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Finite-time Attitude Tracking Control Method of Microsatellite Based on Adaptive Robustness and Neural Network Approximation
21st IFAC Symposium on Automatic Control in Aerospace 21st IFAC Symposium on Automatic August 27-30, 2019. Cranfield, UK Control in Aerospace 21st IFAC Symposium on Automatic Control inonline Aerospace at www.sciencedirect.com August 27-30, 2019. Cranfield, UK Available 21st IFAC Symposium on Automatic August 27-30, 2019. Cranfield, UK Control in Aerospace August 27-30, 2019. Cranfield, UK
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IFAC PapersOnLine 52-12 (2019) 85–90
Finite-time Attitude Tracking Control Method of Microsatellite Finite-time Attitude Tracking Control Method of Microsatellite Finite-time Attitude Trackingand Control Method of Microsatellite Based on Adaptive Robustness Neural Network Approximation Finite-time Attitude Trackingand Control Method of Microsatellite Based on Adaptive Robustness Neural Network Approximation Based on Adaptive Robustness and Neural Network Approximation Based on Adaptive Robustness and Neural Network Approximation Zhang Kaicheng*. Wang Feng**
Zhang Kaicheng*. Wang Feng** Zhang Kaicheng*. Wang Feng** Zhang Kaicheng*. Wang Feng** *Research Centre of Satellite Technology, Institute of Technology, Harbin, China Harbin *Research Centre of Satellite Technology, Harbin Institute of Technology, Harbin, China (e-mail: [email protected]) *Research Centre of Satellite Technology, Harbin Institute of Technology, Harbin, China (e-mail: [email protected]) *Research Centre of Satellite Technology, Harbin Institute **Research Centre of Satellite Technology, Harbin Instituteof ofTechnology, Technology,Harbin, Harbin,China China (e-mail: [email protected]) **Research Centre of Satellite Technology, Harbin Institute of Technology, Harbin, China (e-mail: [email protected]) (e-mail: [email protected]) **Research Centre of Satellite(e-mail: Technology, Harbin Institute of Technology, Harbin, China [email protected]) **Research Centre of Satellite(e-mail: Technology, Harbin Institute of Technology, Harbin, China [email protected]) (e-mail: [email protected]) Abstract: Aiming at the attitude tracking control of microsatellites, a finite-time attitude tracking control Abstract:based Aiming at the attitude tracking microsatellites, a finite-time attitude tracking control method on adaptive robustness andcontrol neural of network approximation is proposed. Specifically, based Abstract:based Aiming at the attitude tracking control of microsatellites, a finite-time attitude tracking control method on adaptive robustness and neural network approximation is proposed. Specifically, based Abstract: Aiming at the attitude tracking control of microsatellites, a finite-time attitude tracking control on quaternion and Newton-Euler equation, the model of microsatellite attitude tracking control is method based on adaptive robustness and neural network approximation is proposed. Specifically, based on quaternion and Newton-Euler equation, the model All of terms microsatellite attitude tracking control is method based on adaptive robustness and neural network approximation is proposed. Specifically, based established. Then, a terminal sliding surface is proposed. except control torque are regarded as on quaternion anda terminal Newton-Euler model All of terms microsatellite attitude tracking control as is established. Then, slidingequation, surface isthe proposed. except control torque are on quaternion and Newton-Euler equation, the model of terms microsatellite attitude tracking control as is total disturbance, which is estimated by three groups of adaptive variables and approximated byregarded Chebyshev established. Then, a terminal sliding surface is proposed. All except control torque are regarded total disturbance, which is estimated by threeneural groups of adaptive variables and approximated byaregarded Chebyshev established. Then, a terminal sliding surface is proposed. All terms except control torque are as neural network. Adaptive robustness and network approximation are combined by switching total disturbance, which is estimated byand threeneural groupsnetwork of adaptive variables and neural network. robustness approximation areapproximated combined byby aChebyshev switching total disturbance, which is estimated byand three groups of adaptive variables and function and the Adaptive controller is designed based on it.network After that, the finite-time stability of the is neural network. Adaptive robustness neural approximation areapproximated combined bybycontroller aChebyshev switching function and the controller is designed based on it. After that, the finite-time stability of the controller is neural network. Adaptive robustness and neural network approximation are combined by a switching proved by finite-time Lyapunov stability theorem. Finally, the numerical simulations are carried out and functionbyand the controller is