7] H Komurcugil ldquoNon-singular terminal sliding-mode controlof DC-DC buck convertersrdquo Control Engineering Practice vol21 no 3 pp 321ndash332 2013

[8] X Yu M Zhihong and Y Wu ldquoTerminal sliding modes withfast transient performancerdquo in Proceedings of the 1997 36th IEEEConference on Decision and Control Part 1 (of 5) vol 2 pp 962ndash963 San Diego Calif USA December 1997

[9] S Mobayen ldquoFast terminal sliding mode controller design fornonlinear second-order systems with time-varying uncertain-tiesrdquo Complexity vol 21 no 2 pp 239ndash244 2015

[10] Z He C Liu Y Zhan H Li X Huang and Z ZhangldquoNonsingular fast terminal sliding mode control with extendedstate observer and tracking differentiator for uncertain nonlin-ear systemsrdquo Mathematical Problems in Engineering vol 2014Article ID 639707 16 pages 2014

[11] L Liu Q M Zhu L Cheng et al ldquoApplied methods and tech-niques for mechatronic systems modelling identification and

controlrdquo Lecture Notes in Control and Information Sciences vol452 pp 79ndash97 2014

[12] X H Yu and Z H Man ldquoOn finite time mechanism terminalsliding modesrdquo in Proceedings of the IEEE International Work-shop on Variable Structure Systems pp 164ndash167 Tokyo Japan1996

[13] L Y Wang T Y Chai and L F Zhai ldquoNeural-network-basedterminal sliding-mode control of robotic manipulators includ-ing actuator dynamicsrdquo IEEE Transactions on Industrial Elec-tronics vol 56 no 9 pp 3296ndash3304 2009

[14] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[15] A Al-Ghanimi J Zheng and Z Man ldquoRobust and fast non-singular terminal sliding mode control for piezoelectric actua-torsrdquo IET ControlTheory ampApplications vol 9 no 18 pp 2678ndash2687 2015

[16] S Mobayen ldquoFinite-time tracking control of chained-formnonholonomic systems with external disturbances based onrecursive terminal sliding mode methodrdquo Nonlinear Dynamicsvol 80 no 1-2 pp 669ndash683 2015

[17] S Mobayen ldquoFast terminal sliding mode tracking of non-holonomic systems with exponential decay raterdquo IET ControlTheory amp Applications vol 9 no 8 pp 1294ndash1301 2015

[18] Y Wu and J Wang ldquoContinuous recursive sliding modecontrol for hypersonic flight vehicle with extended disturbanceobserverrdquo Mathematical Problems in Engineering vol 2015Article ID 506906 26 pages 2015

[19] S Mobayen ldquoAn adaptive fast terminal sliding mode controlcombined with global slidingmode scheme for tracking controlof uncertain nonlinear third-order systemsrdquoNonlinear Dynam-ics vol 82 no 1-2 pp 599ndash610 2015

[20] N Boonsatit and C Pukdeboon ldquoAdaptive fast terminal slidingmode control of magnetic levitation systemrdquo Journal of ControlAutomation and Electrical Systems vol 27 no 4 pp 359ndash3672016

[21] L Y Fang T S Li Z F Li and R Li ldquoAdaptive terminal slidingmode control for anti-synchronization of uncertain chaoticsystemsrdquoNonlinear Dynamics vol 74 no 4 pp 991ndash1002 2013

[22] C-C Yang ldquoSynchronization of second-order chaotic systemsvia adaptive terminal sliding mode control with input nonlin-earityrdquo Journal of the Franklin Institute Engineering and AppliedMathematics vol 349 no 6 pp 2019ndash2032 2012

[23] C-C Yang and C-J Ou ldquoAdaptive terminal sliding mode con-trol subject to input nonlinearity for synchronization of chaoticgyrosrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 3 pp 682ndash691 2013

[24] J-J Kim J-J Lee K-B Park and M-J Youn ldquoDesign ofnew time-varying sliding surface for robot manipulator usingvariable structure controllerrdquo Electronics Letters vol 29 no 2pp 195ndash196 1993

[25] P Li and Z-Q Zheng ldquoRobust adaptive second-order sliding-mode control with fast transient performancerdquo IET ControlTheory amp Applications vol 6 no 2 pp 305ndash312 2012

[26] M Zhihong M OrsquoDay and X Yu ldquoA robust adaptive terminalsliding mode control for rigid robotic manipulatorsrdquo Journal ofIntelligent and Robotic SystemsTheory and Applications vol 24no 1 pp 23ndash41 1999

[27] Z Zhu Y Xia and M Fu ldquoAdaptive sliding mode control forattitude stabilization with actuator saturationrdquo IEEE Transac-tions on Industrial Electronics vol 58 no 10 pp 4898ndash49072011

12 Mathematical Problems in Engineering

[28] H K Khalil Nonlinear Systems Prentice Hall EnglewoodCliffs NJ USA 3rd edition 2002

[29] Y Shtessel M Taleb and F Plestan ldquoA novel adaptive-gainsupertwisting sliding mode controller methodology and appli-cationrdquo Automatica vol 48 no 5 pp 759ndash769 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

o

u

x

m

R(t)

Figure 3 Variable-length pendulum model

where the sat(sdot) function and the adaptive laws are taken as

sat (1199041) = sign (1199041) 100381610038161003816100381611990411003816100381610038161003816 gt 1198741199041119874

100381610038161003816100381611990411003816100381610038161003816 le 119874

1198880 = 1198960 100381610038161003816100381611990411003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 119874) 0 gt 12057211988801198960 0 le 1205721198880

1198881 = 1198961 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 119874) 1 gt 12057211988811198961 1 le 1205721198881

1198882 = 1198962 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 119874) 2 gt 12057211988821198962 2 le 1205721198882

(31)

where a positive constant 119874 denotes the boundary lay 1198960 sim1198962 and 1205721198880 sim 1205721198882 are positive constantsAccording toTheorem 6 the sliding surface 1199041 converges

to zero asymptotically Thus it converges into |1199041| le 119874 infinite time Then the adaptive gains begin to decrease and2 becomes sign indefinite As soon as the sliding surfaceescapes from the region |1199041| le 119874 the adaptive gains increaseagain so that the sliding surface is driven back into the region|1199041| le 119874 Finally the sliding surface 1199041 remains in a largerregion than |1199041| le 119874which is the so-called real sliding surfaceThe second expressions of the modified adaptive laws ensurethe positive values of the adaptive gains and they are valid ina short time

Remark 7 For the tracking control issues it is necessary tomodify the adaptive laws of 1 and 2 to reduce the chatteringamplitude However it is not necessary when the states1199091 1199092 converge to zero4 Simulation

In this section the designed robust and adaptive NTSMcontrollers were tested for the tracking control of a variable-length pendulum [25] (Figure 3) formulated as

= minus2 (119905)119877 (119905) minus 119892

119877 (119905) sin (119909) + 11198981198772 (119905)119906 (32)

The angular coordinate 119909 is impelled to track a referencesignal 119909119888 by the torque 119906 A known mass 119898 moves alonethe rod without friction and the distance 119877(119905) between thecenter 119900 and the mass 119898 is unmeasured It is assumed thatthe states [119909 ]119879 can be measured for the controller designThe parameters of (32) and the reference signal are taken as

119877 (119905) = 1 + 025 sin (4119905) + 05 cos (119905)119898 = 1 kg119892 = 981ms2

119909 (0) = 12119909119888 = 05 sin (05119905)

(33)

Defining the tracking error as 1198901 = 119909 minus 119909119888 the trackingcontrol is converted to the stabilization of the following errorsystem

1198901 = 11989021198902 = minus2 (119905)

119877 (119905) minus 119892119877 (119905) sin (119909) minus 119888 + 1

1198981198772 (119905)119906(34)

41 Robust Control In this subsection three robust control-lers (A B and C) were evaluated According to Theorem 2controllers A and B (without EDSG) were designed using theFNTSM of [11] and the proposed FNTSM (6) respectivelyThe controller C was the proposed FNTSM-EDSG controllerin Theorem 2

Controller A (based on FNTSM of [11])

119906 = minus16 (021199041198651 + 1205721 sign (1199041198651)

+ 790119890572 sign (1199041198651) sign (1198902)

+ 49450119890251 119890572 sign (1199041198651) sign (119890251 1198902))

1199041198651 = 1198901 + 119890751 + 119890972 1205721 = 04 + 04 100381610038161003816100381611990911003816100381610038161003816 + 04 100381610038161003816100381611990921003816100381610038161003816

(35)

Controller B (based on FNTSM (6))

119906 = minus16(021199041198652 + 1205721 sign (1199041198652) + 11199021003816100381610038161003816119890210038161003816100381610038162minus119902 sign (1199041198652))

1199041198652 = 1198901 + 100381610038161003816100381611989021003816100381610038161003816119902 sign (1198902) 119902 = 97 + (1 minus 9

7) sign (100381610038161003816100381611990911003816100381610038161003816 minus 1) (36)

Controller C (based on FNTSM (6) + EDSG)

119906 = minus16(021199041198652 + 119860 (119905) 1205721 sign (1199041198652)

+ 11199021003816100381610038161003816119890210038161003816100381610038162minus119902 sign (1199041198652)) 119860 (119905) = 1 + 119890minus2119905 1205721 = 02 + 02 100381610038161003816100381611990911003816100381610038161003816 + 02 100381610038161003816100381611990921003816100381610038161003816

(37)

The simulation results are given in Figure 4 As shownin Figures 4(a) and 4(b) the angular coordinate converged

Mathematical Problems in Engineering 7

Controller B Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad) Controller A Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad)

Controller C Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad)

(a) Angular coordinate

2 3 4

0

002

Controller A

1 2 3 4 5 6 7 8 9 100Time (s)

0

06

12

Trac

king

erro

r (ra

d)

Controller B

2 3 4

0

002

0

06

12

Trac

king

erro

r (ra

d)

1 2 3 4 5 6 7 8 9 100Time (s)

2 3 4

0

002

Controller C

0

06

12Tr

acki

ng er

ror (

rad)

1 2 3 4 5 6 7 8 9 100Time (s)

