Research Article Sampled-Data Control of Spacecraft Rendezvous with...

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Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 814271, 10 pages http://dx.doi.org/10.1155/2013/814271 Research Article Sampled-Data Control of Spacecraft Rendezvous with Discontinuous Lyapunov Approach Zhuoshi Li, 1 Ming Liu, 1 Hamid Reza Karimi, 2 and Xibin Cao 1 1 Research Center of Satellite Technology, Harbin Institute of Technology, Harbin 150001, China 2 Department of Engineering, Faculty of Technology and Science, University of Agder, 4898 Grimstad, Norway Correspondence should be addressed to Ming Liu; [email protected] Received 23 May 2013; Accepted 19 August 2013 Academic Editor: Rongni Yang Copyright © 2013 Zhuoshi Li 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 investigates the sampled-data stabilization problem of spacecraſt relative positional holding with improved Lyapunov function approach. e classical Clohessy-Wiltshire equation is adopted to describe the relative dynamic model. e relative position holding problem is converted into an output tracking control problem using sampling signals. A time-dependent discontinuous Lyapunov functionals approach is developed, which will lead to essentially less conservative results for the stability analysis and controller design of the corresponding closed-loop system. Sufficient conditions for the exponential stability analysis and the existence of the proposed controller are provided, respectively. Finally, a simulation result is established to illustrate the effectiveness of the proposed control scheme. 1. Introduction In the research area of advanced astronautic missions, space- craſt autonomous rendezvous and docking has been one of the most significant problems over the past few decades. Gen- erally speaking, before completing the final docking action between the chaser spacecraſt and the target spacecraſt, the chaser spacecraſt is forbidden to maintain at a fixed relative position around the target and prepares to implement the final docking action and/or other special missions [1–3]. is procedure is the so-called “relative position holding.” In recent years, a few results have been developed in the existing literatures to investigate the problems of autonomous relative motion for multiple spacecraſts [4, 5]. Among these existing approaches, one of the most popular mathematical tool is the so-called Clohessy-Wiltshire (C-W) equation. is model is introduced by Clohessy and Wiltshire in 1960 [6] and has been broadly applied to address the problem of relative motion between two neighboring spacecraſts [7, 8]. On the other hand, in modern industrial process, one or more computers are always employed as digital controllers to control continuous-time plant [9–12]. Under this set- ting, digital computers are required to sample and quantize a continuous-time analog signal to evaluate a discrete-time sampled-data signal [13–15], based on which a discrete- time control input signal is designed [16–19]. is discrete- time signal is further converted into a continuous-time signal based on a zero-order hold equipment in the actua- tor side [20–24]. Subsequently, the corresponding sampled- data systems contain both continuous-time and discrete- time signals in the continuous-time domain [25–28]. Due to its significance, over the past decade the problems of stabilization and filtering of sampled-data systems have been investigated extensively; see for instance, [29, 30] and the reference therein. It is worth pointing out that there exist three main methodologies for the stabilization and filtering problems of sampled-data systems [31, 32], which are liſting technique method, impulsive model method, and input delay method, respectively. Among these existing approaches, input delay approach is the latest established approach, wherein the sampled-data system is modeled as a continuous-time system with time-varying delay existing in the control input. A great number of results based on this approach have been proposed in the existing literatures [33, 34]. Among these existing results, the time-independent Lyapunov function approach

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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 814271 10 pageshttpdxdoiorg1011552013814271

Research ArticleSampled-Data Control of Spacecraft Rendezvous withDiscontinuous Lyapunov Approach

Zhuoshi Li1 Ming Liu1 Hamid Reza Karimi2 and Xibin Cao1

1 Research Center of Satellite Technology Harbin Institute of Technology Harbin 150001 China2Department of Engineering Faculty of Technology and Science University of Agder 4898 Grimstad Norway

Correspondence should be addressed to Ming Liu robustcontrol23gmailcom

Received 23 May 2013 Accepted 19 August 2013

Academic Editor Rongni Yang

Copyright copy 2013 Zhuoshi Li 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 investigates the sampled-data stabilization problem of spacecraft relative positional holding with improved Lyapunovfunction approach The classical Clohessy-Wiltshire equation is adopted to describe the relative dynamic model The relativeposition holding problem is converted into an output tracking control problem using sampling signals A time-dependentdiscontinuous Lyapunov functionals approach is developed which will lead to essentially less conservative results for the stabilityanalysis and controller design of the corresponding closed-loop system Sufficient conditions for the exponential stability analysisand the existence of the proposed controller are provided respectively Finally a simulation result is established to illustrate theeffectiveness of the proposed control scheme

1 Introduction

In the research area of advanced astronautic missions space-craft autonomous rendezvous and docking has been one ofthemost significant problems over the past few decades Gen-erally speaking before completing the final docking actionbetween the chaser spacecraft and the target spacecraft thechaser spacecraft is forbidden to maintain at a fixed relativeposition around the target and prepares to implement thefinal docking action andor other special missions [1ndash3]This procedure is the so-called ldquorelative position holdingrdquo Inrecent years a few results have been developed in the existingliteratures to investigate the problems of autonomous relativemotion for multiple spacecrafts [4 5] Among these existingapproaches one of the most popular mathematical tool isthe so-called Clohessy-Wiltshire (C-W) equationThismodelis introduced by Clohessy and Wiltshire in 1960 [6] andhas been broadly applied to address the problem of relativemotion between two neighboring spacecrafts [7 8]

On the other hand in modern industrial process one ormore computers are always employed as digital controllersto control continuous-time plant [9ndash12] Under this set-ting digital computers are required to sample and quantize

a continuous-time analog signal to evaluate a discrete-timesampled-data signal [13ndash15] based on which a discrete-time control input signal is designed [16ndash19] This discrete-time signal is further converted into a continuous-timesignal based on a zero-order hold equipment in the actua-tor side [20ndash24] Subsequently the corresponding sampled-data systems contain both continuous-time and discrete-time signals in the continuous-time domain [25ndash28] Dueto its significance over the past decade the problems ofstabilization and filtering of sampled-data systems have beeninvestigated extensively see for instance [29 30] and thereference therein

It is worth pointing out that there exist three mainmethodologies for the stabilization and filtering problems ofsampled-data systems [31 32] which are lifting techniquemethod impulsive model method and input delay methodrespectively Among these existing approaches input delayapproach is the latest established approach wherein thesampled-data system ismodeled as a continuous-time systemwith time-varying delay existing in the control input A greatnumber of results based on this approach have been proposedin the existing literatures [33 34] Among these existingresults the time-independent Lyapunov function approach

2 Mathematical Problems in Engineering

presented by Fridman et al [34] has been proved to beeffective to deal with the input delay Based on the time-independent Lyapunov approach an improved Lyapunovapproach namely time-dependent Lyapunov approach hasbeen also developed by Fridman [33] and Naghshtabriziet al [35] Compared with the time-independent Lyapunovapproach the structure characteristic of the time-dependentLyapunov approach can guarantee that its value does notincrease in each sampling time despite the jumping behaviorof the sampled signals and thus can reduce the design conser-vativenessThismethod is later extended to deal with filteringproblem of Ito stochastic systems with signal sampling andquantization [36]

It should be pointed out that in practical sampled-datasystems of spacecraft rendezvous there also exists jumpingbehavior of the sampled signal at each sampling time instantand the values of the sampled signal increase abruptly atthese time instants Therefore it is desirable to reduce theconservativeness of the controller design of sampled-dataspacecraft rendezvous systems by developing new controlschemes Motivated by the aforementioned time-dependentLyapunov approach in this paper we consider the sampled-data control problem for relative position holding of space-craft rendezvous In this study signal sampling control inputlimitation and 119867

infinperformance are considered together in

a unified framework The discontinuous Lyapunov functionsapproach is adopted for the controller design also we intro-duce free-weight matrices in the Lyapunov functions and itwill be shown that the obtained results are less conservativethen those in the existing work

The remainder of this paper is organized as follows InSection 2 the problem of sampled-data control of relativeposition holding of spacecraft rendezvous is formulated InSection 3 the sufficient condition for the exponential stabilityof the sampled-data closed-loop system is provided basedon which the sufficient for the existence of the sampled-datacontroller is given Finally a numerical example is presentedin Section 4 to demonstrate the feasibility of the proposedmethod

2 Problem Formulation

In the orbital coordinate frame of chaser and target space-crafts it is well defined that the origin attaches to the masscenter of the target spacecraft 119909-axis is along the vector fromthe Earth center to the origin 119910-axis is along the target orbitcircumference 119911-axis completes the right-handed frame 119903

1

denotes the radius of the target circular orbit 120583 denotes thegravitational parameter of the earth 119908 denotes the anglevelocity of the target and 119908 = (1205831199033

1)12

Based on theNewtonmotion theory the relative dynamicmodel of chaser and target spacecrafts can be modeled by thefollowing C-W equations

119888(119905) = 2119908 119910

119888(119905) + 119908

2

(1199031+ 119909119888(119905))

minus120583 (1199031+ 119909119888(119905))

[(1199031+ 119909119888(119905))2

+ 1199102119888(119905) + 119911

2

119888(119905)]32

+ 119886119909

119910119888(119905) = minus2119908

119888(119905) + 119908

2

119910119888(119905)

minus120583119910119888(119905)

[(1199031+ 119909119888(119905))2

+ 1199102119888(119905) + 119911

2

119888(119905)]32

+ 119886119910

119888(119905) = minus

120583119910119888(119905)

[(1199031+ 119909119888(119905))2

+ 1199102119888(119905) + 119911

2

119888(119905)]32

+ 119886119911

(1)

where 119909119888(119905) 119910

119888(119905) and 119911

119888(119905) are the components of the

relative position in corresponding axes and 119886119894(119894 = 119909 119910 119911)

is the 119894th component of the control input vector By thetaylor expansion and linearization approach the linearizedequation around the null solution for (1) can be given by

119888(119905) minus 2119908 119910

119888(119905) minus 3119908

2

119909119888(119905) =

1

119898(119879119909+ 119889119909)

119910119888(119905) + 2119908

119888(119905) =

1

119898(119879119910+ 119889119910)

119888(119905) + 119908

2

119911119888(119905) =

1

119898(119879119911+ 119889119911)

(2)

In system (2) 119898 is the mass of the chaser 119879119894(119894 = 119909 119910 119911)

is the relative motion dynamic and 119889119894 119894 = 119909 119910 119911 is the

119894th component of the external disturbances We define thefollowing vector and matrices

119909 (119905) =

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

[

119909119888(119905)

119910119888(119905)

119911119888(119905)

119888(119905)

119910119888(119905)

119888(119905)

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

]

119906 (119905) =

[[[

[

119879119909(119905)

119879119910(119905)

119879119911(119905)

]]]

]

119889 (119905) =

[[[

[

119889119909(119905)

119889119910(119905)

119889119911(119905)

]]]

]

119910 (119905) =

[[[

[

119909119888(119905)

119910119888(119905)

119911119888(119905)

]]]

]

(3)

119860 =

[[[[[[[

[

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

31199082

0 0 0 2119908 0

0 0 0 minus2119908 0 0

0 0 minus1199082

0 0 0

]]]]]]]

]

Mathematical Problems in Engineering 3

119861 =1

119898

[[[[[[[

[

0 0 0

0 0 0

0 0 0

1 0 0

0 1 0

0 0 1

]]]]]]]

]

(4)

Then the C-W equation (2) can be described equivalently asfollows

(119905) = 119860119909 (119905) + 119861119906 (119905) + 119861119889 (119905)

119910 (119905) = 119862119909 (119905)

(5)

As a result the specific relative motion of the chaser andtarget can be realized via designing effective control strategy119906(119905) for system (5)

In this paper we consider the case that the signals aresampled before transmitted to the controller side DefineT ≜

1199051 1199052 1199053 as a strictly increasing sequence of sampling

times in (1199050infin) for initial time 119905

0= 0 Suppose that

the sampling internals between any two sequent samplinginstants are bounded by

119905119896+1

minus 119905119896le ℎ 119896 = 1 2 (6)

where ℎ gt 0 is a known constant Then the control vector isassumed to be of the following form at the sampling instant119905119896with zero-order hold (ZOH)

119906 (119905) = 119870119909 (119905119896) (7)

During the orbital transfer process the thrust constraint canbe described as

1003816100381610038161003816119906119894 (119905119896)1003816100381610038161003816 ⩽ 119906119894max (119894 = 119909 119910 119911) (8)

where 119906119894(119905119896) is the control input thrust along the 119894th axis at the

sampling instant 119896 and 119906119894max is the maximum allowed thrust

along the 119894th axisOn the other hand the desired holding position can be

treated as a reference output 119910119903of (5) Therefore our object

is thus to propose a sampled-data control strategy (7) suchthat the closed-loop system for plant (5) is exponentiallystable and the output 119910(119905) tracks the reference signal 119910

119903

without steady-state error Subsequently the problem ofrelative control holding is converted into an output trackingcontrol problem To achieve

lim119905rarr+infin

119910 (119905) = 119910119903 (9)

we define the output error

119890 (119905) = int

119905

0

119910119890(119905) 119889119905 (10)

Also we define the following augmented system

120585 (119905) = 119860120585 (119905) + 119861119906 (119905119896) + 1198611119889 (119905) (11)

with

120585 (119905) = [119909 (119905)

119890 (119905)] 119889 (119905) = [

119889 (119905)

119910119903

]

119860 = [119860 06times3

119862 03times3

] 119861 = [119861

03times3

]

1198611= [

119861 06times3

03times3

minus1198683times3

]

1198621= [119862 0

3times3] 119862

2= [03times6

1198683times3]

(12)

The control scheme for augmented system (11) is thus de-signed as

119906 (119905119896) = 119870120585 (119905

119896) = [119870 119870

119890] [

119909 (119905119896)

119890 (119905119896)] (13)

where119870 is defined as in (2) and119870119890isin R119898times119899 is to be designed

Substituting (13) into (11) yields the following augmentedclosed-loop system

120585 (119905) = 119860120585 (119905) + 119861119870120585 (119905119896) + 1198611119889 (119905)

119910 (119905) = 1198621120585 (119905)

119911 (119905) = 1198622120585 (119905)

(14)

If system (14) is exponentially stable then it implies that therelative position error integral action 119890(119905) rarr 0 and furtherimplies that lim

119905rarrinfin= 119910119903

In the following discussion we adopt the delay-inputapproach to deal with the sampled-data control problem Bydefining

120591 (119905) = 119905119896+1

minus 119905119896 (15)

the sampling intervals can be written as

119905119896= 119905 minus (119905

119896+1minus 119905119896) = 119905 minus 120591 (119905) (16)

Therefore the sampled augmented state can be written as

120585 (119905119896) = 120585 (119905 minus 120591 (119905)) (17)

and the control input vector can be transformed into

119906 (119905119896) = 119870120585 (119905

119896) = 119870120585 (119905 minus 120591 (119905)) (18)

Hence the augmented closed-loop system (14) can be writtenby

120585 (119905) = 119860120585 (119905) + 119861119870120585 (119905 minus 120591 (119905)) + 1198611119889 (119905)

119910 (119905) = 1198621120585 (119905)

119911 (119905) = 1198622120585 (119905)

(19)

Now the aim of this paper can be formulated as follows

Problem 1 Design a sampled-data control law 119906(119905119896) such that

the following conditions are met

4 Mathematical Problems in Engineering

(1) The closed-loop system (19) is exponentially stablewith a given decay rate 120572 gt 0 In other words thechaser holds at a relative position

(2) The control input satisfies the given upper bound (8)

(3) For closed-loop system (19) the following119867infinperfor-

mance index is achieved

119890119911(119905) = 119911 (119905)

2minus 12057410038171003817100381710038171003817119889 (119905)

100381710038171003817100381710038172⩽ 0 (20)

3 Main Results

In this section we will first perform the stability analysis andcontrol scheme design for system (19) with 119889(119905) = 0Then wewill consider the internal stability of (19) with the prescribed119867infin

performance index (20)

31 Stabilization Analysis

Theorem 2 Consider the closed-loop system (19) with 119889(119905) =0 given a decay rate 120572 gt 0 if there exist positive definitematrices 119875

1 1198752 and 119875

3isin R9times9 and matrices 119879

1 1198792 1198781 1198782

and 1198783isin R9times9 such that the following conditions hold

Φ = [

[

Φ11Φ12Φ13

⋆ Φ22Φ23

⋆ ⋆ Φ33

]

]

lt 0

Φ =

[[[

[

Φ11Φ12Φ13Φ14

⋆ Φ22Φ23

0

⋆ ⋆ Φ33Φ34

⋆ ⋆ ⋆ Φ44

]]]

]

lt 0

(21)

120588119870119879

119877119879

119894119877119894119870 minus 119906

2

119894max1198751 lt 0 (22)

where 120588 is defined as in (47) and

Φ11= 2120572119875

1minus 1198753+ 2120572ℎ119875

3minus 119879119879

1119860 minus 119860

119879

1198791minus 1198781minus 119878119879

1

Φ12= 1198751+ 119879119879

1minus ℎ1198753

Φ13= 1198753minus 2120572ℎ119875

3minus 119879119879

1119861119870 + 119878

119879

1

Φ22= ℎ1198752+ 1198792+ 119879119879

2

Φ23= minusℎ119875

3minus 119879119879

2119861119870 + 119878

2+ 119878119879

2

Φ33= minus1198753+ 2120572ℎ119875

3+ 1198783+ 119878119879

3

Φ11= 2120572119875

1minus 1198753minus 119879119879

1119860 minus 119860

119879

1198791minus 1198781minus 119878119879

1

Φ12= 1198751+ 119879119879

1

Φ13= 1198753minus 119879119879

1119861119870 + 119878

119879

1

Φ14= ℎ119878119879

1

Φ22= 119879119879

2+ 119879119879

2

Φ23= minus119879119879

2119861119870 + 119878

2+ 119878119879

2

Φ33= minus1198753+ 1198783+ 119878119879

3

Φ34= ℎ119878119879

3

Φ44= minusℎ119890

minus2120572ℎ

1198752

(23)

Then system (19) with 119889(119905) = 0 is exponentially stable with thedecay rate 120572 gt 0

Proof For system (19) consider the following Lyapunovfunctional

119881 (119905) = 1198811(119905) + 119881

2(119905) + 119881

3(119905) (24)

where

1198811(119905) = 120585

119879

(119905) 1198751120585 (119905)

1198812(119905) = (ℎ minus 120591 (119905)) int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 119890minus2120572(119905minus119904)

119889119904

1198813(119905) = (ℎ minus 120591 (119905)) (120585(119905

119896) minus 120585 (119905))

119879

1198753(120585 (119905119896) minus 120585 (119905))

(25)

In function (3) 119875119894gt 0 (119894 = 1 2 3) are Lyapunov matrices to

be designed and 120572 gt 0 is the decay rate as previously definedIt can be calculated that

1(119905) + 2120572119881

1(119905) = 2 120585

119879

(119905) 1198751120585 (119905) + 2120572120585

119879

(119905) 1198751120585 (119905)

2(119905) = minus119890

minus2120572119905

int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

+ (ℎ minus 120591 (119905)) 119890minus2120572119905

(minus2120572)

times int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

= (ℎ minus 120591 (119905)) 119890minus2120572119905

times ( 120585119879

(119905) 1198752

120585 (119905) 1198902120572119905

minus 120585119879

(119905119896) 1198752

120585 (119905119896) 1198902120572119905119896)

(26)

Notice that 120585119879

(119905119896)1198752

120585(119905119896)1198902120572119905119896 = 0 thus

2(119905) = minus119890

minus2120572119905

int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

minus 2120572 (ℎ minus 120591 (119905)) 119890minus2120572119905

int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

(27)

Mathematical Problems in Engineering 5

Therefore it is derived that

2(119905) + 2120572119881

2(119905) = minus119890

minus2120572119905

int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

minus 2120572 (ℎ minus 120591 (119905)) 119890minus2120572119905

times int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

+ 2120572 (ℎ minus 120591 (119905)) 119890minus2120572119905

times int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

= minus119890minus2120572119905

int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

(28)

Furthermore for 1198813(119905) it is calculated directly that

3(119905) = minus(120585(119905

119896) minus 120585 (119905))

119879

1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) 120585119879

(119905) 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) (120585 (119905119896) minus 120585 (119905))

119879

1198753

120585 (119905)

3(119905) + 2120572119881

3(119905) = minus(120585 (119905

119896) minus 120585 (119905))

119879

times 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) 120585119879

(119905)

times 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905))

times (120585(119905119896) minus 120585 (119905))

119879

1198753

120585 (119905)

+ 2120572 (ℎ minus 120591 (119905)) (120585(119905119896) minus 120585(119905))

119879

times 1198753(120585 (119905119896) minus 120585 (119905))

(29)

As a result we have

(119905) + 2120572119881 (119905) = 1(119905) + 2120572119881

1(119905)

+ 2(119905) + 2120572119881

2(119905)

+ 3(119905) + 2120572119881

3(119905)

⩽ 2 120585119879

(119905) 1198751120585 (119905) + 2120572120585

119879

(119905) 1198751120585 (119905)

minus 119890minus2120572119905

int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

minus (120585(119905119896) minus 120585 (119905))

119879

times 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) 120585119879

(119905)

times 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) (120585 (119905119896) minus 120585 (119905))

119879

times 1198753

120585 (119905) + 2120572 (ℎ minus 120591 (119905))

times (120585 (119905119896) minus 120585 (119905))

119879

1198753(120585 (119905119896) minus 120585 (119905))

(30)

We define the following variable

120601120585(119905) =

1

120591 (119905)int

119905

119905119896

120585 (119904) 119889119904 (31)

and it is derived that

119890minus2120572119905

int

119905

119905119896

1198902120572119904 120585119879

(119904) 1198752

120585 (119904) 119889119904

⩾ 119890minus2120572119905

1198902120572119905119896 int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904)

= 119890minus2120572(119905minus119905119896) int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904)

= 119890minus2120572120591(119905)

int

119905

119905minus120591(119905)

120585119879

(119904) 1198752

120585 (119904)

(32)

By Jensenrsquos inequality it is also shown that

int

119905

119905minus120591(119905)

120585119879

(119904) 1198752

120585 (119904) 119889119904

⩽1

120591 (119905)int

119905

119905minus120591(119905)

120585119879

(119904) 119889119904

times 1198752int

119905

119905minus120591(119905)

120585 (119904) 119889119904

= 120591 (119905) (1

120591 (119905)int

119905

119905minus120591(119905)

120585119879

(119904) 119889119904)

times 1198752(

1

120591 (119905)int

119905

119905minus120591(119905)

120585 (119904) 119889119904)

= 120591 (119905) 120601119879

120585(119905) 1198752120601120585(119905)

(33)

In particular for 120591(119905) = 0 it can be verified that the followingholds

1

120591 (119905)int

119905

119905minus120591(119905)

120585 (119904) 119889119904|120591(119905)=0

= lim120591(119905)rarr0

1

120591 (119905)int

119905

119905minus120591(119905)

120585 (119904) 119889119904 = 120585 (119905)

(34)

6 Mathematical Problems in Engineering

Therefore it is shown from (30) that

(119905) + 2120572119881 (119905) ⩽ 2 120585119879

(119905) 1198751120585 (119905)

+ 2120572120585119879

(119905) 1198751120585 (119905)

minus 119890minus2120572120591(119905)

120591 (119905) 120601119879

120585(119905) 1198752120601120585(119905)

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

minus (120585 (119905119896) minus 120585 (119905))

119879

1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) 120585119879

(119905) 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) (120585(119905119896) minus 120585 (119905))

119879

1198753

120585 (119905)

+ 2120572 (ℎ minus 120591 (119905)) (120585 (119905119896) minus 120585 (119905))

119879

times 1198753(120585 (119905119896) minus 120585 (119905))

(35)

We now insert free-weighting matrices by introducing thefollowing zero terms

2 [120585119879

(119905) 119879119879

1+ 120585119879

(119905) 119879119879

2]

times [ 120585119879

(119905) minus 119860120585 (119905) minus 119861119870120585 (119905119896)] = 0

(36)

2 [120585119879

(119905) 119878119879

1+ 120585119879

(119905) 119878119879

2+ 120585119879

(119905119896) 119878119879

3]

times [minus120585 (119905) + 120585 (119905119896) + 120591 (119905) 120601

120585(119905)] = 0

(37)

where the free-weighting matrices 1198791 1198792 1198781 1198782 and 119878

3are to

be designed Then we have

(119905) + 2120572119881 (119905)

⩽ 2 120585119879

(119905) 1198751120585 (119905) + 2120572120585

119879

(119905) 1198751120585 (119905)

minus 119890minus2120572120591(119905)

120591 (119905) 120601119879

120585(119905) 1198752120601120585(119905)

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

minus (120585(119905119896) minus 120585 (119905))

119879

1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) 120585119879

(119905) 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) (120585(119905119896) minus 120585 (119905))

119879

1198753

120585 (119905)

+ 2120572 (ℎ minus 120591 (119905)) (120585(119905119896) minus 120585(119905))

119879

times 1198753(120585 (119905119896) minus 120585 (119905))

+ 2 [120585119879

(119905) 119879119879

1+ 120585119879

(119905) 119879119879

2]

times [ 120585119879

(119905) minus 119860120585 (119905) minus 119861119870120585 (119905119896)]

times 2 [120585119879

(119905) 119878119879

1+ 120585119879

(119905) 119878119879

2+ 120585119879

(119905119896) 119878119879

3]

times [minus120585 (119905) + 120585 (119905119896) + 120591 (119905) 120601

120585(119905)]

(38)

Defining the following new augmented variables

120585 (119905) = [120585119879

(119905) 120585119879

(119905) 120585119879

(119905119896) 120601119879

(119905)]119879

120585 (119905) = [120585119879

(119905) 120585119879

(119905) 120585119879

(119905119896)]119879

(39)

it is calculated that

(119905) + 2120572119881 (119905) ⩽ 120585119879

(119905) Π (119905) 120585 (119905) (40)

with

Π (119905) =

[[[

[

Π11(119905) Π

12(119905) Π

13(119905) Π

14(119905)

⋆ Π22(119905) Π

23(119905) 0

⋆ ⋆ Π33(119905) Π

34(119905)

⋆ ⋆ ⋆ Π44(119905)

]]]

]

Π11(119905) = 2120572119875

1minus 1198753+ 2120572 (ℎ minus 120591 (119905)) 119875

3

minus 119879119879

1119860 minus 119860

119879

1198791minus 1198781minus 119878119879

1

Π12(119905) = 119875

1+ 119879119879

1minus (ℎ minus 120591 (119905)) 119875

3

Π13(119905) = 119875

3minus 2120572 (ℎ minus 120591 (119905)) 119875

3minus 119879119879

1119861119870 + 119878

119879

1

Π14(119905) = 120591 (119905) 119878

119879

1

Π22(119905) = (ℎ minus 120591 (119905)) 119875

2+ 1198792+ 119879119879

2

Π23(119905) = minus (ℎ minus 120591 (119905)) 119875

3minus 119879119879

2119861119870 + 119878

2+ 119878119879

2

Π33(119905) = minus119875

3+ 2120572 (ℎ minus 120591 (119905)) 119875

3+ 1198783+ 119878119879

3

Π34(119905) = 120591 (119905) 119878

119879

3

Π44(119905) = minus119890

minus2120572120591(119905)

120591 (119905) 1198752

(41)

We now consider the matrices Φ and Φ defined in (21) Infact it can be verified thatΦ is obtained fromΠ(119905)|

120591(119905)=0with

deleting the last row and last column and Φ = Π(119905)|120591(119905)=ℎ

Similar to the proof of Theorem 1 of [37] it can be provedthat the system (19) with 119889(119905) = 0 is exponentially stable

Next we deal with the input constraint along each axisWe define

119877119909= [

[

1

0

0

]

]

[1 0 0] = [

[

1 0 0

0 0 0

0 0 0

]

]

119877119910= [

[

0

1

0

]

]

[0 1 0] = [

[

0 0 0

0 1 0

0 0 0

]

]

119877119911= [

[

0

0

1

]

]

[0 0 1] = [

[

0 0 0

0 0 0

0 0 1

]

]

(42)

Then the thrust along each axis can be described as1003816100381610038161003816119906119894 (119905119896)

1003816100381610038161003816 =1003817100381710038171003817119877119894119906 (119905119896)

1003817100381710038171003817 ⩽ 119906119894max 119894 = 119909 119910 119911 (43)

Mathematical Problems in Engineering 7

Therefore the thrust constraint (8) can be rewritten as1003817100381710038171003817119877119894119906 (119905119896)

1003817100381710038171003817 ⩽ 119906119894max 119894 = 119909 119910 119911 (44)

which is equivalent to

(119877119894119906 (119905119896))119879

119877119894119906 (119905119896) ⩽ 1199062

119894max (45)

Furthermore formula (45) is equal to

120585119879

(119905119896)119870119877119879

119894119877119894119870120585 (119905119896) ⩽ 1199062

119894max (46)

Based on the previous analysis it is known that (119905) lt 0 isguaranteed if the matrix conditions (21) hold Subsequently119881(119905) lt 119881(0) holds for any 119905 gt 0 It is therefore reasonable toassume that there exists a scalar 120588 gt 0 such that

119881 (0) ⩽ 120588 (47)