designed based on it. Finally, After that, the finite-time stability are of the controller is proved finite-time Lyapunov stability theorem. the numerical simulations carried out and function the controller isand designed based on After that, the finite-time stability of the controller is have verified correctness effectiveness of it. theFinally, method, indicates that the controller has proved byand finite-time Lyapunov stability theorem. thewhich numerical simulations are carried outquick and have verified the correctness and effectiveness of the method, which indicates that the controller has quick proved by finite-time Lyapunov stability theorem. Finally, the numerical simulations are carried out and convergence speed and excellent control precision. have verified speed the correctness and effectiveness of the method, which indicates that the controller has quick convergence and excellent control precision. have verified speed the correctness and effectiveness of the method, which indicates that the controller has quick convergence and excellent control precision. Copyright © attitude 2019. Thecontrol, Authors. Published by Elsevier Ltd. All rights reserved. Keywords: sliding mode, adaptive control, sliding surfaces, neural network convergence speed and excellent control precision. Keywords: attitude control, sliding mode, adaptive control, sliding surfaces, neural network Keywords: attitude control, sliding mode, adaptive control, sliding surfaces, neural network Keywords: attitude control, sliding mode, adaptive control, sliding surfaces, neural network disturbance. In this paper, three groups of adaptive variables 1. INTRODUCTION disturbance. Intothis paper,the three groups variables are introduced estimate upper boundof ofadaptive total disturbance, 1. INTRODUCTION disturbance. Intothis paper,the three groups ofof adaptive variables are introduced estimate upper bound total disturbance, 1. INTRODUCTION disturbance. Intothis paper, three variables and Chebyshev neural network isgroups introduced to approximate With the development of aerospace technology and the and are introduced estimate the upper boundofofadaptive total disturbance, 1. INTRODUCTION Chebyshev neural network is introduced to approximate With the development of aerospace technology and the are introduced to estimate the upper bound of total disturbance, total disturbance function. Control based on adaptive expansion of satellite applications, and Chebyshev neural networkControl is introduced to on approximate With the development of aerospace high-precision technology andspace the total disturbance function. basedprecision adaptive expansion of satellite applications, high-precision space and Chebyshev neural network is introduced to approximate robustness converges quickly, but control after With theand development of payloads aerospacerequire technology andspace the robustness missions higher demands total disturbance function. Control basedprecision on adaptive expansion ofhigh-precision satellite applications, high-precision converges quickly, butcontrol control after missions and high-precision payloads require higher demands total disturbance function. Control based on adaptive convergence is poor. However, based on neural expansion ofhigh-precision satellite applications, high-precision space convergence on satellite attitude control systems. order to meetdemands mission robustness converges butcontrol controlbased precision after missions and payloadsIn higher is poor. quickly, However, on neural on satellite attitude control systems. Inrequire order to meet mission robustness converges quickly, butcontrol controlbased precision after network approximation has slow convergence speed and missions and high-precision payloads require higher demands requirements, high-precision attitude control is required. In convergence is poor. However, on neural on satellite attitude control systems. Incontrol order toismeet mission approximation has slow convergence speed and requirements, high-precision attitude required. In network convergence is poor. However, control based on neural excellent control precision after convergence, which is on satellite attitude control systems. In order to meet mission recent years, a series of researches have been made on satellite network approximation has slow convergence speed and requirements, attitude is required. In excellent control precision afterTherefore, convergence, which isa recent years, a high-precision series of researches havecontrol been made on satellite network approximation has slow convergence speed and opposite of adaptive robustness. in this paper, requirements, high-precision attitude control is required. In attitude control by scholars, including optimal control (Wang excellent ofcontrol precision afterTherefore, convergence, recent years, a series of researches haveoptimal been made on satellite adaptive robustness. in thiswhich paper, is a attitude control by scholars, including control (Wang opposite