Control Law Design for Twin Rotor MIMO System with Nonlinear ...

Research ArticleControl Law Design for Twin Rotor MIMO System withNonlinear Control Strategy

M. Ilyas,1 N. Abbas,2 M. UbaidUllah,2 Waqas A. Imtiaz,1 M. A. Q. Shah,2 and K. Mahmood1

1Department of Electrical Engineering, Iqra National University, Peshawar 25000, Pakistan2Department of Electrical Engineering, CIIT, Islamabad 45550, Pakistan

Correspondence should be addressed to Waqas A. Imtiaz; [email protected]

Received 13 May 2016; Revised 10 June 2016; Accepted 15 June 2016

Academic Editor: Jean J. Loiseau

Copyright © 2016 M. Ilyas et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Modeling of complex air vehicles is a challenging task due to high nonlinear behavior and significant coupling effect between rotors.Twin rotor multi-input multioutput system (TRMS) is a laboratory setup designed for control experiments, which resembles ahelicopter with unstable, nonlinear, and coupled dynamics. This paper focuses on the design and analysis of sliding mode control(SMC) and backstepping controller for pitch and yaw angle control ofmain and tail rotor of theTRMSunder parametric uncertainty.The proposed control strategy with SMC and backstepping achieves all mentioned limitations of TRMS. Result analysis of SMCand backstepping control schemes elucidates that backstepping provides efficient behavior with the parametric uncertainty for twinrotor system. Chattering and oscillating behaviors of SMC are removed with the backstepping control scheme considering the pitchand yaw angle for TRMS.

1. Introduction

Recent times have witnessed the evolution of variousapproaches for proper flight of air vehicles such as helicopter.Modeling of air vehicles dynamics is difficult owing to the sig-nificant coupling effect among rotors and the unavailability ofsome system states. The laboratory setup, twin rotor MIMOsystem (TRMS), is readily utilized, which resembles the flightof a helicopter [1]. It has gained much importance among thecontrol community by serving as a tool for different experi-ments and providing real time environment of an air vehicle.

It is difficult to design a controller for TRMS due to itsnonlinear behavior between two axes [2, 3]. TRMS consists ofa beamwith two rotors connected at its ends which are drivenby separate DC motors and the beam is counterbalanced byan arm having weight at its end [4]. It has two degrees offreedom, which facilitate movements in both horizontal andvertical direction. TRMS is basically a prototype model of ahelicopter; however, there is significant difference in controlof helicopter and its prototype. In order to control TRMS in adesired way, the speed of rotors is altered, while in helicopterit is done by changing the angles of rotors. There is no

cylindrical control in TRMS while in helicopter it is used indirectional control [5].

The control problem of TRMS has gainedmuch attention,owing to the high coupling effect between two propellers,unstable and nonlinear dynamics. Several techniques likeobserver based and hybrid adaptive fuzzy output feedbackcontrol approaches are developed to solve the nonlinearMIMO system with unknown control direction and deadzones [6, 7]. Genetic algorithms to control the unstable andnonlinear dynamics in TRMS are designed using PID control[8]. Adaptive fuzzy sliding mode control is developed for aclass of MIMO nonlinear system which estimates the statesfrom a semi high gain observer to construct the output feed-back fuzzy controller by incorporating the dynamic slidingmode [9]. References [10, 11] developed the observer basedadaptive fuzzy backstepping dynamic surface control (DSC)for nonlinearMIMOwith immeasurable states. [12] performsa comparative analysis between intelligent control and classi-cal control for TRMS. Adaptive fuzzy, neural network, andfeedback linearization based controllers are also designedfor the tracking of yaw and pitch angles in TRMS [4, 13–22]. However, the proposed state variables are assumed

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2016, Article ID 2952738, 10 pageshttp://dx.doi.org/10.1155/2016/2952738

2 Discrete Dynamics in Nature and Society

𝜑

I2 MB𝜑 + MR

I1

MFG + MΨ + MG

Ψ

Figure 1: TRMS laboratory setup.

measurable, which is practically not feasible to control thepitch and yaw angle of TRMS.