(b) Tracking error

1 2 3 4 5 6 7 8 9 100Time (s)

minus10123

Slid

ing

surfa

ces

Controller A

Controller B

minus10123

Slid

ing

surfa

ces

1 2 3 4 5 6 7 8 9 100Time (s)

Controller C

minus10123

Slid

ing

surfa

ces

1 2 3 4 5 6 7 8 9 100Time (s)

(c) Sliding surfaces

1 2 3 4 5 6 7 8 9 100Time (s)

Controller A

minus30

minus10

10

30

Con

trol t

orqu

es (N

)

Controller B

minus30

minus10

10

30

Con

trol t

orqu

es (N

)

1 2 3 4 5 6 7 8 9 100Time (s)

Controller C

minus30

minus10

10

30

Con

trol t

orqu

es (N

)

1 2 3 4 5 6 7 8 9 100Time (s)

(d) Control torque

Figure 4 Simulation result of the robust control

8 Mathematical Problems in Engineering

to the reference signal within 119905 = 4 s The sliding surfacesconverged to zero within 119905 = 15 s (Figure 4(c)) It is clear thatcontroller B cost less convergence time than the controllerA while the control inputs of these controllers had slightdifference (Figures 4(b) and 4(d)) This fact implies thatthe controller designed by the proposed FNTSM was moreefficient As expected the chattering amplitude of controllerC (plusmn5) was half of controller B (plusmn10) (Figure 4(d)) Howeverthe performance of controllerChad small differencewith thatof controller BThe result confirms that the EDSG is effectivefor chattering suppression without deteriorating the systemrobustness

42 Adaptive Control In this subsection the proposed adap-tive NTSM control scheme ((23) and (25)) was verified forthe tracking control As a comparison the adaptive methodin [25] was adopted to design the adaptive controller 119906119886 Thecontrollers were designed as follows

Controller D (Theorem 6)

119906119899 = minus11199021003816100381610038161003816119890210038161003816100381610038162minus119902 sign (1198902) minus 251199042 minus 10038161003816100381610038161199042100381610038161003816100381679 sign (1199042)

119906119886 = minus16 (051199041 + 10038161003816100381610038161199041100381610038161003816100381679 sign (1199041)+ 100381610038161003816100381610038165119890minus5119905 (1198902 (0) + 119911 (0))10038161003816100381610038161003816 sign (1199041)+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sign (1199041))

1199041 = 1198902 + 119911 minus 119890minus5119905 (1198902 (0) + 119911 (0)) = minus119906119899119911 (0) = 01198880 = 100381610038161003816100381611990411003816100381610038161003816 1198881 = 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 1198882 = 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 0 (0) = 1 (0) = 2 (0) = 001

(38)

Controller E (method in [25])

119906119886 = minus5119890minus5119905 (1198902 (0) + 119911 (0)) minus 0 sign (1199041) 1198880 = 5 100381610038161003816100381611990411003816100381610038161003816

(39)

ControllersD and E shared the sameNTSMcontroller 119906119899and the sliding surface 1199042 was taken as 1199041198652 of controller B

The simulation results are given in Figures 5 and 6 It isclear that both controllers D and E achieved fast tracking ofthe reference signal Controller D had shorter convergencetime and better transient performance than controller Eeven though the maximum control torque and the chatteringamplitude of controller D were smaller Meanwhile thesliding surfaces in Figure 6 converged to zero within 119905 =3 s In Figure 6(a) the motions of ISM started from zeroand there was a short time that ISM did not remain zerodue to the affection of uncertainty It is observed that the

FNTSM converged to zero after the ISM and the stagnationof FNTSM never happened It verifies that the designedadaptive NTSM control scheme is effective Since the twocontrollers shared the same NTSM controller 119906119899 the simula-tion results confirm that the proposed adaptive method hadbetter transient performance and was more efficient than themethod in [25]

Here themodified adaptive controller ((30) and (31)) waschecked for chattering reduction In the following controllerF the boundary layer approach and the modified adaptivelaws were both employed As a comparison controller Gonly employed the boundary layer approach In order toavoid the unbounded growing of adaptive gains inside theboundary layer the switching adaptive laws were adoptedfor controller G In addition controllers F and G used thesame boundary layer and NTSM controller 119906119899 of controllerD Since it is assumed that the states can be measured forthe controller design themeasurement noisemay deterioratethe performance of controller in real implementations Toillustrate the robustness of the controller random noise wastaken for the states [119909 ]119879 (mean value 0 and standarddeviation 001)

Controller F

119906119886 = minus16 (051199041 + 10038161003816100381610038161199041100381610038161003816100381679 sign (1199041)+ 100381610038161003816100381610038165119890minus5119905 (1198902 (0) + 119911 (0))10038161003816100381610038161003816 sat (1199041)+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sat (1199041))

sat (1199041) = sign (1199041) 100381610038161003816100381611990411003816100381610038161003816 gt 0041199041

004100381610038161003816100381611990411003816100381610038161003816 le 004

1198880 = 2 100381610038161003816100381611990411003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 004) 0 gt 0011 0 le 001

1198881 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 005) 1 gt 0011 1 le 001

1198882 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 004) 2 gt 0011 2 le 001

(40)

Controller G

119906119886 = minus16 (051199041 + 10038161003816100381610038161199041100381610038161003816100381679 sign (1199041)+ 100381610038161003816100381610038165119890minus5119905 (1198902 (0) + 119911 (0))10038161003816100381610038161003816 sat (1199041)+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sat (1199041))

1198880 = 2 100381610038161003816100381611990411003816100381610038161003816 1199041 gt 0040 1199041 le 004

Mathematical Problems in Engineering 9

Ang

ular

coor

dina

te (r

ad)

Controller DController E

Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus08

minus06

minus04

minus02

0

02

04

06

08

1

(a) Angular coordinate

Trac

king

erro

r (ra

d)

Controller DController E

minus02

0

02

04

06

08

1

1 2 3 4 5 6 7 8 9 100Time (s)

(b) Tracking error

Adaptive gain 1

Adaptive gain 1Adaptive gain 2

Adaptive gain 3

Controller D

Controller E

1 2 3 4 5 6 7 8 9 100Time (s)

0

01

02

Adap

tive g

ains

0

5

10

15

Adap

tive g

ains

1 2 3 4 5 6 7 8 9 100Time (s)

(c) Adaptive gains

0

Controller E

Controller D

minus20

20

Con

trol t

orqu

es (N

)

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus20

0

20C

ontro

l tor

ques

(N)

(d) Control torque

Figure 5 Simulation result of the adaptive control

1198881 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 1199041 gt 0040 1199041 le 004

1198882 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 1199041 gt 0040 1199041 le 004

(41)

The simulation result of controller F is given in Figure 7It is shown that the angular coordinate tracked the referencesignal with a high precision (Figure 7(a)) Due to the adop-tion of the boundary layer and themodified adaptive laws thesliding surface 1199041 converged to a small domain as |1199041| le 01

(Figure 7(b))Moreover the adaptive gains decreased to a lowlevel with approximate reduction ratios as 5108 8479and 1615 respectively (Figure 7(c)) As a result there was amarked reduction in the chattering of controller F comparedto that of controller G in Figure 7(d) Thus the effectivenessof the modified adaptive controller is verified

5 Conclusions

This paper studies the NTSM control for a class of second-order systems in which the coefficient of control input isunknown In this paper a new FNTSM was designed tointegrate the advantages of LSM and NTSM Based onthis FNTSM new forms of robust controller and adaptive

10 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 10minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

Time (s)

ISM

Controller DController E

(a) ISM

0 1 2 3 4 5 6 7 8 9 10minus2

minus15

minus1

minus05

0

05

1

15

2

25

3

Time (s)

FNTS

M

Controller DController E

(b) FNTSM

Figure 6 The response of sliding surfaces

Controller FReference signal

10 20 30 40 500Time (s)

minus1

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad)

(a) Angular coordinate

minus02

0

02

04

06

08

0 10 20 30 40 50Time (s)

Slid

ing

surfa

ces

ISMFNTSM

(b) Sliding surfaces

0

005

01

015

02

025

Adap

tive g

ains

Adaptive gain 1Adaptive gain 2

Adaptive gain 3

10 20 30 40 500Time (s)

(c) Adaptive gains

0 10 20 30 40 50Time (s)

Controller G

Controller F

minus10

0

10

Con

trol t

orqu

es (N

)

minus10

0

10

Con

trol t

orqu

es (N

)

10 20 30 40 500Time (s)

(d) Control torques

Figure 7 Simulation result of the modified adaptive control

Mathematical Problems in Engineering 11

controller were proposed for the discussed uncertain systemThe simulation verified the effectiveness of the proposed con-trollersThe proposed FNTSM shows faster convergence ratethan the NTSM and chattering reduction was achieved bythe EDSG The adaptive controller presented better transientperformance and more efficiency than the adaptive methodin [25] Moreover the modified adaptive controller achievedeffective chattering reduction

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this article

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (no 61375100 no 61472037 and no61433003)

References

[1] J M Zhang C Y Sun R M Zhang and C Qian ldquoAdaptivesliding mode control for re-entry attitude of near space hyper-sonic vehicle based on backstepping designrdquo IEEECAA Journalof Automatica Sinica vol 2 no 1 pp 94ndash101 2015

[2] J Liu Y Zhao B Geng and B Xiao ldquoAdaptive second ordersliding mode control of a fuel cell hybrid system for electricvehicle applicationsrdquo Mathematical Problems in Engineeringvol 2015 Article ID 370424 14 pages 2015

[3] M Zak ldquoTerminal attractors in neural networksrdquo NeuralNetworks vol 2 no 4 pp 259ndash274 1989

[4] S T Venkataraman and S Gulati ldquoTerminal sliding modes anew approach to nonlinear control synthesisrdquo in Proceedings ofthe 5th International Conference onAdvanced Robotics vol 1 pp443ndash448 Pisa Italy June 1991

[5] D Y Zhao S Y Li and F Gao ldquoA new terminal sliding modecontrol for robotic manipulatorsrdquo International Journal of Con-trol vol 82 no 10 pp 1804ndash1813 2009