excellent precision afterTherefore, convergence, switching function is introduced to combine adaptive recent years, a adaptive series of researches haveoptimal been onfeedback satellite & Ge 2002)., control (Wang et al. made 2015), opposite ofcontrol adaptive robustness. in thiswhich paper, isa attitude control by scholars, including control (Wang switching function is introduced to combine adaptive & Ge 2002)., control (Wang et al. 2015), feedback opposite adaptive Therefore, in and this aadaptive paper, robustness and neuralrobustness. network approximation controla attitude control by 1994), scholars, including control control (Jing etadaptive al. sliding modeoptimal control (Deng &(Wang Song robustness switchingofand function isnetwork introduced to combine & Ge 2002)., adaptive control (Wang et al. 2015), feedback neural approximation andhas aadaptive control control (Jing et al. 1994), sliding mode control (Deng & Song switching function is introduced to combine method based on that is developed, which quick & Ge 2002)., control al. 2015), H2 / H (Wang(Wang et al.et 2009), andfeedback various 2013), robustnessbased and neural network approximation a control al.control control (Jing etadaptive 1994), sliding mode control (Deng & Song method on and that is developed, whichand has quick robustness and neural network approximation and a control convergence speed excellent control precision. H / H control (Wang et al. 2009), and various 2013), 2 control (Jing et al. methods 1994), sliding (Deng &2015). Song convergence method based on and thatexcellent is developed, which has quick integrated (Zou mode etet al.al.control 2010, Li et speed control precision. H 22control / H control (Wang 2009), andal.various 2013), based on and thatexcellent is developed, which has quick integrated methods (Zou etet al.al. 2010, Li et 2015). method H 2control / methods H control (Wang 2009), andal.various 2013), convergence speed control precision. The control above can make the closed-loop system sectionsspeed in this paper are arranged as follows. In Section integrated control methods (Zou et al.the 2010, Li et al. system 2015). Other convergence and excellent control precision. The control methods above can make closed-loop in this are arranged In attitude Section integrated (Zou et al.the 2010, Li et al. system 2015). stable, but control most ofmethods them merely ensure asymptotically stable, Other 2, thesections kinematic andpaper dynamic model as forfollows. satellite The control methods above can make closed-loop Other sections in this paper are arranged as follows. In attitude Section stable, but most oftheoretically them merely ensure asymptotically stable, 2, the kinematic and dynamic model for satellite The control methods above can make the closed-loop system which means that it takes an infinitely long time Other sections in this paper are arranged as follows. In attitude Section tracking is established. Then some definitions and lemmas are stable, but most oftheoretically them merelyitensure asymptotically stable, 2, the kinematic and dynamic model for satellite which means that takes an infinitely long time tracking is established. Then someterminal definitions and lemmas are stable, but most oftheoretically them merelyitensure asymptotically stable, for the means control error to converge to zeroan and as a result, some 2, the kinematic and dynamic model for satellite attitude introduced. In Section 3, the sliding surface is which that takes infinitely long time tracking is established. Then someterminal definitions and lemmas are for the means control error to zeroanand as aemergence result, In Section 3, the sliding surface is which theoretically it to takes infinitely longsome time on-orbit tasks that cannot beconverge well accomplished. The of introduced. tracking is established. Then some definitions and lemmas are designed. In Section 4, adaptive robustness and neural network for the control error to converge to zero and as a result, some introduced. In Section 3, therobustness terminal and sliding surface is on-orbit tasks cannot bemethod well accomplished. of designed. In Section 4, adaptive neural network for the control error to converge to zero andThe as aemergence result, some the finite-time control makes it possible to overcome In are Section 3, theand terminal sliding is approximation introduced combined by a surface switching on-orbit tasks cannot bemethod well accomplished. The emergence of introduced. designed. In Section 4, adaptive robustness and neural network the finite-time control makes it possible to overcome approximation are introduced and combined by a switching on-orbit tasks cannot be well accomplished. The emergence of the disadvantage that the convergence time is too long and designed. In Section 4, adaptive robustness and neural network function, then a finite-time controller based on that is designed the finite-time control method makes it possible to overcome approximation are introduced and combined by aisswitching the disadvantage that the convergence time is too long and function, thenInaSection finite-time controller based on that designed4 the finite-time method makes it possible to overcome therefore ensurecontrol the timeliness of the task. are introduced and combined by in aisswitching and proved. 