This paper proposes the first-order sliding mode controland backstepping control scheme for the nonlinear TRMS.In our proposed methodology, the mathematical model ofTRMS is linearized and the cross coupling effect between themain rotors is considered as disturbance.Themain advantageof SMC is that it mitigates the parametric uncertainty presentin TRMS while backstepping control algorithm performsbetter in case of external disturbance, which in this case is thecoupling effect of the rotors, because of its recursive structure.The proposed approaches are investigated for TRMS keepingin view the need for cancelling the strong coupling betweenrotors and finally providing the desired tracking response ofboth the controllers. Simulation results show the effectivenessof control algorithms but comparatively the backsteppingscheme gives the best performance in terms of stability andreference tracking.

The remainder of the paper is arranged as follows. In Sec-tion 2, themathematicalmodel of TRMS system is introducedand the parameters of the system are specified.The proposedSMC and backstepping controller along with their simulationresults are given in Sections 3 and 4, respectively. Comparisonof proposed controllers is introduced in Section 5 followed bythe concluding remarks.

2. Mathematical Modeling

Figure 1 shows the TRMS laboratory setup, which is used todevelop the mathematical model to compare the operation ofSMC and backstepping controllers.

TRMS system is designed with two rotors (main rotorand tail rotor) as shown in Figure 1 encompassing the effectof forces like gravitational, propulsion, centrifugal, frictional,and disturbance torque on movement of the propellers. Toovercome the effects of these forces we provide controlinput through motors. In the given case, only the pitch andazimuth angles are the measureable outputs and its stabilityis the main objective of designing the controller. As far asthe mechanical unit is concerned the following nonlinear

momentum equations can be derived for the pitchmovementof TRMS [1]. Consider

𝐼1Ψ = 𝑀

1−𝑀𝐹𝐺−𝑀𝐵Ψ−𝑀𝐺, (1)

where

𝑀1= 𝑎1𝜏2

1+ 𝑏1𝜏1, (2)

𝑀𝐹𝐺= 𝑀𝑔sin𝜓, (3)

𝑀𝐵𝜓= 𝐵1ΨΨ + 𝐵

2Ψsin (2Ψ) ��2, (4)

𝑀𝐺= 𝐾𝑔𝑦𝑀1�� cos (Ψ) , (5)

where 𝑎1and 𝑏1are constants. Equation (5) is derived based

on law of conservation of angular momentum of main rotor:𝐼2�� = 𝑀

2−𝑀𝐵𝜑−𝑀𝑅. (6)

Themomentum equations in the vertical plane of motion arewritten as

𝑀2= 𝑎2𝜏2

2+ 𝑏2𝜏2, (7)

𝑀𝐵𝜑= 𝐵1𝜑��, (8)

𝑀𝑅= 𝐾𝑐

𝑇𝑜𝑠 + 1

𝑇𝑝𝑠 + 1

𝑀1. (9)

Similar momentum equation can be used for the horizontalplane motion as well.

Equation (9) is derived based on law of angular conserva-tion of momentum of main rotor. The state space equationsare as follows:

(i) For main motor,

𝜏1=𝑇10

𝑇11

𝜏1+𝑘1

𝑇11

𝑢1. (10)

(ii) For tail motor,

𝜏2=𝑇20

𝑇21

𝜏2+𝑘2

𝑇21

𝑢2, (11)

where 𝑘1and 𝑘

2are the motor gain and 𝑇

20, 𝑇21, 𝑇10, and

𝑇11

are the motor parameter. 𝜏1and 𝜏

2are momentum of

mainmotor and tail motor.The linearization of the nonlinearmodel of TRMS is given in the following section.