[6] Z K Song H X Li and K B Sun ldquoFinite-time control fornonlinear spacecraft attitude based on terminal sliding modetechniquerdquo ISA Transactions vol 53 no 1 pp 117ndash124 2014

[7] H Komurcugil ldquoNon-singular terminal sliding-mode controlof DC-DC buck convertersrdquo Control Engineering Practice vol21 no 3 pp 321ndash332 2013

[8] X Yu M Zhihong and Y Wu ldquoTerminal sliding modes withfast transient performancerdquo in Proceedings of the 1997 36th IEEEConference on Decision and Control Part 1 (of 5) vol 2 pp 962ndash963 San Diego Calif USA December 1997

[9] S Mobayen ldquoFast terminal sliding mode controller design fornonlinear second-order systems with time-varying uncertain-tiesrdquo Complexity vol 21 no 2 pp 239ndash244 2015

[10] Z He C Liu Y Zhan H Li X Huang and Z ZhangldquoNonsingular fast terminal sliding mode control with extendedstate observer and tracking differentiator for uncertain nonlin-ear systemsrdquo Mathematical Problems in Engineering vol 2014Article ID 639707 16 pages 2014

[11] L Liu Q M Zhu L Cheng et al ldquoApplied methods and tech-niques for mechatronic systems modelling identification and

controlrdquo Lecture Notes in Control and Information Sciences vol452 pp 79ndash97 2014

[12] X H Yu and Z H Man ldquoOn finite time mechanism terminalsliding modesrdquo in Proceedings of the IEEE International Work-shop on Variable Structure Systems pp 164ndash167 Tokyo Japan1996

[13] L Y Wang T Y Chai and L F Zhai ldquoNeural-network-basedterminal sliding-mode control of robotic manipulators includ-ing actuator dynamicsrdquo IEEE Transactions on Industrial Elec-tronics vol 56 no 9 pp 3296ndash3304 2009

[14] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[15] A Al-Ghanimi J Zheng and Z Man ldquoRobust and fast non-singular terminal sliding mode control for piezoelectric actua-torsrdquo IET ControlTheory ampApplications vol 9 no 18 pp 2678ndash2687 2015

[16] S Mobayen ldquoFinite-time tracking control of chained-formnonholonomic systems with external disturbances based onrecursive terminal sliding mode methodrdquo Nonlinear Dynamicsvol 80 no 1-2 pp 669ndash683 2015

[17] S Mobayen ldquoFast terminal sliding mode tracking of non-holonomic systems with exponential decay raterdquo IET ControlTheory amp Applications vol 9 no 8 pp 1294ndash1301 2015

[18] Y Wu and J Wang ldquoContinuous recursive sliding modecontrol for hypersonic flight vehicle with extended disturbanceobserverrdquo Mathematical Problems in Engineering vol 2015Article ID 506906 26 pages 2015

[19] S Mobayen ldquoAn adaptive fast terminal sliding mode controlcombined with global slidingmode scheme for tracking controlof uncertain nonlinear third-order systemsrdquoNonlinear Dynam-ics vol 82 no 1-2 pp 599ndash610 2015

[20] N Boonsatit and C Pukdeboon ldquoAdaptive fast terminal slidingmode control of magnetic levitation systemrdquo Journal of ControlAutomation and Electrical Systems vol 27 no 4 pp 359ndash3672016

[21] L Y Fang T S Li Z F Li and R Li ldquoAdaptive terminal slidingmode control for anti-synchronization of uncertain chaoticsystemsrdquoNonlinear Dynamics vol 74 no 4 pp 991ndash1002 2013

[22] C-C Yang ldquoSynchronization of second-order chaotic systemsvia adaptive terminal sliding mode control with input nonlin-earityrdquo Journal of the Franklin Institute Engineering and AppliedMathematics vol 349 no 6 pp 2019ndash2032 2012

[23] C-C Yang and C-J Ou ldquoAdaptive terminal sliding mode con-trol subject to input nonlinearity for synchronization of chaoticgyrosrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 3 pp 682ndash691 2013

[24] J-J Kim J-J Lee K-B Park and M-J Youn ldquoDesign ofnew time-varying sliding surface for robot manipulator usingvariable structure controllerrdquo Electronics Letters vol 29 no 2pp 195ndash196 1993

[25] P Li and Z-Q Zheng ldquoRobust adaptive second-order sliding-mode control with fast transient performancerdquo IET ControlTheory amp Applications vol 6 no 2 pp 305ndash312 2012

[26] M Zhihong M OrsquoDay and X Yu ldquoA robust adaptive terminalsliding mode control for rigid robotic manipulatorsrdquo Journal ofIntelligent and Robotic SystemsTheory and Applications vol 24no 1 pp 23ndash41 1999

[27] Z Zhu Y Xia and M Fu ldquoAdaptive sliding mode control forattitude stabilization with actuator saturationrdquo IEEE Transac-tions on Industrial Electronics vol 58 no 10 pp 4898ndash49072011

12 Mathematical Problems in Engineering

[28] H K Khalil Nonlinear Systems Prentice Hall EnglewoodCliffs NJ USA 3rd edition 2002

[29] Y Shtessel M Taleb and F Plestan ldquoA novel adaptive-gainsupertwisting sliding mode controller methodology and appli-cationrdquo Automatica vol 48 no 5 pp 759ndash769 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Controller B Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad) Controller A Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad)

Controller C Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad)

(a) Angular coordinate

2 3 4

0

002

Controller A

1 2 3 4 5 6 7 8 9 100Time (s)

0

06

12

Trac

king

erro

r (ra

d)

Controller B

2 3 4

0

002

0

06

12

Trac

king

erro

r (ra

d)

1 2 3 4 5 6 7 8 9 100Time (s)

2 3 4

0

002

Controller C

0

06

12Tr

acki

ng er

ror (

rad)

1 2 3 4 5 6 7 8 9 100Time (s)

(b) Tracking error

1 2 3 4 5 6 7 8 9 100Time (s)

minus10123

Slid

ing

surfa

ces

Controller A

Controller B

minus10123

Slid

ing

surfa

ces

1 2 3 4 5 6 7 8 9 100Time (s)

Controller C

minus10123

Slid

ing

surfa

ces

1 2 3 4 5 6 7 8 9 100Time (s)

(c) Sliding surfaces

1 2 3 4 5 6 7 8 9 100Time (s)

Controller A

minus30

minus10

10

30

Con

trol t

orqu

es (N

)

Controller B

minus30

minus10

10

30

Con

trol t

orqu

es (N

)

1 2 3 4 5 6 7 8 9 100Time (s)

Controller C

minus30

minus10

10

30

Con

trol t

orqu

es (N

)

1 2 3 4 5 6 7 8 9 100Time (s)

(d) Control torque

Figure 4 Simulation result of the robust control

8 Mathematical Problems in Engineering

to the reference signal within 119905 = 4 s The sliding surfacesconverged to zero within 119905 = 15 s (Figure 4(c)) It is clear thatcontroller B cost less convergence time than the controllerA while the control inputs of these controllers had slightdifference (Figures 4(b) and 4(d)) This fact implies thatthe controller designed by the proposed FNTSM was moreefficient As expected the chattering amplitude of controllerC (plusmn5) was half of controller B (plusmn10) (Figure 4(d)) Howeverthe performance of controllerChad small differencewith thatof controller BThe result confirms that the EDSG is effectivefor chattering suppression without deteriorating the systemrobustness

42 Adaptive Control In this subsection the proposed adap-tive NTSM control scheme ((23) and (25)) was verified forthe tracking control As a comparison the adaptive methodin [25] was adopted to design the adaptive controller 119906119886 Thecontrollers were designed as follows

Controller D (Theorem 6)

119906119899 = minus11199021003816100381610038161003816119890210038161003816100381610038162minus119902 sign (1198902) minus 251199042 minus 10038161003816100381610038161199042100381610038161003816100381679 sign (1199042)

119906119886 = minus16 (051199041 + 10038161003816100381610038161199041100381610038161003816100381679 sign (1199041)+ 100381610038161003816100381610038165119890minus5119905 (1198902 (0) + 119911 (0))10038161003816100381610038161003816 sign (1199041)+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sign (1199041))

1199041 = 1198902 + 119911 minus 119890minus5119905 (1198902 (0) + 119911 (0)) = minus119906119899119911 (0) = 01198880 = 100381610038161003816100381611990411003816100381610038161003816 1198881 = 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 1198882 = 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 0 (0) = 1 (0) = 2 (0) = 001

(38)

Controller E (method in [25])

119906119886 = minus5119890minus5119905 (1198902 (0) + 119911 (0)) minus 0 sign (1199041) 1198880 = 5 100381610038161003816100381611990411003816100381610038161003816

(39)

ControllersD and E shared the sameNTSMcontroller 119906119899and the sliding surface 1199042 was taken as 1199041198652 of controller B

The simulation results are given in Figures 5 and 6 It isclear that both controllers D and E achieved fast tracking ofthe reference signal Controller D had shorter convergencetime and better transient performance than controller Eeven though the maximum control torque and the chatteringamplitude of controller D were smaller Meanwhile thesliding surfaces in Figure 6 converged to zero within 119905 =3 s In Figure 6(a) the motions of ISM started from zeroand there was a short time that ISM did not remain zerodue to the affection of uncertainty It is observed that the

FNTSM converged to zero after the ISM and the stagnationof FNTSM never happened It verifies that the designedadaptive NTSM control scheme is effective Since the twocontrollers shared the same NTSM controller 119906119899 the simula-tion results confirm that the proposed adaptive method hadbetter transient performance and was more efficient than themethod in [25]

Here themodified adaptive controller ((30) and (31)) waschecked for chattering reduction In the following controllerF the boundary layer approach and the modified adaptivelaws were both employed As a comparison controller Gonly employed the boundary layer approach In order toavoid the unbounded growing of adaptive gains inside theboundary layer the switching adaptive laws were adoptedfor controller G In addition controllers F and G used thesame boundary layer and NTSM controller 119906119899 of controllerD Since it is assumed that the states can be measured forthe controller design themeasurement noisemay deterioratethe performance of controller in real implementations Toillustrate the robustness of the controller random noise wastaken for the states [119909 ]119879 (mean value 0 and standarddeviation 001)