5, the controller designed Section the disadvantage that the convergence time is too long and approximation function, then a finite-time controller based on that designed4 therefore ensure the timeliness of the task. and proved. In Section 5, the controller designed in Section the disadvantage that the convergence time is too long and function, then finite-time controller based on thatinisSection designed4 is applied to In a amicrosatellite. therefore ensure the timeliness of the task. and proved. Section 5, the controller designed Different from previous literature, in this paper, a finite-time is applied to In a microsatellite. therefore ensure the timeliness of the task. and proved. Section 5, the controller designed in Section 4 Different from previous literature, in this paper, a finite-time attitude tracking control literature, method using terminal mode is applied to a microsatellite. Different from previous in this paper, sliding a finite-time is applied to a microsatellite. 2. SATELLITE MODEL AND CONTROL THEORY attitude tracking control method using terminal sliding mode Different fromTerminal previous literature, in this a finite-time is developed. isterminal anpaper, important part of 2. SATELLITE MODEL AND CONTROL THEORY attitude tracking controlsliding methodmode using sliding mode is developed. Terminal sliding mode isterminal an important part of 2. SATELLITE MODEL AND CONTROL THEORY attitude tracking control method using sliding mode finite-time control, which contains a fractional power term and is developed. Terminal sliding mode is an important part of 2. SATELLITE finite-time control, which contains a fractional power term and 2.1 Attitude Model ofMODEL SatelliteAND CONTROL THEORY is developed. Terminal sliding mode is an important part of can converge in finite time. finite-time control, which contains a fractional power term and 2.1 Attitude Model of Satellite can converge in finite time. finite-time control, which contains a fractional power term and Attitude Model of Satellite can converge in finite time. In all terms except control torque are 2.1 2.1 Attitude of kinematic Satellite equations based on quaternion In this paper,Model attitude cansatellite convergedynamics, in finite time. In satellite dynamics, all terms except control torque are In this paper, attitude kinematic equations based on quaternion regarded as total disturbance, which includes external T T T In satelliteasdynamics, all terms except control torque are In this paper, attitude kinematic equations T based Q q q q0 on q1 quaternion q2 q3 TT are used. Quaternion regarded total disturbance, which includes external T T 0 In satellite dynamics, all terms except control torque are disturbance and parameter uncertainties and many coupling T In this paper, attitude kinematic equations based on quaternion regarded asand total disturbance, which and includes external are used. Quaternion Q q00 q T T q00 q11 q22 q33 T disturbance parameter uncertainties many coupling regarded asand total disturbance, which includes external describes terms. Adaptive robustness and neural network approximation q0 q T Tand q0 isq1 theq2angular q3 T are used. the Quaternion attitude ofQthe satellite, disturbance parameter uncertainties and many coupling terms. Adaptive robustness and neural network approximation Q q0 q and q0 isq1 theq2angular q3 are used. Quaternion describesofthe attitude ofThe theattitude disturbance andthat parameter uncertainties and many coupling velocity are two ways are often used to deal with unknown satellite, the satellite. kinematic equations are terms. Adaptive robustness and neural network approximation describes the attitude of the satellite, and is the angular are two ways that are often used to deal with unknown theattitude satellite.ofThe kinematic are terms. Adaptive robustness and neural network approximation is the angular describesofthe theattitude satellite, and equations are two ways that are often used to deal with unknown velocity velocity of the satellite. The attitude kinematic equations are are two ways that are often used to deal with unknown CopyrightCopyright © 2019 IFAC 85rights velocity of the satellite. The attitude kinematic equations are 2405-8963 © 2019. The Authors. Published by Elsevier Ltd. All reserved. Copyright 2019responsibility IFAC 85 Control. Peer review©under of International Federation of Automatic Copyright © 2019 IFAC 85 10.1016/j.ifacol.2019.11.074 Copyright © 2019 IFAC 85
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1 q 2 q q0 E3 q 1 q T 0 2 0 q3 where q 0 q3 q2 q1 matrix of vector q .
satisfy which make the system x f x V x V x 0 , the system will be finite-time stable.