Notice that 119881(119905) = 1198811(119905) + 119881

2(119905) + 119881

3(119905) le 119881(0) ⩽ 120588 and

1198812(119905) ge 0 119881

3(119905) ge 0 Therefore it is true that

119881 (119905) le 1198811(119905) lt 119881 (0) ⩽ 120588 (48)

which implies that

1198811(119905) lt 120588 (49)

Therefore 120585119879(119905)1198751120585(119905) lt 120588 holds for any 119905 gt 0 In particular

let 119905 = 119905119896 we have

120585119879

(119905119896) 1198751120585 (119905119896) lt 120588 (50)

Then (46) can be guaranteed if (22) holdsWe prove this factIn fact if (22) holds we have

120585119879

(119905119896) 120588119870119879

119877119879

119894119877119894119870120585 (119905119896) lt 120585119879

(119905119896) 1199062

119894max1198751120585 (119905119896) (51)

By (51) we have

120585119879

(119905119896) 120588119870119879

119877119879

119894119877119894119870120585 (119905119896)

lt 120585119879

(119905119896) 1199062

119894max1198751120585 (119905119896) lt 1199062

119894max120588

(52)

which implies that

120585119879

(119905119896)119870119879

119877119879

119894119877119894119870120585 (119905119896) lt 1199062

119894max (53)

As a result the thrust limitation (8) can be guaranteed by (22)This completes the proof

32 Disturbance Attenuation In this subsection we willinvestigate analysis of the internal stability and 119867

infinperfor-

mance index for the system (19) Defining a new augmentedvector 120595(119905) ≜ [120585

119879

(119905)119889119879

(119905)]119879 we reconsider the system (19)

with 119889(119905) = 0 We introduce the performance index

119869 (119905) = int

119905

0

[119911119879

(119904) 119911 (119904) minus 120574119889119879

(119904) 119889 (119904)] 119889119904 (54)

Then (20) can be guaranteed by 119869(119905) lt 0

0 5 10 15 20 25 30

0

5

10

minus20

minus15

minus10

minus5

x1(t)

x2(t)

x3(t)

x4(t)

x5(t)

x6(t)

Figure 1 Trajectories of 119909(119905)

Theorem 3 Consider the closed-loop system (19) given aperformance index parameter 120574 gt 0 let the decay rate 120572 = 0 ifthere exist positive definite matrices 119875

1 1198752 and 119875

3isin R9times9 and

matrices 1198791 1198792 1198781 1198782 and 119878

3isin R9times9 such that the matrix

conditions (21) (22) and the following matrices conditionshold

[Φ|120572=0

Γ1

⋆ diag minus1205742 minus1198689] lt 0

[Φ|120572=0

Γ2

⋆ diag minus1205742 minus1198689] lt 0

(55)

Γ1=

[[[[[

[

minus119879119879

1119861119879

1119862119879

2

minus119879119879

2119861119879

10

0 0

]]]]]

]

Γ2=

[[[[[[[

[

minus119879119879

1119861119879

1119862119879

2

minus119879119879

2119861119879

10

0 0

0 0

]]]]]]]

]

(56)

Then the system (19) is internally stable and the 119867infin

perfor-mance index (20) is achieved

Proof It is well known that (119905) lt 0 implies thatlim119905rarrinfin

119881(119905) = 0 We consider a new defined function

119881119869(119905) = (119905) + 119911

119879

(119905) 119911 (119905) minus 1205742

119889119879

(119905) 119889 (119905) (57)

If 119881119869(119905) lt 0 it is shown that

119911119879

(119905) 119911 (119905) minus 1205742

119889119879

(119905) 119889 (119905) lt minus (119905) (58)

8 Mathematical Problems in Engineering

0 5 10 15 20 25 30

0

200

400

600

800

minus600

minus400

minus200

Time (s)

u1(t)

u2(t)

u3(t)

Figure 2 Trajectories of 119906(119905)

0 5 10 15 20 25 300

100

200

300

400

500

600

700

Time (s)

V(t)

Figure 3 Lyapunov function 119881(119905)

and it further implies that 119869(infin) = intinfin

0

(119911119879

(119904)119911(119904) minus 1205742

119889119879

(119904)119889(119904))119889119904 lt minus intinfin

0

(119904)119889119904 = 119881(infin) minus 119881(0) = minus119881(0) lt 0Thatmeans in the case120572 = 0 if (119905) lt 0 and119881

119869(119905) lt 0hold it

can be guaranteed that the system (19) is internally stable andthe119867infinperformance index (20) can be achieved successfully

Notice that (57) can be rewritten as

(119905) + 120585119879

(119905) 119862119879

21198622120585 (119905) minus 120574

2

119889119879

(119905) 119889 (119905) lt 0 (59)Recall the zero terms (37) and notice that the term (37) with119889(119905) = 0 becomes

2 [120585119879

(119905) 119879119879

1+ 120585119879

(119905) 119879119879

2]

times [ 120585119879

(119905) minus 119860120585 (119905) minus 119861119870120585 (119905119896) minus 1198611119889 (119905)] = 0

2 [120585119879

(119905) 119878119879

1+ 120585119879

(119905) 119878119879

2+ 120585119879

(119905119896) 119878119879

3]

times [minus120585 (119905) + 120585 (119905119896) + 120591 (119905) 120601

120585(119905)] = 0

(60)

By similar derivation of Theorem 2 it can be proved thatsystem (19) is internally stable if the conditions (55) hold Onthe other hand it has been proved inTheorem 2 that (119905) lt 0is guaranteed by (21)-(22) This completes the proof

4 Simulation

In this section simulation results will be shown to illustratethe usefulness and advantage of the proposed sampled-datacontrol approach We consider the numerical example in [3]suppose that themass of the chaser is 200 kg that is 119898 = 200and the target is moving along a geosynchronous orbit withradius 119903 = 42241 km with an orbital period of 24 hThe anglevelocity 119908 = 1117 times 103 rads The maximum thrust alongeach axis is 1000N The disturbance vector is described as119889(119905) = [2 sin(2119905) 2 sin(2119905) 2 sin(2119905)]119879 The upper boundof the sampling period is given as ℎ = 05 and the decayrate is selected as 120572 = 002 The disturbance attenuation levelis given as 120574 = 04 In the coordinate based on the targetframe assume that the initial state is [10 minus20 10 0 0 0] andthe desired holding position is (0 minus1 0) Design the statefeedback gain 119870 as

119870 = [

[

minus343807 01490 0 minus973889 00051 0 minus87062 00254 0

minus01490 minus343801 0 minus00051 minus973880 0 minus00254 minus87062 0

0 0 minus578745 0 0 minus1100009 0 0 minus165843

]

]

(61)

such that (119860 + 119861119870) is Hurwitz Suppose that 1199051= 0 which

means that the first sampling time instant is the initial timeThe designed parameter 120588 is chosen as 120588 = 950 Solvingthe conditions (21)-(22) and (55) it is found that there existsfeasible solution Then it can be derived that 119881(0) = 119881

1(0) =

119909(0)119879

1198751119909(0) = 8359423 It is obvious the condition (47)

holds which means that our obtained solution is correct andfeasible

The simulation results are provided in Figures 1ndash3 whichshow the effectiveness of the proposed methods The tra-jectories of state variable 119909(119905) with 119889(119905) = 0 are shown inFigure 1 the trajectories of sampled-data control input 119906(119905)with 119889(119905) = 0 are shown in Figure 2 the trajectories ofLyapunov functions 119881(119905) with 119889(119905) = 0 are shown inFigure 3 It can be seen that the obtained performance isdesirable

Mathematical Problems in Engineering 9

5 Conclusion

In this paper we have addressed the problem of sampled-data stabilization for spacecraft rendezvous with a time-dependent Lyapunov approach A discontinuous Lyapunovfunctional has been constructed the value of which does notincrease in each sampling time instant The sufficient condi-tions for the stability analysis and controller design are estab-lished in terms of matrix inequality technique Future workwill be focused onmore complicated case that is the effects ofnetwork-induced phenomenon such as communication delayand packet losses are taken into account simultaneously

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China 61104101 the FundamentalResearch Funds for the Central Universities the China Post-doctoral Science Foundation funded project (2011M500058)the Special Chinese National Postdoctoral Science Founda-tion under Grant 2012T50356 and the Heilongjiang Postdoc-toral Fund (LBH-Z11144)

References

[1] U Ahsun D W Miller and J L Ramirez ldquoControl of electro-magnetic satellite formations in near-Earth orbitsrdquo Journal ofGuidance Control and Dynamics vol 33 no 6 pp 1883ndash18912010

[2] H Schaub ldquoStabilization of satellite motion relative to aCoulomb spacecraft formationrdquo Journal of Guidance Controland Dynamics vol 28 no 6 pp 1231ndash1239 2005

[3] X Yang X Cao and H Gao ldquoSampled-data control forrelative position holding of spacecraft rendezvous with thrustnonlinearityrdquo IEEE Transactions on Industrial Electronics vol59 no 2 pp 1146ndash1153 2011

[4] P Gurfil ldquoNonlinear feedback control of low-thrust orbitaltransfer in a central gravitational fieldrdquo Acta Astronautica vol60 no 8-9 pp 631ndash648 2007

[5] S Kurnaz O Cetin and O Kaynak ldquoFuzzy logic basedapproach to design of flight control and navigation tasks forautonomous unmanned aerial vehiclesrdquo Journal of Intelligentand Robotic Systems vol 54 no 1ndash3 pp 229ndash244 2009

[6] W Clohessy and R Wiltshire ldquoTerminal guidance system forsatellite rendezvousrdquo Journal of the Aerospace Sciences vol 27no 9 pp 653ndash658 1960

[7] Y Luo and G Tang ldquoSpacecraft optimal rendezvous controllerdesign using simulated annealingrdquo Aerospace Science and Tech-nology vol 9 no 8 pp 732ndash737 2005

[8] Y Luo G Tang and Y Lei ldquoOptimal multi-objective lin-earized impulsive rendezvousrdquo Journal of Guidance Controland Dynamics vol 30 no 2 pp 383ndash389 2007

[9] Z Mao and B Jiang ldquoFault identification and fault-tolerantcontrol for a class of networked control systemsrdquo InternationalJournal of Innovative Computing Information and Control vol3 no 5 pp 1121ndash1130 2007

[10] X Su P Shi L Wu and Y Song ldquoA novel control design ondiscrete-time Takagi-Sugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems vol 21 no 4 pp655ndash671 2012

[11] X Su L Wu P Shi and Y Song ldquoHinfin

model reduction of T-Sfuzzy stochastic systemsrdquo IEEE Transactions on Systems Manand Cybernetics B vol 42 no 6 pp 1574ndash1585 2012

[12] S Yin S Ding and H Luo ldquoReal-time implementation of faulttolerant control system with performance optimizationrdquo IEEETransactions on Industrial Electronics 2013

[13] R Yang Z Zhang and P Shi ldquoExponential stability on stochas-tic neural networks with discrete interval and distributeddelaysrdquo IEEE Transactions on Neural Networks vol 21 no 1 pp169ndash175 2010

[14] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[15] R Yang P Shi and G-P Liu ldquoFiltering for discrete-timenetworked nonlinear systems with mixed random delays andpacket dropoutsrdquo IEEE Transactions on Automatic Control vol56 no 11 pp 2655ndash2660 2011

[16] Y Shu L Yu and Z Wenan ldquoEstimator-based control of net-worked systems with packet-dropoutsrdquo International Journal ofInnovative Computing Information andControl vol 6 no 6 pp2737ndash2748 2010

[17] C Jiang Q Zhang and D Zou ldquoDelay-dependent robust filter-ing for networked control systemswith polytopic uncertaintiesrdquoInternational Journal of Innovative Computing Information andControl vol 6 no 11 pp 4857ndash4868 2010

[18] S Shi Z Yuan and Q Zhang ldquoFault-tolerant Hinfin

filter designof a class of switched systems with sensor failuresrdquo InternationalJournal of Innovative Computing Information and Control vol5 no 11 pp 3827ndash3838 2009

[19] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[20] J Lu D W C Ho and J Cao ldquoA unified synchronizationcriterion for impulsive dynamical networksrdquo Automatica vol46 no 7 pp 1215ndash1221 2010

[21] J Lu D W C Ho J Cao and J Kurths ldquoExponential synchro-nization of linearly coupled neural networks with impulsivedisturbancesrdquo IEEE Transactions on Neural Networks vol 22no 2 pp 329ndash336 2011

[22] H Wu and M Bai ldquoStochastic stability analysis and synthesisfor nonlinear fault tolerant control systems based on the T-Sfuzzy modelrdquo International Journal of Innovative ComputingInformation and Control vol 6 no 9 pp 3989ndash4000 2010

[23] S Yin Data-Driven Design of Fault Diagnosis Systems VDIDuesseldorf Germany 2012

[24] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[25] L Wu and W X Zheng ldquoPassivity-based sliding mode controlof uncertain singular time-delay systemsrdquo Automatica vol 45no 9 pp 2120ndash2127 2009

[26] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo IEEETransactions on Automatic Control vol 55 no 5 pp 1213ndash12192010

[27] L Zhang and E-K Boukas ldquoStability and stabilization ofMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo Automatica vol 45 no 2 pp 463ndash468 2009

10 Mathematical Problems in Engineering

[28] L Zhang and E-K Boukas ldquoMode-dependent Hinfin

filteringfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo Automatica vol 45 no 6pp 1462ndash1467 2009

[29] L Zhou and G Lu ldquoQuantized feedback stabilization fornetworked control systems with nonlinear perturbationrdquo Inter-national Journal of Innovative Computing Information andControl vol 6 no 6 pp 2485ndash2496 2010

[30] Z-G Wu J H Park H Su and J Chu ldquoDiscontinuousLyapunov functional approach to synchronization of time-delayneural networks using sampled-datardquoNonlinear Dynamics vol69 no 4 pp 2021ndash2030 2012

[31] H Li H Liu H Gao and P Shi ldquoReliable fuzzy control foractive suspension systems with actuator delay and faultrdquo IEEETransactions on Fuzzy Systems vol 20 no 2 pp 342ndash357 2012

[32] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding modecontrol for nonlinear active suspension systems using T-S fuzzymodelrdquo IEEE Transactions on Industrial Electronics vol 60 no8 pp 3328ndash3338 2012

[33] E Fridman ldquoA refined input delay approach to sampled-datacontrolrdquo Automatica vol 46 no 2 pp 421ndash427 2010

[34] E Fridman A Seuret and J-P Richard ldquoRobust sampled-data stabilization of linear systems an input delay approachrdquoAutomatica vol 40 no 8 pp 1441ndash1446 2004

[35] P Naghshtabrizi J P Hespanha and A R Teel ldquoExponentialstability of impulsive systems with application to uncertainsampled-data systemsrdquo Systems amp Control Letters vol 57 no5 pp 378ndash385 2008

[36] M Liu J You and X Ma ldquoHinfin

filtering for sampled-datastochastic systems with limited capacity channelrdquo Signal Pro-cessing vol 91 no 8 pp 1826ndash1837 2011

[37] M Liu D W C Ho and Y Niu ldquoObserver-based controllerdesign for linear systems with limited communication capacityvia a descriptor augmentation methodrdquo IET Control Theory ampApplications vol 6 no 3 pp 437ndash447 2012

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Page 2: Research Article Sampled-Data Control of Spacecraft Rendezvous with …downloads.hindawi.com/journals/mpe/2013/814271.pdf · 2019-07-31 · Research Article Sampled-Data Control of

2 Mathematical Problems in Engineering

presented by Fridman et al [34] has been proved to beeffective to deal with the input delay Based on the time-independent Lyapunov approach an improved Lyapunovapproach namely time-dependent Lyapunov approach hasbeen also developed by Fridman [33] and Naghshtabriziet al [35] Compared with the time-independent Lyapunovapproach the structure characteristic of the time-dependentLyapunov approach can guarantee that its value does notincrease in each sampling time despite the jumping behaviorof the sampled signals and thus can reduce the design conser-vativenessThismethod is later extended to deal with filteringproblem of Ito stochastic systems with signal sampling andquantization [36]

It should be pointed out that in practical sampled-datasystems of spacecraft rendezvous there also exists jumpingbehavior of the sampled signal at each sampling time instantand the values of the sampled signal increase abruptly atthese time instants Therefore it is desirable to reduce theconservativeness of the controller design of sampled-dataspacecraft rendezvous systems by developing new controlschemes Motivated by the aforementioned time-dependentLyapunov approach in this paper we consider the sampled-data control problem for relative position holding of space-craft rendezvous In this study signal sampling control inputlimitation and 119867

infinperformance are considered together in

a unified framework The discontinuous Lyapunov functionsapproach is adopted for the controller design also we intro-duce free-weight matrices in the Lyapunov functions and itwill be shown that the obtained results are less conservativethen those in the existing work

The remainder of this paper is organized as follows InSection 2 the problem of sampled-data control of relativeposition holding of spacecraft rendezvous is formulated InSection 3 the sufficient condition for the exponential stabilityof the sampled-data closed-loop system is provided basedon which the sufficient for the existence of the sampled-datacontroller is given Finally a numerical example is presentedin Section 4 to demonstrate the feasibility of the proposedmethod

2 Problem Formulation

In the orbital coordinate frame of chaser and target space-crafts it is well defined that the origin attaches to the masscenter of the target spacecraft 119909-axis is along the vector fromthe Earth center to the origin 119910-axis is along the target orbitcircumference 119911-axis completes the right-handed frame 119903

1

denotes the radius of the target circular orbit 120583 denotes thegravitational parameter of the earth 119908 denotes the anglevelocity of the target and 119908 = (1205831199033

1)12

Based on theNewtonmotion theory the relative dynamicmodel of chaser and target spacecrafts can be modeled by thefollowing C-W equations

119888(119905) = 2119908 119910

119888(119905) + 119908

2

(1199031+ 119909119888(119905))

minus120583 (1199031+ 119909119888(119905))

[(1199031+ 119909119888(119905))2

+ 1199102119888(119905) + 119911

2

119888(119905)]32

+ 119886119909

119910119888(119905) = minus2119908

119888(119905) + 119908

2

119910119888(119905)

minus120583119910119888(119905)

[(1199031+ 119909119888(119905))2

+ 1199102119888(119905) + 119911

2

119888(119905)]32

+ 119886119910

119888(119905) = minus

120583119910119888(119905)

[(1199031+ 119909119888(119905))2

+ 1199102119888(119905) + 119911

2

119888(119905)]32

+ 119886119911

(1)

where 119909119888(119905) 119910

119888(119905) and 119911

119888(119905) are the components of the

relative position in corresponding axes and 119886119894(119894 = 119909 119910 119911)

is the 119894th component of the control input vector By thetaylor expansion and linearization approach the linearizedequation around the null solution for (1) can be given by

119888(119905) minus 2119908 119910

119888(119905) minus 3119908

2

119909119888(119905) =

1

119898(119879119909+ 119889119909)

119910119888(119905) + 2119908

119888(119905) =

1

119898(119879119910+ 119889119910)

119888(119905) + 119908

2

119911119888(119905) =

1

119898(119879119911+ 119889119911)

(2)

In system (2) 119898 is the mass of the chaser 119879119894(119894 = 119909 119910 119911)

is the relative motion dynamic and 119889119894 119894 = 119909 119910 119911 is the

119894th component of the external disturbances We define thefollowing vector and matrices

119909 (119905) =

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

[

119909119888(119905)

119910119888(119905)

119911119888(119905)

119888(119905)

119910119888(119905)

119888(119905)

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

]

119906 (119905) =

[[[

[

119879119909(119905)

119879119910(119905)

119879119911(119905)

]]]

]

119889 (119905) =

[[[

[

119889119909(119905)

119889119910(119905)

119889119911(119905)

]]]

]

119910 (119905) =

[[[

[

119909119888(119905)

119910119888(119905)

119911119888(119905)

]]]

]

(3)

119860 =

[[[[[[[

[

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

31199082

0 0 0 2119908 0

0 0 0 minus2119908 0 0

0 0 minus1199082

0 0 0

]]]]]]]

]

Mathematical Problems in Engineering 3

119861 =1

119898

[[[[[[[

[

0 0 0

0 0 0

0 0 0

1 0 0

0 1 0

0 0 1

]]]]]]]

]

(4)

Then the C-W equation (2) can be described equivalently asfollows

(119905) = 119860119909 (119905) + 119861119906 (119905) + 119861119889 (119905)

119910 (119905) = 119862119909 (119905)

(5)

As a result the specific relative motion of the chaser andtarget can be realized via designing effective control strategy119906(119905) for system (5)

In this paper we consider the case that the signals aresampled before transmitted to the controller side DefineT ≜

1199051 1199052 1199053 as a strictly increasing sequence of sampling

times in (1199050infin) for initial time 119905

0= 0 Suppose that

the sampling internals between any two sequent samplinginstants are bounded by

119905119896+1

minus 119905119896le ℎ 119896 = 1 2 (6)

where ℎ gt 0 is a known constant Then the control vector isassumed to be of the following form at the sampling instant119905119896with zero-order hold (ZOH)

119906 (119905) = 119870119909 (119905119896) (7)

During the orbital transfer process the thrust constraint canbe described as

1003816100381610038161003816119906119894 (119905119896)1003816100381610038161003816 ⩽ 119906119894max (119894 = 119909 119910 119911) (8)

where 119906119894(119905119896) is the control input thrust along the 119894th axis at the

sampling instant 119896 and 119906119894max is the maximum allowed thrust

along the 119894th axisOn the other hand the desired holding position can be

treated as a reference output 119910119903of (5) Therefore our object

is thus to propose a sampled-data control strategy (7) suchthat the closed-loop system for plant (5) is exponentiallystable and the output 119910(119905) tracks the reference signal 119910

119903

without steady-state error Subsequently the problem ofrelative control holding is converted into an output trackingcontrol problem To achieve

lim119905rarr+infin

119910 (119905) = 119910119903 (9)

we define the output error

119890 (119905) = int

119905

0

119910119890(119905) 119889119905 (10)

Also we define the following augmented system

120585 (119905) = 119860120585 (119905) + 119861119906 (119905119896) + 1198611119889 (119905) (11)

with

120585 (119905) = [119909 (119905)

119890 (119905)] 119889 (119905) = [

119889 (119905)

119910119903

]

119860 = [119860 06times3

119862 03times3

] 119861 = [119861

03times3

]

1198611= [

119861 06times3

03times3

minus1198683times3

]

1198621= [119862 0

3times3] 119862

2= [03times6

1198683times3]

(12)

The control scheme for augmented system (11) is thus de-signed as

119906 (119905119896) = 119870120585 (119905

119896) = [119870 119870

119890] [

119909 (119905119896)

119890 (119905119896)] (13)

where119870 is defined as in (2) and119870119890isin R119898times119899 is to be designed

Substituting (13) into (11) yields the following augmentedclosed-loop system

120585 (119905) = 119860120585 (119905) + 119861119870120585 (119905119896) + 1198611119889 (119905)

119910 (119905) = 1198621120585 (119905)

119911 (119905) = 1198622120585 (119905)

(14)

If system (14) is exponentially stable then it implies that therelative position error integral action 119890(119905) rarr 0 and furtherimplies that lim

119905rarrinfin= 119910119903

In the following discussion we adopt the delay-inputapproach to deal with the sampled-data control problem Bydefining

120591 (119905) = 119905119896+1

minus 119905119896 (15)

the sampling intervals can be written as

119905119896= 119905 minus (119905

119896+1minus 119905119896) = 119905 minus 120591 (119905) (16)

Therefore the sampled augmented state can be written as

120585 (119905119896) = 120585 (119905 minus 120591 (119905)) (17)

and the control input vector can be transformed into

119906 (119905119896) = 119870120585 (119905

119896) = 119870120585 (119905 minus 120591 (119905)) (18)

Hence the augmented closed-loop system (14) can be writtenby

120585 (119905) = 119860120585 (119905) + 119861119870120585 (119905 minus 120591 (119905)) + 1198611119889 (119905)

119910 (119905) = 1198621120585 (119905)

119911 (119905) = 1198622120585 (119905)

(19)

Now the aim of this paper can be formulated as follows

Problem 1 Design a sampled-data control law 119906(119905119896) such that

the following conditions are met

4 Mathematical Problems in Engineering

(1) The closed-loop system (19) is exponentially stablewith a given decay rate 120572 gt 0 In other words thechaser holds at a relative position

(2) The control input satisfies the given upper bound (8)

(3) For closed-loop system (19) the following119867infinperfor-

mance index is achieved

119890119911(119905) = 119911 (119905)

2minus 12057410038171003817100381710038171003817119889 (119905)

100381710038171003817100381710038172⩽ 0 (20)

3 Main Results

In this section we will first perform the stability analysis andcontrol scheme design for system (19) with 119889(119905) = 0Then wewill consider the internal stability of (19) with the prescribed119867infin

performance index (20)

31 Stabilization Analysis

Theorem 2 Consider the closed-loop system (19) with 119889(119905) =0 given a decay rate 120572 gt 0 if there exist positive definitematrices 119875

1 1198752 and 119875

3isin R9times9 and matrices 119879

1 1198792 1198781 1198782

and 1198783isin R9times9 such that the following conditions hold

Φ = [

[

Φ11Φ12Φ13

⋆ Φ22Φ23

⋆ ⋆ Φ33

]

]

lt 0

Φ =

[[[

[

Φ11Φ12Φ13Φ14

⋆ Φ22Φ23

0

⋆ ⋆ Φ33Φ34

⋆ ⋆ ⋆ Φ44

]]]

]

lt 0

(21)

120588119870119879

119877119879

119894119877119894119870 minus 119906

2

119894max1198751 lt 0 (22)

where 120588 is defined as in (47) and

Φ11= 2120572119875

1minus 1198753+ 2120572ℎ119875

3minus 119879119879

1119860 minus 119860

119879

1198791minus 1198781minus 119878119879

1

Φ12= 1198751+ 119879119879

1minus ℎ1198753

Φ13= 1198753minus 2120572ℎ119875

3minus 119879119879

1119861119870 + 119878

119879

1

Φ22= ℎ1198752+ 1198792+ 119879119879

2

Φ23= minusℎ119875

3minus 119879119879

2119861119870 + 119878

2+ 119878119879

2

Φ33= minus1198753+ 2120572ℎ119875

3+ 1198783+ 119878119879

3

Φ11= 2120572119875

1minus 1198753minus 119879119879

1119860 minus 119860

119879

1198791minus 1198781minus 119878119879

1

Φ12= 1198751+ 119879119879

1

Φ13= 1198753minus 119879119879

1119861119870 + 119878

119879

1

Φ14= ℎ119878119879

1

Φ22= 119879119879

2+ 119879119879

2

Φ23= minus119879119879

2119861119870 + 119878

2+ 119878119879

2

Φ33= minus1198753+ 1198783+ 119878119879

3

Φ34= ℎ119878119879

3

Φ44= minusℎ119890

minus2120572ℎ

1198752

(23)

Then system (19) with 119889(119905) = 0 is exponentially stable with thedecay rate 120572 gt 0

Proof For system (19) consider the following Lyapunovfunctional

119881 (119905) = 1198811(119905) + 119881

2(119905) + 119881

3(119905) (24)

where

1198811(119905) = 120585

119879

(119905) 1198751120585 (119905)

1198812(119905) = (ℎ minus 120591 (119905)) int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 119890minus2120572(119905minus119904)

119889119904

1198813(119905) = (ℎ minus 120591 (119905)) (120585(119905

119896) minus 120585 (119905))

119879

1198753(120585 (119905119896) minus 120585 (119905))

(25)

In function (3) 119875119894gt 0 (119894 = 1 2 3) are Lyapunov matrices to

be designed and 120572 gt 0 is the decay rate as previously definedIt can be calculated that

1(119905) + 2120572119881

1(119905) = 2 120585

119879

(119905) 1198751120585 (119905) + 2120572120585

119879

(119905) 1198751120585 (119905)

2(119905) = minus119890

minus2120572119905

int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

+ (ℎ minus 120591 (119905)) 119890minus2120572119905

(minus2120572)

times int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

= (ℎ minus 120591 (119905)) 119890minus2120572119905

times ( 120585119879

(119905) 1198752

120585 (119905) 1198902120572119905

minus 120585119879

(119905119896) 1198752

120585 (119905119896) 1198902120572119905119896)

(26)

Notice that 120585119879

(119905119896)1198752

120585(119905119896)1198902120572119905119896 = 0 thus

2(119905) = minus119890

minus2120572119905

int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

minus 2120572 (ℎ minus 120591 (119905)) 119890minus2120572119905

int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

(27)

Mathematical Problems in Engineering 5

Therefore it is derived that

2(119905) + 2120572119881

2(119905) = minus119890

minus2120572119905

int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

minus 2120572 (ℎ minus 120591 (119905)) 119890minus2120572119905

times int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

+ 2120572 (ℎ minus 120591 (119905)) 119890minus2120572119905

times int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

= minus119890minus2120572119905

int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

(28)

Furthermore for 1198813(119905) it is calculated directly that

3(119905) = minus(120585(119905

119896) minus 120585 (119905))

119879

1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) 120585119879

(119905) 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) (120585 (119905119896) minus 120585 (119905))

119879

1198753

120585 (119905)