Controller F

119906119886 = minus16 (051199041 + 10038161003816100381610038161199041100381610038161003816100381679 sign (1199041)+ 100381610038161003816100381610038165119890minus5119905 (1198902 (0) + 119911 (0))10038161003816100381610038161003816 sat (1199041)+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sat (1199041))

sat (1199041) = sign (1199041) 100381610038161003816100381611990411003816100381610038161003816 gt 0041199041

004100381610038161003816100381611990411003816100381610038161003816 le 004

1198880 = 2 100381610038161003816100381611990411003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 004) 0 gt 0011 0 le 001

1198881 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 005) 1 gt 0011 1 le 001

1198882 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 004) 2 gt 0011 2 le 001

(40)

Controller G

119906119886 = minus16 (051199041 + 10038161003816100381610038161199041100381610038161003816100381679 sign (1199041)+ 100381610038161003816100381610038165119890minus5119905 (1198902 (0) + 119911 (0))10038161003816100381610038161003816 sat (1199041)+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sat (1199041))

1198880 = 2 100381610038161003816100381611990411003816100381610038161003816 1199041 gt 0040 1199041 le 004

Mathematical Problems in Engineering 9

Ang

ular

coor

dina

te (r

ad)

Controller DController E

Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus08

minus06

minus04

minus02

0

02

04

06

08

1

(a) Angular coordinate

Trac

king

erro

r (ra

d)

Controller DController E

minus02

0

02

04

06

08

1

1 2 3 4 5 6 7 8 9 100Time (s)

(b) Tracking error

Adaptive gain 1

Adaptive gain 1Adaptive gain 2

Adaptive gain 3

Controller D

Controller E

1 2 3 4 5 6 7 8 9 100Time (s)

0

01

02

Adap

tive g

ains

0

5

10

15

Adap

tive g

ains

1 2 3 4 5 6 7 8 9 100Time (s)

(c) Adaptive gains

0

Controller E

Controller D

minus20

20

Con

trol t

orqu

es (N

)

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus20

0

20C

ontro

l tor

ques

(N)

(d) Control torque

Figure 5 Simulation result of the adaptive control

1198881 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 1199041 gt 0040 1199041 le 004

1198882 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 1199041 gt 0040 1199041 le 004

(41)

The simulation result of controller F is given in Figure 7It is shown that the angular coordinate tracked the referencesignal with a high precision (Figure 7(a)) Due to the adop-tion of the boundary layer and themodified adaptive laws thesliding surface 1199041 converged to a small domain as |1199041| le 01

(Figure 7(b))Moreover the adaptive gains decreased to a lowlevel with approximate reduction ratios as 5108 8479and 1615 respectively (Figure 7(c)) As a result there was amarked reduction in the chattering of controller F comparedto that of controller G in Figure 7(d) Thus the effectivenessof the modified adaptive controller is verified

5 Conclusions

This paper studies the NTSM control for a class of second-order systems in which the coefficient of control input isunknown In this paper a new FNTSM was designed tointegrate the advantages of LSM and NTSM Based onthis FNTSM new forms of robust controller and adaptive

10 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 10minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

Time (s)

ISM

Controller DController E

(a) ISM

0 1 2 3 4 5 6 7 8 9 10minus2

minus15

minus1

minus05

0

05

1

15

2

25

3

Time (s)

FNTS

M

Controller DController E

(b) FNTSM

Figure 6 The response of sliding surfaces

Controller FReference signal

10 20 30 40 500Time (s)

minus1

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad)

(a) Angular coordinate

minus02

0

02

04

06

08

0 10 20 30 40 50Time (s)

Slid

ing

surfa

ces

ISMFNTSM

(b) Sliding surfaces

0

005

01

015

02

025

Adap

tive g

ains

Adaptive gain 1Adaptive gain 2

Adaptive gain 3

10 20 30 40 500Time (s)

(c) Adaptive gains

0 10 20 30 40 50Time (s)

Controller G

Controller F

minus10

0

10

Con

trol t

orqu

es (N

)

minus10

0

10

Con

trol t

orqu

es (N

)

10 20 30 40 500Time (s)

(d) Control torques

Figure 7 Simulation result of the modified adaptive control

Mathematical Problems in Engineering 11

controller were proposed for the discussed uncertain systemThe simulation verified the effectiveness of the proposed con-trollersThe proposed FNTSM shows faster convergence ratethan the NTSM and chattering reduction was achieved bythe EDSG The adaptive controller presented better transientperformance and more efficiency than the adaptive methodin [25] Moreover the modified adaptive controller achievedeffective chattering reduction

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this article

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (no 61375100 no 61472037 and no61433003)

References

[1] J M Zhang C Y Sun R M Zhang and C Qian ldquoAdaptivesliding mode control for re-entry attitude of near space hyper-sonic vehicle based on backstepping designrdquo IEEECAA Journalof Automatica Sinica vol 2 no 1 pp 94ndash101 2015

[2] J Liu Y Zhao B Geng and B Xiao ldquoAdaptive second ordersliding mode control of a fuel cell hybrid system for electricvehicle applicationsrdquo Mathematical Problems in Engineeringvol 2015 Article ID 370424 14 pages 2015

[3] M Zak ldquoTerminal attractors in neural networksrdquo NeuralNetworks vol 2 no 4 pp 259ndash274 1989

[4] S T Venkataraman and S Gulati ldquoTerminal sliding modes anew approach to nonlinear control synthesisrdquo in Proceedings ofthe 5th International Conference onAdvanced Robotics vol 1 pp443ndash448 Pisa Italy June 1991

[5] D Y Zhao S Y Li and F Gao ldquoA new terminal sliding modecontrol for robotic manipulatorsrdquo International Journal of Con-trol vol 82 no 10 pp 1804ndash1813 2009

[6] Z K Song H X Li and K B Sun ldquoFinite-time control fornonlinear spacecraft attitude based on terminal sliding modetechniquerdquo ISA Transactions vol 53 no 1 pp 117ndash124 2014

[7] H Komurcugil ldquoNon-singular terminal sliding-mode controlof DC-DC buck convertersrdquo Control Engineering Practice vol21 no 3 pp 321ndash332 2013

[8] X Yu M Zhihong and Y Wu ldquoTerminal sliding modes withfast transient performancerdquo in Proceedings of the 1997 36th IEEEConference on Decision and Control Part 1 (of 5) vol 2 pp 962ndash963 San Diego Calif USA December 1997

[9] S Mobayen ldquoFast terminal sliding mode controller design fornonlinear second-order systems with time-varying uncertain-tiesrdquo Complexity vol 21 no 2 pp 239ndash244 2015

[10] Z He C Liu Y Zhan H Li X Huang and Z ZhangldquoNonsingular fast terminal sliding mode control with extendedstate observer and tracking differentiator for uncertain nonlin-ear systemsrdquo Mathematical Problems in Engineering vol 2014Article ID 639707 16 pages 2014

[11] L Liu Q M Zhu L Cheng et al ldquoApplied methods and tech-niques for mechatronic systems modelling identification and

controlrdquo Lecture Notes in Control and Information Sciences vol452 pp 79ndash97 2014

[12] X H Yu and Z H Man ldquoOn finite time mechanism terminalsliding modesrdquo in Proceedings of the IEEE International Work-shop on Variable Structure Systems pp 164ndash167 Tokyo Japan1996

[13] L Y Wang T Y Chai and L F Zhai ldquoNeural-network-basedterminal sliding-mode control of robotic manipulators includ-ing actuator dynamicsrdquo IEEE Transactions on Industrial Elec-tronics vol 56 no 9 pp 3296ndash3304 2009

[14] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[15] A Al-Ghanimi J Zheng and Z Man ldquoRobust and fast non-singular terminal sliding mode control for piezoelectric actua-torsrdquo IET ControlTheory ampApplications vol 9 no 18 pp 2678ndash2687 2015

[16] S Mobayen ldquoFinite-time tracking control of chained-formnonholonomic systems with external disturbances based onrecursive terminal sliding mode methodrdquo Nonlinear Dynamicsvol 80 no 1-2 pp 669ndash683 2015

[17] S Mobayen ldquoFast terminal sliding mode tracking of non-holonomic systems with exponential decay raterdquo IET ControlTheory amp Applications vol 9 no 8 pp 1294ndash1301 2015

[18] Y Wu and J Wang ldquoContinuous recursive sliding modecontrol for hypersonic flight vehicle with extended disturbanceobserverrdquo Mathematical Problems in Engineering vol 2015Article ID 506906 26 pages 2015

[19] S Mobayen ldquoAn adaptive fast terminal sliding mode controlcombined with global slidingmode scheme for tracking controlof uncertain nonlinear third-order systemsrdquoNonlinear Dynam-ics vol 82 no 1-2 pp 599ndash610 2015

[20] N Boonsatit and C Pukdeboon ldquoAdaptive fast terminal slidingmode control of magnetic levitation systemrdquo Journal of ControlAutomation and Electrical Systems vol 27 no 4 pp 359ndash3672016

[21] L Y Fang T S Li Z F Li and R Li ldquoAdaptive terminal slidingmode control for anti-synchronization of uncertain chaoticsystemsrdquoNonlinear Dynamics vol 74 no 4 pp 991ndash1002 2013

[22] C-C Yang ldquoSynchronization of second-order chaotic systemsvia adaptive terminal sliding mode control with input nonlin-earityrdquo Journal of the Franklin Institute Engineering and AppliedMathematics vol 349 no 6 pp 2019ndash2032 2012

[23] C-C Yang and C-J Ou ldquoAdaptive terminal sliding mode con-trol subject to input nonlinearity for synchronization of chaoticgyrosrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 3 pp 682ndash691 2013

[24] J-J Kim J-J Lee K-B Park and M-J Youn ldquoDesign ofnew time-varying sliding surface for robot manipulator usingvariable structure controllerrdquo Electronics Letters vol 29 no 2pp 195ndash196 1993

[25] P Li and Z-Q Zheng ldquoRobust adaptive second-order sliding-mode control with fast transient performancerdquo IET ControlTheory amp Applications vol 6 no 2 pp 305ndash312 2012