(1)
Definition 2 For the dynamic system x f x , x R n ,if
q2 q1 is the cross-multiplication 0
there is 0 and T 0 , which make x t for all t T , the system will be actual finite-time stable.
Lemma 2 If there is a continuous differentiable positive definite function V x , real number 0 , 0 1 and
0 , which make the system x f x satisfy V x V x , the system will be actual finite-time
In this paper, we use the Newton-Euler equation to establish the attitude dynamic equation
J J u d
(2)
stable.
where J 33 is the inertia matrix of the satellite, and u31 , d 31 , 31 are the control torque, the external disturbance torque and the disturbance torque caused by actuator saturation respectively. We assume that disturbance d and are both bounded. Further we assume that there are m0i 0 , m2i 0 and i max 0 , so that their components satisfy 2
d i m0i m2i , i 1, 2,3
(3)
i i max , i 1, 2,3
(4)
In addition, the following theorem is used in the following proof process. Lemma 3 For any real number xi , if 0 w 1 , the following inequality holds.
3 2 xi xi i 1 i 1 3
1 w
1 w 2
3. SLIDING SURFACE DESIGN
For the attitude tracking problem, the target attitude is represented by quaternion Qd , then attitude tracking error
The sliding surface is designed as follows.
quaternion Qe =Qd1 Q represents the error between the satellite attitude quaternion and the target quaternion. The Qe is direction cosine matrix decided by
where
s e kf qe ,
T e
T e e
e0 e
and s e ksig qe , 1
Assume the target angular velocity is d , the error angular velocity and its derivative can be described as
e C d e eCd C d
1 2 , 2 1 , 2 T
f qe , f qe1 , f qe 2 , f qe3 , . is a small
(5)
positive number.
(6)
Since
Then, the attitude model based on tracking error quaternion is 1 qe 2 qe qe 0 E3 e q 1 q T e e e 0 2 J e J J e C d C d u d
(10)
si 0 or qei qei sgn q ei , (11) f qei , 2 1 sgn qei qei 2 qei , si 0 and qei
C q q qe E3 2q q 2q q . 2 e0
(9)
sliding
e kdiag qei sgn qei
surface
(10)
satisfies
when s 0 , we select the
Lyapunov function V qeT qe 1 qe 0 and its derivative 1 2
(7)
V1 2qeT qe 2qe 0 1 qe 0 qeT e . According to Lemma 3, 1
(8)
1 3 2 2 V1 k qei k qeT qe 2 . Since V1 2qeT qe , we i 1 can finally get the inequality
2.2 Finite-time Control Theory Definition 1 For the dynamic system x f x , x R n , if
1
1
1 2 V1 kV1 2 0 (12) 2 According to Lemma 1, we can get the following conclusion.
there is a constant T 0 , which makes lim x t 0 and t T
x t 0 for all t T , the system will be finite-time stable.
Theorem 1 For the microsatellite attitude tracking control system described by (7) and (8), if the system state maintains on the sliding surface (10), the system state can converge to the equilibrium point along the sliding surface in finite time.