(A) Linearization of TRMS. The state space equation ofnonlinear system along with the parameters is given by thefollowing equations:

𝑑Ψ

𝑑𝑡=𝑎1

𝐼1

𝜏2

1+𝑏1

𝐼1

𝜏1−

𝑀𝑔

𝐼1

sin (Ψ)

+0.0326

2𝐼1

sin (2Ψ) ��2 −𝐵1ΨI1Ψ

−

𝑘𝑔𝑦

𝐼1

cos (Ψ) 𝜑 (𝑎1𝜏2

1+ 𝑏1𝜏1) ,

𝑑��

𝑑𝑡=𝑎2

𝐼2

𝜏2

2+𝑏2

𝐼2

𝜏2−

𝐵1𝜑

𝐼2

�� −𝑘𝑐

𝐼2

1.75 ( 𝑎1𝜏2

1+ 𝑏1𝜏1) ,

Discrete Dynamics in Nature and Society 3

Table 1: Values of tuning parameters for SMC.

Pitch angle(rad)

Yaw angle(rad)

Case 1𝐶1 = 2.102𝐶2 = 3.5𝐾1 = 1.5

𝐶3 = 6.305𝐶4 = 0.0875𝐾2 = 4.9

Case 2𝐶1 = 1.1𝐶2 = 3.5𝐾1 = 1.95

𝐶3 = 1.3𝐶4 = 0.088𝐾2 = 2.897

Case 3𝐶1 = 2.5𝐶2 = 3.5𝐾1 = 1.915

𝐶3 = 3.3𝐶4 = 0.088𝐾2 = 2.89

𝑑𝜏1

𝑑𝑡= −

𝑇10

𝑇11

𝜏1+𝑘1

𝑇11

𝑢1,

𝑑𝜏2

𝑑𝑡= −

𝑇20

𝑇21

𝜏2+𝑘2

𝑇21

𝑢2.

(12)

The state and output vectors are given by

x = [Ψ Ψ 𝜑 ��𝜏1𝜏2]𝑇

,

y = [Ψ 𝜑]𝑇

,

(13)

where variables are as follows (Table 1):𝜓: pitch (elevation) angle.𝜙: yaw (azimuth) angle.𝜏1: momentum of main rotor.

𝜏2: momentum of tail rotor.

Here all the variables of system are expressed in term of“𝑥.” So

𝑥1= Ψ,

𝑥2= Ψ,

𝑥3= 𝜑,

𝑥4= ��,

𝑥5= 𝜏1,

𝑥6= 𝜏2.

(14)

Now the state space of the system in term of variable “𝑥” willbecome

��1= 𝑥2,

��2=𝑎1

𝐼1

𝑥2

5+𝑏1

𝐼1

𝑥5−

𝑀𝑔

𝐼1

sin (𝑥1)

+0.0326

2𝐼1

sin (2𝑥1) 𝑥2

4−𝐵1Ψ

𝐼1

��1

−

𝑘𝑔𝑦

𝐼1

cos (𝑥1) 𝑥3( 𝑎1𝑥2

5+ 𝑏1𝑥5) ,

��3= 𝑥4,

��4=𝑎2

𝐼2

𝑥2

6+𝑏2

𝐼2

𝑥6−

𝐵1𝜑

𝐼2

𝑥5−𝑘𝑐

𝐼2

1.75 ( 𝑎1𝑥2

5+ 𝑏1𝑥5) ,

��5= −

𝑇10

𝑇11

𝑥5+𝑘1

𝑇11

𝑢1,

��6= −

𝑇20

𝑇21

𝑥6+𝑘2

𝑇21

𝑢2.

(15)

To linearize the system, let the system be represented as

�� = A𝑥 + B𝑢,

𝑦 = C𝑥,(16)

where 𝑥 ∈ 𝑅 as states, 𝑢 ∈ 𝑅 as the control input, and 𝑦 ∈ 𝑅as the measured output.

Consider

𝑓1= ��1,

𝑓2= ��2,

𝑓3= ��3,

𝑓4= ��4,

𝑓5= ��5,

𝑓6= ��6.