[26] M Zhihong M OrsquoDay and X Yu ldquoA robust adaptive terminalsliding mode control for rigid robotic manipulatorsrdquo Journal ofIntelligent and Robotic SystemsTheory and Applications vol 24no 1 pp 23ndash41 1999

[27] Z Zhu Y Xia and M Fu ldquoAdaptive sliding mode control forattitude stabilization with actuator saturationrdquo IEEE Transac-tions on Industrial Electronics vol 58 no 10 pp 4898ndash49072011

12 Mathematical Problems in Engineering

[28] H K Khalil Nonlinear Systems Prentice Hall EnglewoodCliffs NJ USA 3rd edition 2002

[29] Y Shtessel M Taleb and F Plestan ldquoA novel adaptive-gainsupertwisting sliding mode controller methodology and appli-cationrdquo Automatica vol 48 no 5 pp 759ndash769 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

to the reference signal within 119905 = 4 s The sliding surfacesconverged to zero within 119905 = 15 s (Figure 4(c)) It is clear thatcontroller B cost less convergence time than the controllerA while the control inputs of these controllers had slightdifference (Figures 4(b) and 4(d)) This fact implies thatthe controller designed by the proposed FNTSM was moreefficient As expected the chattering amplitude of controllerC (plusmn5) was half of controller B (plusmn10) (Figure 4(d)) Howeverthe performance of controllerChad small differencewith thatof controller BThe result confirms that the EDSG is effectivefor chattering suppression without deteriorating the systemrobustness

42 Adaptive Control In this subsection the proposed adap-tive NTSM control scheme ((23) and (25)) was verified forthe tracking control As a comparison the adaptive methodin [25] was adopted to design the adaptive controller 119906119886 Thecontrollers were designed as follows

Controller D (Theorem 6)

119906119899 = minus11199021003816100381610038161003816119890210038161003816100381610038162minus119902 sign (1198902) minus 251199042 minus 10038161003816100381610038161199042100381610038161003816100381679 sign (1199042)

119906119886 = minus16 (051199041 + 10038161003816100381610038161199041100381610038161003816100381679 sign (1199041)+ 100381610038161003816100381610038165119890minus5119905 (1198902 (0) + 119911 (0))10038161003816100381610038161003816 sign (1199041)+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sign (1199041))

1199041 = 1198902 + 119911 minus 119890minus5119905 (1198902 (0) + 119911 (0)) = minus119906119899119911 (0) = 01198880 = 100381610038161003816100381611990411003816100381610038161003816 1198881 = 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 1198882 = 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 0 (0) = 1 (0) = 2 (0) = 001

(38)

Controller E (method in [25])

119906119886 = minus5119890minus5119905 (1198902 (0) + 119911 (0)) minus 0 sign (1199041) 1198880 = 5 100381610038161003816100381611990411003816100381610038161003816

(39)

ControllersD and E shared the sameNTSMcontroller 119906119899and the sliding surface 1199042 was taken as 1199041198652 of controller B

The simulation results are given in Figures 5 and 6 It isclear that both controllers D and E achieved fast tracking ofthe reference signal Controller D had shorter convergencetime and better transient performance than controller Eeven though the maximum control torque and the chatteringamplitude of controller D were smaller Meanwhile thesliding surfaces in Figure 6 converged to zero within 119905 =3 s In Figure 6(a) the motions of ISM started from zeroand there was a short time that ISM did not remain zerodue to the affection of uncertainty It is observed that the

FNTSM converged to zero after the ISM and the stagnationof FNTSM never happened It verifies that the designedadaptive NTSM control scheme is effective Since the twocontrollers shared the same NTSM controller 119906119899 the simula-tion results confirm that the proposed adaptive method hadbetter transient performance and was more efficient than themethod in [25]

Here themodified adaptive controller ((30) and (31)) waschecked for chattering reduction In the following controllerF the boundary layer approach and the modified adaptivelaws were both employed As a comparison controller Gonly employed the boundary layer approach In order toavoid the unbounded growing of adaptive gains inside theboundary layer the switching adaptive laws were adoptedfor controller G In addition controllers F and G used thesame boundary layer and NTSM controller 119906119899 of controllerD Since it is assumed that the states can be measured forthe controller design themeasurement noisemay deterioratethe performance of controller in real implementations Toillustrate the robustness of the controller random noise wastaken for the states [119909 ]119879 (mean value 0 and standarddeviation 001)

Controller F

119906119886 = minus16 (051199041 + 10038161003816100381610038161199041100381610038161003816100381679 sign (1199041)+ 100381610038161003816100381610038165119890minus5119905 (1198902 (0) + 119911 (0))10038161003816100381610038161003816 sat (1199041)+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sat (1199041))

sat (1199041) = sign (1199041) 100381610038161003816100381611990411003816100381610038161003816 gt 0041199041

004100381610038161003816100381611990411003816100381610038161003816 le 004

1198880 = 2 100381610038161003816100381611990411003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 004) 0 gt 0011 0 le 001

1198881 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 005) 1 gt 0011 1 le 001

1198882 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 sign (100381610038161003816100381611990411003816100381610038161003816 minus 004) 2 gt 0011 2 le 001

(40)

Controller G

119906119886 = minus16 (051199041 + 10038161003816100381610038161199041100381610038161003816100381679 sign (1199041)+ 100381610038161003816100381610038165119890minus5119905 (1198902 (0) + 119911 (0))10038161003816100381610038161003816 sat (1199041)+ (0 + 1 100381610038161003816100381611990911003816100381610038161003816 + 2 100381610038161003816100381611990921003816100381610038161003816) sat (1199041))

1198880 = 2 100381610038161003816100381611990411003816100381610038161003816 1199041 gt 0040 1199041 le 004

Mathematical Problems in Engineering 9

Ang

ular

coor

dina

te (r

ad)

Controller DController E

Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus08

minus06

minus04

minus02

0

02

04

06

08

1

(a) Angular coordinate

Trac

king

erro

r (ra

d)

Controller DController E

minus02

0

02

04

06

08

1

1 2 3 4 5 6 7 8 9 100Time (s)

(b) Tracking error

Adaptive gain 1

Adaptive gain 1Adaptive gain 2

Adaptive gain 3

Controller D

Controller E

1 2 3 4 5 6 7 8 9 100Time (s)

0

01

02

Adap

tive g

ains

0

5

10

15

Adap

tive g

ains

1 2 3 4 5 6 7 8 9 100Time (s)

(c) Adaptive gains

0

Controller E

Controller D

minus20

20

Con

trol t

orqu

es (N

)

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus20

0

20C

ontro

l tor

ques

(N)

(d) Control torque

Figure 5 Simulation result of the adaptive control

1198881 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 1199041 gt 0040 1199041 le 004

1198882 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 1199041 gt 0040 1199041 le 004

(41)

The simulation result of controller F is given in Figure 7It is shown that the angular coordinate tracked the referencesignal with a high precision (Figure 7(a)) Due to the adop-tion of the boundary layer and themodified adaptive laws thesliding surface 1199041 converged to a small domain as |1199041| le 01

(Figure 7(b))Moreover the adaptive gains decreased to a lowlevel with approximate reduction ratios as 5108 8479and 1615 respectively (Figure 7(c)) As a result there was amarked reduction in the chattering of controller F comparedto that of controller G in Figure 7(d) Thus the effectivenessof the modified adaptive controller is verified

5 Conclusions

This paper studies the NTSM control for a class of second-order systems in which the coefficient of control input isunknown In this paper a new FNTSM was designed tointegrate the advantages of LSM and NTSM Based onthis FNTSM new forms of robust controller and adaptive

10 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 10minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

Time (s)

ISM

Controller DController E

(a) ISM

0 1 2 3 4 5 6 7 8 9 10minus2

minus15

minus1

minus05

0

05

1

15

2

25

3

Time (s)

FNTS

M

Controller DController E

(b) FNTSM

Figure 6 The response of sliding surfaces

Controller FReference signal

10 20 30 40 500Time (s)

minus1

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad)

(a) Angular coordinate

minus02

0

02

04

06

08

0 10 20 30 40 50Time (s)

Slid

ing

surfa

ces

ISMFNTSM

(b) Sliding surfaces

0

005

01

015

02

025

Adap

tive g

ains

Adaptive gain 1Adaptive gain 2

Adaptive gain 3

10 20 30 40 500Time (s)

(c) Adaptive gains

0 10 20 30 40 50Time (s)

Controller G

Controller F

minus10

0

10

Con

trol t

orqu

es (N

)

minus10

0

10

Con

trol t

orqu

es (N

)

10 20 30 40 500Time (s)

(d) Control torques

Figure 7 Simulation result of the modified adaptive control

Mathematical Problems in Engineering 11

controller were proposed for the discussed uncertain systemThe simulation verified the effectiveness of the proposed con-trollersThe proposed FNTSM shows faster convergence ratethan the NTSM and chattering reduction was achieved bythe EDSG The adaptive controller presented better transientperformance and more efficiency than the adaptive methodin [25] Moreover the modified adaptive controller achievedeffective chattering reduction

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this article

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (no 61375100 no 61472037 and no61433003)

References

[1] J M Zhang C Y Sun R M Zhang and C Qian ldquoAdaptivesliding mode control for re-entry attitude of near space hyper-sonic vehicle based on backstepping designrdquo IEEECAA Journalof Automatica Sinica vol 2 no 1 pp 94ndash101 2015

[2] J Liu Y Zhao B Geng and B Xiao ldquoAdaptive second ordersliding mode control of a fuel cell hybrid system for electricvehicle applicationsrdquo Mathematical Problems in Engineeringvol 2015 Article ID 370424 14 pages 2015

[3] M Zak ldquoTerminal attractors in neural networksrdquo NeuralNetworks vol 2 no 4 pp 259ndash274 1989

[4] S T Venkataraman and S Gulati ldquoTerminal sliding modes anew approach to nonlinear control synthesisrdquo in Proceedings ofthe 5th International Conference onAdvanced Robotics vol 1 pp443ndash448 Pisa Italy June 1991

[5] D Y Zhao S Y Li and F Gao ldquoA new terminal sliding modecontrol for robotic manipulatorsrdquo International Journal of Con-trol vol 82 no 10 pp 1804ndash1813 2009