3(119905) + 2120572119881

3(119905) = minus(120585 (119905

119896) minus 120585 (119905))

119879

times 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) 120585119879

(119905)

times 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905))

times (120585(119905119896) minus 120585 (119905))

119879

1198753

120585 (119905)

+ 2120572 (ℎ minus 120591 (119905)) (120585(119905119896) minus 120585(119905))

119879

times 1198753(120585 (119905119896) minus 120585 (119905))

(29)

As a result we have

(119905) + 2120572119881 (119905) = 1(119905) + 2120572119881

1(119905)

+ 2(119905) + 2120572119881

2(119905)

+ 3(119905) + 2120572119881

3(119905)

⩽ 2 120585119879

(119905) 1198751120585 (119905) + 2120572120585

119879

(119905) 1198751120585 (119905)

minus 119890minus2120572119905

int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

minus (120585(119905119896) minus 120585 (119905))

119879

times 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) 120585119879

(119905)

times 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) (120585 (119905119896) minus 120585 (119905))

119879

times 1198753

120585 (119905) + 2120572 (ℎ minus 120591 (119905))

times (120585 (119905119896) minus 120585 (119905))

119879

1198753(120585 (119905119896) minus 120585 (119905))

(30)

We define the following variable

120601120585(119905) =

1

120591 (119905)int

119905

119905119896

120585 (119904) 119889119904 (31)

and it is derived that

119890minus2120572119905

int

119905

119905119896

1198902120572119904 120585119879

(119904) 1198752

120585 (119904) 119889119904

⩾ 119890minus2120572119905

1198902120572119905119896 int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904)

= 119890minus2120572(119905minus119905119896) int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904)

= 119890minus2120572120591(119905)

int

119905

119905minus120591(119905)

120585119879

(119904) 1198752

120585 (119904)

(32)

By Jensenrsquos inequality it is also shown that

int

119905

119905minus120591(119905)

120585119879

(119904) 1198752

120585 (119904) 119889119904

⩽1

120591 (119905)int

119905

119905minus120591(119905)

120585119879

(119904) 119889119904

times 1198752int

119905

119905minus120591(119905)

120585 (119904) 119889119904

= 120591 (119905) (1

120591 (119905)int

119905

119905minus120591(119905)

120585119879

(119904) 119889119904)

times 1198752(

1

120591 (119905)int

119905

119905minus120591(119905)

120585 (119904) 119889119904)

= 120591 (119905) 120601119879

120585(119905) 1198752120601120585(119905)

(33)

In particular for 120591(119905) = 0 it can be verified that the followingholds

1

120591 (119905)int

119905

119905minus120591(119905)

120585 (119904) 119889119904|120591(119905)=0

= lim120591(119905)rarr0

1

120591 (119905)int

119905

119905minus120591(119905)

120585 (119904) 119889119904 = 120585 (119905)

(34)

6 Mathematical Problems in Engineering

Therefore it is shown from (30) that

(119905) + 2120572119881 (119905) ⩽ 2 120585119879

(119905) 1198751120585 (119905)

+ 2120572120585119879

(119905) 1198751120585 (119905)

minus 119890minus2120572120591(119905)

120591 (119905) 120601119879

120585(119905) 1198752120601120585(119905)

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

minus (120585 (119905119896) minus 120585 (119905))

119879

1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) 120585119879

(119905) 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) (120585(119905119896) minus 120585 (119905))

119879

1198753

120585 (119905)

+ 2120572 (ℎ minus 120591 (119905)) (120585 (119905119896) minus 120585 (119905))

119879

times 1198753(120585 (119905119896) minus 120585 (119905))

(35)

We now insert free-weighting matrices by introducing thefollowing zero terms

2 [120585119879

(119905) 119879119879

1+ 120585119879

(119905) 119879119879

2]

times [ 120585119879

(119905) minus 119860120585 (119905) minus 119861119870120585 (119905119896)] = 0

(36)

2 [120585119879

(119905) 119878119879

1+ 120585119879

(119905) 119878119879

2+ 120585119879

(119905119896) 119878119879

3]

times [minus120585 (119905) + 120585 (119905119896) + 120591 (119905) 120601

120585(119905)] = 0

(37)

where the free-weighting matrices 1198791 1198792 1198781 1198782 and 119878

3are to

be designed Then we have

(119905) + 2120572119881 (119905)

⩽ 2 120585119879

(119905) 1198751120585 (119905) + 2120572120585

119879

(119905) 1198751120585 (119905)

minus 119890minus2120572120591(119905)

120591 (119905) 120601119879

120585(119905) 1198752120601120585(119905)

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

minus (120585(119905119896) minus 120585 (119905))

119879

1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) 120585119879

(119905) 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) (120585(119905119896) minus 120585 (119905))

119879

1198753

120585 (119905)

+ 2120572 (ℎ minus 120591 (119905)) (120585(119905119896) minus 120585(119905))

119879

times 1198753(120585 (119905119896) minus 120585 (119905))

+ 2 [120585119879

(119905) 119879119879

1+ 120585119879

(119905) 119879119879

2]

times [ 120585119879

(119905) minus 119860120585 (119905) minus 119861119870120585 (119905119896)]

times 2 [120585119879

(119905) 119878119879

1+ 120585119879

(119905) 119878119879

2+ 120585119879

(119905119896) 119878119879

3]

times [minus120585 (119905) + 120585 (119905119896) + 120591 (119905) 120601

120585(119905)]

(38)

Defining the following new augmented variables

120585 (119905) = [120585119879

(119905) 120585119879

(119905) 120585119879

(119905119896) 120601119879

(119905)]119879

120585 (119905) = [120585119879

(119905) 120585119879

(119905) 120585119879

(119905119896)]119879

(39)

it is calculated that

(119905) + 2120572119881 (119905) ⩽ 120585119879

(119905) Π (119905) 120585 (119905) (40)

with

Π (119905) =

[[[

[

Π11(119905) Π

12(119905) Π

13(119905) Π

14(119905)

⋆ Π22(119905) Π

23(119905) 0

⋆ ⋆ Π33(119905) Π

34(119905)

⋆ ⋆ ⋆ Π44(119905)

]]]

]

Π11(119905) = 2120572119875

1minus 1198753+ 2120572 (ℎ minus 120591 (119905)) 119875

3

minus 119879119879

1119860 minus 119860

119879

1198791minus 1198781minus 119878119879

1

Π12(119905) = 119875

1+ 119879119879

1minus (ℎ minus 120591 (119905)) 119875

3

Π13(119905) = 119875

3minus 2120572 (ℎ minus 120591 (119905)) 119875

3minus 119879119879

1119861119870 + 119878

119879

1

Π14(119905) = 120591 (119905) 119878

119879

1

Π22(119905) = (ℎ minus 120591 (119905)) 119875

2+ 1198792+ 119879119879

2

Π23(119905) = minus (ℎ minus 120591 (119905)) 119875

3minus 119879119879

2119861119870 + 119878

2+ 119878119879

2

Π33(119905) = minus119875

3+ 2120572 (ℎ minus 120591 (119905)) 119875

3+ 1198783+ 119878119879

3

Π34(119905) = 120591 (119905) 119878

119879

3

Π44(119905) = minus119890

minus2120572120591(119905)

120591 (119905) 1198752

(41)

We now consider the matrices Φ and Φ defined in (21) Infact it can be verified thatΦ is obtained fromΠ(119905)|

120591(119905)=0with

deleting the last row and last column and Φ = Π(119905)|120591(119905)=ℎ

Similar to the proof of Theorem 1 of [37] it can be provedthat the system (19) with 119889(119905) = 0 is exponentially stable

Next we deal with the input constraint along each axisWe define

119877119909= [

[

1

0

0

]

]

[1 0 0] = [

[

1 0 0

0 0 0

0 0 0

]

]

119877119910= [

[

0

1

0

]

]

[0 1 0] = [

[

0 0 0

0 1 0

0 0 0

]

]

119877119911= [

[

0

0

1

]

]

[0 0 1] = [

[

0 0 0

0 0 0

0 0 1

]

]

(42)

Then the thrust along each axis can be described as1003816100381610038161003816119906119894 (119905119896)

1003816100381610038161003816 =1003817100381710038171003817119877119894119906 (119905119896)

1003817100381710038171003817 ⩽ 119906119894max 119894 = 119909 119910 119911 (43)

Mathematical Problems in Engineering 7

Therefore the thrust constraint (8) can be rewritten as1003817100381710038171003817119877119894119906 (119905119896)

1003817100381710038171003817 ⩽ 119906119894max 119894 = 119909 119910 119911 (44)

which is equivalent to

(119877119894119906 (119905119896))119879

119877119894119906 (119905119896) ⩽ 1199062

119894max (45)

Furthermore formula (45) is equal to

120585119879

(119905119896)119870119877119879

119894119877119894119870120585 (119905119896) ⩽ 1199062

119894max (46)

Based on the previous analysis it is known that (119905) lt 0 isguaranteed if the matrix conditions (21) hold Subsequently119881(119905) lt 119881(0) holds for any 119905 gt 0 It is therefore reasonable toassume that there exists a scalar 120588 gt 0 such that

119881 (0) ⩽ 120588 (47)

Notice that 119881(119905) = 1198811(119905) + 119881

2(119905) + 119881

3(119905) le 119881(0) ⩽ 120588 and

1198812(119905) ge 0 119881

3(119905) ge 0 Therefore it is true that

119881 (119905) le 1198811(119905) lt 119881 (0) ⩽ 120588 (48)

which implies that

1198811(119905) lt 120588 (49)

Therefore 120585119879(119905)1198751120585(119905) lt 120588 holds for any 119905 gt 0 In particular

let 119905 = 119905119896 we have

120585119879

(119905119896) 1198751120585 (119905119896) lt 120588 (50)

Then (46) can be guaranteed if (22) holdsWe prove this factIn fact if (22) holds we have

120585119879

(119905119896) 120588119870119879

119877119879

119894119877119894119870120585 (119905119896) lt 120585119879

(119905119896) 1199062

119894max1198751120585 (119905119896) (51)

By (51) we have

120585119879

(119905119896) 120588119870119879

119877119879

119894119877119894119870120585 (119905119896)

lt 120585119879

(119905119896) 1199062

119894max1198751120585 (119905119896) lt 1199062

119894max120588

(52)

which implies that

120585119879

(119905119896)119870119879

119877119879

119894119877119894119870120585 (119905119896) lt 1199062

119894max (53)

As a result the thrust limitation (8) can be guaranteed by (22)This completes the proof

32 Disturbance Attenuation In this subsection we willinvestigate analysis of the internal stability and 119867

infinperfor-

mance index for the system (19) Defining a new augmentedvector 120595(119905) ≜ [120585

119879

(119905)119889119879

(119905)]119879 we reconsider the system (19)

with 119889(119905) = 0 We introduce the performance index

119869 (119905) = int

119905

0

[119911119879

(119904) 119911 (119904) minus 120574119889119879

(119904) 119889 (119904)] 119889119904 (54)

Then (20) can be guaranteed by 119869(119905) lt 0

0 5 10 15 20 25 30

0

5

10

minus20

minus15

minus10

minus5

x1(t)

x2(t)

x3(t)

x4(t)

x5(t)

x6(t)

Figure 1 Trajectories of 119909(119905)

Theorem 3 Consider the closed-loop system (19) given aperformance index parameter 120574 gt 0 let the decay rate 120572 = 0 ifthere exist positive definite matrices 119875

1 1198752 and 119875

3isin R9times9 and

matrices 1198791 1198792 1198781 1198782 and 119878

3isin R9times9 such that the matrix

conditions (21) (22) and the following matrices conditionshold

[Φ|120572=0

Γ1

⋆ diag minus1205742 minus1198689] lt 0

[Φ|120572=0

Γ2

⋆ diag minus1205742 minus1198689] lt 0

(55)

Γ1=

[[[[[

[

minus119879119879

1119861119879

1119862119879

2

minus119879119879

2119861119879

10

0 0

]]]]]

]

Γ2=

[[[[[[[

[

minus119879119879

1119861119879

1119862119879

2

minus119879119879

2119861119879

10

0 0

0 0

]]]]]]]

]

(56)

Then the system (19) is internally stable and the 119867infin

perfor-mance index (20) is achieved

Proof It is well known that (119905) lt 0 implies thatlim119905rarrinfin

119881(119905) = 0 We consider a new defined function

119881119869(119905) = (119905) + 119911

119879

(119905) 119911 (119905) minus 1205742

119889119879

(119905) 119889 (119905) (57)

If 119881119869(119905) lt 0 it is shown that

119911119879

(119905) 119911 (119905) minus 1205742

119889119879

(119905) 119889 (119905) lt minus (119905) (58)

8 Mathematical Problems in Engineering

0 5 10 15 20 25 30

0

200

400

600

800

minus600

minus400

minus200

Time (s)

u1(t)

u2(t)

u3(t)

Figure 2 Trajectories of 119906(119905)

0 5 10 15 20 25 300

100

200

300

400

500

600

700

Time (s)

V(t)

Figure 3 Lyapunov function 119881(119905)

and it further implies that 119869(infin) = intinfin

0

(119911119879

(119904)119911(119904) minus 1205742

119889119879

(119904)119889(119904))119889119904 lt minus intinfin

0

(119904)119889119904 = 119881(infin) minus 119881(0) = minus119881(0) lt 0Thatmeans in the case120572 = 0 if (119905) lt 0 and119881

119869(119905) lt 0hold it

can be guaranteed that the system (19) is internally stable andthe119867infinperformance index (20) can be achieved successfully

Notice that (57) can be rewritten as

(119905) + 120585119879

(119905) 119862119879

21198622120585 (119905) minus 120574

2

119889119879

(119905) 119889 (119905) lt 0 (59)Recall the zero terms (37) and notice that the term (37) with119889(119905) = 0 becomes

2 [120585119879

(119905) 119879119879

1+ 120585119879

(119905) 119879119879

2]

times [ 120585119879

(119905) minus 119860120585 (119905) minus 119861119870120585 (119905119896) minus 1198611119889 (119905)] = 0

2 [120585119879

(119905) 119878119879

1+ 120585119879

(119905) 119878119879

2+ 120585119879

(119905119896) 119878119879

3]

times [minus120585 (119905) + 120585 (119905119896) + 120591 (119905) 120601

120585(119905)] = 0

(60)

By similar derivation of Theorem 2 it can be proved thatsystem (19) is internally stable if the conditions (55) hold Onthe other hand it has been proved inTheorem 2 that (119905) lt 0is guaranteed by (21)-(22) This completes the proof

4 Simulation

In this section simulation results will be shown to illustratethe usefulness and advantage of the proposed sampled-datacontrol approach We consider the numerical example in [3]suppose that themass of the chaser is 200 kg that is 119898 = 200and the target is moving along a geosynchronous orbit withradius 119903 = 42241 km with an orbital period of 24 hThe anglevelocity 119908 = 1117 times 103 rads The maximum thrust alongeach axis is 1000N The disturbance vector is described as119889(119905) = [2 sin(2119905) 2 sin(2119905) 2 sin(2119905)]119879 The upper boundof the sampling period is given as ℎ = 05 and the decayrate is selected as 120572 = 002 The disturbance attenuation levelis given as 120574 = 04 In the coordinate based on the targetframe assume that the initial state is [10 minus20 10 0 0 0] andthe desired holding position is (0 minus1 0) Design the statefeedback gain 119870 as

119870 = [

[

minus343807 01490 0 minus973889 00051 0 minus87062 00254 0

minus01490 minus343801 0 minus00051 minus973880 0 minus00254 minus87062 0

0 0 minus578745 0 0 minus1100009 0 0 minus165843

]

]

(61)

such that (119860 + 119861119870) is Hurwitz Suppose that 1199051= 0 which

means that the first sampling time instant is the initial timeThe designed parameter 120588 is chosen as 120588 = 950 Solvingthe conditions (21)-(22) and (55) it is found that there existsfeasible solution Then it can be derived that 119881(0) = 119881

1(0) =

119909(0)119879

1198751119909(0) = 8359423 It is obvious the condition (47)

holds which means that our obtained solution is correct andfeasible

The simulation results are provided in Figures 1ndash3 whichshow the effectiveness of the proposed methods The tra-jectories of state variable 119909(119905) with 119889(119905) = 0 are shown inFigure 1 the trajectories of sampled-data control input 119906(119905)with 119889(119905) = 0 are shown in Figure 2 the trajectories ofLyapunov functions 119881(119905) with 119889(119905) = 0 are shown inFigure 3 It can be seen that the obtained performance isdesirable

Mathematical Problems in Engineering 9

5 Conclusion

In this paper we have addressed the problem of sampled-data stabilization for spacecraft rendezvous with a time-dependent Lyapunov approach A discontinuous Lyapunovfunctional has been constructed the value of which does notincrease in each sampling time instant The sufficient condi-tions for the stability analysis and controller design are estab-lished in terms of matrix inequality technique Future workwill be focused onmore complicated case that is the effects ofnetwork-induced phenomenon such as communication delayand packet losses are taken into account simultaneously

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China 61104101 the FundamentalResearch Funds for the Central Universities the China Post-doctoral Science Foundation funded project (2011M500058)the Special Chinese National Postdoctoral Science Founda-tion under Grant 2012T50356 and the Heilongjiang Postdoc-toral Fund (LBH-Z11144)

References

[1] U Ahsun D W Miller and J L Ramirez ldquoControl of electro-magnetic satellite formations in near-Earth orbitsrdquo Journal ofGuidance Control and Dynamics vol 33 no 6 pp 1883ndash18912010

[2] H Schaub ldquoStabilization of satellite motion relative to aCoulomb spacecraft formationrdquo Journal of Guidance Controland Dynamics vol 28 no 6 pp 1231ndash1239 2005

[3] X Yang X Cao and H Gao ldquoSampled-data control forrelative position holding of spacecraft rendezvous with thrustnonlinearityrdquo IEEE Transactions on Industrial Electronics vol59 no 2 pp 1146ndash1153 2011

[4] P Gurfil ldquoNonlinear feedback control of low-thrust orbitaltransfer in a central gravitational fieldrdquo Acta Astronautica vol60 no 8-9 pp 631ndash648 2007

[5] S Kurnaz O Cetin and O Kaynak ldquoFuzzy logic basedapproach to design of flight control and navigation tasks forautonomous unmanned aerial vehiclesrdquo Journal of Intelligentand Robotic Systems vol 54 no 1ndash3 pp 229ndash244 2009

[6] W Clohessy and R Wiltshire ldquoTerminal guidance system forsatellite rendezvousrdquo Journal of the Aerospace Sciences vol 27no 9 pp 653ndash658 1960

[7] Y Luo and G Tang ldquoSpacecraft optimal rendezvous controllerdesign using simulated annealingrdquo Aerospace Science and Tech-nology vol 9 no 8 pp 732ndash737 2005

[8] Y Luo G Tang and Y Lei ldquoOptimal multi-objective lin-earized impulsive rendezvousrdquo Journal of Guidance Controland Dynamics vol 30 no 2 pp 383ndash389 2007

[9] Z Mao and B Jiang ldquoFault identification and fault-tolerantcontrol for a class of networked control systemsrdquo InternationalJournal of Innovative Computing Information and Control vol3 no 5 pp 1121ndash1130 2007

[10] X Su P Shi L Wu and Y Song ldquoA novel control design ondiscrete-time Takagi-Sugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems vol 21 no 4 pp655ndash671 2012

[11] X Su L Wu P Shi and Y Song ldquoHinfin

model reduction of T-Sfuzzy stochastic systemsrdquo IEEE Transactions on Systems Manand Cybernetics B vol 42 no 6 pp 1574ndash1585 2012

[12] S Yin S Ding and H Luo ldquoReal-time implementation of faulttolerant control system with performance optimizationrdquo IEEETransactions on Industrial Electronics 2013

[13] R Yang Z Zhang and P Shi ldquoExponential stability on stochas-tic neural networks with discrete interval and distributeddelaysrdquo IEEE Transactions on Neural Networks vol 21 no 1 pp169ndash175 2010

[14] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[15] R Yang P Shi and G-P Liu ldquoFiltering for discrete-timenetworked nonlinear systems with mixed random delays andpacket dropoutsrdquo IEEE Transactions on Automatic Control vol56 no 11 pp 2655ndash2660 2011

[16] Y Shu L Yu and Z Wenan ldquoEstimator-based control of net-worked systems with packet-dropoutsrdquo International Journal ofInnovative Computing Information andControl vol 6 no 6 pp2737ndash2748 2010

[17] C Jiang Q Zhang and D Zou ldquoDelay-dependent robust filter-ing for networked control systemswith polytopic uncertaintiesrdquoInternational Journal of Innovative Computing Information andControl vol 6 no 11 pp 4857ndash4868 2010

[18] S Shi Z Yuan and Q Zhang ldquoFault-tolerant Hinfin

filter designof a class of switched systems with sensor failuresrdquo InternationalJournal of Innovative Computing Information and Control vol5 no 11 pp 3827ndash3838 2009

[19] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[20] J Lu D W C Ho and J Cao ldquoA unified synchronizationcriterion for impulsive dynamical networksrdquo Automatica vol46 no 7 pp 1215ndash1221 2010

[21] J Lu D W C Ho J Cao and J Kurths ldquoExponential synchro-nization of linearly coupled neural networks with impulsivedisturbancesrdquo IEEE Transactions on Neural Networks vol 22no 2 pp 329ndash336 2011

[22] H Wu and M Bai ldquoStochastic stability analysis and synthesisfor nonlinear fault tolerant control systems based on the T-Sfuzzy modelrdquo International Journal of Innovative ComputingInformation and Control vol 6 no 9 pp 3989ndash4000 2010

[23] S Yin Data-Driven Design of Fault Diagnosis Systems VDIDuesseldorf Germany 2012

[24] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[25] L Wu and W X Zheng ldquoPassivity-based sliding mode controlof uncertain singular time-delay systemsrdquo Automatica vol 45no 9 pp 2120ndash2127 2009

[26] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo IEEETransactions on Automatic Control vol 55 no 5 pp 1213ndash12192010

[27] L Zhang and E-K Boukas ldquoStability and stabilization ofMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo Automatica vol 45 no 2 pp 463ndash468 2009

10 Mathematical Problems in Engineering

[28] L Zhang and E-K Boukas ldquoMode-dependent Hinfin

filteringfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo Automatica vol 45 no 6pp 1462ndash1467 2009

[29] L Zhou and G Lu ldquoQuantized feedback stabilization fornetworked control systems with nonlinear perturbationrdquo Inter-national Journal of Innovative Computing Information andControl vol 6 no 6 pp 2485ndash2496 2010

[30] Z-G Wu J H Park H Su and J Chu ldquoDiscontinuousLyapunov functional approach to synchronization of time-delayneural networks using sampled-datardquoNonlinear Dynamics vol69 no 4 pp 2021ndash2030 2012

[31] H Li H Liu H Gao and P Shi ldquoReliable fuzzy control foractive suspension systems with actuator delay and faultrdquo IEEETransactions on Fuzzy Systems vol 20 no 2 pp 342ndash357 2012

[32] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding modecontrol for nonlinear active suspension systems using T-S fuzzymodelrdquo IEEE Transactions on Industrial Electronics vol 60 no8 pp 3328ndash3338 2012

[33] E Fridman ldquoA refined input delay approach to sampled-datacontrolrdquo Automatica vol 46 no 2 pp 421ndash427 2010

[34] E Fridman A Seuret and J-P Richard ldquoRobust sampled-data stabilization of linear systems an input delay approachrdquoAutomatica vol 40 no 8 pp 1441ndash1446 2004

[35] P Naghshtabrizi J P Hespanha and A R Teel ldquoExponentialstability of impulsive systems with application to uncertainsampled-data systemsrdquo Systems amp Control Letters vol 57 no5 pp 378ndash385 2008

[36] M Liu J You and X Ma ldquoHinfin

filtering for sampled-datastochastic systems with limited capacity channelrdquo Signal Pro-cessing vol 91 no 8 pp 1826ndash1837 2011

[37] M Liu D W C Ho and Y Niu ldquoObserver-based controllerdesign for linear systems with limited communication capacityvia a descriptor augmentation methodrdquo IET Control Theory ampApplications vol 6 no 3 pp 437ndash447 2012

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 3: Research Article Sampled-Data Control of Spacecraft Rendezvous with …downloads.hindawi.com/journals/mpe/2013/814271.pdf · 2019-07-31 · Research Article Sampled-Data Control of

Mathematical Problems in Engineering 3

119861 =1

119898

[[[[[[[

[

0 0 0

0 0 0

0 0 0

1 0 0

0 1 0

0 0 1

]]]]]]]

]

(4)

Then the C-W equation (2) can be described equivalently asfollows

(119905) = 119860119909 (119905) + 119861119906 (119905) + 119861119889 (119905)

119910 (119905) = 119862119909 (119905)

(5)

As a result the specific relative motion of the chaser andtarget can be realized via designing effective control strategy119906(119905) for system (5)

In this paper we consider the case that the signals aresampled before transmitted to the controller side DefineT ≜

1199051 1199052 1199053 as a strictly increasing sequence of sampling

times in (1199050infin) for initial time 119905

0= 0 Suppose that

the sampling internals between any two sequent samplinginstants are bounded by

119905119896+1

minus 119905119896le ℎ 119896 = 1 2 (6)

where ℎ gt 0 is a known constant Then the control vector isassumed to be of the following form at the sampling instant119905119896with zero-order hold (ZOH)

119906 (119905) = 119870119909 (119905119896) (7)

During the orbital transfer process the thrust constraint canbe described as

1003816100381610038161003816119906119894 (119905119896)1003816100381610038161003816 ⩽ 119906119894max (119894 = 119909 119910 119911) (8)

where 119906119894(119905119896) is the control input thrust along the 119894th axis at the

sampling instant 119896 and 119906119894max is the maximum allowed thrust

along the 119894th axisOn the other hand the desired holding position can be

treated as a reference output 119910119903of (5) Therefore our object

is thus to propose a sampled-data control strategy (7) suchthat the closed-loop system for plant (5) is exponentiallystable and the output 119910(119905) tracks the reference signal 119910

119903

without steady-state error Subsequently the problem ofrelative control holding is converted into an output trackingcontrol problem To achieve

lim119905rarr+infin

119910 (119905) = 119910119903 (9)

we define the output error

119890 (119905) = int

119905

0

119910119890(119905) 119889119905 (10)

Also we define the following augmented system

120585 (119905) = 119860120585 (119905) + 119861119906 (119905119896) + 1198611119889 (119905) (11)

with

120585 (119905) = [119909 (119905)

119890 (119905)] 119889 (119905) = [

119889 (119905)

119910119903

]

119860 = [119860 06times3

119862 03times3

] 119861 = [119861

03times3

]

1198611= [

119861 06times3

03times3

minus1198683times3

]

1198621= [119862 0

3times3] 119862

2= [03times6

1198683times3]

(12)

The control scheme for augmented system (11) is thus de-signed as

119906 (119905119896) = 119870120585 (119905

119896) = [119870 119870

119890] [

119909 (119905119896)

119890 (119905119896)] (13)

where119870 is defined as in (2) and119870119890isin R119898times119899 is to be designed

Substituting (13) into (11) yields the following augmentedclosed-loop system

120585 (119905) = 119860120585 (119905) + 119861119870120585 (119905119896) + 1198611119889 (119905)

119910 (119905) = 1198621120585 (119905)

119911 (119905) = 1198622120585 (119905)

(14)

If system (14) is exponentially stable then it implies that therelative position error integral action 119890(119905) rarr 0 and furtherimplies that lim

119905rarrinfin= 119910119903

In the following discussion we adopt the delay-inputapproach to deal with the sampled-data control problem Bydefining

120591 (119905) = 119905119896+1

minus 119905119896 (15)

the sampling intervals can be written as

119905119896= 119905 minus (119905

119896+1minus 119905119896) = 119905 minus 120591 (119905) (16)

Therefore the sampled augmented state can be written as

120585 (119905119896) = 120585 (119905 minus 120591 (119905)) (17)

and the control input vector can be transformed into

119906 (119905119896) = 119870120585 (119905

119896) = 119870120585 (119905 minus 120591 (119905)) (18)

Hence the augmented closed-loop system (14) can be writtenby

120585 (119905) = 119860120585 (119905) + 119861119870120585 (119905 minus 120591 (119905)) + 1198611119889 (119905)

119910 (119905) = 1198621120585 (119905)

119911 (119905) = 1198622120585 (119905)

(19)

Now the aim of this paper can be formulated as follows

Problem 1 Design a sampled-data control law 119906(119905119896) such that

the following conditions are met

4 Mathematical Problems in Engineering

(1) The closed-loop system (19) is exponentially stablewith a given decay rate 120572 gt 0 In other words thechaser holds at a relative position

(2) The control input satisfies the given upper bound (8)

(3) For closed-loop system (19) the following119867infinperfor-

mance index is achieved

119890119911(119905) = 119911 (119905)

2minus 12057410038171003817100381710038171003817119889 (119905)

100381710038171003817100381710038172⩽ 0 (20)

3 Main Results

In this section we will first perform the stability analysis andcontrol scheme design for system (19) with 119889(119905) = 0Then wewill consider the internal stability of (19) with the prescribed119867infin

performance index (20)