(17)

After taking Jacobean and putting point (0, 0), then resultingsystem matrices are given below. Consider

A=

[[[[[[[[[[[[[[[[[[[[[

[

0 1 0 0 0 0

−

𝑀𝑔

𝐼1

−𝐵1Ψ

𝐼1

0 0𝑏1

𝐼1

0

0 0 0 1 0 0

0 0 0 −

𝐵1𝜑

𝐼2

−𝑘𝑐

𝐼2

1.75𝑏2

𝐼2

0 0 0 0 −𝑇10

𝑇11

0

0 0 0 0 0 −𝑇20

𝑇21

]]]]]]]]]]]]]]]]]]]]]

]

,

B =

[[[[[[[[[[[[[[[

[

0 0

0 0

0 0

0 0

𝑘1

𝑇11

0

0𝑘2

𝑇21

]]]]]]]]]]]]]]]

]

,

C = [

1 0 0 0 0 0

0 0 1 0 0 0] .

(18)


(B) State Space Equations of Linearized Model. Values ofconstants are given in the Abbreviation section [1]. By puttingvalues of all these constants, the state space equations can begiven as

��1= 𝑥2,

��2= −4.7059𝑥

1− 0.0882𝑥

2+ 1.3588𝑥

5,

��3= 𝑥4,

��4= −5𝑥

4+ 1.617𝑥

5+ 4.5𝑥

6,

��5= −0.9091𝑥

5+ 𝑢1,

��6= −𝑥6+ 0.8𝑢

2.

(19)

3. Proposed SMC Controller

(A) Choosing Sliding Surface. SMC is a nonlinear controltechnique, which deals with the capability of controlling theuncertainties of nonlinear systems [23, 24]. The primaryadvantage of the SMC technique is the low sensitivity tosystem disturbances. Moreover, it accredits the decoupling ofthe lower dimensions, and consequently, it scales down thecomplication of feedback design [25]. SMC generally consistof two phases: reaching phase and the sliding phase. Thereaching phase converges the system states to desired surfaceand sliding phase handles the oscillations. Sliding surface canbe designed as

𝑆 = 𝑐𝑇𝑥 = 𝑐1𝑥1+ 𝑐2𝑥2+ ⋅ ⋅ ⋅ + 𝑐

𝑛−1𝑥𝑛−1

+ 𝑥𝑛= 0. (20)

Now the control input consist of two parts:(i) Equivalent controller, 𝑈eq.

(ii) Discontinuous controller, ��.Consequently, the required controller can be determines as

𝑈 = 𝑈eq + ��. (21)

(B) SMC Design for Linearized Model. This section outlinesthe SMC design for linearized model. Sliding surface of thesystem is designed at first to facilitate the process. TRMS is aMIMO system so we will design two sliding surfaces.

Sliding surface for the vertical plane is as follows:

𝑠1= 𝑥5+ 𝑐2𝑥2+ 𝑐1𝑒1,

𝑒1= 𝑥1− 𝑥1𝑑.

(22)

Lyapunov condition is satisfied as

��1= −𝑠2

1− 𝑘1sign (𝑠

1) . (23)

Sliding surface for horizontal plane is as follows:

𝑠2= 𝑥6+ 𝑐4𝑥4+ 𝑐3𝑒2,

𝑒2= 𝑥3− 𝑥2𝑑,

(24)

where we have the following.Lyapunov condition is satisfied as

��2= −𝑠2

2− 𝑘2sign (𝑠

2) . (25)

2 4 6 8 10 12 14 16 18 200Time (s)

−0.2

0

0.2

0.4

0.6

0.8

1

Pitc

h an

gle (

rad)

SMC controller

Pitch angle (rad)

Figure 2: Pitch angle for linearized system using SMC.

SMC controller

Yaw angle (rad)

2 4 6 8 10 12 14 16 18 200Time (s)

−0.2

0

0.2

0.4

0.6

0.8

1

Yaw

angl

e (ra

d)

Figure 3: Yaw angle for linearized system using SMC.