[6] Z K Song H X Li and K B Sun ldquoFinite-time control fornonlinear spacecraft attitude based on terminal sliding modetechniquerdquo ISA Transactions vol 53 no 1 pp 117ndash124 2014

[7] H Komurcugil ldquoNon-singular terminal sliding-mode controlof DC-DC buck convertersrdquo Control Engineering Practice vol21 no 3 pp 321ndash332 2013

[8] X Yu M Zhihong and Y Wu ldquoTerminal sliding modes withfast transient performancerdquo in Proceedings of the 1997 36th IEEEConference on Decision and Control Part 1 (of 5) vol 2 pp 962ndash963 San Diego Calif USA December 1997

[9] S Mobayen ldquoFast terminal sliding mode controller design fornonlinear second-order systems with time-varying uncertain-tiesrdquo Complexity vol 21 no 2 pp 239ndash244 2015

[10] Z He C Liu Y Zhan H Li X Huang and Z ZhangldquoNonsingular fast terminal sliding mode control with extendedstate observer and tracking differentiator for uncertain nonlin-ear systemsrdquo Mathematical Problems in Engineering vol 2014Article ID 639707 16 pages 2014

[11] L Liu Q M Zhu L Cheng et al ldquoApplied methods and tech-niques for mechatronic systems modelling identification and

controlrdquo Lecture Notes in Control and Information Sciences vol452 pp 79ndash97 2014

[12] X H Yu and Z H Man ldquoOn finite time mechanism terminalsliding modesrdquo in Proceedings of the IEEE International Work-shop on Variable Structure Systems pp 164ndash167 Tokyo Japan1996

[13] L Y Wang T Y Chai and L F Zhai ldquoNeural-network-basedterminal sliding-mode control of robotic manipulators includ-ing actuator dynamicsrdquo IEEE Transactions on Industrial Elec-tronics vol 56 no 9 pp 3296ndash3304 2009

[14] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[15] A Al-Ghanimi J Zheng and Z Man ldquoRobust and fast non-singular terminal sliding mode control for piezoelectric actua-torsrdquo IET ControlTheory ampApplications vol 9 no 18 pp 2678ndash2687 2015

[16] S Mobayen ldquoFinite-time tracking control of chained-formnonholonomic systems with external disturbances based onrecursive terminal sliding mode methodrdquo Nonlinear Dynamicsvol 80 no 1-2 pp 669ndash683 2015

[17] S Mobayen ldquoFast terminal sliding mode tracking of non-holonomic systems with exponential decay raterdquo IET ControlTheory amp Applications vol 9 no 8 pp 1294ndash1301 2015

[18] Y Wu and J Wang ldquoContinuous recursive sliding modecontrol for hypersonic flight vehicle with extended disturbanceobserverrdquo Mathematical Problems in Engineering vol 2015Article ID 506906 26 pages 2015

[19] S Mobayen ldquoAn adaptive fast terminal sliding mode controlcombined with global slidingmode scheme for tracking controlof uncertain nonlinear third-order systemsrdquoNonlinear Dynam-ics vol 82 no 1-2 pp 599ndash610 2015

[20] N Boonsatit and C Pukdeboon ldquoAdaptive fast terminal slidingmode control of magnetic levitation systemrdquo Journal of ControlAutomation and Electrical Systems vol 27 no 4 pp 359ndash3672016

[21] L Y Fang T S Li Z F Li and R Li ldquoAdaptive terminal slidingmode control for anti-synchronization of uncertain chaoticsystemsrdquoNonlinear Dynamics vol 74 no 4 pp 991ndash1002 2013

[22] C-C Yang ldquoSynchronization of second-order chaotic systemsvia adaptive terminal sliding mode control with input nonlin-earityrdquo Journal of the Franklin Institute Engineering and AppliedMathematics vol 349 no 6 pp 2019ndash2032 2012

[23] C-C Yang and C-J Ou ldquoAdaptive terminal sliding mode con-trol subject to input nonlinearity for synchronization of chaoticgyrosrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 3 pp 682ndash691 2013

[24] J-J Kim J-J Lee K-B Park and M-J Youn ldquoDesign ofnew time-varying sliding surface for robot manipulator usingvariable structure controllerrdquo Electronics Letters vol 29 no 2pp 195ndash196 1993

[25] P Li and Z-Q Zheng ldquoRobust adaptive second-order sliding-mode control with fast transient performancerdquo IET ControlTheory amp Applications vol 6 no 2 pp 305ndash312 2012

[26] M Zhihong M OrsquoDay and X Yu ldquoA robust adaptive terminalsliding mode control for rigid robotic manipulatorsrdquo Journal ofIntelligent and Robotic SystemsTheory and Applications vol 24no 1 pp 23ndash41 1999

[27] Z Zhu Y Xia and M Fu ldquoAdaptive sliding mode control forattitude stabilization with actuator saturationrdquo IEEE Transac-tions on Industrial Electronics vol 58 no 10 pp 4898ndash49072011

12 Mathematical Problems in Engineering

[28] H K Khalil Nonlinear Systems Prentice Hall EnglewoodCliffs NJ USA 3rd edition 2002

[29] Y Shtessel M Taleb and F Plestan ldquoA novel adaptive-gainsupertwisting sliding mode controller methodology and appli-cationrdquo Automatica vol 48 no 5 pp 759ndash769 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Ang

ular

coor

dina

te (r

ad)

Controller DController E

Reference signal

1 2 3 4 5 6 7 8 9 100Time (s)

minus08

minus06

minus04

minus02

0

02

04

06

08

1

(a) Angular coordinate

Trac

king

erro

r (ra

d)

Controller DController E

minus02

0

02

04

06

08

1

1 2 3 4 5 6 7 8 9 100Time (s)

(b) Tracking error

Adaptive gain 1

Adaptive gain 1Adaptive gain 2

Adaptive gain 3

Controller D

Controller E

1 2 3 4 5 6 7 8 9 100Time (s)

0

01

02

Adap

tive g

ains

0

5

10

15

Adap

tive g

ains

1 2 3 4 5 6 7 8 9 100Time (s)

(c) Adaptive gains

0

Controller E

Controller D

minus20

20

Con

trol t

orqu

es (N

)

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus20

0

20C

ontro

l tor

ques

(N)

(d) Control torque

Figure 5 Simulation result of the adaptive control

1198881 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990911003816100381610038161003816 1199041 gt 0040 1199041 le 004

1198882 = 2 100381610038161003816100381611990411003816100381610038161003816 100381610038161003816100381611990921003816100381610038161003816 1199041 gt 0040 1199041 le 004

(41)

The simulation result of controller F is given in Figure 7It is shown that the angular coordinate tracked the referencesignal with a high precision (Figure 7(a)) Due to the adop-tion of the boundary layer and themodified adaptive laws thesliding surface 1199041 converged to a small domain as |1199041| le 01

(Figure 7(b))Moreover the adaptive gains decreased to a lowlevel with approximate reduction ratios as 5108 8479and 1615 respectively (Figure 7(c)) As a result there was amarked reduction in the chattering of controller F comparedto that of controller G in Figure 7(d) Thus the effectivenessof the modified adaptive controller is verified

5 Conclusions

This paper studies the NTSM control for a class of second-order systems in which the coefficient of control input isunknown In this paper a new FNTSM was designed tointegrate the advantages of LSM and NTSM Based onthis FNTSM new forms of robust controller and adaptive

10 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 10minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

Time (s)

ISM

Controller DController E

(a) ISM

0 1 2 3 4 5 6 7 8 9 10minus2

minus15

minus1

minus05

0

05

1

15

2

25

3

Time (s)

FNTS

M

Controller DController E

(b) FNTSM

Figure 6 The response of sliding surfaces

Controller FReference signal

10 20 30 40 500Time (s)

minus1

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad)

(a) Angular coordinate

minus02

0

02

04

06

08

0 10 20 30 40 50Time (s)

Slid

ing

surfa

ces

ISMFNTSM

(b) Sliding surfaces

0

005

01

015

02

025

Adap

tive g

ains

Adaptive gain 1Adaptive gain 2

Adaptive gain 3

10 20 30 40 500Time (s)

(c) Adaptive gains

0 10 20 30 40 50Time (s)

Controller G

Controller F

minus10

0

10

Con

trol t

orqu

es (N

)

minus10

0

10

Con

trol t

orqu

es (N

)

10 20 30 40 500Time (s)

(d) Control torques

Figure 7 Simulation result of the modified adaptive control

Mathematical Problems in Engineering 11

controller were proposed for the discussed uncertain systemThe simulation verified the effectiveness of the proposed con-trollersThe proposed FNTSM shows faster convergence ratethan the NTSM and chattering reduction was achieved bythe EDSG The adaptive controller presented better transientperformance and more efficiency than the adaptive methodin [25] Moreover the modified adaptive controller achievedeffective chattering reduction

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this article

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (no 61375100 no 61472037 and no61433003)

References

[1] J M Zhang C Y Sun R M Zhang and C Qian ldquoAdaptivesliding mode control for re-entry attitude of near space hyper-sonic vehicle based on backstepping designrdquo IEEECAA Journalof Automatica Sinica vol 2 no 1 pp 94ndash101 2015

[2] J Liu Y Zhao B Geng and B Xiao ldquoAdaptive second ordersliding mode control of a fuel cell hybrid system for electricvehicle applicationsrdquo Mathematical Problems in Engineeringvol 2015 Article ID 370424 14 pages 2015

[3] M Zak ldquoTerminal attractors in neural networksrdquo NeuralNetworks vol 2 no 4 pp 259ndash274 1989

[4] S T Venkataraman and S Gulati ldquoTerminal sliding modes anew approach to nonlinear control synthesisrdquo in Proceedings ofthe 5th International Conference onAdvanced Robotics vol 1 pp443ndash448 Pisa Italy June 1991

[5] D Y Zhao S Y Li and F Gao ldquoA new terminal sliding modecontrol for robotic manipulatorsrdquo International Journal of Con-trol vol 82 no 10 pp 1804ndash1813 2009