31 Stabilization Analysis

Theorem 2 Consider the closed-loop system (19) with 119889(119905) =0 given a decay rate 120572 gt 0 if there exist positive definitematrices 119875

1 1198752 and 119875

3isin R9times9 and matrices 119879

1 1198792 1198781 1198782

and 1198783isin R9times9 such that the following conditions hold

Φ = [

[

Φ11Φ12Φ13

⋆ Φ22Φ23

⋆ ⋆ Φ33

]

]

lt 0

Φ =

[[[

[

Φ11Φ12Φ13Φ14

⋆ Φ22Φ23

0

⋆ ⋆ Φ33Φ34

⋆ ⋆ ⋆ Φ44

]]]

]

lt 0

(21)

120588119870119879

119877119879

119894119877119894119870 minus 119906

2

119894max1198751 lt 0 (22)

where 120588 is defined as in (47) and

Φ11= 2120572119875

1minus 1198753+ 2120572ℎ119875

3minus 119879119879

1119860 minus 119860

119879

1198791minus 1198781minus 119878119879

1

Φ12= 1198751+ 119879119879

1minus ℎ1198753

Φ13= 1198753minus 2120572ℎ119875

3minus 119879119879

1119861119870 + 119878

119879

1

Φ22= ℎ1198752+ 1198792+ 119879119879

2

Φ23= minusℎ119875

3minus 119879119879

2119861119870 + 119878

2+ 119878119879

2

Φ33= minus1198753+ 2120572ℎ119875

3+ 1198783+ 119878119879

3

Φ11= 2120572119875

1minus 1198753minus 119879119879

1119860 minus 119860

119879

1198791minus 1198781minus 119878119879

1

Φ12= 1198751+ 119879119879

1

Φ13= 1198753minus 119879119879

1119861119870 + 119878

119879

1

Φ14= ℎ119878119879

1

Φ22= 119879119879

2+ 119879119879

2

Φ23= minus119879119879

2119861119870 + 119878

2+ 119878119879

2

Φ33= minus1198753+ 1198783+ 119878119879

3

Φ34= ℎ119878119879

3

Φ44= minusℎ119890

minus2120572ℎ

1198752

(23)

Then system (19) with 119889(119905) = 0 is exponentially stable with thedecay rate 120572 gt 0

Proof For system (19) consider the following Lyapunovfunctional

119881 (119905) = 1198811(119905) + 119881

2(119905) + 119881

3(119905) (24)

where

1198811(119905) = 120585

119879

(119905) 1198751120585 (119905)

1198812(119905) = (ℎ minus 120591 (119905)) int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 119890minus2120572(119905minus119904)

119889119904

1198813(119905) = (ℎ minus 120591 (119905)) (120585(119905

119896) minus 120585 (119905))

119879

1198753(120585 (119905119896) minus 120585 (119905))

(25)

In function (3) 119875119894gt 0 (119894 = 1 2 3) are Lyapunov matrices to

be designed and 120572 gt 0 is the decay rate as previously definedIt can be calculated that

1(119905) + 2120572119881

1(119905) = 2 120585

119879

(119905) 1198751120585 (119905) + 2120572120585

119879

(119905) 1198751120585 (119905)

2(119905) = minus119890

minus2120572119905

int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

+ (ℎ minus 120591 (119905)) 119890minus2120572119905

(minus2120572)

times int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

= (ℎ minus 120591 (119905)) 119890minus2120572119905

times ( 120585119879

(119905) 1198752

120585 (119905) 1198902120572119905

minus 120585119879

(119905119896) 1198752

120585 (119905119896) 1198902120572119905119896)

(26)

Notice that 120585119879

(119905119896)1198752

120585(119905119896)1198902120572119905119896 = 0 thus

2(119905) = minus119890

minus2120572119905

int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

minus 2120572 (ℎ minus 120591 (119905)) 119890minus2120572119905

int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

(27)

Mathematical Problems in Engineering 5

Therefore it is derived that

2(119905) + 2120572119881

2(119905) = minus119890

minus2120572119905

int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

minus 2120572 (ℎ minus 120591 (119905)) 119890minus2120572119905

times int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

+ 2120572 (ℎ minus 120591 (119905)) 119890minus2120572119905

times int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

= minus119890minus2120572119905

int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

(28)

Furthermore for 1198813(119905) it is calculated directly that

3(119905) = minus(120585(119905

119896) minus 120585 (119905))

119879

1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) 120585119879

(119905) 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) (120585 (119905119896) minus 120585 (119905))

119879

1198753

120585 (119905)

3(119905) + 2120572119881

3(119905) = minus(120585 (119905

119896) minus 120585 (119905))

119879

times 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) 120585119879

(119905)

times 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905))

times (120585(119905119896) minus 120585 (119905))

119879

1198753

120585 (119905)

+ 2120572 (ℎ minus 120591 (119905)) (120585(119905119896) minus 120585(119905))

119879

times 1198753(120585 (119905119896) minus 120585 (119905))

(29)

As a result we have

(119905) + 2120572119881 (119905) = 1(119905) + 2120572119881

1(119905)

+ 2(119905) + 2120572119881

2(119905)

+ 3(119905) + 2120572119881

3(119905)

⩽ 2 120585119879

(119905) 1198751120585 (119905) + 2120572120585

119879

(119905) 1198751120585 (119905)

minus 119890minus2120572119905

int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

minus (120585(119905119896) minus 120585 (119905))

119879

times 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) 120585119879

(119905)

times 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) (120585 (119905119896) minus 120585 (119905))

119879

times 1198753

120585 (119905) + 2120572 (ℎ minus 120591 (119905))

times (120585 (119905119896) minus 120585 (119905))

119879

1198753(120585 (119905119896) minus 120585 (119905))

(30)

We define the following variable

120601120585(119905) =

1

120591 (119905)int

119905

119905119896

120585 (119904) 119889119904 (31)

and it is derived that

119890minus2120572119905

int

119905

119905119896

1198902120572119904 120585119879

(119904) 1198752

120585 (119904) 119889119904

⩾ 119890minus2120572119905

1198902120572119905119896 int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904)

= 119890minus2120572(119905minus119905119896) int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904)

= 119890minus2120572120591(119905)

int

119905

119905minus120591(119905)

120585119879

(119904) 1198752

120585 (119904)

(32)

By Jensenrsquos inequality it is also shown that

int

119905

119905minus120591(119905)

120585119879

(119904) 1198752

120585 (119904) 119889119904

⩽1

120591 (119905)int

119905

119905minus120591(119905)

120585119879

(119904) 119889119904

times 1198752int

119905

119905minus120591(119905)

120585 (119904) 119889119904

= 120591 (119905) (1

120591 (119905)int

119905

119905minus120591(119905)

120585119879

(119904) 119889119904)

times 1198752(

1

120591 (119905)int

119905

119905minus120591(119905)

120585 (119904) 119889119904)

= 120591 (119905) 120601119879

120585(119905) 1198752120601120585(119905)

(33)

In particular for 120591(119905) = 0 it can be verified that the followingholds

1

120591 (119905)int

119905

119905minus120591(119905)

120585 (119904) 119889119904|120591(119905)=0

= lim120591(119905)rarr0

1

120591 (119905)int

119905

119905minus120591(119905)

120585 (119904) 119889119904 = 120585 (119905)

(34)

6 Mathematical Problems in Engineering

Therefore it is shown from (30) that

(119905) + 2120572119881 (119905) ⩽ 2 120585119879

(119905) 1198751120585 (119905)

+ 2120572120585119879

(119905) 1198751120585 (119905)

minus 119890minus2120572120591(119905)

120591 (119905) 120601119879

120585(119905) 1198752120601120585(119905)

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

minus (120585 (119905119896) minus 120585 (119905))

119879

1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) 120585119879

(119905) 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) (120585(119905119896) minus 120585 (119905))

119879

1198753

120585 (119905)

+ 2120572 (ℎ minus 120591 (119905)) (120585 (119905119896) minus 120585 (119905))

119879

times 1198753(120585 (119905119896) minus 120585 (119905))

(35)

We now insert free-weighting matrices by introducing thefollowing zero terms

2 [120585119879

(119905) 119879119879

1+ 120585119879

(119905) 119879119879

2]

times [ 120585119879

(119905) minus 119860120585 (119905) minus 119861119870120585 (119905119896)] = 0

(36)

2 [120585119879

(119905) 119878119879

1+ 120585119879

(119905) 119878119879

2+ 120585119879

(119905119896) 119878119879

3]

times [minus120585 (119905) + 120585 (119905119896) + 120591 (119905) 120601

120585(119905)] = 0

(37)

where the free-weighting matrices 1198791 1198792 1198781 1198782 and 119878

3are to

be designed Then we have

(119905) + 2120572119881 (119905)

⩽ 2 120585119879

(119905) 1198751120585 (119905) + 2120572120585

119879

(119905) 1198751120585 (119905)

minus 119890minus2120572120591(119905)

120591 (119905) 120601119879

120585(119905) 1198752120601120585(119905)

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

minus (120585(119905119896) minus 120585 (119905))

119879

1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) 120585119879

(119905) 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) (120585(119905119896) minus 120585 (119905))

119879

1198753

120585 (119905)

+ 2120572 (ℎ minus 120591 (119905)) (120585(119905119896) minus 120585(119905))

119879

times 1198753(120585 (119905119896) minus 120585 (119905))

+ 2 [120585119879

(119905) 119879119879

1+ 120585119879

(119905) 119879119879

2]

times [ 120585119879

(119905) minus 119860120585 (119905) minus 119861119870120585 (119905119896)]

times 2 [120585119879

(119905) 119878119879

1+ 120585119879

(119905) 119878119879

2+ 120585119879

(119905119896) 119878119879

3]

times [minus120585 (119905) + 120585 (119905119896) + 120591 (119905) 120601

120585(119905)]

(38)

Defining the following new augmented variables

120585 (119905) = [120585119879

(119905) 120585119879

(119905) 120585119879

(119905119896) 120601119879

(119905)]119879

120585 (119905) = [120585119879

(119905) 120585119879

(119905) 120585119879

(119905119896)]119879

(39)

it is calculated that

(119905) + 2120572119881 (119905) ⩽ 120585119879

(119905) Π (119905) 120585 (119905) (40)

with

Π (119905) =

[[[

[

Π11(119905) Π

12(119905) Π

13(119905) Π

14(119905)

⋆ Π22(119905) Π

23(119905) 0

⋆ ⋆ Π33(119905) Π

34(119905)

⋆ ⋆ ⋆ Π44(119905)

]]]

]

Π11(119905) = 2120572119875

1minus 1198753+ 2120572 (ℎ minus 120591 (119905)) 119875

3

minus 119879119879

1119860 minus 119860

119879

1198791minus 1198781minus 119878119879

1

Π12(119905) = 119875

1+ 119879119879

1minus (ℎ minus 120591 (119905)) 119875

3

Π13(119905) = 119875

3minus 2120572 (ℎ minus 120591 (119905)) 119875

3minus 119879119879

1119861119870 + 119878

119879

1

Π14(119905) = 120591 (119905) 119878

119879

1

Π22(119905) = (ℎ minus 120591 (119905)) 119875

2+ 1198792+ 119879119879

2

Π23(119905) = minus (ℎ minus 120591 (119905)) 119875

3minus 119879119879

2119861119870 + 119878

2+ 119878119879

2

Π33(119905) = minus119875

3+ 2120572 (ℎ minus 120591 (119905)) 119875

3+ 1198783+ 119878119879

3

Π34(119905) = 120591 (119905) 119878

119879

3

Π44(119905) = minus119890

minus2120572120591(119905)

120591 (119905) 1198752

(41)

We now consider the matrices Φ and Φ defined in (21) Infact it can be verified thatΦ is obtained fromΠ(119905)|

120591(119905)=0with

deleting the last row and last column and Φ = Π(119905)|120591(119905)=ℎ

Similar to the proof of Theorem 1 of [37] it can be provedthat the system (19) with 119889(119905) = 0 is exponentially stable

Next we deal with the input constraint along each axisWe define

119877119909= [

[

1

0

0

]

]

[1 0 0] = [

[

1 0 0

0 0 0

0 0 0

]

]

119877119910= [

[

0

1

0

]

]

[0 1 0] = [

[

0 0 0

0 1 0

0 0 0

]

]

119877119911= [

[

0

0

1

]

]

[0 0 1] = [

[

0 0 0

0 0 0

0 0 1

]

]

(42)

Then the thrust along each axis can be described as1003816100381610038161003816119906119894 (119905119896)

1003816100381610038161003816 =1003817100381710038171003817119877119894119906 (119905119896)

1003817100381710038171003817 ⩽ 119906119894max 119894 = 119909 119910 119911 (43)

Mathematical Problems in Engineering 7

Therefore the thrust constraint (8) can be rewritten as1003817100381710038171003817119877119894119906 (119905119896)

1003817100381710038171003817 ⩽ 119906119894max 119894 = 119909 119910 119911 (44)

which is equivalent to

(119877119894119906 (119905119896))119879

119877119894119906 (119905119896) ⩽ 1199062

119894max (45)

Furthermore formula (45) is equal to

120585119879

(119905119896)119870119877119879

119894119877119894119870120585 (119905119896) ⩽ 1199062

119894max (46)

Based on the previous analysis it is known that (119905) lt 0 isguaranteed if the matrix conditions (21) hold Subsequently119881(119905) lt 119881(0) holds for any 119905 gt 0 It is therefore reasonable toassume that there exists a scalar 120588 gt 0 such that

119881 (0) ⩽ 120588 (47)

Notice that 119881(119905) = 1198811(119905) + 119881

2(119905) + 119881

3(119905) le 119881(0) ⩽ 120588 and

1198812(119905) ge 0 119881

3(119905) ge 0 Therefore it is true that

119881 (119905) le 1198811(119905) lt 119881 (0) ⩽ 120588 (48)

which implies that

1198811(119905) lt 120588 (49)

Therefore 120585119879(119905)1198751120585(119905) lt 120588 holds for any 119905 gt 0 In particular

let 119905 = 119905119896 we have

120585119879

(119905119896) 1198751120585 (119905119896) lt 120588 (50)

Then (46) can be guaranteed if (22) holdsWe prove this factIn fact if (22) holds we have

120585119879

(119905119896) 120588119870119879

119877119879

119894119877119894119870120585 (119905119896) lt 120585119879

(119905119896) 1199062

119894max1198751120585 (119905119896) (51)

By (51) we have

120585119879

(119905119896) 120588119870119879

119877119879

119894119877119894119870120585 (119905119896)

lt 120585119879

(119905119896) 1199062

119894max1198751120585 (119905119896) lt 1199062

119894max120588

(52)

which implies that

120585119879

(119905119896)119870119879

119877119879

119894119877119894119870120585 (119905119896) lt 1199062

119894max (53)

As a result the thrust limitation (8) can be guaranteed by (22)This completes the proof

32 Disturbance Attenuation In this subsection we willinvestigate analysis of the internal stability and 119867

infinperfor-

mance index for the system (19) Defining a new augmentedvector 120595(119905) ≜ [120585

119879

(119905)119889119879

(119905)]119879 we reconsider the system (19)

with 119889(119905) = 0 We introduce the performance index

119869 (119905) = int

119905

0

[119911119879

(119904) 119911 (119904) minus 120574119889119879

(119904) 119889 (119904)] 119889119904 (54)

Then (20) can be guaranteed by 119869(119905) lt 0

0 5 10 15 20 25 30

0

5

10

minus20

minus15

minus10

minus5

x1(t)

x2(t)

x3(t)

x4(t)

x5(t)

x6(t)

Figure 1 Trajectories of 119909(119905)

Theorem 3 Consider the closed-loop system (19) given aperformance index parameter 120574 gt 0 let the decay rate 120572 = 0 ifthere exist positive definite matrices 119875

1 1198752 and 119875

3isin R9times9 and

matrices 1198791 1198792 1198781 1198782 and 119878

3isin R9times9 such that the matrix

conditions (21) (22) and the following matrices conditionshold

[Φ|120572=0

Γ1

⋆ diag minus1205742 minus1198689] lt 0

[Φ|120572=0

Γ2

⋆ diag minus1205742 minus1198689] lt 0

(55)

Γ1=

[[[[[

[

minus119879119879

1119861119879

1119862119879

2

minus119879119879

2119861119879

10

0 0

]]]]]

]

Γ2=

[[[[[[[

[

minus119879119879

1119861119879

1119862119879

2

minus119879119879

2119861119879

10

0 0

0 0

]]]]]]]

]

(56)

Then the system (19) is internally stable and the 119867infin

perfor-mance index (20) is achieved

Proof It is well known that (119905) lt 0 implies thatlim119905rarrinfin

119881(119905) = 0 We consider a new defined function

119881119869(119905) = (119905) + 119911

119879

(119905) 119911 (119905) minus 1205742

119889119879

(119905) 119889 (119905) (57)

If 119881119869(119905) lt 0 it is shown that

119911119879

(119905) 119911 (119905) minus 1205742

119889119879

(119905) 119889 (119905) lt minus (119905) (58)

8 Mathematical Problems in Engineering

0 5 10 15 20 25 30

0

200

400

600

800

minus600

minus400

minus200

Time (s)

u1(t)

u2(t)

u3(t)

Figure 2 Trajectories of 119906(119905)

0 5 10 15 20 25 300

100

200

300

400

500

600

700

Time (s)

V(t)

Figure 3 Lyapunov function 119881(119905)

and it further implies that 119869(infin) = intinfin

0

(119911119879

(119904)119911(119904) minus 1205742

119889119879

(119904)119889(119904))119889119904 lt minus intinfin

0

(119904)119889119904 = 119881(infin) minus 119881(0) = minus119881(0) lt 0Thatmeans in the case120572 = 0 if (119905) lt 0 and119881

119869(119905) lt 0hold it

can be guaranteed that the system (19) is internally stable andthe119867infinperformance index (20) can be achieved successfully

Notice that (57) can be rewritten as

(119905) + 120585119879

(119905) 119862119879

21198622120585 (119905) minus 120574

2

119889119879

(119905) 119889 (119905) lt 0 (59)Recall the zero terms (37) and notice that the term (37) with119889(119905) = 0 becomes

2 [120585119879

(119905) 119879119879

1+ 120585119879

(119905) 119879119879

2]

times [ 120585119879

(119905) minus 119860120585 (119905) minus 119861119870120585 (119905119896) minus 1198611119889 (119905)] = 0

2 [120585119879

(119905) 119878119879

1+ 120585119879

(119905) 119878119879

2+ 120585119879

(119905119896) 119878119879

3]

times [minus120585 (119905) + 120585 (119905119896) + 120591 (119905) 120601

120585(119905)] = 0

(60)

By similar derivation of Theorem 2 it can be proved thatsystem (19) is internally stable if the conditions (55) hold Onthe other hand it has been proved inTheorem 2 that (119905) lt 0is guaranteed by (21)-(22) This completes the proof

4 Simulation

In this section simulation results will be shown to illustratethe usefulness and advantage of the proposed sampled-datacontrol approach We consider the numerical example in [3]suppose that themass of the chaser is 200 kg that is 119898 = 200and the target is moving along a geosynchronous orbit withradius 119903 = 42241 km with an orbital period of 24 hThe anglevelocity 119908 = 1117 times 103 rads The maximum thrust alongeach axis is 1000N The disturbance vector is described as119889(119905) = [2 sin(2119905) 2 sin(2119905) 2 sin(2119905)]119879 The upper boundof the sampling period is given as ℎ = 05 and the decayrate is selected as 120572 = 002 The disturbance attenuation levelis given as 120574 = 04 In the coordinate based on the targetframe assume that the initial state is [10 minus20 10 0 0 0] andthe desired holding position is (0 minus1 0) Design the statefeedback gain 119870 as

119870 = [

[

minus343807 01490 0 minus973889 00051 0 minus87062 00254 0

minus01490 minus343801 0 minus00051 minus973880 0 minus00254 minus87062 0

0 0 minus578745 0 0 minus1100009 0 0 minus165843

]

]

(61)

such that (119860 + 119861119870) is Hurwitz Suppose that 1199051= 0 which

means that the first sampling time instant is the initial timeThe designed parameter 120588 is chosen as 120588 = 950 Solvingthe conditions (21)-(22) and (55) it is found that there existsfeasible solution Then it can be derived that 119881(0) = 119881

1(0) =

119909(0)119879

1198751119909(0) = 8359423 It is obvious the condition (47)

holds which means that our obtained solution is correct andfeasible

The simulation results are provided in Figures 1ndash3 whichshow the effectiveness of the proposed methods The tra-jectories of state variable 119909(119905) with 119889(119905) = 0 are shown inFigure 1 the trajectories of sampled-data control input 119906(119905)with 119889(119905) = 0 are shown in Figure 2 the trajectories ofLyapunov functions 119881(119905) with 119889(119905) = 0 are shown inFigure 3 It can be seen that the obtained performance isdesirable

Mathematical Problems in Engineering 9

5 Conclusion

In this paper we have addressed the problem of sampled-data stabilization for spacecraft rendezvous with a time-dependent Lyapunov approach A discontinuous Lyapunovfunctional has been constructed the value of which does notincrease in each sampling time instant The sufficient condi-tions for the stability analysis and controller design are estab-lished in terms of matrix inequality technique Future workwill be focused onmore complicated case that is the effects ofnetwork-induced phenomenon such as communication delayand packet losses are taken into account simultaneously

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China 61104101 the FundamentalResearch Funds for the Central Universities the China Post-doctoral Science Foundation funded project (2011M500058)the Special Chinese National Postdoctoral Science Founda-tion under Grant 2012T50356 and the Heilongjiang Postdoc-toral Fund (LBH-Z11144)

References

[1] U Ahsun D W Miller and J L Ramirez ldquoControl of electro-magnetic satellite formations in near-Earth orbitsrdquo Journal ofGuidance Control and Dynamics vol 33 no 6 pp 1883ndash18912010

[2] H Schaub ldquoStabilization of satellite motion relative to aCoulomb spacecraft formationrdquo Journal of Guidance Controland Dynamics vol 28 no 6 pp 1231ndash1239 2005

[3] X Yang X Cao and H Gao ldquoSampled-data control forrelative position holding of spacecraft rendezvous with thrustnonlinearityrdquo IEEE Transactions on Industrial Electronics vol59 no 2 pp 1146ndash1153 2011

[4] P Gurfil ldquoNonlinear feedback control of low-thrust orbitaltransfer in a central gravitational fieldrdquo Acta Astronautica vol60 no 8-9 pp 631ndash648 2007

[5] S Kurnaz O Cetin and O Kaynak ldquoFuzzy logic basedapproach to design of flight control and navigation tasks forautonomous unmanned aerial vehiclesrdquo Journal of Intelligentand Robotic Systems vol 54 no 1ndash3 pp 229ndash244 2009

[6] W Clohessy and R Wiltshire ldquoTerminal guidance system forsatellite rendezvousrdquo Journal of the Aerospace Sciences vol 27no 9 pp 653ndash658 1960

[7] Y Luo and G Tang ldquoSpacecraft optimal rendezvous controllerdesign using simulated annealingrdquo Aerospace Science and Tech-nology vol 9 no 8 pp 732ndash737 2005

[8] Y Luo G Tang and Y Lei ldquoOptimal multi-objective lin-earized impulsive rendezvousrdquo Journal of Guidance Controland Dynamics vol 30 no 2 pp 383ndash389 2007

[9] Z Mao and B Jiang ldquoFault identification and fault-tolerantcontrol for a class of networked control systemsrdquo InternationalJournal of Innovative Computing Information and Control vol3 no 5 pp 1121ndash1130 2007

[10] X Su P Shi L Wu and Y Song ldquoA novel control design ondiscrete-time Takagi-Sugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems vol 21 no 4 pp655ndash671 2012

[11] X Su L Wu P Shi and Y Song ldquoHinfin

model reduction of T-Sfuzzy stochastic systemsrdquo IEEE Transactions on Systems Manand Cybernetics B vol 42 no 6 pp 1574ndash1585 2012

[12] S Yin S Ding and H Luo ldquoReal-time implementation of faulttolerant control system with performance optimizationrdquo IEEETransactions on Industrial Electronics 2013

[13] R Yang Z Zhang and P Shi ldquoExponential stability on stochas-tic neural networks with discrete interval and distributeddelaysrdquo IEEE Transactions on Neural Networks vol 21 no 1 pp169ndash175 2010

[14] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[15] R Yang P Shi and G-P Liu ldquoFiltering for discrete-timenetworked nonlinear systems with mixed random delays andpacket dropoutsrdquo IEEE Transactions on Automatic Control vol56 no 11 pp 2655ndash2660 2011

[16] Y Shu L Yu and Z Wenan ldquoEstimator-based control of net-worked systems with packet-dropoutsrdquo International Journal ofInnovative Computing Information andControl vol 6 no 6 pp2737ndash2748 2010

[17] C Jiang Q Zhang and D Zou ldquoDelay-dependent robust filter-ing for networked control systemswith polytopic uncertaintiesrdquoInternational Journal of Innovative Computing Information andControl vol 6 no 11 pp 4857ndash4868 2010

[18] S Shi Z Yuan and Q Zhang ldquoFault-tolerant Hinfin

filter designof a class of switched systems with sensor failuresrdquo InternationalJournal of Innovative Computing Information and Control vol5 no 11 pp 3827ndash3838 2009

[19] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[20] J Lu D W C Ho and J Cao ldquoA unified synchronizationcriterion for impulsive dynamical networksrdquo Automatica vol46 no 7 pp 1215ndash1221 2010

[21] J Lu D W C Ho J Cao and J Kurths ldquoExponential synchro-nization of linearly coupled neural networks with impulsivedisturbancesrdquo IEEE Transactions on Neural Networks vol 22no 2 pp 329ndash336 2011

[22] H Wu and M Bai ldquoStochastic stability analysis and synthesisfor nonlinear fault tolerant control systems based on the T-Sfuzzy modelrdquo International Journal of Innovative ComputingInformation and Control vol 6 no 9 pp 3989ndash4000 2010

[23] S Yin Data-Driven Design of Fault Diagnosis Systems VDIDuesseldorf Germany 2012

[24] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[25] L Wu and W X Zheng ldquoPassivity-based sliding mode controlof uncertain singular time-delay systemsrdquo Automatica vol 45no 9 pp 2120ndash2127 2009

[26] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo IEEETransactions on Automatic Control vol 55 no 5 pp 1213ndash12192010

[27] L Zhang and E-K Boukas ldquoStability and stabilization ofMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo Automatica vol 45 no 2 pp 463ndash468 2009

10 Mathematical Problems in Engineering

[28] L Zhang and E-K Boukas ldquoMode-dependent Hinfin

filteringfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo Automatica vol 45 no 6pp 1462ndash1467 2009

[29] L Zhou and G Lu ldquoQuantized feedback stabilization fornetworked control systems with nonlinear perturbationrdquo Inter-national Journal of Innovative Computing Information andControl vol 6 no 6 pp 2485ndash2496 2010

[30] Z-G Wu J H Park H Su and J Chu ldquoDiscontinuousLyapunov functional approach to synchronization of time-delayneural networks using sampled-datardquoNonlinear Dynamics vol69 no 4 pp 2021ndash2030 2012

[31] H Li H Liu H Gao and P Shi ldquoReliable fuzzy control foractive suspension systems with actuator delay and faultrdquo IEEETransactions on Fuzzy Systems vol 20 no 2 pp 342ndash357 2012

[32] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding modecontrol for nonlinear active suspension systems using T-S fuzzymodelrdquo IEEE Transactions on Industrial Electronics vol 60 no8 pp 3328ndash3338 2012

[33] E Fridman ldquoA refined input delay approach to sampled-datacontrolrdquo Automatica vol 46 no 2 pp 421ndash427 2010

[34] E Fridman A Seuret and J-P Richard ldquoRobust sampled-data stabilization of linear systems an input delay approachrdquoAutomatica vol 40 no 8 pp 1441ndash1446 2004

[35] P Naghshtabrizi J P Hespanha and A R Teel ldquoExponentialstability of impulsive systems with application to uncertainsampled-data systemsrdquo Systems amp Control Letters vol 57 no5 pp 378ndash385 2008

[36] M Liu J You and X Ma ldquoHinfin

filtering for sampled-datastochastic systems with limited capacity channelrdquo Signal Pro-cessing vol 91 no 8 pp 1826ndash1837 2011

[37] M Liu D W C Ho and Y Niu ldquoObserver-based controllerdesign for linear systems with limited communication capacityvia a descriptor augmentation methodrdquo IET Control Theory ampApplications vol 6 no 3 pp 437ndash447 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 4: Research Article Sampled-Data Control of Spacecraft Rendezvous with …downloads.hindawi.com/journals/mpe/2013/814271.pdf · 2019-07-31 · Research Article Sampled-Data Control of

4 Mathematical Problems in Engineering

(1) The closed-loop system (19) is exponentially stablewith a given decay rate 120572 gt 0 In other words thechaser holds at a relative position

(2) The control input satisfies the given upper bound (8)

(3) For closed-loop system (19) the following119867infinperfor-

mance index is achieved

119890119911(119905) = 119911 (119905)

2minus 12057410038171003817100381710038171003817119889 (119905)

100381710038171003817100381710038172⩽ 0 (20)