3.1. Simulation Results of SMC. Figures 2 and 3 show the pitchand yaw position of TRMS obtained after implementation ofSMC in Simulink MATLAB on linearized model. It is clearthat the desired objective of regulating the system for twodegrees of freedom has been achieved under the robust con-trol action of SMC. It is shown that settling time for pitch andyaw angles is under 3 and 5 seconds, respectively. Moreover,it is observed that steady state error is approximately zero.Therefore, the proposed TRMS attains the equilibrium posi-tion with respect to pitch and yaw movement under appliedcontrol action.

The control inputs 𝑢𝑎and 𝑢

𝑏for pitch and yaw move-

ments of TRMS, respectively, are in volts and provided by twoindependent DC motors connected to corresponding rotors.The control inputs contain two types of control action, that is,the equivalent and discontinuous control. It is evident fromFigures 2 and 3 that the corresponding equivalent controlefforts successfully drive the system dynamics to correspond-ing sliding surfaces in a short period of time.

Moreover, the discontinuous control parts efficientlymaintain the system states on sliding manifolds for all


SMC controller

2 4 6 8 10 12 14 16 18 200Time (s)

−4

−2

02468

10121416

Con

trol i

nput

(V) f

or p

itch

angl

e

u1

Figure 4: Control input for pitch angle.

SMC controller

2 4 6 8 10 12 14 16 18 200Time (s)

−10

−5

0

5

10

15

Con

trol i

nput

(V) f

or y

aw an

gle

u2

Figure 5: Control input for yaw angle.

subsequent times and are responsible for system robustnessagainst uncertainties. However, chattering in control inputs“𝑢𝑎” and “𝑢

𝑏” can be clearly seen from Figures 4 and 5,

which arises due to fast switching of discontinuous controlaction around the sliding manifolds. Since the amplitude ofchattering is small, both yaw and pitch movement of TRMSare not affected by this undesired phenomenon.

The sliding manifolds 𝑠1and 𝑠2have been designed by

linearly combining system states for regulation purpose ofTRMS under system uncertainties and significant couplingbut in the absence of external perturbation. The tuningparameters have been suitably adjusted for sliding surface.Figures 6 and 7 show chattering phenomenon of sliding sur-face for pitch and yaw angle of TRMS. It is observed that thechattering phenomenon in the sliding surface is miniscule.Moreover, the settling time of sliding surface for both verticaland horizontal planes is under 1 second, which is desirablefor the system under consideration. Now another discussionis provided about the implementation of SMC with tracking.

8 9 10

2 4 6 8 10 12 14 16 18 200Time (s)

−0.1

0

0.1

0.2

0.3

0.4

0.5

Slid

ing

surfa

ce

SMC controller

−10

−5

05×10−3

Sliding surface for vertical plane

Figure 6: Chattering in sliding surface for pitch angle.

8 9 10

2 4 6 8 10 12 14 16 18 200Time (s)

−0.5

0

0.5

1

1.5

2

2.5

Slid

ing

surfa

ce fo

r yaw

angl

e

SMC controller

0.01

0

−0.01

Sliding surface for horizontal plane

Figure 7: Chattering in sliding surface for yaw angle.

3.2. Simulation Results of TRMS with Tracking. The pitch andyaw angles are obtained after implementation of SMC on aTRMS in Simulink MATLAB. Figure 8 shows the responseof pitch angle at different values of tuning parameters andFigure 9 shows the response of yaw angle at different values oftuning parameters. Figure 8 shows that the tuning parameterfor case 1 has approximately 20% overshoot from the desiredposition. Case 3 is showing undershoot from the desiredposition due the difference tuning parameters. It is observedthat case 2 is the most suitable set of tuning parametersfor achieving the desired results without showing over- andundershoot. Thus, the desired objective of regulating thesystem for two degrees of freedom has been achieved underthe robust control action of SMC. Same phenomenon isobtained for yaw angle in Figure 9, where case 2 is mostpreferable with desired tuning parameters.

Different values of tuning parameters show how we canobtain different responses of pitch and yaw angles accordingto requirements. Overshoot problem is faced in case of sharpresponse, and if we need a slower response then settling timeis increased.