Lemma 1 If there is a continuous differentiable positive definite function V x , real number 0 and 0 1 ,
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control method. In order to realize smooth switching between adaptive control method and NN control method, define the switching function (Wu et al., 2013)
4. CONTROL LAW DESIGN 4.1 Proposal of the Control Law
0, e a e 2 a 2 n n sin h e 1 cos 2 b2 a 2 2 1, e b
Substituting the derivative of (10) into (8) (13)
J s D u
where
1 1 si 0 or qei qe qe0 E 3 , 2 Jkdiag qei R q 1 J 2 diag sgn qei qei 2 qe qe 0 E3 , 2 1 si 0 and qei
and
D J J C d C d R e d
Theorem 2 For the microsatellite attitude tracking control system described by (7) and (8), the sliding surface is selected as (10). If we use the control law
total disturbance. In the perspective of adaptive control, based on the assumption provided by (3) and (4), the components of D satisfy
JC
3
ij
d CQ ij
J ij C d Qij j 1
ai bi ci
m2i
u k s sig s 1 h ωe uN h ωe ua
(18)
uN Wˆ Φ(ω) εˆ sgn s
(19)
2 ua aˆ bˆ ω cˆ ω sgn s
(20)
m0i i max
J
2
Wˆ W 1 1 h ωe sΦT ω W 2Wˆ εˆ 1 1 h ωe s 2 εˆ
(14)
(22) (23) (24) (25)
the satellite attitude will converge to a small neighbourhood near zero in finite time, where k s 0 , 0 1 , a1 0 , a 2 0 , b1 0 , b 2 0 , c1 0 , c 2 0 , W 1 0 ,
In the perspective of neural network control, the unknown function D can be approximated by neural network. In this paper, Chebyshev neural network is used, which does not require extra parameters and is faster to calculate. Chebyshev neural network is based on Chebyshev polynomials, which can be obtained by using the two-term recursive formula (Zou et al. 2010) Ti 1 x 2 xTi x Ti 1 x
(21)
aˆ a1 h ωe s a 2 aˆ bˆ b1 h ωe s ω b 2 bˆ 2 cˆ c1 h ωe s ω c 2 cˆ
Therefore, three groups of adaptive variables ( a31 , b31 and c31 ) are introduced to estimate the upper bound of the total disturbance. aˆ , bˆ , cˆ are their estimations and a a aˆ , b b bˆ , c c cˆ are their estimation errors respectively.
W 2 0 ,
1 0 ,
(15)
s s1 s2 s3
2 0 ,
sig s s1 sgn s1
s2 sgn s2
T
,
T
s3 sgn s3 .
The block diagram of the controller is illustrated in Figure 1.
where T0 x 1 , T1 x x . An enhanced pattern using
Negative Feedback
Chebyshev polynomials for a vector X x1 ,, xm R m is T
TSM
given by qe
T
X 1, T1 x1 ,, Tn x1 ,, T1 xm ,, Tn xm (16) Thus, total disturbance can be approximated by D ω W ω
and the update law
2
*
, a e b
Combining two control methods with the switching function, the control law is proposed below.
is the
e
3 Di j 1
87
(17)
Kinematics
Qd
s h e
Adaptive Control NN Control
ks sig s
h ωe ua
+
1 h ω u
+
e
+ N
u
Switching Function
Dynamics
ω
ωe
+
ωd -
where W * is the optimal weight matrix and is the approximation error. Wˆ , ˆ are their estimations and W * Wˆ , ˆ are their estimation errors W respectively.
Figure 1 Controller block diagram 4.2 Proof of the Control Law Due to the switching function, (13) is rewritten as
The control method to be proposed in this paper is a combination of adaptive control method and neural network
Js 1 h ωe W *Φ(ω) ε h ωe D u 87
(26)
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The Lyapunov function is selected as
If
1 T 1 1 T tr W TW V2 s Js ε ε 2 2W 1 2 1
1 T 1 T a T a b b c c 2a1 2b1 2c1
Its derivative is
2
1
a1
W 1W 2
1
1 T ε T εˆ 1 h ωe 1 s a aˆ h ωe a1 s a1
1 T ˆ 1 T b b h ωe b1 s c cˆ h ωe c1 s b1 c1
1
1 2
1 s T Js 2
1 2
(30)
, and J max is the maximum eigenvalue
is satisfied.
1 T 1 T b b c c 2b1 2c1
2
2kW
2
W 2 2 kW tr W TW 2
(32)
2
Therefore, the following inequality holds. 2
(33)
According to Lemma 2, the system is finite-time stable and we have completed the proof of Theorem 2.
(31)
5. NUMERICAL SIMULATION AND DISCUSSIONS
1 2
i 0
1
1
2
V2 V2
2 kW 2 W2 tr W TW W W 2 tr W TWˆ
1
where W 0 0 a 0 b 0 c 0 .
1
where W
2 kW
1 1 1 T 1 T tr W TW s T Js ε ε a a 2 2 2 2 1 W1 a1
Rewrite it as
1
2 kW W 2 tr W TWˆ W 2 tr W TW W 2 tr W TW
and
1 1 1 2 2 2 1 1 1 T T T ε ε tr W W V2 s Js 2 2W 1 2 1 1 1 1 1 T 2 1 T 2 1 T 2 a a b b c c 2a1 2b1 2c1 W 0 0 a 0 b 0 c 0
For any 0 kW 2 ,
W 0
2
The first item in the right side satisfies
1 2 J max 2 of the inertia matrix J .