3 Main Results

In this section we will first perform the stability analysis andcontrol scheme design for system (19) with 119889(119905) = 0Then wewill consider the internal stability of (19) with the prescribed119867infin

performance index (20)

31 Stabilization Analysis

Theorem 2 Consider the closed-loop system (19) with 119889(119905) =0 given a decay rate 120572 gt 0 if there exist positive definitematrices 119875

1 1198752 and 119875

3isin R9times9 and matrices 119879

1 1198792 1198781 1198782

and 1198783isin R9times9 such that the following conditions hold

Φ = [

[

Φ11Φ12Φ13

⋆ Φ22Φ23

⋆ ⋆ Φ33

]

]

lt 0

Φ =

[[[

[

Φ11Φ12Φ13Φ14

⋆ Φ22Φ23

0

⋆ ⋆ Φ33Φ34

⋆ ⋆ ⋆ Φ44

]]]

]

lt 0

(21)

120588119870119879

119877119879

119894119877119894119870 minus 119906

2

119894max1198751 lt 0 (22)

where 120588 is defined as in (47) and

Φ11= 2120572119875

1minus 1198753+ 2120572ℎ119875

3minus 119879119879

1119860 minus 119860

119879

1198791minus 1198781minus 119878119879

1

Φ12= 1198751+ 119879119879

1minus ℎ1198753

Φ13= 1198753minus 2120572ℎ119875

3minus 119879119879

1119861119870 + 119878

119879

1

Φ22= ℎ1198752+ 1198792+ 119879119879

2

Φ23= minusℎ119875

3minus 119879119879

2119861119870 + 119878

2+ 119878119879

2

Φ33= minus1198753+ 2120572ℎ119875

3+ 1198783+ 119878119879

3

Φ11= 2120572119875

1minus 1198753minus 119879119879

1119860 minus 119860

119879

1198791minus 1198781minus 119878119879

1

Φ12= 1198751+ 119879119879

1

Φ13= 1198753minus 119879119879

1119861119870 + 119878

119879

1

Φ14= ℎ119878119879

1

Φ22= 119879119879

2+ 119879119879

2

Φ23= minus119879119879

2119861119870 + 119878

2+ 119878119879

2

Φ33= minus1198753+ 1198783+ 119878119879

3

Φ34= ℎ119878119879

3

Φ44= minusℎ119890

minus2120572ℎ

1198752

(23)

Then system (19) with 119889(119905) = 0 is exponentially stable with thedecay rate 120572 gt 0

Proof For system (19) consider the following Lyapunovfunctional

119881 (119905) = 1198811(119905) + 119881

2(119905) + 119881

3(119905) (24)

where

1198811(119905) = 120585

119879

(119905) 1198751120585 (119905)

1198812(119905) = (ℎ minus 120591 (119905)) int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 119890minus2120572(119905minus119904)

119889119904

1198813(119905) = (ℎ minus 120591 (119905)) (120585(119905

119896) minus 120585 (119905))

119879

1198753(120585 (119905119896) minus 120585 (119905))

(25)

In function (3) 119875119894gt 0 (119894 = 1 2 3) are Lyapunov matrices to

be designed and 120572 gt 0 is the decay rate as previously definedIt can be calculated that

1(119905) + 2120572119881

1(119905) = 2 120585

119879

(119905) 1198751120585 (119905) + 2120572120585

119879

(119905) 1198751120585 (119905)

2(119905) = minus119890

minus2120572119905

int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

+ (ℎ minus 120591 (119905)) 119890minus2120572119905

(minus2120572)

times int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

= (ℎ minus 120591 (119905)) 119890minus2120572119905

times ( 120585119879

(119905) 1198752

120585 (119905) 1198902120572119905

minus 120585119879

(119905119896) 1198752

120585 (119905119896) 1198902120572119905119896)

(26)

Notice that 120585119879

(119905119896)1198752

120585(119905119896)1198902120572119905119896 = 0 thus

2(119905) = minus119890

minus2120572119905

int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

minus 2120572 (ℎ minus 120591 (119905)) 119890minus2120572119905

int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

(27)

Mathematical Problems in Engineering 5

Therefore it is derived that

2(119905) + 2120572119881

2(119905) = minus119890

minus2120572119905

int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

minus 2120572 (ℎ minus 120591 (119905)) 119890minus2120572119905

times int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

+ 2120572 (ℎ minus 120591 (119905)) 119890minus2120572119905

times int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

= minus119890minus2120572119905

int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

(28)

Furthermore for 1198813(119905) it is calculated directly that

3(119905) = minus(120585(119905

119896) minus 120585 (119905))

119879

1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) 120585119879

(119905) 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) (120585 (119905119896) minus 120585 (119905))

119879

1198753

120585 (119905)

3(119905) + 2120572119881

3(119905) = minus(120585 (119905

119896) minus 120585 (119905))

119879

times 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) 120585119879

(119905)

times 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905))

times (120585(119905119896) minus 120585 (119905))

119879

1198753

120585 (119905)

+ 2120572 (ℎ minus 120591 (119905)) (120585(119905119896) minus 120585(119905))

119879

times 1198753(120585 (119905119896) minus 120585 (119905))

(29)

As a result we have

(119905) + 2120572119881 (119905) = 1(119905) + 2120572119881

1(119905)

+ 2(119905) + 2120572119881

2(119905)

+ 3(119905) + 2120572119881

3(119905)

⩽ 2 120585119879

(119905) 1198751120585 (119905) + 2120572120585

119879

(119905) 1198751120585 (119905)

minus 119890minus2120572119905

int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

minus (120585(119905119896) minus 120585 (119905))

119879

times 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) 120585119879

(119905)

times 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) (120585 (119905119896) minus 120585 (119905))

119879

times 1198753

120585 (119905) + 2120572 (ℎ minus 120591 (119905))

times (120585 (119905119896) minus 120585 (119905))

119879

1198753(120585 (119905119896) minus 120585 (119905))

(30)

We define the following variable

120601120585(119905) =

1

120591 (119905)int

119905

119905119896

120585 (119904) 119889119904 (31)

and it is derived that

119890minus2120572119905

int

119905

119905119896

1198902120572119904 120585119879

(119904) 1198752

120585 (119904) 119889119904

⩾ 119890minus2120572119905

1198902120572119905119896 int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904)

= 119890minus2120572(119905minus119905119896) int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904)

= 119890minus2120572120591(119905)

int

119905

119905minus120591(119905)

120585119879

(119904) 1198752

120585 (119904)

(32)

By Jensenrsquos inequality it is also shown that

int

119905

119905minus120591(119905)

120585119879

(119904) 1198752

120585 (119904) 119889119904

⩽1

120591 (119905)int

119905

119905minus120591(119905)

120585119879

(119904) 119889119904

times 1198752int

119905

119905minus120591(119905)

120585 (119904) 119889119904

= 120591 (119905) (1

120591 (119905)int

119905

119905minus120591(119905)

120585119879

(119904) 119889119904)

times 1198752(

1

120591 (119905)int

119905

119905minus120591(119905)

120585 (119904) 119889119904)

= 120591 (119905) 120601119879

120585(119905) 1198752120601120585(119905)

(33)

In particular for 120591(119905) = 0 it can be verified that the followingholds

1

120591 (119905)int

119905

119905minus120591(119905)

120585 (119904) 119889119904|120591(119905)=0

= lim120591(119905)rarr0

1

120591 (119905)int

119905

119905minus120591(119905)

120585 (119904) 119889119904 = 120585 (119905)

(34)

6 Mathematical Problems in Engineering

Therefore it is shown from (30) that

(119905) + 2120572119881 (119905) ⩽ 2 120585119879

(119905) 1198751120585 (119905)

+ 2120572120585119879

(119905) 1198751120585 (119905)

minus 119890minus2120572120591(119905)

120591 (119905) 120601119879

120585(119905) 1198752120601120585(119905)

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

minus (120585 (119905119896) minus 120585 (119905))

119879

1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) 120585119879

(119905) 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) (120585(119905119896) minus 120585 (119905))

119879

1198753

120585 (119905)

+ 2120572 (ℎ minus 120591 (119905)) (120585 (119905119896) minus 120585 (119905))

119879

times 1198753(120585 (119905119896) minus 120585 (119905))

(35)

We now insert free-weighting matrices by introducing thefollowing zero terms

2 [120585119879

(119905) 119879119879

1+ 120585119879

(119905) 119879119879

2]

times [ 120585119879

(119905) minus 119860120585 (119905) minus 119861119870120585 (119905119896)] = 0

(36)

2 [120585119879

(119905) 119878119879

1+ 120585119879

(119905) 119878119879

2+ 120585119879

(119905119896) 119878119879

3]

times [minus120585 (119905) + 120585 (119905119896) + 120591 (119905) 120601

120585(119905)] = 0

(37)

where the free-weighting matrices 1198791 1198792 1198781 1198782 and 119878

3are to

be designed Then we have

(119905) + 2120572119881 (119905)

⩽ 2 120585119879

(119905) 1198751120585 (119905) + 2120572120585

119879

(119905) 1198751120585 (119905)

minus 119890minus2120572120591(119905)

120591 (119905) 120601119879

120585(119905) 1198752120601120585(119905)

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

minus (120585(119905119896) minus 120585 (119905))

119879

1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) 120585119879

(119905) 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) (120585(119905119896) minus 120585 (119905))

119879

1198753

120585 (119905)

+ 2120572 (ℎ minus 120591 (119905)) (120585(119905119896) minus 120585(119905))

119879

times 1198753(120585 (119905119896) minus 120585 (119905))

+ 2 [120585119879

(119905) 119879119879

1+ 120585119879

(119905) 119879119879

2]

times [ 120585119879

(119905) minus 119860120585 (119905) minus 119861119870120585 (119905119896)]

times 2 [120585119879

(119905) 119878119879

1+ 120585119879

(119905) 119878119879

2+ 120585119879

(119905119896) 119878119879

3]

times [minus120585 (119905) + 120585 (119905119896) + 120591 (119905) 120601

120585(119905)]

(38)

Defining the following new augmented variables

120585 (119905) = [120585119879

(119905) 120585119879

(119905) 120585119879

(119905119896) 120601119879

(119905)]119879

120585 (119905) = [120585119879

(119905) 120585119879

(119905) 120585119879

(119905119896)]119879

(39)

it is calculated that

(119905) + 2120572119881 (119905) ⩽ 120585119879

(119905) Π (119905) 120585 (119905) (40)

with

Π (119905) =

[[[

[

Π11(119905) Π

12(119905) Π

13(119905) Π

14(119905)

⋆ Π22(119905) Π

23(119905) 0

⋆ ⋆ Π33(119905) Π

34(119905)

⋆ ⋆ ⋆ Π44(119905)

]]]

]

Π11(119905) = 2120572119875

1minus 1198753+ 2120572 (ℎ minus 120591 (119905)) 119875

3

minus 119879119879

1119860 minus 119860

119879

1198791minus 1198781minus 119878119879

1

Π12(119905) = 119875

1+ 119879119879

1minus (ℎ minus 120591 (119905)) 119875

3

Π13(119905) = 119875

3minus 2120572 (ℎ minus 120591 (119905)) 119875

3minus 119879119879

1119861119870 + 119878

119879

1

Π14(119905) = 120591 (119905) 119878

119879

1

Π22(119905) = (ℎ minus 120591 (119905)) 119875

2+ 1198792+ 119879119879

2

Π23(119905) = minus (ℎ minus 120591 (119905)) 119875

3minus 119879119879

2119861119870 + 119878

2+ 119878119879

2

Π33(119905) = minus119875

3+ 2120572 (ℎ minus 120591 (119905)) 119875

3+ 1198783+ 119878119879

3

Π34(119905) = 120591 (119905) 119878

119879

3

Π44(119905) = minus119890

minus2120572120591(119905)

120591 (119905) 1198752

(41)

We now consider the matrices Φ and Φ defined in (21) Infact it can be verified thatΦ is obtained fromΠ(119905)|

120591(119905)=0with

deleting the last row and last column and Φ = Π(119905)|120591(119905)=ℎ

Similar to the proof of Theorem 1 of [37] it can be provedthat the system (19) with 119889(119905) = 0 is exponentially stable

Next we deal with the input constraint along each axisWe define

119877119909= [

[

1

0

0

]

]

[1 0 0] = [

[

1 0 0

0 0 0

0 0 0

]

]

119877119910= [

[

0

1

0

]

]

[0 1 0] = [

[

0 0 0

0 1 0

0 0 0

]

]

119877119911= [

[

0

0

1

]

]

[0 0 1] = [

[

0 0 0

0 0 0

0 0 1

]

]

(42)

Then the thrust along each axis can be described as1003816100381610038161003816119906119894 (119905119896)

1003816100381610038161003816 =1003817100381710038171003817119877119894119906 (119905119896)

1003817100381710038171003817 ⩽ 119906119894max 119894 = 119909 119910 119911 (43)

Mathematical Problems in Engineering 7

Therefore the thrust constraint (8) can be rewritten as1003817100381710038171003817119877119894119906 (119905119896)

1003817100381710038171003817 ⩽ 119906119894max 119894 = 119909 119910 119911 (44)

which is equivalent to

(119877119894119906 (119905119896))119879

119877119894119906 (119905119896) ⩽ 1199062

119894max (45)

Furthermore formula (45) is equal to

120585119879

(119905119896)119870119877119879

119894119877119894119870120585 (119905119896) ⩽ 1199062

119894max (46)

Based on the previous analysis it is known that (119905) lt 0 isguaranteed if the matrix conditions (21) hold Subsequently119881(119905) lt 119881(0) holds for any 119905 gt 0 It is therefore reasonable toassume that there exists a scalar 120588 gt 0 such that

119881 (0) ⩽ 120588 (47)

Notice that 119881(119905) = 1198811(119905) + 119881

2(119905) + 119881

3(119905) le 119881(0) ⩽ 120588 and

1198812(119905) ge 0 119881

3(119905) ge 0 Therefore it is true that

119881 (119905) le 1198811(119905) lt 119881 (0) ⩽ 120588 (48)

which implies that

1198811(119905) lt 120588 (49)

Therefore 120585119879(119905)1198751120585(119905) lt 120588 holds for any 119905 gt 0 In particular

let 119905 = 119905119896 we have

120585119879

(119905119896) 1198751120585 (119905119896) lt 120588 (50)

Then (46) can be guaranteed if (22) holdsWe prove this factIn fact if (22) holds we have

120585119879

(119905119896) 120588119870119879

119877119879

119894119877119894119870120585 (119905119896) lt 120585119879

(119905119896) 1199062

119894max1198751120585 (119905119896) (51)

By (51) we have

120585119879

(119905119896) 120588119870119879

119877119879

119894119877119894119870120585 (119905119896)

lt 120585119879

(119905119896) 1199062

119894max1198751120585 (119905119896) lt 1199062

119894max120588

(52)

which implies that

120585119879

(119905119896)119870119879

119877119879

119894119877119894119870120585 (119905119896) lt 1199062

119894max (53)

As a result the thrust limitation (8) can be guaranteed by (22)This completes the proof

32 Disturbance Attenuation In this subsection we willinvestigate analysis of the internal stability and 119867

infinperfor-

mance index for the system (19) Defining a new augmentedvector 120595(119905) ≜ [120585

119879

(119905)119889119879

(119905)]119879 we reconsider the system (19)

with 119889(119905) = 0 We introduce the performance index

119869 (119905) = int

119905

0

[119911119879

(119904) 119911 (119904) minus 120574119889119879

(119904) 119889 (119904)] 119889119904 (54)

Then (20) can be guaranteed by 119869(119905) lt 0

0 5 10 15 20 25 30

0

5

10

minus20

minus15

minus10

minus5

x1(t)

x2(t)

x3(t)

x4(t)

x5(t)

x6(t)

Figure 1 Trajectories of 119909(119905)

Theorem 3 Consider the closed-loop system (19) given aperformance index parameter 120574 gt 0 let the decay rate 120572 = 0 ifthere exist positive definite matrices 119875

1 1198752 and 119875

3isin R9times9 and

matrices 1198791 1198792 1198781 1198782 and 119878

3isin R9times9 such that the matrix

conditions (21) (22) and the following matrices conditionshold

[Φ|120572=0

Γ1

⋆ diag minus1205742 minus1198689] lt 0

[Φ|120572=0

Γ2

⋆ diag minus1205742 minus1198689] lt 0

(55)

Γ1=

[[[[[

[

minus119879119879

1119861119879

1119862119879

2

minus119879119879

2119861119879

10

0 0

]]]]]

]

Γ2=

[[[[[[[

[

minus119879119879

1119861119879

1119862119879

2

minus119879119879

2119861119879

10

0 0

0 0

]]]]]]]

]

(56)

Then the system (19) is internally stable and the 119867infin

perfor-mance index (20) is achieved

Proof It is well known that (119905) lt 0 implies thatlim119905rarrinfin

119881(119905) = 0 We consider a new defined function

119881119869(119905) = (119905) + 119911

119879

(119905) 119911 (119905) minus 1205742

119889119879

(119905) 119889 (119905) (57)

If 119881119869(119905) lt 0 it is shown that

119911119879

(119905) 119911 (119905) minus 1205742

119889119879

(119905) 119889 (119905) lt minus (119905) (58)

8 Mathematical Problems in Engineering

0 5 10 15 20 25 30

0

200

400

600

800

minus600

minus400

minus200

Time (s)

u1(t)

u2(t)

u3(t)

Figure 2 Trajectories of 119906(119905)

0 5 10 15 20 25 300

100

200

300

400

500

600

700

Time (s)

V(t)

Figure 3 Lyapunov function 119881(119905)

and it further implies that 119869(infin) = intinfin

0

(119911119879

(119904)119911(119904) minus 1205742

119889119879

(119904)119889(119904))119889119904 lt minus intinfin

0

(119904)119889119904 = 119881(infin) minus 119881(0) = minus119881(0) lt 0Thatmeans in the case120572 = 0 if (119905) lt 0 and119881

119869(119905) lt 0hold it

can be guaranteed that the system (19) is internally stable andthe119867infinperformance index (20) can be achieved successfully

Notice that (57) can be rewritten as

(119905) + 120585119879

(119905) 119862119879

21198622120585 (119905) minus 120574

2

119889119879

(119905) 119889 (119905) lt 0 (59)Recall the zero terms (37) and notice that the term (37) with119889(119905) = 0 becomes

2 [120585119879

(119905) 119879119879

1+ 120585119879

(119905) 119879119879

2]

times [ 120585119879

(119905) minus 119860120585 (119905) minus 119861119870120585 (119905119896) minus 1198611119889 (119905)] = 0

2 [120585119879

(119905) 119878119879

1+ 120585119879

(119905) 119878119879

2+ 120585119879

(119905119896) 119878119879

3]

times [minus120585 (119905) + 120585 (119905119896) + 120591 (119905) 120601

120585(119905)] = 0

(60)

By similar derivation of Theorem 2 it can be proved thatsystem (19) is internally stable if the conditions (55) hold Onthe other hand it has been proved inTheorem 2 that (119905) lt 0is guaranteed by (21)-(22) This completes the proof

4 Simulation

In this section simulation results will be shown to illustratethe usefulness and advantage of the proposed sampled-datacontrol approach We consider the numerical example in [3]suppose that themass of the chaser is 200 kg that is 119898 = 200and the target is moving along a geosynchronous orbit withradius 119903 = 42241 km with an orbital period of 24 hThe anglevelocity 119908 = 1117 times 103 rads The maximum thrust alongeach axis is 1000N The disturbance vector is described as119889(119905) = [2 sin(2119905) 2 sin(2119905) 2 sin(2119905)]119879 The upper boundof the sampling period is given as ℎ = 05 and the decayrate is selected as 120572 = 002 The disturbance attenuation levelis given as 120574 = 04 In the coordinate based on the targetframe assume that the initial state is [10 minus20 10 0 0 0] andthe desired holding position is (0 minus1 0) Design the statefeedback gain 119870 as

119870 = [

[

minus343807 01490 0 minus973889 00051 0 minus87062 00254 0

minus01490 minus343801 0 minus00051 minus973880 0 minus00254 minus87062 0

0 0 minus578745 0 0 minus1100009 0 0 minus165843

]

]

(61)

such that (119860 + 119861119870) is Hurwitz Suppose that 1199051= 0 which

means that the first sampling time instant is the initial timeThe designed parameter 120588 is chosen as 120588 = 950 Solvingthe conditions (21)-(22) and (55) it is found that there existsfeasible solution Then it can be derived that 119881(0) = 119881

1(0) =

119909(0)119879

1198751119909(0) = 8359423 It is obvious the condition (47)

holds which means that our obtained solution is correct andfeasible

The simulation results are provided in Figures 1ndash3 whichshow the effectiveness of the proposed methods The tra-jectories of state variable 119909(119905) with 119889(119905) = 0 are shown inFigure 1 the trajectories of sampled-data control input 119906(119905)with 119889(119905) = 0 are shown in Figure 2 the trajectories ofLyapunov functions 119881(119905) with 119889(119905) = 0 are shown inFigure 3 It can be seen that the obtained performance isdesirable

Mathematical Problems in Engineering 9

5 Conclusion

In this paper we have addressed the problem of sampled-data stabilization for spacecraft rendezvous with a time-dependent Lyapunov approach A discontinuous Lyapunovfunctional has been constructed the value of which does notincrease in each sampling time instant The sufficient condi-tions for the stability analysis and controller design are estab-lished in terms of matrix inequality technique Future workwill be focused onmore complicated case that is the effects ofnetwork-induced phenomenon such as communication delayand packet losses are taken into account simultaneously

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China 61104101 the FundamentalResearch Funds for the Central Universities the China Post-doctoral Science Foundation funded project (2011M500058)the Special Chinese National Postdoctoral Science Founda-tion under Grant 2012T50356 and the Heilongjiang Postdoc-toral Fund (LBH-Z11144)

References

[1] U Ahsun D W Miller and J L Ramirez ldquoControl of electro-magnetic satellite formations in near-Earth orbitsrdquo Journal ofGuidance Control and Dynamics vol 33 no 6 pp 1883ndash18912010

[2] H Schaub ldquoStabilization of satellite motion relative to aCoulomb spacecraft formationrdquo Journal of Guidance Controland Dynamics vol 28 no 6 pp 1231ndash1239 2005

[3] X Yang X Cao and H Gao ldquoSampled-data control forrelative position holding of spacecraft rendezvous with thrustnonlinearityrdquo IEEE Transactions on Industrial Electronics vol59 no 2 pp 1146ndash1153 2011

[4] P Gurfil ldquoNonlinear feedback control of low-thrust orbitaltransfer in a central gravitational fieldrdquo Acta Astronautica vol60 no 8-9 pp 631ndash648 2007

[5] S Kurnaz O Cetin and O Kaynak ldquoFuzzy logic basedapproach to design of flight control and navigation tasks forautonomous unmanned aerial vehiclesrdquo Journal of Intelligentand Robotic Systems vol 54 no 1ndash3 pp 229ndash244 2009

[6] W Clohessy and R Wiltshire ldquoTerminal guidance system forsatellite rendezvousrdquo Journal of the Aerospace Sciences vol 27no 9 pp 653ndash658 1960

[7] Y Luo and G Tang ldquoSpacecraft optimal rendezvous controllerdesign using simulated annealingrdquo Aerospace Science and Tech-nology vol 9 no 8 pp 732ndash737 2005

[8] Y Luo G Tang and Y Lei ldquoOptimal multi-objective lin-earized impulsive rendezvousrdquo Journal of Guidance Controland Dynamics vol 30 no 2 pp 383ndash389 2007

[9] Z Mao and B Jiang ldquoFault identification and fault-tolerantcontrol for a class of networked control systemsrdquo InternationalJournal of Innovative Computing Information and Control vol3 no 5 pp 1121ndash1130 2007

[10] X Su P Shi L Wu and Y Song ldquoA novel control design ondiscrete-time Takagi-Sugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems vol 21 no 4 pp655ndash671 2012

[11] X Su L Wu P Shi and Y Song ldquoHinfin

model reduction of T-Sfuzzy stochastic systemsrdquo IEEE Transactions on Systems Manand Cybernetics B vol 42 no 6 pp 1574ndash1585 2012

[12] S Yin S Ding and H Luo ldquoReal-time implementation of faulttolerant control system with performance optimizationrdquo IEEETransactions on Industrial Electronics 2013

[13] R Yang Z Zhang and P Shi ldquoExponential stability on stochas-tic neural networks with discrete interval and distributeddelaysrdquo IEEE Transactions on Neural Networks vol 21 no 1 pp169ndash175 2010

[14] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[15] R Yang P Shi and G-P Liu ldquoFiltering for discrete-timenetworked nonlinear systems with mixed random delays andpacket dropoutsrdquo IEEE Transactions on Automatic Control vol56 no 11 pp 2655ndash2660 2011

[16] Y Shu L Yu and Z Wenan ldquoEstimator-based control of net-worked systems with packet-dropoutsrdquo International Journal ofInnovative Computing Information andControl vol 6 no 6 pp2737ndash2748 2010

[17] C Jiang Q Zhang and D Zou ldquoDelay-dependent robust filter-ing for networked control systemswith polytopic uncertaintiesrdquoInternational Journal of Innovative Computing Information andControl vol 6 no 11 pp 4857ndash4868 2010

[18] S Shi Z Yuan and Q Zhang ldquoFault-tolerant Hinfin

filter designof a class of switched systems with sensor failuresrdquo InternationalJournal of Innovative Computing Information and Control vol5 no 11 pp 3827ndash3838 2009

[19] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[20] J Lu D W C Ho and J Cao ldquoA unified synchronizationcriterion for impulsive dynamical networksrdquo Automatica vol46 no 7 pp 1215ndash1221 2010

[21] J Lu D W C Ho J Cao and J Kurths ldquoExponential synchro-nization of linearly coupled neural networks with impulsivedisturbancesrdquo IEEE Transactions on Neural Networks vol 22no 2 pp 329ndash336 2011

[22] H Wu and M Bai ldquoStochastic stability analysis and synthesisfor nonlinear fault tolerant control systems based on the T-Sfuzzy modelrdquo International Journal of Innovative ComputingInformation and Control vol 6 no 9 pp 3989ndash4000 2010

[23] S Yin Data-Driven Design of Fault Diagnosis Systems VDIDuesseldorf Germany 2012

[24] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[25] L Wu and W X Zheng ldquoPassivity-based sliding mode controlof uncertain singular time-delay systemsrdquo Automatica vol 45no 9 pp 2120ndash2127 2009

[26] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo IEEETransactions on Automatic Control vol 55 no 5 pp 1213ndash12192010

[27] L Zhang and E-K Boukas ldquoStability and stabilization ofMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo Automatica vol 45 no 2 pp 463ndash468 2009

10 Mathematical Problems in Engineering

[28] L Zhang and E-K Boukas ldquoMode-dependent Hinfin

filteringfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo Automatica vol 45 no 6pp 1462ndash1467 2009

[29] L Zhou and G Lu ldquoQuantized feedback stabilization fornetworked control systems with nonlinear perturbationrdquo Inter-national Journal of Innovative Computing Information andControl vol 6 no 6 pp 2485ndash2496 2010

[30] Z-G Wu J H Park H Su and J Chu ldquoDiscontinuousLyapunov functional approach to synchronization of time-delayneural networks using sampled-datardquoNonlinear Dynamics vol69 no 4 pp 2021ndash2030 2012

[31] H Li H Liu H Gao and P Shi ldquoReliable fuzzy control foractive suspension systems with actuator delay and faultrdquo IEEETransactions on Fuzzy Systems vol 20 no 2 pp 342ndash357 2012

[32] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding modecontrol for nonlinear active suspension systems using T-S fuzzymodelrdquo IEEE Transactions on Industrial Electronics vol 60 no8 pp 3328ndash3338 2012

[33] E Fridman ldquoA refined input delay approach to sampled-datacontrolrdquo Automatica vol 46 no 2 pp 421ndash427 2010

[34] E Fridman A Seuret and J-P Richard ldquoRobust sampled-data stabilization of linear systems an input delay approachrdquoAutomatica vol 40 no 8 pp 1441ndash1446 2004

[35] P Naghshtabrizi J P Hespanha and A R Teel ldquoExponentialstability of impulsive systems with application to uncertainsampled-data systemsrdquo Systems amp Control Letters vol 57 no5 pp 378ndash385 2008

[36] M Liu J You and X Ma ldquoHinfin

filtering for sampled-datastochastic systems with limited capacity channelrdquo Signal Pro-cessing vol 91 no 8 pp 1826ndash1837 2011

[37] M Liu D W C Ho and Y Niu ldquoObserver-based controllerdesign for linear systems with limited communication capacityvia a descriptor augmentation methodrdquo IET Control Theory ampApplications vol 6 no 3 pp 437ndash447 2012

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Page 5: Research Article Sampled-Data Control of Spacecraft Rendezvous with …downloads.hindawi.com/journals/mpe/2013/814271.pdf · 2019-07-31 · Research Article Sampled-Data Control of

Mathematical Problems in Engineering 5

Therefore it is derived that

2(119905) + 2120572119881

2(119905) = minus119890

minus2120572119905

int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

minus 2120572 (ℎ minus 120591 (119905)) 119890minus2120572119905

times int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

+ 2120572 (ℎ minus 120591 (119905)) 119890minus2120572119905

times int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

= minus119890minus2120572119905

int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

(28)