SMC controller

Pitch angle (rad) for case 1Pitch angle (rad) for case 2Pitch angle (rad) for case 3

3010 15 20 2550Time (s)

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Pitc

h an

gle (

rad)

Figure 8: Response of pitch angle for TRMS.

SMC controller

Yaw angle (rad) for case 3Yaw angle (rad) for case 2Yaw angle (rad) for case 1

5 10 15 20 25 300 Time (s)

0

0.2

0.4

0.6

0.8

1

1.2

Yaw

angl

e (ra

d)

Figure 9: Response of yaw angle for TRMS.

4. Backstepping Controller

In control system theory, backstepping controller schemeis introduced by Krstic in 1995 and his companions fordesigning stability control system for a special class of linearand nonlinear dynamical system [26]. Backstepping is asystematic, Lyapunov-based method for nonlinear controlwhich refers to the recursive nature of the design procedurewhich starts at the scalar equation separated by the largestnumber of integrations from the control input and steps backtowards the control input [27, 28].

In the theory of Ordinary Differential Equations (ODEs),Lyapunov functions are scalar functions that may be usedto prove the stability of equilibrium of an ODE. The basicidea behind the Lyapunov function method consists of (I)choosing a radially unbounded positive definite Lyapunovfunction candidate 𝑉(𝑥) and (II) evaluating its derivative

𝑉(𝑥) along system dynamics and checking its negativenessfor stability analysis [25, 26].

The recursitivity terminates when the final control phaseis reached. The process which receives its stability throughrecursitivity is called backstepping [27, 28]. Backstepping canbe used for tracking and regulation problem. With the aidof Lyapunov stability, this control approach for asymptotictracking can be achieved.

4.1. Design Steps. The controller is designed using backstep-ping control technique on the proposed control problem.The standard backstepping control is based on step-by-step construction of Lyapunov function. Here we designcontroller, based on Lyapunov function.

(A) For Vertical Plane. First of all in 1st step we introduce anew state

𝑧1= 𝑥1− 𝑥1𝑑, (26)

where “𝑧1” is the new state and “𝑥

1” is state variable.

Lyapunov candidate function (LCF) for new state is

𝑉1= (

1

2) 𝑧2

1, (27)

where 𝑉 > 0, ∀𝑥 = 0, and 𝑉(0) = 0.By taking time derivative of Lyapunov function we get

��1= 𝑧1��1, (28)

��1= 𝑧2+ 𝑎1− ��1𝑑, (29)

where “𝑎1” is virtual control input to control the system and

𝑧2is the new state

𝑎1= −𝑐1𝑧1+ 𝑥1𝑑. (30)

Now another new state is introduced that is given by “𝑧2”:

𝑧2= 𝑥2− 𝑎1, (31)

where “𝑥2” is the second state variable of the system. By

putting values in (28) we get

��1= −𝑧2

1+ 𝑧1𝑧2. (32)

Now again we repeat the previous step to calculate the virtualcontrol input.

So,

𝑧2= 𝑥2− 𝑎1. (33)

We know

��1= −𝑐1��1+ ��1𝑑. (34)

New Lyapunov candidate function (LCF) is as follows:

𝑉2= 𝑉1+ (

1

2) 𝑧2

2. (35)


Taking derivative of (35),

��2= ��1+ 𝑧2��2. (36)

By putting the value of ��1and ��2we get

��2= −𝑧2

1− 𝑧2

2+ 𝑧2𝑧3, (37)

where

��2= ��2− ��1,

��1= −𝑧2

1+ 𝑧1𝑧2.

(38)

Now again we introduce new state

𝑧3= 𝑥5− 𝑎2. (39)

By taking derivative we get

��3= ��5− ��2. (40)

Now LCF will be as

𝑉3= 𝑉2+ (

1

2) 𝑧2

3. (41)

Taking derivative of above equation, we get

��3= ��2+ ��3𝑧3. (42)

By putting the values in above equation we obtain anothervirtual control input for the second state of the system asgiven below:

𝑎2= (

1

1.3588)

⋅ [−𝑧1+ 4.7051𝑥

1+ 0.0882𝑥

2− 𝑐1��1+ ��1𝑑] .