2
1 where i 0 max i 2 i T i ,1 i 2 i T iˆ , i1i 2 , and i 2 ki 2 ki refers to , a , b and c .
(29)
a 2 a T aˆ b 2 b T bˆ c 2 c T cˆ
ks
Then (29) can be rewritten as
V2 k s s T sig s W 2 tr W TWˆ 2 ε T εˆ
where
then
1
1 T i i i 2 i T iˆ 2i1
Then considering the update law (21)~(25), we get
3 2 ks s T sig s ks si i 1
,
Define
Follow the similar process, we can get
1 k s s T sig s tr W T Wˆ 1 h ωe W 1 sΦ T (ω) V2 =
1
T
.
2
Taking control law (18) into (28)
tr W TW 1 , W 1 W 2 tr W TWˆ
1 tr W TW W 2 tr W Wˆ 2W 1
c1
W 1
1 T ˆ 1 T b b c cˆ
b1
W 0 max W 2 tr W TW ,1 W 2 tr W TWˆ 2kW
s T 1 h ω W *Φ(ω) ε h ω D u V e e 2 1 1 1 T T T (28) tr W Wˆ ε εˆ a aˆ
W 1
tr W TW 1 , W W 2 tr W TW ; and if 2kW
2
W 2 2 kW
(27)
1
W 2 2 kW
W 2 tr W TWˆ .
In order to illustrate the performance of the controller designed in Section 4, a model was built in Simulink, and the controller was applied to the microsatellite attitude model. According to the microsatellite parameters provided in the literature (Lu & Xia 2013), the parameters used in the simulations are shown in Table 1. 88
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Inertia matrix Initial attitude
Target attitude
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89
Table 1 Simulation Parameters 20 1.2 0.9 J 1.2 17 1.4 0.9 1.4 15 0 Q0
0.06
0.8832
0.04 0.05 , T
0.3 0.2 0.3
T
sin 0.01t 4 rad / s , d 0.05 sin 0.03t 6 sin 0.02t
(b) Angular velocity error
Qd 1 0 0 0
T
Sliding surface parameters
k 1 , 0.5 , 0.1
6 cos 10x t 8sin 30 t 20 Disturbance d 3sin 2 y t 6 cos 50 t 30 104 torque ( Nm ) 6sin 10z t 16sin 40 t 20 Controller k s 20 , 0.1 parameters W 1 20 , W 2 0.2 , 10 , 1 a1 b1 c1 ˆ Update law 0.1 , W 0 03301 , 2 a2 b2 c2 parameters T ˆ 0 0 0 0 , aˆ 0 bˆ 0 cˆ 0
0.01
(c) Sliding variable
0.01 0.01
T
(d) Control torque Figure 2 Simulation results According to (18), if we set h ωe 1 , the controller (18) will
ua 0.5N m ， ub 0.5N m Torque limit Measure Power 4 10 14 s 2 , bias 10 6 rad / s noise Switching a 0.01 ， b 0.02 function The simulation result is presented in Figure 2.
become an “adaptive-only” controller. Similarly, if h ωe 0 , the controller (18) will become an “NN-only” controller. In order to further illustrate the performance of the proposed controller (18), it is compared with “adaptive-only” and “NNonly” controllers.
It can be observed from Figure 2 that both the attitude error and angular velocity error are bounded and will converge to sets around the origin in finite time, which indicates that the tracking objective is accomplished with the proposed controller based on adaptive robustness and neural network.
Two more simulations are made using “adaptive-only” controller and “NN-only” controller. Simulation parameters used in each simulation are the same of those shown in Table 1. Comparisons between controllers are shown in Figure 3.