Furthermore for 1198813(119905) it is calculated directly that

3(119905) = minus(120585(119905

119896) minus 120585 (119905))

119879

1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) 120585119879

(119905) 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) (120585 (119905119896) minus 120585 (119905))

119879

1198753

120585 (119905)

3(119905) + 2120572119881

3(119905) = minus(120585 (119905

119896) minus 120585 (119905))

119879

times 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) 120585119879

(119905)

times 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905))

times (120585(119905119896) minus 120585 (119905))

119879

1198753

120585 (119905)

+ 2120572 (ℎ minus 120591 (119905)) (120585(119905119896) minus 120585(119905))

119879

times 1198753(120585 (119905119896) minus 120585 (119905))

(29)

As a result we have

(119905) + 2120572119881 (119905) = 1(119905) + 2120572119881

1(119905)

+ 2(119905) + 2120572119881

2(119905)

+ 3(119905) + 2120572119881

3(119905)

⩽ 2 120585119879

(119905) 1198751120585 (119905) + 2120572120585

119879

(119905) 1198751120585 (119905)

minus 119890minus2120572119905

int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904) 1198902120572119904

119889119904

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

minus (120585(119905119896) minus 120585 (119905))

119879

times 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) 120585119879

(119905)

times 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) (120585 (119905119896) minus 120585 (119905))

119879

times 1198753

120585 (119905) + 2120572 (ℎ minus 120591 (119905))

times (120585 (119905119896) minus 120585 (119905))

119879

1198753(120585 (119905119896) minus 120585 (119905))

(30)

We define the following variable

120601120585(119905) =

1

120591 (119905)int

119905

119905119896

120585 (119904) 119889119904 (31)

and it is derived that

119890minus2120572119905

int

119905

119905119896

1198902120572119904 120585119879

(119904) 1198752

120585 (119904) 119889119904

⩾ 119890minus2120572119905

1198902120572119905119896 int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904)

= 119890minus2120572(119905minus119905119896) int

119905

119905119896

120585119879

(119904) 1198752

120585 (119904)

= 119890minus2120572120591(119905)

int

119905

119905minus120591(119905)

120585119879

(119904) 1198752

120585 (119904)

(32)

By Jensenrsquos inequality it is also shown that

int

119905

119905minus120591(119905)

120585119879

(119904) 1198752

120585 (119904) 119889119904

⩽1

120591 (119905)int

119905

119905minus120591(119905)

120585119879

(119904) 119889119904

times 1198752int

119905

119905minus120591(119905)

120585 (119904) 119889119904

= 120591 (119905) (1

120591 (119905)int

119905

119905minus120591(119905)

120585119879

(119904) 119889119904)

times 1198752(

1

120591 (119905)int

119905

119905minus120591(119905)

120585 (119904) 119889119904)

= 120591 (119905) 120601119879

120585(119905) 1198752120601120585(119905)

(33)

In particular for 120591(119905) = 0 it can be verified that the followingholds

1

120591 (119905)int

119905

119905minus120591(119905)

120585 (119904) 119889119904|120591(119905)=0

= lim120591(119905)rarr0

1

120591 (119905)int

119905

119905minus120591(119905)

120585 (119904) 119889119904 = 120585 (119905)

(34)

6 Mathematical Problems in Engineering

Therefore it is shown from (30) that

(119905) + 2120572119881 (119905) ⩽ 2 120585119879

(119905) 1198751120585 (119905)

+ 2120572120585119879

(119905) 1198751120585 (119905)

minus 119890minus2120572120591(119905)

120591 (119905) 120601119879

120585(119905) 1198752120601120585(119905)

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

minus (120585 (119905119896) minus 120585 (119905))

119879

1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) 120585119879

(119905) 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) (120585(119905119896) minus 120585 (119905))

119879

1198753

120585 (119905)

+ 2120572 (ℎ minus 120591 (119905)) (120585 (119905119896) minus 120585 (119905))

119879

times 1198753(120585 (119905119896) minus 120585 (119905))

(35)

We now insert free-weighting matrices by introducing thefollowing zero terms

2 [120585119879

(119905) 119879119879

1+ 120585119879

(119905) 119879119879

2]

times [ 120585119879

(119905) minus 119860120585 (119905) minus 119861119870120585 (119905119896)] = 0

(36)

2 [120585119879

(119905) 119878119879

1+ 120585119879

(119905) 119878119879

2+ 120585119879

(119905119896) 119878119879

3]

times [minus120585 (119905) + 120585 (119905119896) + 120591 (119905) 120601

120585(119905)] = 0

(37)

where the free-weighting matrices 1198791 1198792 1198781 1198782 and 119878

3are to

be designed Then we have

(119905) + 2120572119881 (119905)

⩽ 2 120585119879

(119905) 1198751120585 (119905) + 2120572120585

119879

(119905) 1198751120585 (119905)

minus 119890minus2120572120591(119905)

120591 (119905) 120601119879

120585(119905) 1198752120601120585(119905)

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

minus (120585(119905119896) minus 120585 (119905))

119879

1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) 120585119879

(119905) 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) (120585(119905119896) minus 120585 (119905))

119879

1198753

120585 (119905)

+ 2120572 (ℎ minus 120591 (119905)) (120585(119905119896) minus 120585(119905))

119879

times 1198753(120585 (119905119896) minus 120585 (119905))

+ 2 [120585119879

(119905) 119879119879

1+ 120585119879

(119905) 119879119879

2]

times [ 120585119879

(119905) minus 119860120585 (119905) minus 119861119870120585 (119905119896)]

times 2 [120585119879

(119905) 119878119879

1+ 120585119879

(119905) 119878119879

2+ 120585119879

(119905119896) 119878119879

3]

times [minus120585 (119905) + 120585 (119905119896) + 120591 (119905) 120601

120585(119905)]

(38)

Defining the following new augmented variables

120585 (119905) = [120585119879

(119905) 120585119879

(119905) 120585119879

(119905119896) 120601119879

(119905)]119879

120585 (119905) = [120585119879

(119905) 120585119879

(119905) 120585119879

(119905119896)]119879

(39)

it is calculated that

(119905) + 2120572119881 (119905) ⩽ 120585119879

(119905) Π (119905) 120585 (119905) (40)

with

Π (119905) =

[[[

[

Π11(119905) Π

12(119905) Π

13(119905) Π

14(119905)

⋆ Π22(119905) Π

23(119905) 0

⋆ ⋆ Π33(119905) Π

34(119905)

⋆ ⋆ ⋆ Π44(119905)

]]]

]

Π11(119905) = 2120572119875

1minus 1198753+ 2120572 (ℎ minus 120591 (119905)) 119875

3

minus 119879119879

1119860 minus 119860

119879

1198791minus 1198781minus 119878119879

1

Π12(119905) = 119875

1+ 119879119879

1minus (ℎ minus 120591 (119905)) 119875

3

Π13(119905) = 119875

3minus 2120572 (ℎ minus 120591 (119905)) 119875

3minus 119879119879

1119861119870 + 119878

119879

1

Π14(119905) = 120591 (119905) 119878

119879

1

Π22(119905) = (ℎ minus 120591 (119905)) 119875

2+ 1198792+ 119879119879

2

Π23(119905) = minus (ℎ minus 120591 (119905)) 119875

3minus 119879119879

2119861119870 + 119878

2+ 119878119879

2

Π33(119905) = minus119875

3+ 2120572 (ℎ minus 120591 (119905)) 119875

3+ 1198783+ 119878119879

3

Π34(119905) = 120591 (119905) 119878

119879

3

Π44(119905) = minus119890

minus2120572120591(119905)

120591 (119905) 1198752

(41)

We now consider the matrices Φ and Φ defined in (21) Infact it can be verified thatΦ is obtained fromΠ(119905)|

120591(119905)=0with

deleting the last row and last column and Φ = Π(119905)|120591(119905)=ℎ

Similar to the proof of Theorem 1 of [37] it can be provedthat the system (19) with 119889(119905) = 0 is exponentially stable

Next we deal with the input constraint along each axisWe define

119877119909= [

[

1

0

0

]

]

[1 0 0] = [

[

1 0 0

0 0 0

0 0 0

]

]

119877119910= [

[

0

1

0

]

]

[0 1 0] = [

[

0 0 0

0 1 0

0 0 0

]

]

119877119911= [

[

0

0

1

]

]

[0 0 1] = [

[

0 0 0

0 0 0

0 0 1

]

]

(42)

Then the thrust along each axis can be described as1003816100381610038161003816119906119894 (119905119896)

1003816100381610038161003816 =1003817100381710038171003817119877119894119906 (119905119896)

1003817100381710038171003817 ⩽ 119906119894max 119894 = 119909 119910 119911 (43)

Mathematical Problems in Engineering 7

Therefore the thrust constraint (8) can be rewritten as1003817100381710038171003817119877119894119906 (119905119896)

1003817100381710038171003817 ⩽ 119906119894max 119894 = 119909 119910 119911 (44)

which is equivalent to

(119877119894119906 (119905119896))119879

119877119894119906 (119905119896) ⩽ 1199062

119894max (45)

Furthermore formula (45) is equal to

120585119879

(119905119896)119870119877119879

119894119877119894119870120585 (119905119896) ⩽ 1199062

119894max (46)

Based on the previous analysis it is known that (119905) lt 0 isguaranteed if the matrix conditions (21) hold Subsequently119881(119905) lt 119881(0) holds for any 119905 gt 0 It is therefore reasonable toassume that there exists a scalar 120588 gt 0 such that

119881 (0) ⩽ 120588 (47)

Notice that 119881(119905) = 1198811(119905) + 119881

2(119905) + 119881

3(119905) le 119881(0) ⩽ 120588 and

1198812(119905) ge 0 119881

3(119905) ge 0 Therefore it is true that

119881 (119905) le 1198811(119905) lt 119881 (0) ⩽ 120588 (48)

which implies that

1198811(119905) lt 120588 (49)

Therefore 120585119879(119905)1198751120585(119905) lt 120588 holds for any 119905 gt 0 In particular

let 119905 = 119905119896 we have

120585119879

(119905119896) 1198751120585 (119905119896) lt 120588 (50)

Then (46) can be guaranteed if (22) holdsWe prove this factIn fact if (22) holds we have

120585119879

(119905119896) 120588119870119879

119877119879

119894119877119894119870120585 (119905119896) lt 120585119879

(119905119896) 1199062

119894max1198751120585 (119905119896) (51)

By (51) we have

120585119879

(119905119896) 120588119870119879

119877119879

119894119877119894119870120585 (119905119896)

lt 120585119879

(119905119896) 1199062

119894max1198751120585 (119905119896) lt 1199062

119894max120588

(52)

which implies that

120585119879

(119905119896)119870119879

119877119879

119894119877119894119870120585 (119905119896) lt 1199062

119894max (53)

As a result the thrust limitation (8) can be guaranteed by (22)This completes the proof

32 Disturbance Attenuation In this subsection we willinvestigate analysis of the internal stability and 119867

infinperfor-

mance index for the system (19) Defining a new augmentedvector 120595(119905) ≜ [120585

119879

(119905)119889119879

(119905)]119879 we reconsider the system (19)

with 119889(119905) = 0 We introduce the performance index

119869 (119905) = int

119905

0

[119911119879

(119904) 119911 (119904) minus 120574119889119879

(119904) 119889 (119904)] 119889119904 (54)

Then (20) can be guaranteed by 119869(119905) lt 0

0 5 10 15 20 25 30

0

5

10

minus20

minus15

minus10

minus5

x1(t)

x2(t)

x3(t)

x4(t)

x5(t)

x6(t)

Figure 1 Trajectories of 119909(119905)

Theorem 3 Consider the closed-loop system (19) given aperformance index parameter 120574 gt 0 let the decay rate 120572 = 0 ifthere exist positive definite matrices 119875

1 1198752 and 119875

3isin R9times9 and

matrices 1198791 1198792 1198781 1198782 and 119878

3isin R9times9 such that the matrix

conditions (21) (22) and the following matrices conditionshold

[Φ|120572=0

Γ1

⋆ diag minus1205742 minus1198689] lt 0

[Φ|120572=0

Γ2

⋆ diag minus1205742 minus1198689] lt 0

(55)

Γ1=

[[[[[

[

minus119879119879

1119861119879

1119862119879

2

minus119879119879

2119861119879

10

0 0

]]]]]

]

Γ2=

[[[[[[[

[

minus119879119879

1119861119879

1119862119879

2

minus119879119879

2119861119879

10

0 0

0 0

]]]]]]]

]

(56)

Then the system (19) is internally stable and the 119867infin

perfor-mance index (20) is achieved

Proof It is well known that (119905) lt 0 implies thatlim119905rarrinfin

119881(119905) = 0 We consider a new defined function

119881119869(119905) = (119905) + 119911

119879

(119905) 119911 (119905) minus 1205742

119889119879

(119905) 119889 (119905) (57)

If 119881119869(119905) lt 0 it is shown that

119911119879

(119905) 119911 (119905) minus 1205742

119889119879

(119905) 119889 (119905) lt minus (119905) (58)

8 Mathematical Problems in Engineering

0 5 10 15 20 25 30

0

200

400

600

800

minus600

minus400

minus200

Time (s)

u1(t)

u2(t)

u3(t)

Figure 2 Trajectories of 119906(119905)

0 5 10 15 20 25 300

100

200

300

400

500

600

700

Time (s)

V(t)

Figure 3 Lyapunov function 119881(119905)

and it further implies that 119869(infin) = intinfin

0

(119911119879

(119904)119911(119904) minus 1205742

119889119879

(119904)119889(119904))119889119904 lt minus intinfin

0

(119904)119889119904 = 119881(infin) minus 119881(0) = minus119881(0) lt 0Thatmeans in the case120572 = 0 if (119905) lt 0 and119881

119869(119905) lt 0hold it

can be guaranteed that the system (19) is internally stable andthe119867infinperformance index (20) can be achieved successfully

Notice that (57) can be rewritten as

(119905) + 120585119879

(119905) 119862119879

21198622120585 (119905) minus 120574

2

119889119879

(119905) 119889 (119905) lt 0 (59)Recall the zero terms (37) and notice that the term (37) with119889(119905) = 0 becomes

2 [120585119879

(119905) 119879119879

1+ 120585119879

(119905) 119879119879

2]

times [ 120585119879

(119905) minus 119860120585 (119905) minus 119861119870120585 (119905119896) minus 1198611119889 (119905)] = 0

2 [120585119879

(119905) 119878119879

1+ 120585119879

(119905) 119878119879

2+ 120585119879

(119905119896) 119878119879

3]

times [minus120585 (119905) + 120585 (119905119896) + 120591 (119905) 120601

120585(119905)] = 0

(60)

By similar derivation of Theorem 2 it can be proved thatsystem (19) is internally stable if the conditions (55) hold Onthe other hand it has been proved inTheorem 2 that (119905) lt 0is guaranteed by (21)-(22) This completes the proof

4 Simulation

In this section simulation results will be shown to illustratethe usefulness and advantage of the proposed sampled-datacontrol approach We consider the numerical example in [3]suppose that themass of the chaser is 200 kg that is 119898 = 200and the target is moving along a geosynchronous orbit withradius 119903 = 42241 km with an orbital period of 24 hThe anglevelocity 119908 = 1117 times 103 rads The maximum thrust alongeach axis is 1000N The disturbance vector is described as119889(119905) = [2 sin(2119905) 2 sin(2119905) 2 sin(2119905)]119879 The upper boundof the sampling period is given as ℎ = 05 and the decayrate is selected as 120572 = 002 The disturbance attenuation levelis given as 120574 = 04 In the coordinate based on the targetframe assume that the initial state is [10 minus20 10 0 0 0] andthe desired holding position is (0 minus1 0) Design the statefeedback gain 119870 as

119870 = [

[

minus343807 01490 0 minus973889 00051 0 minus87062 00254 0

minus01490 minus343801 0 minus00051 minus973880 0 minus00254 minus87062 0

0 0 minus578745 0 0 minus1100009 0 0 minus165843

]

]

(61)

such that (119860 + 119861119870) is Hurwitz Suppose that 1199051= 0 which

means that the first sampling time instant is the initial timeThe designed parameter 120588 is chosen as 120588 = 950 Solvingthe conditions (21)-(22) and (55) it is found that there existsfeasible solution Then it can be derived that 119881(0) = 119881

1(0) =

119909(0)119879

1198751119909(0) = 8359423 It is obvious the condition (47)

holds which means that our obtained solution is correct andfeasible

The simulation results are provided in Figures 1ndash3 whichshow the effectiveness of the proposed methods The tra-jectories of state variable 119909(119905) with 119889(119905) = 0 are shown inFigure 1 the trajectories of sampled-data control input 119906(119905)with 119889(119905) = 0 are shown in Figure 2 the trajectories ofLyapunov functions 119881(119905) with 119889(119905) = 0 are shown inFigure 3 It can be seen that the obtained performance isdesirable

Mathematical Problems in Engineering 9

5 Conclusion

In this paper we have addressed the problem of sampled-data stabilization for spacecraft rendezvous with a time-dependent Lyapunov approach A discontinuous Lyapunovfunctional has been constructed the value of which does notincrease in each sampling time instant The sufficient condi-tions for the stability analysis and controller design are estab-lished in terms of matrix inequality technique Future workwill be focused onmore complicated case that is the effects ofnetwork-induced phenomenon such as communication delayand packet losses are taken into account simultaneously

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China 61104101 the FundamentalResearch Funds for the Central Universities the China Post-doctoral Science Foundation funded project (2011M500058)the Special Chinese National Postdoctoral Science Founda-tion under Grant 2012T50356 and the Heilongjiang Postdoc-toral Fund (LBH-Z11144)

References

[1] U Ahsun D W Miller and J L Ramirez ldquoControl of electro-magnetic satellite formations in near-Earth orbitsrdquo Journal ofGuidance Control and Dynamics vol 33 no 6 pp 1883ndash18912010

[2] H Schaub ldquoStabilization of satellite motion relative to aCoulomb spacecraft formationrdquo Journal of Guidance Controland Dynamics vol 28 no 6 pp 1231ndash1239 2005

[3] X Yang X Cao and H Gao ldquoSampled-data control forrelative position holding of spacecraft rendezvous with thrustnonlinearityrdquo IEEE Transactions on Industrial Electronics vol59 no 2 pp 1146ndash1153 2011

[4] P Gurfil ldquoNonlinear feedback control of low-thrust orbitaltransfer in a central gravitational fieldrdquo Acta Astronautica vol60 no 8-9 pp 631ndash648 2007

[5] S Kurnaz O Cetin and O Kaynak ldquoFuzzy logic basedapproach to design of flight control and navigation tasks forautonomous unmanned aerial vehiclesrdquo Journal of Intelligentand Robotic Systems vol 54 no 1ndash3 pp 229ndash244 2009

[6] W Clohessy and R Wiltshire ldquoTerminal guidance system forsatellite rendezvousrdquo Journal of the Aerospace Sciences vol 27no 9 pp 653ndash658 1960

[7] Y Luo and G Tang ldquoSpacecraft optimal rendezvous controllerdesign using simulated annealingrdquo Aerospace Science and Tech-nology vol 9 no 8 pp 732ndash737 2005

[8] Y Luo G Tang and Y Lei ldquoOptimal multi-objective lin-earized impulsive rendezvousrdquo Journal of Guidance Controland Dynamics vol 30 no 2 pp 383ndash389 2007

[9] Z Mao and B Jiang ldquoFault identification and fault-tolerantcontrol for a class of networked control systemsrdquo InternationalJournal of Innovative Computing Information and Control vol3 no 5 pp 1121ndash1130 2007

[10] X Su P Shi L Wu and Y Song ldquoA novel control design ondiscrete-time Takagi-Sugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems vol 21 no 4 pp655ndash671 2012

[11] X Su L Wu P Shi and Y Song ldquoHinfin

model reduction of T-Sfuzzy stochastic systemsrdquo IEEE Transactions on Systems Manand Cybernetics B vol 42 no 6 pp 1574ndash1585 2012

[12] S Yin S Ding and H Luo ldquoReal-time implementation of faulttolerant control system with performance optimizationrdquo IEEETransactions on Industrial Electronics 2013

[13] R Yang Z Zhang and P Shi ldquoExponential stability on stochas-tic neural networks with discrete interval and distributeddelaysrdquo IEEE Transactions on Neural Networks vol 21 no 1 pp169ndash175 2010

[14] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[15] R Yang P Shi and G-P Liu ldquoFiltering for discrete-timenetworked nonlinear systems with mixed random delays andpacket dropoutsrdquo IEEE Transactions on Automatic Control vol56 no 11 pp 2655ndash2660 2011

[16] Y Shu L Yu and Z Wenan ldquoEstimator-based control of net-worked systems with packet-dropoutsrdquo International Journal ofInnovative Computing Information andControl vol 6 no 6 pp2737ndash2748 2010

[17] C Jiang Q Zhang and D Zou ldquoDelay-dependent robust filter-ing for networked control systemswith polytopic uncertaintiesrdquoInternational Journal of Innovative Computing Information andControl vol 6 no 11 pp 4857ndash4868 2010

[18] S Shi Z Yuan and Q Zhang ldquoFault-tolerant Hinfin

filter designof a class of switched systems with sensor failuresrdquo InternationalJournal of Innovative Computing Information and Control vol5 no 11 pp 3827ndash3838 2009

[19] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[20] J Lu D W C Ho and J Cao ldquoA unified synchronizationcriterion for impulsive dynamical networksrdquo Automatica vol46 no 7 pp 1215ndash1221 2010

[21] J Lu D W C Ho J Cao and J Kurths ldquoExponential synchro-nization of linearly coupled neural networks with impulsivedisturbancesrdquo IEEE Transactions on Neural Networks vol 22no 2 pp 329ndash336 2011

[22] H Wu and M Bai ldquoStochastic stability analysis and synthesisfor nonlinear fault tolerant control systems based on the T-Sfuzzy modelrdquo International Journal of Innovative ComputingInformation and Control vol 6 no 9 pp 3989ndash4000 2010

[23] S Yin Data-Driven Design of Fault Diagnosis Systems VDIDuesseldorf Germany 2012

[24] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[25] L Wu and W X Zheng ldquoPassivity-based sliding mode controlof uncertain singular time-delay systemsrdquo Automatica vol 45no 9 pp 2120ndash2127 2009

[26] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo IEEETransactions on Automatic Control vol 55 no 5 pp 1213ndash12192010

[27] L Zhang and E-K Boukas ldquoStability and stabilization ofMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo Automatica vol 45 no 2 pp 463ndash468 2009

10 Mathematical Problems in Engineering

[28] L Zhang and E-K Boukas ldquoMode-dependent Hinfin

filteringfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo Automatica vol 45 no 6pp 1462ndash1467 2009

[29] L Zhou and G Lu ldquoQuantized feedback stabilization fornetworked control systems with nonlinear perturbationrdquo Inter-national Journal of Innovative Computing Information andControl vol 6 no 6 pp 2485ndash2496 2010

[30] Z-G Wu J H Park H Su and J Chu ldquoDiscontinuousLyapunov functional approach to synchronization of time-delayneural networks using sampled-datardquoNonlinear Dynamics vol69 no 4 pp 2021ndash2030 2012

[31] H Li H Liu H Gao and P Shi ldquoReliable fuzzy control foractive suspension systems with actuator delay and faultrdquo IEEETransactions on Fuzzy Systems vol 20 no 2 pp 342ndash357 2012

[32] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding modecontrol for nonlinear active suspension systems using T-S fuzzymodelrdquo IEEE Transactions on Industrial Electronics vol 60 no8 pp 3328ndash3338 2012

[33] E Fridman ldquoA refined input delay approach to sampled-datacontrolrdquo Automatica vol 46 no 2 pp 421ndash427 2010

[34] E Fridman A Seuret and J-P Richard ldquoRobust sampled-data stabilization of linear systems an input delay approachrdquoAutomatica vol 40 no 8 pp 1441ndash1446 2004

[35] P Naghshtabrizi J P Hespanha and A R Teel ldquoExponentialstability of impulsive systems with application to uncertainsampled-data systemsrdquo Systems amp Control Letters vol 57 no5 pp 378ndash385 2008

[36] M Liu J You and X Ma ldquoHinfin

filtering for sampled-datastochastic systems with limited capacity channelrdquo Signal Pro-cessing vol 91 no 8 pp 1826ndash1837 2011

[37] M Liu D W C Ho and Y Niu ldquoObserver-based controllerdesign for linear systems with limited communication capacityvia a descriptor augmentation methodrdquo IET Control Theory ampApplications vol 6 no 3 pp 437ndash447 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

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Research Article Sampled-Data Control of Spacecraft Rendezvous with …downloads.hindawi.com/journals/mpe/2013/814271.pdf · 2019-07-31 · Research Article Sampled-Data Control of

6 Mathematical Problems in Engineering

Therefore it is shown from (30) that

(119905) + 2120572119881 (119905) ⩽ 2 120585119879

(119905) 1198751120585 (119905)

+ 2120572120585119879

(119905) 1198751120585 (119905)

minus 119890minus2120572120591(119905)

120591 (119905) 120601119879

120585(119905) 1198752120601120585(119905)

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

minus (120585 (119905119896) minus 120585 (119905))

119879

1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) 120585119879

(119905) 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) (120585(119905119896) minus 120585 (119905))

119879

1198753

120585 (119905)

+ 2120572 (ℎ minus 120591 (119905)) (120585 (119905119896) minus 120585 (119905))

119879

times 1198753(120585 (119905119896) minus 120585 (119905))

(35)

We now insert free-weighting matrices by introducing thefollowing zero terms

2 [120585119879

(119905) 119879119879

1+ 120585119879

(119905) 119879119879

2]

times [ 120585119879

(119905) minus 119860120585 (119905) minus 119861119870120585 (119905119896)] = 0

(36)

2 [120585119879

(119905) 119878119879

1+ 120585119879

(119905) 119878119879

2+ 120585119879

(119905119896) 119878119879

3]

times [minus120585 (119905) + 120585 (119905119896) + 120591 (119905) 120601

120585(119905)] = 0

(37)

where the free-weighting matrices 1198791 1198792 1198781 1198782 and 119878

3are to

be designed Then we have

(119905) + 2120572119881 (119905)

⩽ 2 120585119879

(119905) 1198751120585 (119905) + 2120572120585

119879

(119905) 1198751120585 (119905)

minus 119890minus2120572120591(119905)

120591 (119905) 120601119879

120585(119905) 1198752120601120585(119905)

+ (ℎ minus 120591 (119905)) 120585119879

(119905) 1198752

120585 (119905)

minus (120585(119905119896) minus 120585 (119905))

119879

1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) 120585119879

(119905) 1198753(120585 (119905119896) minus 120585 (119905))

minus (ℎ minus 120591 (119905)) (120585(119905119896) minus 120585 (119905))

119879

1198753

120585 (119905)

+ 2120572 (ℎ minus 120591 (119905)) (120585(119905119896) minus 120585(119905))

119879

times 1198753(120585 (119905119896) minus 120585 (119905))

+ 2 [120585119879

(119905) 119879119879

1+ 120585119879

(119905) 119879119879

2]

times [ 120585119879

(119905) minus 119860120585 (119905) minus 119861119870120585 (119905119896)]

times 2 [120585119879

(119905) 119878119879

1+ 120585119879

(119905) 119878119879

2+ 120585119879

(119905119896) 119878119879

3]

times [minus120585 (119905) + 120585 (119905119896) + 120591 (119905) 120601

120585(119905)]

(38)

Defining the following new augmented variables

120585 (119905) = [120585119879

(119905) 120585119879

(119905) 120585119879

(119905119896) 120601119879

(119905)]119879

120585 (119905) = [120585119879

(119905) 120585119879

(119905) 120585119879

(119905119896)]119879

(39)

it is calculated that

(119905) + 2120572119881 (119905) ⩽ 120585119879

(119905) Π (119905) 120585 (119905) (40)

with

Π (119905) =

[[[

[

Π11(119905) Π

12(119905) Π

13(119905) Π

14(119905)

⋆ Π22(119905) Π

23(119905) 0

⋆ ⋆ Π33(119905) Π

34(119905)

⋆ ⋆ ⋆ Π44(119905)

]]]

]

Π11(119905) = 2120572119875

1minus 1198753+ 2120572 (ℎ minus 120591 (119905)) 119875

3

minus 119879119879

1119860 minus 119860

119879

1198791minus 1198781minus 119878119879

1

Π12(119905) = 119875

1+ 119879119879

1minus (ℎ minus 120591 (119905)) 119875

3

Π13(119905) = 119875

3minus 2120572 (ℎ minus 120591 (119905)) 119875

3minus 119879119879

1119861119870 + 119878

119879

1

Π14(119905) = 120591 (119905) 119878

119879

1

Π22(119905) = (ℎ minus 120591 (119905)) 119875

2+ 1198792+ 119879119879

2

Π23(119905) = minus (ℎ minus 120591 (119905)) 119875

3minus 119879119879

2119861119870 + 119878

2+ 119878119879

2

Π33(119905) = minus119875

3+ 2120572 (ℎ minus 120591 (119905)) 119875

3+ 1198783+ 119878119879

3

Π34(119905) = 120591 (119905) 119878

119879

3

Π44(119905) = minus119890

minus2120572120591(119905)