[6] Z K Song H X Li and K B Sun ldquoFinite-time control fornonlinear spacecraft attitude based on terminal sliding modetechniquerdquo ISA Transactions vol 53 no 1 pp 117ndash124 2014

[7] H Komurcugil ldquoNon-singular terminal sliding-mode controlof DC-DC buck convertersrdquo Control Engineering Practice vol21 no 3 pp 321ndash332 2013

[8] X Yu M Zhihong and Y Wu ldquoTerminal sliding modes withfast transient performancerdquo in Proceedings of the 1997 36th IEEEConference on Decision and Control Part 1 (of 5) vol 2 pp 962ndash963 San Diego Calif USA December 1997

[9] S Mobayen ldquoFast terminal sliding mode controller design fornonlinear second-order systems with time-varying uncertain-tiesrdquo Complexity vol 21 no 2 pp 239ndash244 2015

[10] Z He C Liu Y Zhan H Li X Huang and Z ZhangldquoNonsingular fast terminal sliding mode control with extendedstate observer and tracking differentiator for uncertain nonlin-ear systemsrdquo Mathematical Problems in Engineering vol 2014Article ID 639707 16 pages 2014

[11] L Liu Q M Zhu L Cheng et al ldquoApplied methods and tech-niques for mechatronic systems modelling identification and

controlrdquo Lecture Notes in Control and Information Sciences vol452 pp 79ndash97 2014

[12] X H Yu and Z H Man ldquoOn finite time mechanism terminalsliding modesrdquo in Proceedings of the IEEE International Work-shop on Variable Structure Systems pp 164ndash167 Tokyo Japan1996

[13] L Y Wang T Y Chai and L F Zhai ldquoNeural-network-basedterminal sliding-mode control of robotic manipulators includ-ing actuator dynamicsrdquo IEEE Transactions on Industrial Elec-tronics vol 56 no 9 pp 3296ndash3304 2009

[14] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[15] A Al-Ghanimi J Zheng and Z Man ldquoRobust and fast non-singular terminal sliding mode control for piezoelectric actua-torsrdquo IET ControlTheory ampApplications vol 9 no 18 pp 2678ndash2687 2015

[16] S Mobayen ldquoFinite-time tracking control of chained-formnonholonomic systems with external disturbances based onrecursive terminal sliding mode methodrdquo Nonlinear Dynamicsvol 80 no 1-2 pp 669ndash683 2015

[17] S Mobayen ldquoFast terminal sliding mode tracking of non-holonomic systems with exponential decay raterdquo IET ControlTheory amp Applications vol 9 no 8 pp 1294ndash1301 2015

[18] Y Wu and J Wang ldquoContinuous recursive sliding modecontrol for hypersonic flight vehicle with extended disturbanceobserverrdquo Mathematical Problems in Engineering vol 2015Article ID 506906 26 pages 2015

[19] S Mobayen ldquoAn adaptive fast terminal sliding mode controlcombined with global slidingmode scheme for tracking controlof uncertain nonlinear third-order systemsrdquoNonlinear Dynam-ics vol 82 no 1-2 pp 599ndash610 2015

[20] N Boonsatit and C Pukdeboon ldquoAdaptive fast terminal slidingmode control of magnetic levitation systemrdquo Journal of ControlAutomation and Electrical Systems vol 27 no 4 pp 359ndash3672016

[21] L Y Fang T S Li Z F Li and R Li ldquoAdaptive terminal slidingmode control for anti-synchronization of uncertain chaoticsystemsrdquoNonlinear Dynamics vol 74 no 4 pp 991ndash1002 2013

[22] C-C Yang ldquoSynchronization of second-order chaotic systemsvia adaptive terminal sliding mode control with input nonlin-earityrdquo Journal of the Franklin Institute Engineering and AppliedMathematics vol 349 no 6 pp 2019ndash2032 2012

[23] C-C Yang and C-J Ou ldquoAdaptive terminal sliding mode con-trol subject to input nonlinearity for synchronization of chaoticgyrosrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 3 pp 682ndash691 2013

[24] J-J Kim J-J Lee K-B Park and M-J Youn ldquoDesign ofnew time-varying sliding surface for robot manipulator usingvariable structure controllerrdquo Electronics Letters vol 29 no 2pp 195ndash196 1993

[25] P Li and Z-Q Zheng ldquoRobust adaptive second-order sliding-mode control with fast transient performancerdquo IET ControlTheory amp Applications vol 6 no 2 pp 305ndash312 2012

[26] M Zhihong M OrsquoDay and X Yu ldquoA robust adaptive terminalsliding mode control for rigid robotic manipulatorsrdquo Journal ofIntelligent and Robotic SystemsTheory and Applications vol 24no 1 pp 23ndash41 1999

[27] Z Zhu Y Xia and M Fu ldquoAdaptive sliding mode control forattitude stabilization with actuator saturationrdquo IEEE Transac-tions on Industrial Electronics vol 58 no 10 pp 4898ndash49072011

12 Mathematical Problems in Engineering

[28] H K Khalil Nonlinear Systems Prentice Hall EnglewoodCliffs NJ USA 3rd edition 2002

[29] Y Shtessel M Taleb and F Plestan ldquoA novel adaptive-gainsupertwisting sliding mode controller methodology and appli-cationrdquo Automatica vol 48 no 5 pp 759ndash769 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 10minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

Time (s)

ISM

Controller DController E

(a) ISM

0 1 2 3 4 5 6 7 8 9 10minus2

minus15

minus1

minus05

0

05

1

15

2

25

3

Time (s)

FNTS

M

Controller DController E

(b) FNTSM

Figure 6 The response of sliding surfaces

Controller FReference signal

10 20 30 40 500Time (s)

minus1

minus05

0

05

1

Ang

ular

coor

dina

te (r

ad)

(a) Angular coordinate

minus02

0

02

04

06

08

0 10 20 30 40 50Time (s)

Slid

ing

surfa

ces

ISMFNTSM

(b) Sliding surfaces

0

005

01

015

02

025

Adap

tive g

ains

Adaptive gain 1Adaptive gain 2

Adaptive gain 3

10 20 30 40 500Time (s)

(c) Adaptive gains

0 10 20 30 40 50Time (s)

Controller G

Controller F

minus10

0

10

Con

trol t

orqu

es (N

)

minus10

0

10

Con

trol t

orqu

es (N

)

10 20 30 40 500Time (s)

(d) Control torques

Figure 7 Simulation result of the modified adaptive control

Mathematical Problems in Engineering 11

controller were proposed for the discussed uncertain systemThe simulation verified the effectiveness of the proposed con-trollersThe proposed FNTSM shows faster convergence ratethan the NTSM and chattering reduction was achieved bythe EDSG The adaptive controller presented better transientperformance and more efficiency than the adaptive methodin [25] Moreover the modified adaptive controller achievedeffective chattering reduction

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this article

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (no 61375100 no 61472037 and no61433003)

References

[1] J M Zhang C Y Sun R M Zhang and C Qian ldquoAdaptivesliding mode control for re-entry attitude of near space hyper-sonic vehicle based on backstepping designrdquo IEEECAA Journalof Automatica Sinica vol 2 no 1 pp 94ndash101 2015

[2] J Liu Y Zhao B Geng and B Xiao ldquoAdaptive second ordersliding mode control of a fuel cell hybrid system for electricvehicle applicationsrdquo Mathematical Problems in Engineeringvol 2015 Article ID 370424 14 pages 2015

[3] M Zak ldquoTerminal attractors in neural networksrdquo NeuralNetworks vol 2 no 4 pp 259ndash274 1989

[4] S T Venkataraman and S Gulati ldquoTerminal sliding modes anew approach to nonlinear control synthesisrdquo in Proceedings ofthe 5th International Conference onAdvanced Robotics vol 1 pp443ndash448 Pisa Italy June 1991

[5] D Y Zhao S Y Li and F Gao ldquoA new terminal sliding modecontrol for robotic manipulatorsrdquo International Journal of Con-trol vol 82 no 10 pp 1804ndash1813 2009

[6] Z K Song H X Li and K B Sun ldquoFinite-time control fornonlinear spacecraft attitude based on terminal sliding modetechniquerdquo ISA Transactions vol 53 no 1 pp 117ndash124 2014

[7] H Komurcugil ldquoNon-singular terminal sliding-mode controlof DC-DC buck convertersrdquo Control Engineering Practice vol21 no 3 pp 321ndash332 2013

[8] X Yu M Zhihong and Y Wu ldquoTerminal sliding modes withfast transient performancerdquo in Proceedings of the 1997 36th IEEEConference on Decision and Control Part 1 (of 5) vol 2 pp 962ndash963 San Diego Calif USA December 1997

[9] S Mobayen ldquoFast terminal sliding mode controller design fornonlinear second-order systems with time-varying uncertain-tiesrdquo Complexity vol 21 no 2 pp 239ndash244 2015

[10] Z He C Liu Y Zhan H Li X Huang and Z ZhangldquoNonsingular fast terminal sliding mode control with extendedstate observer and tracking differentiator for uncertain nonlin-ear systemsrdquo Mathematical Problems in Engineering vol 2014Article ID 639707 16 pages 2014

[11] L Liu Q M Zhu L Cheng et al ldquoApplied methods and tech-niques for mechatronic systems modelling identification and

controlrdquo Lecture Notes in Control and Information Sciences vol452 pp 79ndash97 2014

[12] X H Yu and Z H Man ldquoOn finite time mechanism terminalsliding modesrdquo in Proceedings of the IEEE International Work-shop on Variable Structure Systems pp 164ndash167 Tokyo Japan1996

[13] L Y Wang T Y Chai and L F Zhai ldquoNeural-network-basedterminal sliding-mode control of robotic manipulators includ-ing actuator dynamicsrdquo IEEE Transactions on Industrial Elec-tronics vol 56 no 9 pp 3296ndash3304 2009

[14] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[15] A Al-Ghanimi J Zheng and Z Man ldquoRobust and fast non-singular terminal sliding mode control for piezoelectric actua-torsrdquo IET ControlTheory ampApplications vol 9 no 18 pp 2678ndash2687 2015

[16] S Mobayen ldquoFinite-time tracking control of chained-formnonholonomic systems with external disturbances based onrecursive terminal sliding mode methodrdquo Nonlinear Dynamicsvol 80 no 1-2 pp 669ndash683 2015