(43)

After differentiation,

��2= (

1

1.3588)

⋅ [−��1+ 4.7051��

1+ 0.0882��

2− 𝑐1��1+...𝑥1𝑑] .

(44)

Now the control input is given by

𝑢1= −𝑐3𝑧3− 𝑧2+ 0.909𝑥

5+ ��2. (45)

We get the control input 𝑢1for vertical plane of TRMS. Now

putting the value of control input we get

��3= −𝑧2

1− 𝑧2

2− 𝑧2

3. (46)

Hence condition is satisfied and system is asymptoticallystable.

(B) For horizontal Plane. First of all in 1st step we introduce anew state

𝑧4= 𝑥3− 𝑥2𝑑, (47)

where 𝑧4is the new state and 𝑥

3is the state variable.

LCF for new state is

𝑉4= (

1

2) 𝑧2

4. (48)

By taking derivative and putting the value of variables we get

��4= −𝑧2

4+ 𝑧4𝑧5. (49)

Now, the abovementioned steps are repeated to calculate theother virtual control input.

Consequently,

��4= 𝑧5+ 𝑎3− ��2𝑑,

𝑎3= −𝑐4𝑧4+ ��2𝑑.

(50)

Here 𝑎3is arbitrary control input to converge the state 𝑧

4

towards stability.Introduce another new state to calculate another arbitrary

control input.So,

𝑧5= 𝑥4− 𝑎3, (51)

where 𝑧5is new state for state variable 𝑥

4. By taking time

derivative,

��5= ��4− ��3. (52)

Also we take time derivative of previous arbitrary controlinput as

��3= −𝑐4��4+ ��2𝑑. (53)

Now LCF will be as

𝑉5= 𝑉4+ (

1

2) 𝑧2

5. (54)

By taking derivative and putting the values of ��5and ��

3we

get

��5= −𝑧2

4+ 𝑧4𝑧5+ 𝑧5[��4− (−𝑐4��4+ ��2𝑑)] . (55)

After some algebraic calculations we get

��5= −𝑧2

4− 𝑧2

5+ 𝑧6𝑧5, (56)

where 𝑧6is the new state:

𝑧6= 𝑥6− 𝑎4. (57)

Now LCF will be as

𝑉6= 𝑉5+ (

1

2) 𝑧2

6. (58)

After taking derivative,

��6= ��5+ ��6𝑧6, (59)


Backstepping controller

Pitch angle (rad)

5 10 150Time (s)

−0.2

0

0.2

0.4

0.6

0.8

1

Pitc

h an

gle (

rad)

Figure 10: Pitch angle (rad) for backstepping.

where

𝑎4= (

1

4.5) [−𝑧4− 𝑐5𝑧5+ 5𝑥4− 1.617𝑥

5− ��3+ ��2𝑑] . (60)

By taking derivative of (60),

��4= (

1

4.5) [−��4− 𝑐5��5+ 5��4− 1.617��

5− ��3+...𝑥2𝑑] . (61)

By putting the values of ��6, ��4, and ��

5we get

��6= −𝑧2

4− 𝑧2

5− 𝑧2

6, (62)

where ��6is negative definite and system will be asymptoti-

cally stable. Nowwe get control input for the horizontal plane.Finally we get the control law, whichwill regulate all the statesof the system to the origin.The system is asymptotically stableby using Backstepping design method:

𝑢2= (

1

0.8) [−𝑐6𝑧6− 𝑧5+ 𝑥6] + (

1

4.5)

⋅ [−��4− 𝑐5��5+ 5��4− 1.617��

5− ��3+...𝑥2𝑑] .