120591 (119905) 1198752

(41)

We now consider the matrices Φ and Φ defined in (21) Infact it can be verified thatΦ is obtained fromΠ(119905)|

120591(119905)=0with

deleting the last row and last column and Φ = Π(119905)|120591(119905)=ℎ

Similar to the proof of Theorem 1 of [37] it can be provedthat the system (19) with 119889(119905) = 0 is exponentially stable

Next we deal with the input constraint along each axisWe define

119877119909= [

[

1

0

0

]

]

[1 0 0] = [

[

1 0 0

0 0 0

0 0 0

]

]

119877119910= [

[

0

1

0

]

]

[0 1 0] = [

[

0 0 0

0 1 0

0 0 0

]

]

119877119911= [

[

0

0

1

]

]

[0 0 1] = [

[

0 0 0

0 0 0

0 0 1

]

]

(42)

Then the thrust along each axis can be described as1003816100381610038161003816119906119894 (119905119896)

1003816100381610038161003816 =1003817100381710038171003817119877119894119906 (119905119896)

1003817100381710038171003817 ⩽ 119906119894max 119894 = 119909 119910 119911 (43)

Mathematical Problems in Engineering 7

Therefore the thrust constraint (8) can be rewritten as1003817100381710038171003817119877119894119906 (119905119896)

1003817100381710038171003817 ⩽ 119906119894max 119894 = 119909 119910 119911 (44)

which is equivalent to

(119877119894119906 (119905119896))119879

119877119894119906 (119905119896) ⩽ 1199062

119894max (45)

Furthermore formula (45) is equal to

120585119879

(119905119896)119870119877119879

119894119877119894119870120585 (119905119896) ⩽ 1199062

119894max (46)

Based on the previous analysis it is known that (119905) lt 0 isguaranteed if the matrix conditions (21) hold Subsequently119881(119905) lt 119881(0) holds for any 119905 gt 0 It is therefore reasonable toassume that there exists a scalar 120588 gt 0 such that

119881 (0) ⩽ 120588 (47)

Notice that 119881(119905) = 1198811(119905) + 119881

2(119905) + 119881

3(119905) le 119881(0) ⩽ 120588 and

1198812(119905) ge 0 119881

3(119905) ge 0 Therefore it is true that

119881 (119905) le 1198811(119905) lt 119881 (0) ⩽ 120588 (48)

which implies that

1198811(119905) lt 120588 (49)

Therefore 120585119879(119905)1198751120585(119905) lt 120588 holds for any 119905 gt 0 In particular

let 119905 = 119905119896 we have

120585119879

(119905119896) 1198751120585 (119905119896) lt 120588 (50)

Then (46) can be guaranteed if (22) holdsWe prove this factIn fact if (22) holds we have

120585119879

(119905119896) 120588119870119879

119877119879

119894119877119894119870120585 (119905119896) lt 120585119879

(119905119896) 1199062

119894max1198751120585 (119905119896) (51)

By (51) we have

120585119879

(119905119896) 120588119870119879

119877119879

119894119877119894119870120585 (119905119896)

lt 120585119879

(119905119896) 1199062

119894max1198751120585 (119905119896) lt 1199062

119894max120588

(52)

which implies that

120585119879

(119905119896)119870119879

119877119879

119894119877119894119870120585 (119905119896) lt 1199062

119894max (53)

As a result the thrust limitation (8) can be guaranteed by (22)This completes the proof

32 Disturbance Attenuation In this subsection we willinvestigate analysis of the internal stability and 119867

infinperfor-

mance index for the system (19) Defining a new augmentedvector 120595(119905) ≜ [120585

119879

(119905)119889119879

(119905)]119879 we reconsider the system (19)

with 119889(119905) = 0 We introduce the performance index

119869 (119905) = int

119905

0

[119911119879

(119904) 119911 (119904) minus 120574119889119879

(119904) 119889 (119904)] 119889119904 (54)

Then (20) can be guaranteed by 119869(119905) lt 0

0 5 10 15 20 25 30

0

5

10

minus20

minus15

minus10

minus5

x1(t)

x2(t)

x3(t)

x4(t)

x5(t)

x6(t)

Figure 1 Trajectories of 119909(119905)

Theorem 3 Consider the closed-loop system (19) given aperformance index parameter 120574 gt 0 let the decay rate 120572 = 0 ifthere exist positive definite matrices 119875

1 1198752 and 119875

3isin R9times9 and

matrices 1198791 1198792 1198781 1198782 and 119878

3isin R9times9 such that the matrix

conditions (21) (22) and the following matrices conditionshold

[Φ|120572=0

Γ1

⋆ diag minus1205742 minus1198689] lt 0

[Φ|120572=0

Γ2

⋆ diag minus1205742 minus1198689] lt 0

(55)

Γ1=

[[[[[

[

minus119879119879

1119861119879

1119862119879

2

minus119879119879

2119861119879

10

0 0

]]]]]

]

Γ2=

[[[[[[[

[

minus119879119879

1119861119879

1119862119879

2

minus119879119879

2119861119879

10

0 0

0 0

]]]]]]]

]

(56)

Then the system (19) is internally stable and the 119867infin

perfor-mance index (20) is achieved

Proof It is well known that (119905) lt 0 implies thatlim119905rarrinfin

119881(119905) = 0 We consider a new defined function

119881119869(119905) = (119905) + 119911

119879

(119905) 119911 (119905) minus 1205742

119889119879

(119905) 119889 (119905) (57)

If 119881119869(119905) lt 0 it is shown that

119911119879

(119905) 119911 (119905) minus 1205742

119889119879

(119905) 119889 (119905) lt minus (119905) (58)

8 Mathematical Problems in Engineering

0 5 10 15 20 25 30

0

200

400

600

800

minus600

minus400

minus200

Time (s)

u1(t)

u2(t)

u3(t)

Figure 2 Trajectories of 119906(119905)

0 5 10 15 20 25 300

100

200

300

400

500

600

700

Time (s)

V(t)

Figure 3 Lyapunov function 119881(119905)

and it further implies that 119869(infin) = intinfin

0

(119911119879

(119904)119911(119904) minus 1205742

119889119879

(119904)119889(119904))119889119904 lt minus intinfin

0

(119904)119889119904 = 119881(infin) minus 119881(0) = minus119881(0) lt 0Thatmeans in the case120572 = 0 if (119905) lt 0 and119881

119869(119905) lt 0hold it

can be guaranteed that the system (19) is internally stable andthe119867infinperformance index (20) can be achieved successfully

Notice that (57) can be rewritten as

(119905) + 120585119879

(119905) 119862119879

21198622120585 (119905) minus 120574

2

119889119879

(119905) 119889 (119905) lt 0 (59)Recall the zero terms (37) and notice that the term (37) with119889(119905) = 0 becomes

2 [120585119879

(119905) 119879119879

1+ 120585119879

(119905) 119879119879

2]

times [ 120585119879

(119905) minus 119860120585 (119905) minus 119861119870120585 (119905119896) minus 1198611119889 (119905)] = 0

2 [120585119879

(119905) 119878119879

1+ 120585119879

(119905) 119878119879

2+ 120585119879

(119905119896) 119878119879

3]

times [minus120585 (119905) + 120585 (119905119896) + 120591 (119905) 120601

120585(119905)] = 0

(60)

By similar derivation of Theorem 2 it can be proved thatsystem (19) is internally stable if the conditions (55) hold Onthe other hand it has been proved inTheorem 2 that (119905) lt 0is guaranteed by (21)-(22) This completes the proof

4 Simulation

In this section simulation results will be shown to illustratethe usefulness and advantage of the proposed sampled-datacontrol approach We consider the numerical example in [3]suppose that themass of the chaser is 200 kg that is 119898 = 200and the target is moving along a geosynchronous orbit withradius 119903 = 42241 km with an orbital period of 24 hThe anglevelocity 119908 = 1117 times 103 rads The maximum thrust alongeach axis is 1000N The disturbance vector is described as119889(119905) = [2 sin(2119905) 2 sin(2119905) 2 sin(2119905)]119879 The upper boundof the sampling period is given as ℎ = 05 and the decayrate is selected as 120572 = 002 The disturbance attenuation levelis given as 120574 = 04 In the coordinate based on the targetframe assume that the initial state is [10 minus20 10 0 0 0] andthe desired holding position is (0 minus1 0) Design the statefeedback gain 119870 as

119870 = [

[

minus343807 01490 0 minus973889 00051 0 minus87062 00254 0

minus01490 minus343801 0 minus00051 minus973880 0 minus00254 minus87062 0

0 0 minus578745 0 0 minus1100009 0 0 minus165843

]

]

(61)

such that (119860 + 119861119870) is Hurwitz Suppose that 1199051= 0 which

means that the first sampling time instant is the initial timeThe designed parameter 120588 is chosen as 120588 = 950 Solvingthe conditions (21)-(22) and (55) it is found that there existsfeasible solution Then it can be derived that 119881(0) = 119881

1(0) =

119909(0)119879

1198751119909(0) = 8359423 It is obvious the condition (47)

holds which means that our obtained solution is correct andfeasible

The simulation results are provided in Figures 1ndash3 whichshow the effectiveness of the proposed methods The tra-jectories of state variable 119909(119905) with 119889(119905) = 0 are shown inFigure 1 the trajectories of sampled-data control input 119906(119905)with 119889(119905) = 0 are shown in Figure 2 the trajectories ofLyapunov functions 119881(119905) with 119889(119905) = 0 are shown inFigure 3 It can be seen that the obtained performance isdesirable

Mathematical Problems in Engineering 9

5 Conclusion

In this paper we have addressed the problem of sampled-data stabilization for spacecraft rendezvous with a time-dependent Lyapunov approach A discontinuous Lyapunovfunctional has been constructed the value of which does notincrease in each sampling time instant The sufficient condi-tions for the stability analysis and controller design are estab-lished in terms of matrix inequality technique Future workwill be focused onmore complicated case that is the effects ofnetwork-induced phenomenon such as communication delayand packet losses are taken into account simultaneously

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China 61104101 the FundamentalResearch Funds for the Central Universities the China Post-doctoral Science Foundation funded project (2011M500058)the Special Chinese National Postdoctoral Science Founda-tion under Grant 2012T50356 and the Heilongjiang Postdoc-toral Fund (LBH-Z11144)

References

[1] U Ahsun D W Miller and J L Ramirez ldquoControl of electro-magnetic satellite formations in near-Earth orbitsrdquo Journal ofGuidance Control and Dynamics vol 33 no 6 pp 1883ndash18912010

[2] H Schaub ldquoStabilization of satellite motion relative to aCoulomb spacecraft formationrdquo Journal of Guidance Controland Dynamics vol 28 no 6 pp 1231ndash1239 2005

[3] X Yang X Cao and H Gao ldquoSampled-data control forrelative position holding of spacecraft rendezvous with thrustnonlinearityrdquo IEEE Transactions on Industrial Electronics vol59 no 2 pp 1146ndash1153 2011

[4] P Gurfil ldquoNonlinear feedback control of low-thrust orbitaltransfer in a central gravitational fieldrdquo Acta Astronautica vol60 no 8-9 pp 631ndash648 2007

[5] S Kurnaz O Cetin and O Kaynak ldquoFuzzy logic basedapproach to design of flight control and navigation tasks forautonomous unmanned aerial vehiclesrdquo Journal of Intelligentand Robotic Systems vol 54 no 1ndash3 pp 229ndash244 2009

[6] W Clohessy and R Wiltshire ldquoTerminal guidance system forsatellite rendezvousrdquo Journal of the Aerospace Sciences vol 27no 9 pp 653ndash658 1960

[7] Y Luo and G Tang ldquoSpacecraft optimal rendezvous controllerdesign using simulated annealingrdquo Aerospace Science and Tech-nology vol 9 no 8 pp 732ndash737 2005

[8] Y Luo G Tang and Y Lei ldquoOptimal multi-objective lin-earized impulsive rendezvousrdquo Journal of Guidance Controland Dynamics vol 30 no 2 pp 383ndash389 2007

[9] Z Mao and B Jiang ldquoFault identification and fault-tolerantcontrol for a class of networked control systemsrdquo InternationalJournal of Innovative Computing Information and Control vol3 no 5 pp 1121ndash1130 2007

[10] X Su P Shi L Wu and Y Song ldquoA novel control design ondiscrete-time Takagi-Sugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems vol 21 no 4 pp655ndash671 2012

[11] X Su L Wu P Shi and Y Song ldquoHinfin

model reduction of T-Sfuzzy stochastic systemsrdquo IEEE Transactions on Systems Manand Cybernetics B vol 42 no 6 pp 1574ndash1585 2012

[12] S Yin S Ding and H Luo ldquoReal-time implementation of faulttolerant control system with performance optimizationrdquo IEEETransactions on Industrial Electronics 2013

[13] R Yang Z Zhang and P Shi ldquoExponential stability on stochas-tic neural networks with discrete interval and distributeddelaysrdquo IEEE Transactions on Neural Networks vol 21 no 1 pp169ndash175 2010

[14] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[15] R Yang P Shi and G-P Liu ldquoFiltering for discrete-timenetworked nonlinear systems with mixed random delays andpacket dropoutsrdquo IEEE Transactions on Automatic Control vol56 no 11 pp 2655ndash2660 2011

[16] Y Shu L Yu and Z Wenan ldquoEstimator-based control of net-worked systems with packet-dropoutsrdquo International Journal ofInnovative Computing Information andControl vol 6 no 6 pp2737ndash2748 2010

[17] C Jiang Q Zhang and D Zou ldquoDelay-dependent robust filter-ing for networked control systemswith polytopic uncertaintiesrdquoInternational Journal of Innovative Computing Information andControl vol 6 no 11 pp 4857ndash4868 2010

[18] S Shi Z Yuan and Q Zhang ldquoFault-tolerant Hinfin

filter designof a class of switched systems with sensor failuresrdquo InternationalJournal of Innovative Computing Information and Control vol5 no 11 pp 3827ndash3838 2009

[19] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[20] J Lu D W C Ho and J Cao ldquoA unified synchronizationcriterion for impulsive dynamical networksrdquo Automatica vol46 no 7 pp 1215ndash1221 2010

[21] J Lu D W C Ho J Cao and J Kurths ldquoExponential synchro-nization of linearly coupled neural networks with impulsivedisturbancesrdquo IEEE Transactions on Neural Networks vol 22no 2 pp 329ndash336 2011

[22] H Wu and M Bai ldquoStochastic stability analysis and synthesisfor nonlinear fault tolerant control systems based on the T-Sfuzzy modelrdquo International Journal of Innovative ComputingInformation and Control vol 6 no 9 pp 3989ndash4000 2010

[23] S Yin Data-Driven Design of Fault Diagnosis Systems VDIDuesseldorf Germany 2012

[24] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[25] L Wu and W X Zheng ldquoPassivity-based sliding mode controlof uncertain singular time-delay systemsrdquo Automatica vol 45no 9 pp 2120ndash2127 2009

[26] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo IEEETransactions on Automatic Control vol 55 no 5 pp 1213ndash12192010

[27] L Zhang and E-K Boukas ldquoStability and stabilization ofMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo Automatica vol 45 no 2 pp 463ndash468 2009

10 Mathematical Problems in Engineering

[28] L Zhang and E-K Boukas ldquoMode-dependent Hinfin

filteringfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo Automatica vol 45 no 6pp 1462ndash1467 2009

[29] L Zhou and G Lu ldquoQuantized feedback stabilization fornetworked control systems with nonlinear perturbationrdquo Inter-national Journal of Innovative Computing Information andControl vol 6 no 6 pp 2485ndash2496 2010

[30] Z-G Wu J H Park H Su and J Chu ldquoDiscontinuousLyapunov functional approach to synchronization of time-delayneural networks using sampled-datardquoNonlinear Dynamics vol69 no 4 pp 2021ndash2030 2012

[31] H Li H Liu H Gao and P Shi ldquoReliable fuzzy control foractive suspension systems with actuator delay and faultrdquo IEEETransactions on Fuzzy Systems vol 20 no 2 pp 342ndash357 2012

[32] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding modecontrol for nonlinear active suspension systems using T-S fuzzymodelrdquo IEEE Transactions on Industrial Electronics vol 60 no8 pp 3328ndash3338 2012

[33] E Fridman ldquoA refined input delay approach to sampled-datacontrolrdquo Automatica vol 46 no 2 pp 421ndash427 2010

[34] E Fridman A Seuret and J-P Richard ldquoRobust sampled-data stabilization of linear systems an input delay approachrdquoAutomatica vol 40 no 8 pp 1441ndash1446 2004

[35] P Naghshtabrizi J P Hespanha and A R Teel ldquoExponentialstability of impulsive systems with application to uncertainsampled-data systemsrdquo Systems amp Control Letters vol 57 no5 pp 378ndash385 2008

[36] M Liu J You and X Ma ldquoHinfin

filtering for sampled-datastochastic systems with limited capacity channelrdquo Signal Pro-cessing vol 91 no 8 pp 1826ndash1837 2011

[37] M Liu D W C Ho and Y Niu ldquoObserver-based controllerdesign for linear systems with limited communication capacityvia a descriptor augmentation methodrdquo IET Control Theory ampApplications vol 6 no 3 pp 437ndash447 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

Page 7: Research Article Sampled-Data Control of Spacecraft Rendezvous with …downloads.hindawi.com/journals/mpe/2013/814271.pdf · 2019-07-31 · Research Article Sampled-Data Control of

Mathematical Problems in Engineering 7

Therefore the thrust constraint (8) can be rewritten as1003817100381710038171003817119877119894119906 (119905119896)

1003817100381710038171003817 ⩽ 119906119894max 119894 = 119909 119910 119911 (44)

which is equivalent to

(119877119894119906 (119905119896))119879

119877119894119906 (119905119896) ⩽ 1199062

119894max (45)

Furthermore formula (45) is equal to

120585119879

(119905119896)119870119877119879

119894119877119894119870120585 (119905119896) ⩽ 1199062

119894max (46)

Based on the previous analysis it is known that (119905) lt 0 isguaranteed if the matrix conditions (21) hold Subsequently119881(119905) lt 119881(0) holds for any 119905 gt 0 It is therefore reasonable toassume that there exists a scalar 120588 gt 0 such that

119881 (0) ⩽ 120588 (47)

Notice that 119881(119905) = 1198811(119905) + 119881

2(119905) + 119881

3(119905) le 119881(0) ⩽ 120588 and

1198812(119905) ge 0 119881

3(119905) ge 0 Therefore it is true that

119881 (119905) le 1198811(119905) lt 119881 (0) ⩽ 120588 (48)

which implies that

1198811(119905) lt 120588 (49)

Therefore 120585119879(119905)1198751120585(119905) lt 120588 holds for any 119905 gt 0 In particular

let 119905 = 119905119896 we have

120585119879

(119905119896) 1198751120585 (119905119896) lt 120588 (50)

Then (46) can be guaranteed if (22) holdsWe prove this factIn fact if (22) holds we have

120585119879

(119905119896) 120588119870119879

119877119879

119894119877119894119870120585 (119905119896) lt 120585119879

(119905119896) 1199062

119894max1198751120585 (119905119896) (51)

By (51) we have

120585119879

(119905119896) 120588119870119879

119877119879

119894119877119894119870120585 (119905119896)

lt 120585119879

(119905119896) 1199062

119894max1198751120585 (119905119896) lt 1199062

119894max120588

(52)

which implies that

120585119879

(119905119896)119870119879

119877119879

119894119877119894119870120585 (119905119896) lt 1199062

119894max (53)

As a result the thrust limitation (8) can be guaranteed by (22)This completes the proof

32 Disturbance Attenuation In this subsection we willinvestigate analysis of the internal stability and 119867

infinperfor-

mance index for the system (19) Defining a new augmentedvector 120595(119905) ≜ [120585

119879

(119905)119889119879

(119905)]119879 we reconsider the system (19)

with 119889(119905) = 0 We introduce the performance index

119869 (119905) = int

119905

0

[119911119879

(119904) 119911 (119904) minus 120574119889119879

(119904) 119889 (119904)] 119889119904 (54)

Then (20) can be guaranteed by 119869(119905) lt 0

0 5 10 15 20 25 30

0

5

10

minus20

minus15

minus10

minus5

x1(t)

x2(t)

x3(t)

x4(t)

x5(t)

x6(t)

Figure 1 Trajectories of 119909(119905)

Theorem 3 Consider the closed-loop system (19) given aperformance index parameter 120574 gt 0 let the decay rate 120572 = 0 ifthere exist positive definite matrices 119875

1 1198752 and 119875

3isin R9times9 and

matrices 1198791 1198792 1198781 1198782 and 119878

3isin R9times9 such that the matrix

conditions (21) (22) and the following matrices conditionshold

[Φ|120572=0

Γ1

⋆ diag minus1205742 minus1198689] lt 0

[Φ|120572=0

Γ2

⋆ diag minus1205742 minus1198689] lt 0

(55)

Γ1=

[[[[[

[

minus119879119879

1119861119879

1119862119879

2

minus119879119879

2119861119879

10

0 0

]]]]]

]

Γ2=

[[[[[[[

[

minus119879119879

1119861119879

1119862119879

2

minus119879119879

2119861119879

10

0 0

0 0

]]]]]]]

]

(56)

Then the system (19) is internally stable and the 119867infin

perfor-mance index (20) is achieved

Proof It is well known that (119905) lt 0 implies thatlim119905rarrinfin

119881(119905) = 0 We consider a new defined function

119881119869(119905) = (119905) + 119911

119879

(119905) 119911 (119905) minus 1205742

119889119879

(119905) 119889 (119905) (57)

If 119881119869(119905) lt 0 it is shown that

119911119879

(119905) 119911 (119905) minus 1205742

119889119879

(119905) 119889 (119905) lt minus (119905) (58)

8 Mathematical Problems in Engineering

0 5 10 15 20 25 30

0

200

400

600

800

minus600

minus400

minus200

Time (s)

u1(t)

u2(t)

u3(t)

Figure 2 Trajectories of 119906(119905)

0 5 10 15 20 25 300

100

200

300

400

500

600

700

Time (s)

V(t)

Figure 3 Lyapunov function 119881(119905)

and it further implies that 119869(infin) = intinfin

0

(119911119879

(119904)119911(119904) minus 1205742

119889119879

(119904)119889(119904))119889119904 lt minus intinfin

0

(119904)119889119904 = 119881(infin) minus 119881(0) = minus119881(0) lt 0Thatmeans in the case120572 = 0 if (119905) lt 0 and119881

119869(119905) lt 0hold it

can be guaranteed that the system (19) is internally stable andthe119867infinperformance index (20) can be achieved successfully

Notice that (57) can be rewritten as

(119905) + 120585119879

(119905) 119862119879

21198622120585 (119905) minus 120574

2

119889119879

(119905) 119889 (119905) lt 0 (59)Recall the zero terms (37) and notice that the term (37) with119889(119905) = 0 becomes

2 [120585119879

(119905) 119879119879

1+ 120585119879

(119905) 119879119879

2]

times [ 120585119879

(119905) minus 119860120585 (119905) minus 119861119870120585 (119905119896) minus 1198611119889 (119905)] = 0

2 [120585119879

(119905) 119878119879

1+ 120585119879

(119905) 119878119879

2+ 120585119879

(119905119896) 119878119879

3]

times [minus120585 (119905) + 120585 (119905119896) + 120591 (119905) 120601

120585(119905)] = 0

(60)

By similar derivation of Theorem 2 it can be proved thatsystem (19) is internally stable if the conditions (55) hold Onthe other hand it has been proved inTheorem 2 that (119905) lt 0is guaranteed by (21)-(22) This completes the proof

4 Simulation

In this section simulation results will be shown to illustratethe usefulness and advantage of the proposed sampled-datacontrol approach We consider the numerical example in [3]suppose that themass of the chaser is 200 kg that is 119898 = 200and the target is moving along a geosynchronous orbit withradius 119903 = 42241 km with an orbital period of 24 hThe anglevelocity 119908 = 1117 times 103 rads The maximum thrust alongeach axis is 1000N The disturbance vector is described as119889(119905) = [2 sin(2119905) 2 sin(2119905) 2 sin(2119905)]119879 The upper boundof the sampling period is given as ℎ = 05 and the decayrate is selected as 120572 = 002 The disturbance attenuation levelis given as 120574 = 04 In the coordinate based on the targetframe assume that the initial state is [10 minus20 10 0 0 0] andthe desired holding position is (0 minus1 0) Design the statefeedback gain 119870 as

119870 = [

[

minus343807 01490 0 minus973889 00051 0 minus87062 00254 0

minus01490 minus343801 0 minus00051 minus973880 0 minus00254 minus87062 0

0 0 minus578745 0 0 minus1100009 0 0 minus165843

]

]

(61)

such that (119860 + 119861119870) is Hurwitz Suppose that 1199051= 0 which

means that the first sampling time instant is the initial timeThe designed parameter 120588 is chosen as 120588 = 950 Solvingthe conditions (21)-(22) and (55) it is found that there existsfeasible solution Then it can be derived that 119881(0) = 119881

1(0) =

119909(0)119879

1198751119909(0) = 8359423 It is obvious the condition (47)

holds which means that our obtained solution is correct andfeasible

The simulation results are provided in Figures 1ndash3 whichshow the effectiveness of the proposed methods The tra-jectories of state variable 119909(119905) with 119889(119905) = 0 are shown inFigure 1 the trajectories of sampled-data control input 119906(119905)with 119889(119905) = 0 are shown in Figure 2 the trajectories ofLyapunov functions 119881(119905) with 119889(119905) = 0 are shown inFigure 3 It can be seen that the obtained performance isdesirable

Mathematical Problems in Engineering 9

5 Conclusion

In this paper we have addressed the problem of sampled-data stabilization for spacecraft rendezvous with a time-dependent Lyapunov approach A discontinuous Lyapunovfunctional has been constructed the value of which does notincrease in each sampling time instant The sufficient condi-tions for the stability analysis and controller design are estab-lished in terms of matrix inequality technique Future workwill be focused onmore complicated case that is the effects ofnetwork-induced phenomenon such as communication delayand packet losses are taken into account simultaneously

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China 61104101 the FundamentalResearch Funds for the Central Universities the China Post-doctoral Science Foundation funded project (2011M500058)the Special Chinese National Postdoctoral Science Founda-tion under Grant 2012T50356 and the Heilongjiang Postdoc-toral Fund (LBH-Z11144)

References

[1] U Ahsun D W Miller and J L Ramirez ldquoControl of electro-magnetic satellite formations in near-Earth orbitsrdquo Journal ofGuidance Control and Dynamics vol 33 no 6 pp 1883ndash18912010

[2] H Schaub ldquoStabilization of satellite motion relative to aCoulomb spacecraft formationrdquo Journal of Guidance Controland Dynamics vol 28 no 6 pp 1231ndash1239 2005

[3] X Yang X Cao and H Gao ldquoSampled-data control forrelative position holding of spacecraft rendezvous with thrustnonlinearityrdquo IEEE Transactions on Industrial Electronics vol59 no 2 pp 1146ndash1153 2011

[4] P Gurfil ldquoNonlinear feedback control of low-thrust orbitaltransfer in a central gravitational fieldrdquo Acta Astronautica vol60 no 8-9 pp 631ndash648 2007

[5] S Kurnaz O Cetin and O Kaynak ldquoFuzzy logic basedapproach to design of flight control and navigation tasks forautonomous unmanned aerial vehiclesrdquo Journal of Intelligentand Robotic Systems vol 54 no 1ndash3 pp 229ndash244 2009

[6] W Clohessy and R Wiltshire ldquoTerminal guidance system forsatellite rendezvousrdquo Journal of the Aerospace Sciences vol 27no 9 pp 653ndash658 1960

[7] Y Luo and G Tang ldquoSpacecraft optimal rendezvous controllerdesign using simulated annealingrdquo Aerospace Science and Tech-nology vol 9 no 8 pp 732ndash737 2005

[8] Y Luo G Tang and Y Lei ldquoOptimal multi-objective lin-earized impulsive rendezvousrdquo Journal of Guidance Controland Dynamics vol 30 no 2 pp 383ndash389 2007

[9] Z Mao and B Jiang ldquoFault identification and fault-tolerantcontrol for a class of networked control systemsrdquo InternationalJournal of Innovative Computing Information and Control vol3 no 5 pp 1121ndash1130 2007

[10] X Su P Shi L Wu and Y Song ldquoA novel control design ondiscrete-time Takagi-Sugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems vol 21 no 4 pp655ndash671 2012

[11] X Su L Wu P Shi and Y Song ldquoHinfin

model reduction of T-Sfuzzy stochastic systemsrdquo IEEE Transactions on Systems Manand Cybernetics B vol 42 no 6 pp 1574ndash1585 2012

[12] S Yin S Ding and H Luo ldquoReal-time implementation of faulttolerant control system with performance optimizationrdquo IEEETransactions on Industrial Electronics 2013

[13] R Yang Z Zhang and P Shi ldquoExponential stability on stochas-tic neural networks with discrete interval and distributeddelaysrdquo IEEE Transactions on Neural Networks vol 21 no 1 pp169ndash175 2010

[14] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[15] R Yang P Shi and G-P Liu ldquoFiltering for discrete-timenetworked nonlinear systems with mixed random delays andpacket dropoutsrdquo IEEE Transactions on Automatic Control vol56 no 11 pp 2655ndash2660 2011