[17] S Mobayen ldquoFast terminal sliding mode tracking of non-holonomic systems with exponential decay raterdquo IET ControlTheory amp Applications vol 9 no 8 pp 1294ndash1301 2015

[18] Y Wu and J Wang ldquoContinuous recursive sliding modecontrol for hypersonic flight vehicle with extended disturbanceobserverrdquo Mathematical Problems in Engineering vol 2015Article ID 506906 26 pages 2015

[19] S Mobayen ldquoAn adaptive fast terminal sliding mode controlcombined with global slidingmode scheme for tracking controlof uncertain nonlinear third-order systemsrdquoNonlinear Dynam-ics vol 82 no 1-2 pp 599ndash610 2015

[20] N Boonsatit and C Pukdeboon ldquoAdaptive fast terminal slidingmode control of magnetic levitation systemrdquo Journal of ControlAutomation and Electrical Systems vol 27 no 4 pp 359ndash3672016

[21] L Y Fang T S Li Z F Li and R Li ldquoAdaptive terminal slidingmode control for anti-synchronization of uncertain chaoticsystemsrdquoNonlinear Dynamics vol 74 no 4 pp 991ndash1002 2013

[22] C-C Yang ldquoSynchronization of second-order chaotic systemsvia adaptive terminal sliding mode control with input nonlin-earityrdquo Journal of the Franklin Institute Engineering and AppliedMathematics vol 349 no 6 pp 2019ndash2032 2012

[23] C-C Yang and C-J Ou ldquoAdaptive terminal sliding mode con-trol subject to input nonlinearity for synchronization of chaoticgyrosrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 3 pp 682ndash691 2013

[24] J-J Kim J-J Lee K-B Park and M-J Youn ldquoDesign ofnew time-varying sliding surface for robot manipulator usingvariable structure controllerrdquo Electronics Letters vol 29 no 2pp 195ndash196 1993

[25] P Li and Z-Q Zheng ldquoRobust adaptive second-order sliding-mode control with fast transient performancerdquo IET ControlTheory amp Applications vol 6 no 2 pp 305ndash312 2012

[26] M Zhihong M OrsquoDay and X Yu ldquoA robust adaptive terminalsliding mode control for rigid robotic manipulatorsrdquo Journal ofIntelligent and Robotic SystemsTheory and Applications vol 24no 1 pp 23ndash41 1999

[27] Z Zhu Y Xia and M Fu ldquoAdaptive sliding mode control forattitude stabilization with actuator saturationrdquo IEEE Transac-tions on Industrial Electronics vol 58 no 10 pp 4898ndash49072011

12 Mathematical Problems in Engineering

[28] H K Khalil Nonlinear Systems Prentice Hall EnglewoodCliffs NJ USA 3rd edition 2002

[29] Y Shtessel M Taleb and F Plestan ldquoA novel adaptive-gainsupertwisting sliding mode controller methodology and appli-cationrdquo Automatica vol 48 no 5 pp 759ndash769 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

controller were proposed for the discussed uncertain systemThe simulation verified the effectiveness of the proposed con-trollersThe proposed FNTSM shows faster convergence ratethan the NTSM and chattering reduction was achieved bythe EDSG The adaptive controller presented better transientperformance and more efficiency than the adaptive methodin [25] Moreover the modified adaptive controller achievedeffective chattering reduction

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this article

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (no 61375100 no 61472037 and no61433003)

References

[1] J M Zhang C Y Sun R M Zhang and C Qian ldquoAdaptivesliding mode control for re-entry attitude of near space hyper-sonic vehicle based on backstepping designrdquo IEEECAA Journalof Automatica Sinica vol 2 no 1 pp 94ndash101 2015

[2] J Liu Y Zhao B Geng and B Xiao ldquoAdaptive second ordersliding mode control of a fuel cell hybrid system for electricvehicle applicationsrdquo Mathematical Problems in Engineeringvol 2015 Article ID 370424 14 pages 2015

[3] M Zak ldquoTerminal attractors in neural networksrdquo NeuralNetworks vol 2 no 4 pp 259ndash274 1989

[4] S T Venkataraman and S Gulati ldquoTerminal sliding modes anew approach to nonlinear control synthesisrdquo in Proceedings ofthe 5th International Conference onAdvanced Robotics vol 1 pp443ndash448 Pisa Italy June 1991

[5] D Y Zhao S Y Li and F Gao ldquoA new terminal sliding modecontrol for robotic manipulatorsrdquo International Journal of Con-trol vol 82 no 10 pp 1804ndash1813 2009

[6] Z K Song H X Li and K B Sun ldquoFinite-time control fornonlinear spacecraft attitude based on terminal sliding modetechniquerdquo ISA Transactions vol 53 no 1 pp 117ndash124 2014

[7] H Komurcugil ldquoNon-singular terminal sliding-mode controlof DC-DC buck convertersrdquo Control Engineering Practice vol21 no 3 pp 321ndash332 2013

[8] X Yu M Zhihong and Y Wu ldquoTerminal sliding modes withfast transient performancerdquo in Proceedings of the 1997 36th IEEEConference on Decision and Control Part 1 (of 5) vol 2 pp 962ndash963 San Diego Calif USA December 1997

[9] S Mobayen ldquoFast terminal sliding mode controller design fornonlinear second-order systems with time-varying uncertain-tiesrdquo Complexity vol 21 no 2 pp 239ndash244 2015

[10] Z He C Liu Y Zhan H Li X Huang and Z ZhangldquoNonsingular fast terminal sliding mode control with extendedstate observer and tracking differentiator for uncertain nonlin-ear systemsrdquo Mathematical Problems in Engineering vol 2014Article ID 639707 16 pages 2014

[11] L Liu Q M Zhu L Cheng et al ldquoApplied methods and tech-niques for mechatronic systems modelling identification and

controlrdquo Lecture Notes in Control and Information Sciences vol452 pp 79ndash97 2014

[12] X H Yu and Z H Man ldquoOn finite time mechanism terminalsliding modesrdquo in Proceedings of the IEEE International Work-shop on Variable Structure Systems pp 164ndash167 Tokyo Japan1996

[13] L Y Wang T Y Chai and L F Zhai ldquoNeural-network-basedterminal sliding-mode control of robotic manipulators includ-ing actuator dynamicsrdquo IEEE Transactions on Industrial Elec-tronics vol 56 no 9 pp 3296ndash3304 2009

[14] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[15] A Al-Ghanimi J Zheng and Z Man ldquoRobust and fast non-singular terminal sliding mode control for piezoelectric actua-torsrdquo IET ControlTheory ampApplications vol 9 no 18 pp 2678ndash2687 2015

[16] S Mobayen ldquoFinite-time tracking control of chained-formnonholonomic systems with external disturbances based onrecursive terminal sliding mode methodrdquo Nonlinear Dynamicsvol 80 no 1-2 pp 669ndash683 2015

[17] S Mobayen ldquoFast terminal sliding mode tracking of non-holonomic systems with exponential decay raterdquo IET ControlTheory amp Applications vol 9 no 8 pp 1294ndash1301 2015

[18] Y Wu and J Wang ldquoContinuous recursive sliding modecontrol for hypersonic flight vehicle with extended disturbanceobserverrdquo Mathematical Problems in Engineering vol 2015Article ID 506906 26 pages 2015

[19] S Mobayen ldquoAn adaptive fast terminal sliding mode controlcombined with global slidingmode scheme for tracking controlof uncertain nonlinear third-order systemsrdquoNonlinear Dynam-ics vol 82 no 1-2 pp 599ndash610 2015

[20] N Boonsatit and C Pukdeboon ldquoAdaptive fast terminal slidingmode control of magnetic levitation systemrdquo Journal of ControlAutomation and Electrical Systems vol 27 no 4 pp 359ndash3672016

[21] L Y Fang T S Li Z F Li and R Li ldquoAdaptive terminal slidingmode control for anti-synchronization of uncertain chaoticsystemsrdquoNonlinear Dynamics vol 74 no 4 pp 991ndash1002 2013

[22] C-C Yang ldquoSynchronization of second-order chaotic systemsvia adaptive terminal sliding mode control with input nonlin-earityrdquo Journal of the Franklin Institute Engineering and AppliedMathematics vol 349 no 6 pp 2019ndash2032 2012

[23] C-C Yang and C-J Ou ldquoAdaptive terminal sliding mode con-trol subject to input nonlinearity for synchronization of chaoticgyrosrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 3 pp 682ndash691 2013

[24] J-J Kim J-J Lee K-B Park and M-J Youn ldquoDesign ofnew time-varying sliding surface for robot manipulator usingvariable structure controllerrdquo Electronics Letters vol 29 no 2pp 195ndash196 1993

[25] P Li and Z-Q Zheng ldquoRobust adaptive second-order sliding-mode control with fast transient performancerdquo IET ControlTheory amp Applications vol 6 no 2 pp 305ndash312 2012

[26] M Zhihong M OrsquoDay and X Yu ldquoA robust adaptive terminalsliding mode control for rigid robotic manipulatorsrdquo Journal ofIntelligent and Robotic SystemsTheory and Applications vol 24no 1 pp 23ndash41 1999

[27] Z Zhu Y Xia and M Fu ldquoAdaptive sliding mode control forattitude stabilization with actuator saturationrdquo IEEE Transac-tions on Industrial Electronics vol 58 no 10 pp 4898ndash49072011

12 Mathematical Problems in Engineering

[28] H K Khalil Nonlinear Systems Prentice Hall EnglewoodCliffs NJ USA 3rd edition 2002

[29] Y Shtessel M Taleb and F Plestan ldquoA novel adaptive-gainsupertwisting sliding mode controller methodology and appli-cationrdquo Automatica vol 48 no 5 pp 759ndash769 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

[28] H K Khalil Nonlinear Systems Prentice Hall EnglewoodCliffs NJ USA 3rd edition 2002

[29] Y Shtessel M Taleb and F Plestan ldquoA novel adaptive-gainsupertwisting sliding mode controller methodology and appli-cationrdquo Automatica vol 48 no 5 pp 759ndash769 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of