(63)

After mathematical calculations of backstepping controller,we use user defined block from MATLAB Simulink library.Required equations are used in this function. On the basisof simulation results we will elaborate the performance ofcontroller which is given below

4.2. Simulation Results of Backstepping Controller. On thebasis of backstepping control design, simulation results inFigures 10 and 11 show the stability response of pitch angleand the control input for pitch angle, respectively. SimilarlyFigures 12 and 13 show the stability of yaw angle and controlinput for the pitch angle, respectively. It is observed thatsettling time for pitch and yaw angle in case of backsteppingtechnique is less as compared to sliding mode control. Thecontroller shows very promising results and it is foundthat backstepping controller is capable of tracking and littlevariation in control inputs of both pitch and yaw angle.

Control input (V)


5 10 150Time (s)

−2

0

2

4

6

8

10

Con

trol i

nput

(V) f

or p

itch

angl

e

Figure 11: Control input for pitch angle.


Yaw angle (rad)

5 10 150Time (s)

−0.2

0

0.2

0.4

0.6

0.8

1

Yaw

angl

e (ra

d)

Figure 12: Yaw angle (rad) for backstepping.

Control input (V)


5 10 150Time (s)

−20

−15

−10

−5

0

5

Con

trol i

nput

(V) f

or y

aw an

gle

Figure 13: Control input for yaw angle.

The performance of SMC is limited due to chattering incontrol inputs but this issue is resolved through backstepping,which is shown in Figures 11 and 13. An extensive overview is


SMC and backstepping

RefSMCBackstepping

5 10 150Time (s)

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Pitc

h an

gle (

rad)

Figure 14: Pitch angle (rad) for SMC and backstepping.

SMC and backstepping

SMCRefBackstepping

5 10 150Time (s)

0

0.2

0.4

0.6

0.8

1

1.2

Yaw

angl

e (ra

d)

Figure 15: Yaw angle (rad) for SMC and backstepping.

given below for comparison of backstepping and SMC on thebasis of simulation results.

5. Comparison of Backstepping andSMC Controller

Figures 14 and 15 show the comparison between SMC andbackstepping controller for pitch and yaw angle of TRMS. Itis observed that backstepping controller shows good resultscompared to SMC in terms of handling oscillation andchattering. By using backstepping controller, the settling timefor the pitch and yaw angle is less as compared to SMC asshown in Figures 14 and 15.

6. Conclusion

This work outlines the design analysis of robust controllertechniques by implementing them in TRMS, which is a non-linear system. SMC and backstepping are implemented andanalyzed for handling the oscillation and chattering in pitchand yaw angles of TRMS. It is observed that backsteppingshows better performance in terms of less settling time andhandling perturbation as compared to SMC, owing to therecursive structure for controller design. This philosophy isthe core idea that has been followed for developing robustcontroller. The controller was implemented in the Simulinkenvironment where the state space model of the controllerwas engaged with system to achieve the desired result.Implementing the sliding mode control via backsteppingcontrol can be considered as a recommended future work,since parametric uncertainty and external disturbance canbe mitigated within a single model, which can stabilize theTRMS in a more robust way.

System Parameters

𝐼1: Moment of inertia of vertical rotor (6.8 × 10−2 kgm2)

𝐼2: Moment of inertia of horizontal rotor (2 × 10−2 kgm2)

𝐵2𝜑: Friction momentum function parameter(1 × 10−2Nms2/rad)

𝑘𝑔𝑦: Gyroscopic momentum parameter (0.05 s/rad)

𝑇20: Motor denominator parameter (1)

𝑘𝑐: Cross reaction momentum gain (2)

𝐵2Ψ: Friction momentum function parameter(1 × 10−3Nms2/rad)

𝐵1Ψ: Friction momentum function parameter(6 × 10−3Nm⋅s/rad)

𝑎1: Static characteristic parameter (0.0135)

𝑏1: Static characteristic parameter (0.0924)

𝑎2: Static characteristic parameter (0.02)

𝑇10: Motor 1 denominator parameter (1)

𝑀𝑔: Gravity momentum (0.32Nm)

𝑘1: Motor 1 gain 1 2 3.5 0.2 (1.1)

𝑘2: Motor 2 gain (0.8)

𝑇11: Motor 1 denominator parameter (1.1).

Competing Interests

The authors declare that they have no competing interests.

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