[16] Y Shu L Yu and Z Wenan ldquoEstimator-based control of net-worked systems with packet-dropoutsrdquo International Journal ofInnovative Computing Information andControl vol 6 no 6 pp2737ndash2748 2010

[17] C Jiang Q Zhang and D Zou ldquoDelay-dependent robust filter-ing for networked control systemswith polytopic uncertaintiesrdquoInternational Journal of Innovative Computing Information andControl vol 6 no 11 pp 4857ndash4868 2010

[18] S Shi Z Yuan and Q Zhang ldquoFault-tolerant Hinfin

filter designof a class of switched systems with sensor failuresrdquo InternationalJournal of Innovative Computing Information and Control vol5 no 11 pp 3827ndash3838 2009

[19] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[20] J Lu D W C Ho and J Cao ldquoA unified synchronizationcriterion for impulsive dynamical networksrdquo Automatica vol46 no 7 pp 1215ndash1221 2010

[21] J Lu D W C Ho J Cao and J Kurths ldquoExponential synchro-nization of linearly coupled neural networks with impulsivedisturbancesrdquo IEEE Transactions on Neural Networks vol 22no 2 pp 329ndash336 2011

[22] H Wu and M Bai ldquoStochastic stability analysis and synthesisfor nonlinear fault tolerant control systems based on the T-Sfuzzy modelrdquo International Journal of Innovative ComputingInformation and Control vol 6 no 9 pp 3989ndash4000 2010

[23] S Yin Data-Driven Design of Fault Diagnosis Systems VDIDuesseldorf Germany 2012

[24] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[25] L Wu and W X Zheng ldquoPassivity-based sliding mode controlof uncertain singular time-delay systemsrdquo Automatica vol 45no 9 pp 2120ndash2127 2009

[26] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo IEEETransactions on Automatic Control vol 55 no 5 pp 1213ndash12192010

[27] L Zhang and E-K Boukas ldquoStability and stabilization ofMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo Automatica vol 45 no 2 pp 463ndash468 2009

10 Mathematical Problems in Engineering

[28] L Zhang and E-K Boukas ldquoMode-dependent Hinfin

filteringfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo Automatica vol 45 no 6pp 1462ndash1467 2009

[29] L Zhou and G Lu ldquoQuantized feedback stabilization fornetworked control systems with nonlinear perturbationrdquo Inter-national Journal of Innovative Computing Information andControl vol 6 no 6 pp 2485ndash2496 2010

[30] Z-G Wu J H Park H Su and J Chu ldquoDiscontinuousLyapunov functional approach to synchronization of time-delayneural networks using sampled-datardquoNonlinear Dynamics vol69 no 4 pp 2021ndash2030 2012

[31] H Li H Liu H Gao and P Shi ldquoReliable fuzzy control foractive suspension systems with actuator delay and faultrdquo IEEETransactions on Fuzzy Systems vol 20 no 2 pp 342ndash357 2012

[32] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding modecontrol for nonlinear active suspension systems using T-S fuzzymodelrdquo IEEE Transactions on Industrial Electronics vol 60 no8 pp 3328ndash3338 2012

[33] E Fridman ldquoA refined input delay approach to sampled-datacontrolrdquo Automatica vol 46 no 2 pp 421ndash427 2010

[34] E Fridman A Seuret and J-P Richard ldquoRobust sampled-data stabilization of linear systems an input delay approachrdquoAutomatica vol 40 no 8 pp 1441ndash1446 2004

[35] P Naghshtabrizi J P Hespanha and A R Teel ldquoExponentialstability of impulsive systems with application to uncertainsampled-data systemsrdquo Systems amp Control Letters vol 57 no5 pp 378ndash385 2008

[36] M Liu J You and X Ma ldquoHinfin

filtering for sampled-datastochastic systems with limited capacity channelrdquo Signal Pro-cessing vol 91 no 8 pp 1826ndash1837 2011

[37] M Liu D W C Ho and Y Niu ldquoObserver-based controllerdesign for linear systems with limited communication capacityvia a descriptor augmentation methodrdquo IET Control Theory ampApplications vol 6 no 3 pp 437ndash447 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

Page 8: Research Article Sampled-Data Control of Spacecraft Rendezvous with …downloads.hindawi.com/journals/mpe/2013/814271.pdf · 2019-07-31 · Research Article Sampled-Data Control of

8 Mathematical Problems in Engineering

0 5 10 15 20 25 30

0

200

400

600

800

minus600

minus400

minus200

Time (s)

u1(t)

u2(t)

u3(t)

Figure 2 Trajectories of 119906(119905)

0 5 10 15 20 25 300

100

200

300

400

500

600

700

Time (s)

V(t)

Figure 3 Lyapunov function 119881(119905)

and it further implies that 119869(infin) = intinfin

0

(119911119879

(119904)119911(119904) minus 1205742

119889119879

(119904)119889(119904))119889119904 lt minus intinfin

0

(119904)119889119904 = 119881(infin) minus 119881(0) = minus119881(0) lt 0Thatmeans in the case120572 = 0 if (119905) lt 0 and119881

119869(119905) lt 0hold it

can be guaranteed that the system (19) is internally stable andthe119867infinperformance index (20) can be achieved successfully

Notice that (57) can be rewritten as

(119905) + 120585119879

(119905) 119862119879

21198622120585 (119905) minus 120574

2

119889119879

(119905) 119889 (119905) lt 0 (59)Recall the zero terms (37) and notice that the term (37) with119889(119905) = 0 becomes

2 [120585119879

(119905) 119879119879

1+ 120585119879

(119905) 119879119879

2]

times [ 120585119879

(119905) minus 119860120585 (119905) minus 119861119870120585 (119905119896) minus 1198611119889 (119905)] = 0

2 [120585119879

(119905) 119878119879

1+ 120585119879

(119905) 119878119879

2+ 120585119879

(119905119896) 119878119879

3]

times [minus120585 (119905) + 120585 (119905119896) + 120591 (119905) 120601

120585(119905)] = 0

(60)

By similar derivation of Theorem 2 it can be proved thatsystem (19) is internally stable if the conditions (55) hold Onthe other hand it has been proved inTheorem 2 that (119905) lt 0is guaranteed by (21)-(22) This completes the proof

4 Simulation

In this section simulation results will be shown to illustratethe usefulness and advantage of the proposed sampled-datacontrol approach We consider the numerical example in [3]suppose that themass of the chaser is 200 kg that is 119898 = 200and the target is moving along a geosynchronous orbit withradius 119903 = 42241 km with an orbital period of 24 hThe anglevelocity 119908 = 1117 times 103 rads The maximum thrust alongeach axis is 1000N The disturbance vector is described as119889(119905) = [2 sin(2119905) 2 sin(2119905) 2 sin(2119905)]119879 The upper boundof the sampling period is given as ℎ = 05 and the decayrate is selected as 120572 = 002 The disturbance attenuation levelis given as 120574 = 04 In the coordinate based on the targetframe assume that the initial state is [10 minus20 10 0 0 0] andthe desired holding position is (0 minus1 0) Design the statefeedback gain 119870 as

119870 = [

[

minus343807 01490 0 minus973889 00051 0 minus87062 00254 0

minus01490 minus343801 0 minus00051 minus973880 0 minus00254 minus87062 0

0 0 minus578745 0 0 minus1100009 0 0 minus165843

]

]

(61)

such that (119860 + 119861119870) is Hurwitz Suppose that 1199051= 0 which

means that the first sampling time instant is the initial timeThe designed parameter 120588 is chosen as 120588 = 950 Solvingthe conditions (21)-(22) and (55) it is found that there existsfeasible solution Then it can be derived that 119881(0) = 119881

1(0) =

119909(0)119879

1198751119909(0) = 8359423 It is obvious the condition (47)

holds which means that our obtained solution is correct andfeasible

The simulation results are provided in Figures 1ndash3 whichshow the effectiveness of the proposed methods The tra-jectories of state variable 119909(119905) with 119889(119905) = 0 are shown inFigure 1 the trajectories of sampled-data control input 119906(119905)with 119889(119905) = 0 are shown in Figure 2 the trajectories ofLyapunov functions 119881(119905) with 119889(119905) = 0 are shown inFigure 3 It can be seen that the obtained performance isdesirable

Mathematical Problems in Engineering 9

5 Conclusion

In this paper we have addressed the problem of sampled-data stabilization for spacecraft rendezvous with a time-dependent Lyapunov approach A discontinuous Lyapunovfunctional has been constructed the value of which does notincrease in each sampling time instant The sufficient condi-tions for the stability analysis and controller design are estab-lished in terms of matrix inequality technique Future workwill be focused onmore complicated case that is the effects ofnetwork-induced phenomenon such as communication delayand packet losses are taken into account simultaneously

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China 61104101 the FundamentalResearch Funds for the Central Universities the China Post-doctoral Science Foundation funded project (2011M500058)the Special Chinese National Postdoctoral Science Founda-tion under Grant 2012T50356 and the Heilongjiang Postdoc-toral Fund (LBH-Z11144)

References

[1] U Ahsun D W Miller and J L Ramirez ldquoControl of electro-magnetic satellite formations in near-Earth orbitsrdquo Journal ofGuidance Control and Dynamics vol 33 no 6 pp 1883ndash18912010

[2] H Schaub ldquoStabilization of satellite motion relative to aCoulomb spacecraft formationrdquo Journal of Guidance Controland Dynamics vol 28 no 6 pp 1231ndash1239 2005

[3] X Yang X Cao and H Gao ldquoSampled-data control forrelative position holding of spacecraft rendezvous with thrustnonlinearityrdquo IEEE Transactions on Industrial Electronics vol59 no 2 pp 1146ndash1153 2011

[4] P Gurfil ldquoNonlinear feedback control of low-thrust orbitaltransfer in a central gravitational fieldrdquo Acta Astronautica vol60 no 8-9 pp 631ndash648 2007

[5] S Kurnaz O Cetin and O Kaynak ldquoFuzzy logic basedapproach to design of flight control and navigation tasks forautonomous unmanned aerial vehiclesrdquo Journal of Intelligentand Robotic Systems vol 54 no 1ndash3 pp 229ndash244 2009

[6] W Clohessy and R Wiltshire ldquoTerminal guidance system forsatellite rendezvousrdquo Journal of the Aerospace Sciences vol 27no 9 pp 653ndash658 1960

[7] Y Luo and G Tang ldquoSpacecraft optimal rendezvous controllerdesign using simulated annealingrdquo Aerospace Science and Tech-nology vol 9 no 8 pp 732ndash737 2005

[8] Y Luo G Tang and Y Lei ldquoOptimal multi-objective lin-earized impulsive rendezvousrdquo Journal of Guidance Controland Dynamics vol 30 no 2 pp 383ndash389 2007

[9] Z Mao and B Jiang ldquoFault identification and fault-tolerantcontrol for a class of networked control systemsrdquo InternationalJournal of Innovative Computing Information and Control vol3 no 5 pp 1121ndash1130 2007

[10] X Su P Shi L Wu and Y Song ldquoA novel control design ondiscrete-time Takagi-Sugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems vol 21 no 4 pp655ndash671 2012

[11] X Su L Wu P Shi and Y Song ldquoHinfin

model reduction of T-Sfuzzy stochastic systemsrdquo IEEE Transactions on Systems Manand Cybernetics B vol 42 no 6 pp 1574ndash1585 2012

[12] S Yin S Ding and H Luo ldquoReal-time implementation of faulttolerant control system with performance optimizationrdquo IEEETransactions on Industrial Electronics 2013

[13] R Yang Z Zhang and P Shi ldquoExponential stability on stochas-tic neural networks with discrete interval and distributeddelaysrdquo IEEE Transactions on Neural Networks vol 21 no 1 pp169ndash175 2010

[14] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[15] R Yang P Shi and G-P Liu ldquoFiltering for discrete-timenetworked nonlinear systems with mixed random delays andpacket dropoutsrdquo IEEE Transactions on Automatic Control vol56 no 11 pp 2655ndash2660 2011

[16] Y Shu L Yu and Z Wenan ldquoEstimator-based control of net-worked systems with packet-dropoutsrdquo International Journal ofInnovative Computing Information andControl vol 6 no 6 pp2737ndash2748 2010

[17] C Jiang Q Zhang and D Zou ldquoDelay-dependent robust filter-ing for networked control systemswith polytopic uncertaintiesrdquoInternational Journal of Innovative Computing Information andControl vol 6 no 11 pp 4857ndash4868 2010

[18] S Shi Z Yuan and Q Zhang ldquoFault-tolerant Hinfin

filter designof a class of switched systems with sensor failuresrdquo InternationalJournal of Innovative Computing Information and Control vol5 no 11 pp 3827ndash3838 2009

[19] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[20] J Lu D W C Ho and J Cao ldquoA unified synchronizationcriterion for impulsive dynamical networksrdquo Automatica vol46 no 7 pp 1215ndash1221 2010

[21] J Lu D W C Ho J Cao and J Kurths ldquoExponential synchro-nization of linearly coupled neural networks with impulsivedisturbancesrdquo IEEE Transactions on Neural Networks vol 22no 2 pp 329ndash336 2011

[22] H Wu and M Bai ldquoStochastic stability analysis and synthesisfor nonlinear fault tolerant control systems based on the T-Sfuzzy modelrdquo International Journal of Innovative ComputingInformation and Control vol 6 no 9 pp 3989ndash4000 2010

[23] S Yin Data-Driven Design of Fault Diagnosis Systems VDIDuesseldorf Germany 2012

[24] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[25] L Wu and W X Zheng ldquoPassivity-based sliding mode controlof uncertain singular time-delay systemsrdquo Automatica vol 45no 9 pp 2120ndash2127 2009

[26] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo IEEETransactions on Automatic Control vol 55 no 5 pp 1213ndash12192010

[27] L Zhang and E-K Boukas ldquoStability and stabilization ofMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo Automatica vol 45 no 2 pp 463ndash468 2009

10 Mathematical Problems in Engineering

[28] L Zhang and E-K Boukas ldquoMode-dependent Hinfin

filteringfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo Automatica vol 45 no 6pp 1462ndash1467 2009

[29] L Zhou and G Lu ldquoQuantized feedback stabilization fornetworked control systems with nonlinear perturbationrdquo Inter-national Journal of Innovative Computing Information andControl vol 6 no 6 pp 2485ndash2496 2010

[30] Z-G Wu J H Park H Su and J Chu ldquoDiscontinuousLyapunov functional approach to synchronization of time-delayneural networks using sampled-datardquoNonlinear Dynamics vol69 no 4 pp 2021ndash2030 2012

[31] H Li H Liu H Gao and P Shi ldquoReliable fuzzy control foractive suspension systems with actuator delay and faultrdquo IEEETransactions on Fuzzy Systems vol 20 no 2 pp 342ndash357 2012

[32] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding modecontrol for nonlinear active suspension systems using T-S fuzzymodelrdquo IEEE Transactions on Industrial Electronics vol 60 no8 pp 3328ndash3338 2012

[33] E Fridman ldquoA refined input delay approach to sampled-datacontrolrdquo Automatica vol 46 no 2 pp 421ndash427 2010

[34] E Fridman A Seuret and J-P Richard ldquoRobust sampled-data stabilization of linear systems an input delay approachrdquoAutomatica vol 40 no 8 pp 1441ndash1446 2004

[35] P Naghshtabrizi J P Hespanha and A R Teel ldquoExponentialstability of impulsive systems with application to uncertainsampled-data systemsrdquo Systems amp Control Letters vol 57 no5 pp 378ndash385 2008

[36] M Liu J You and X Ma ldquoHinfin

filtering for sampled-datastochastic systems with limited capacity channelrdquo Signal Pro-cessing vol 91 no 8 pp 1826ndash1837 2011

[37] M Liu D W C Ho and Y Niu ldquoObserver-based controllerdesign for linear systems with limited communication capacityvia a descriptor augmentation methodrdquo IET Control Theory ampApplications vol 6 no 3 pp 437ndash447 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

Page 9: Research Article Sampled-Data Control of Spacecraft Rendezvous with …downloads.hindawi.com/journals/mpe/2013/814271.pdf · 2019-07-31 · Research Article Sampled-Data Control of

Mathematical Problems in Engineering 9

5 Conclusion

In this paper we have addressed the problem of sampled-data stabilization for spacecraft rendezvous with a time-dependent Lyapunov approach A discontinuous Lyapunovfunctional has been constructed the value of which does notincrease in each sampling time instant The sufficient condi-tions for the stability analysis and controller design are estab-lished in terms of matrix inequality technique Future workwill be focused onmore complicated case that is the effects ofnetwork-induced phenomenon such as communication delayand packet losses are taken into account simultaneously

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China 61104101 the FundamentalResearch Funds for the Central Universities the China Post-doctoral Science Foundation funded project (2011M500058)the Special Chinese National Postdoctoral Science Founda-tion under Grant 2012T50356 and the Heilongjiang Postdoc-toral Fund (LBH-Z11144)

References

[1] U Ahsun D W Miller and J L Ramirez ldquoControl of electro-magnetic satellite formations in near-Earth orbitsrdquo Journal ofGuidance Control and Dynamics vol 33 no 6 pp 1883ndash18912010

[2] H Schaub ldquoStabilization of satellite motion relative to aCoulomb spacecraft formationrdquo Journal of Guidance Controland Dynamics vol 28 no 6 pp 1231ndash1239 2005

[3] X Yang X Cao and H Gao ldquoSampled-data control forrelative position holding of spacecraft rendezvous with thrustnonlinearityrdquo IEEE Transactions on Industrial Electronics vol59 no 2 pp 1146ndash1153 2011

[4] P Gurfil ldquoNonlinear feedback control of low-thrust orbitaltransfer in a central gravitational fieldrdquo Acta Astronautica vol60 no 8-9 pp 631ndash648 2007

[5] S Kurnaz O Cetin and O Kaynak ldquoFuzzy logic basedapproach to design of flight control and navigation tasks forautonomous unmanned aerial vehiclesrdquo Journal of Intelligentand Robotic Systems vol 54 no 1ndash3 pp 229ndash244 2009

[6] W Clohessy and R Wiltshire ldquoTerminal guidance system forsatellite rendezvousrdquo Journal of the Aerospace Sciences vol 27no 9 pp 653ndash658 1960

[7] Y Luo and G Tang ldquoSpacecraft optimal rendezvous controllerdesign using simulated annealingrdquo Aerospace Science and Tech-nology vol 9 no 8 pp 732ndash737 2005

[8] Y Luo G Tang and Y Lei ldquoOptimal multi-objective lin-earized impulsive rendezvousrdquo Journal of Guidance Controland Dynamics vol 30 no 2 pp 383ndash389 2007

[9] Z Mao and B Jiang ldquoFault identification and fault-tolerantcontrol for a class of networked control systemsrdquo InternationalJournal of Innovative Computing Information and Control vol3 no 5 pp 1121ndash1130 2007

[10] X Su P Shi L Wu and Y Song ldquoA novel control design ondiscrete-time Takagi-Sugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems vol 21 no 4 pp655ndash671 2012

[11] X Su L Wu P Shi and Y Song ldquoHinfin

model reduction of T-Sfuzzy stochastic systemsrdquo IEEE Transactions on Systems Manand Cybernetics B vol 42 no 6 pp 1574ndash1585 2012

[12] S Yin S Ding and H Luo ldquoReal-time implementation of faulttolerant control system with performance optimizationrdquo IEEETransactions on Industrial Electronics 2013

[13] R Yang Z Zhang and P Shi ldquoExponential stability on stochas-tic neural networks with discrete interval and distributeddelaysrdquo IEEE Transactions on Neural Networks vol 21 no 1 pp169ndash175 2010

[14] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[15] R Yang P Shi and G-P Liu ldquoFiltering for discrete-timenetworked nonlinear systems with mixed random delays andpacket dropoutsrdquo IEEE Transactions on Automatic Control vol56 no 11 pp 2655ndash2660 2011

[16] Y Shu L Yu and Z Wenan ldquoEstimator-based control of net-worked systems with packet-dropoutsrdquo International Journal ofInnovative Computing Information andControl vol 6 no 6 pp2737ndash2748 2010

[17] C Jiang Q Zhang and D Zou ldquoDelay-dependent robust filter-ing for networked control systemswith polytopic uncertaintiesrdquoInternational Journal of Innovative Computing Information andControl vol 6 no 11 pp 4857ndash4868 2010

[18] S Shi Z Yuan and Q Zhang ldquoFault-tolerant Hinfin

filter designof a class of switched systems with sensor failuresrdquo InternationalJournal of Innovative Computing Information and Control vol5 no 11 pp 3827ndash3838 2009

[19] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[20] J Lu D W C Ho and J Cao ldquoA unified synchronizationcriterion for impulsive dynamical networksrdquo Automatica vol46 no 7 pp 1215ndash1221 2010

[21] J Lu D W C Ho J Cao and J Kurths ldquoExponential synchro-nization of linearly coupled neural networks with impulsivedisturbancesrdquo IEEE Transactions on Neural Networks vol 22no 2 pp 329ndash336 2011

[22] H Wu and M Bai ldquoStochastic stability analysis and synthesisfor nonlinear fault tolerant control systems based on the T-Sfuzzy modelrdquo International Journal of Innovative ComputingInformation and Control vol 6 no 9 pp 3989ndash4000 2010

[23] S Yin Data-Driven Design of Fault Diagnosis Systems VDIDuesseldorf Germany 2012

[24] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[25] L Wu and W X Zheng ldquoPassivity-based sliding mode controlof uncertain singular time-delay systemsrdquo Automatica vol 45no 9 pp 2120ndash2127 2009

[26] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo IEEETransactions on Automatic Control vol 55 no 5 pp 1213ndash12192010

[27] L Zhang and E-K Boukas ldquoStability and stabilization ofMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo Automatica vol 45 no 2 pp 463ndash468 2009

10 Mathematical Problems in Engineering

[28] L Zhang and E-K Boukas ldquoMode-dependent Hinfin

filteringfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo Automatica vol 45 no 6pp 1462ndash1467 2009

[29] L Zhou and G Lu ldquoQuantized feedback stabilization fornetworked control systems with nonlinear perturbationrdquo Inter-national Journal of Innovative Computing Information andControl vol 6 no 6 pp 2485ndash2496 2010

[30] Z-G Wu J H Park H Su and J Chu ldquoDiscontinuousLyapunov functional approach to synchronization of time-delayneural networks using sampled-datardquoNonlinear Dynamics vol69 no 4 pp 2021ndash2030 2012

[31] H Li H Liu H Gao and P Shi ldquoReliable fuzzy control foractive suspension systems with actuator delay and faultrdquo IEEETransactions on Fuzzy Systems vol 20 no 2 pp 342ndash357 2012

[32] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding modecontrol for nonlinear active suspension systems using T-S fuzzymodelrdquo IEEE Transactions on Industrial Electronics vol 60 no8 pp 3328ndash3338 2012

[33] E Fridman ldquoA refined input delay approach to sampled-datacontrolrdquo Automatica vol 46 no 2 pp 421ndash427 2010

[34] E Fridman A Seuret and J-P Richard ldquoRobust sampled-data stabilization of linear systems an input delay approachrdquoAutomatica vol 40 no 8 pp 1441ndash1446 2004

[35] P Naghshtabrizi J P Hespanha and A R Teel ldquoExponentialstability of impulsive systems with application to uncertainsampled-data systemsrdquo Systems amp Control Letters vol 57 no5 pp 378ndash385 2008

[36] M Liu J You and X Ma ldquoHinfin

filtering for sampled-datastochastic systems with limited capacity channelrdquo Signal Pro-cessing vol 91 no 8 pp 1826ndash1837 2011

[37] M Liu D W C Ho and Y Niu ldquoObserver-based controllerdesign for linear systems with limited communication capacityvia a descriptor augmentation methodrdquo IET Control Theory ampApplications vol 6 no 3 pp 437ndash447 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

Page 10: Research Article Sampled-Data Control of Spacecraft Rendezvous with …downloads.hindawi.com/journals/mpe/2013/814271.pdf · 2019-07-31 · Research Article Sampled-Data Control of

10 Mathematical Problems in Engineering

[28] L Zhang and E-K Boukas ldquoMode-dependent Hinfin

filteringfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo Automatica vol 45 no 6pp 1462ndash1467 2009

[29] L Zhou and G Lu ldquoQuantized feedback stabilization fornetworked control systems with nonlinear perturbationrdquo Inter-national Journal of Innovative Computing Information andControl vol 6 no 6 pp 2485ndash2496 2010

[30] Z-G Wu J H Park H Su and J Chu ldquoDiscontinuousLyapunov functional approach to synchronization of time-delayneural networks using sampled-datardquoNonlinear Dynamics vol69 no 4 pp 2021ndash2030 2012

[31] H Li H Liu H Gao and P Shi ldquoReliable fuzzy control foractive suspension systems with actuator delay and faultrdquo IEEETransactions on Fuzzy Systems vol 20 no 2 pp 342ndash357 2012

[32] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding modecontrol for nonlinear active suspension systems using T-S fuzzymodelrdquo IEEE Transactions on Industrial Electronics vol 60 no8 pp 3328ndash3338 2012

[33] E Fridman ldquoA refined input delay approach to sampled-datacontrolrdquo Automatica vol 46 no 2 pp 421ndash427 2010

[34] E Fridman A Seuret and J-P Richard ldquoRobust sampled-data stabilization of linear systems an input delay approachrdquoAutomatica vol 40 no 8 pp 1441ndash1446 2004

[35] P Naghshtabrizi J P Hespanha and A R Teel ldquoExponentialstability of impulsive systems with application to uncertainsampled-data systemsrdquo Systems amp Control Letters vol 57 no5 pp 378ndash385 2008

[36] M Liu J You and X Ma ldquoHinfin

filtering for sampled-datastochastic systems with limited capacity channelrdquo Signal Pro-cessing vol 91 no 8 pp 1826ndash1837 2011

[37] M Liu D W C Ho and Y Niu ldquoObserver-based controllerdesign for linear systems with limited communication capacityvia a descriptor augmentation methodrdquo IET Control Theory ampApplications vol 6 no 3 pp 437ndash447 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

Page 11: Research Article Sampled-Data Control of Spacecraft Rendezvous with …downloads.hindawi.com/journals/mpe/2013/814271.pdf · 2019-07-31 · Research Article Sampled-Data Control 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


Seaside rendezvous

Seaside rendezvous

Rendezvous point

Rendezvous point

TIBCO Rendezvous Concepts

TIBCO Rendezvous Concepts

Travellers app rendezvous

Travellers app rendezvous

Western rendezvous 2013

Western rendezvous 2013

2014 National FCF Rendezvous- Strongheart Award … National Rendezvous/Strongheart...2014 National FCF Rendezvous- Strongheart Award Events ... Rendezvous Games OT 1st Bryan J. Boudreaux

2014 National FCF Rendezvous- Strongheart Award … National Rendezvous/Strongheart...2014 National FCF Rendezvous- Strongheart Award Events ... Rendezvous Games OT 1st Bryan J. Boudreaux

DesCours | Rendezvous

DesCours | Rendezvous

Rendezvous 2012 - University of Colorado Bouldercires1.colorado.edu/events/rendezvous/2012/Rendezvous...Rendezvous 2012 Come celebrate innovation, performance, and outstanding service

Rendezvous 2012 - University of Colorado Bouldercires1.colorado.edu/events/rendezvous/2012/Rendezvous...Rendezvous 2012 Come celebrate innovation, performance, and outstanding service

Major League Rendezvous

Major League Rendezvous

Western Rendezvous 2014

Western Rendezvous 2014

Rendezvous with ray

Rendezvous with ray

The Near Earth Asteroid Rendezvous (NEAR) Rendezvous Burn Anomaly

The Near Earth Asteroid Rendezvous (NEAR) Rendezvous Burn Anomaly

Rendezvous 2014 sale

Rendezvous 2014 sale

Rendezvous 2010 presentation

Rendezvous 2010 presentation

Space rendezvous

Space rendezvous

Explorers Territorial RENDEZVOUS

Explorers Territorial RENDEZVOUS

Lighting Rendezvous - Jaquar

Lighting Rendezvous - Jaquar

Adaptive Finite-Time Control for Spacecraft Rendezvous ...Research Article Adaptive Finite-Time Control for Spacecraft Rendezvous under Unknown System Parameters Yuan Liu,1 Guojian

Adaptive Finite-Time Control for Spacecraft Rendezvous ...Research Article Adaptive Finite-Time Control for Spacecraft Rendezvous under Unknown System Parameters Yuan Liu,1 Guojian

Research Article Controller Design for Asynchronous ...Multirate sampled-data control systems are required when the signals of interest are sampled at di erent rates compared with

Research Article Controller Design for Asynchronous ...Multirate sampled-data control systems are required when the signals of interest are sampled at di erent rates compared with

Research Article Asteroid Rendezvous Mission …downloads.hindawi.com/journals/mpe/2014/823659.pdfResearch Article Asteroid Rendezvous Mission Design Using Multiobjective Particle

Research Article Asteroid Rendezvous Mission …downloads.hindawi.com/journals/mpe/2014/823659.pdfResearch Article Asteroid Rendezvous Mission Design Using Multiobjective Particle