(a) Attitude error comparison ( qe1 )
(a) Attitude error
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Deng, L., and Song, S. (2013). Flexible spacecraft attitude robust tracking control based on fractional order sliding mode. Acta Aeronautica, 34(08), pp. 1915-1923. (in Chinese) Du, H., and Li, S. (2012). Finite-time attitude stabilization for a spacecraft using homogeneous method. Journal of Guidance, Control, and Dynamics, 35(3), pp. 740-748. Jing, W., Huang, W., Wu, Y., and Yang, D. (1994). QuasiEuler-angles feedback control of spacecraft attitude maneuver. Journal of Astronautics, (02), pp. 41-47+107. (in Chinese)
(b) Angular velocity error comparison ( x ) Figure 3 Control comparison Figure 3 shows the variation of some classical variables ( qe1 and x ) during the simulation. It can be concluded the controller proposed in this paper is a perfect combination of “adaptive-only” controller and “NN-only” controller. In the early stage of control process, the convergence speed of the proposed controller is almost the same as that of “adaptiveonly” controller; in the later stage of control process, the control precision of the proposed controller after convergence is almost the same as that of “NN-only” controller. It shows that the controller proposed in this paper has quicker convergence speed and higher control precision. The performance comparison of controllers is shown in Table 2.
Li, L., Hou, J., Shi, X., and Yang, J. (2015). Adaptive sliding mode control for spacecraft attitude tracking system. Electric Machines and Control, 19(02), pp. 96-100+108. (in Chinese) Lu, K., and Xia, Y. (2013). Adaptive attitude tracking control for rigid spacecraft with finite-time convergence. Oxford: Pergamon Press, Inc. Ran, D., Ni, Q., Sheng, T., and Chen, X. (2017). Spacecraft attitude maneuver algorithm design based on adaptive secondorder terminal sliding mode. Journal of National University of Defense Technology, 39(01), pp. 6-10. (in Chinese) Wang, H. (2015). Novel terminal sliding mode based finite time control research for spacecraft application. Master Degree. Civil Aviation University of China. (in Chinese)
Table 2 Control performance comparison Proposed Adaptive NN only controller only qe precision
1 10 5
2 104
1 10 5
e precision ( rad / s )
1 10 5
3 10 5
1 10 5
s precision
5 10 5
5 10 3
5 10 5
Convergence time ( s )
50
50
75
Wang, J., Zeng, M., and Xu, W. (2009). Nonlinear H control of large angle attitude maneuvers for satellite using sum of squares. 2009 International Conference on Mechatronics and Automation. Wang, W., Ren, Y., Li Y., and Luo Y. (2015). Nonlinear robust adaptive attitude maneuver control law for spacecraft. Systems Engineering and Electronics, 37(01), pp. 135-140. (in Chinese) Wang, Z., and Ge, X. (2002). Attitude control research of rigid spacecraft with two momentum wheels. Journal of Beijing Institute of Machinery, 17(03), pp. 1-5. (in Chinese) Wu, J., Chen, W., Zhao, D. and Li, J. (2013). Globally stable direct adaptive backstepping NN control for uncertain nonlinear strict-feedback systems. Neurocomputing, 122, pp.134-147.
6. CONCLUSION In this paper, a finite-time controller based on adaptive robustness and neural network approximation is designed for the problem of microsatellite attitude tracking control. A switching function combines adaptive robustness and neural network approximation. In the controller, the terminal sliding mode is adopted and a fractional power term is introduced in the sliding surface to ensure finite-time convergence. Lyapunov stability theorem is used to prove the stability of the controller. Numerical simulations have shown that the proposed controller has as fast convergence speed as controller based on adaptive robustness and as excellent precision as controller based on neural network approximation.
Zhang, A., Zhang, Z., and Huo, X. (2015). Finite-time Spacecraft Attitude Stabilization Control Subject to Parameter Uncertainties. Information and Control, 44(03), pp. 303308+315. (in Chinese) Zhou, Y., Zhu, W., and Du, H. (2017). Global finite-time attitude regulation using bounded feedback for a rigid spacecraft. Control Theory and Technology, 15(1), pp. 26-33. Zou, A., Kumar, K., and Hou, Z. (2010). Quaternion-Based Adaptive Output Feedback Attitude Control of Spacecraft Using Chebyshev Neural Networks. IEEE Transactions on Neural Networks, 21(9), pp.1457-1471.
REFERENCES Bhat, S., Bernstein D. (2000). Finite-time Stability of Continuous Autonomous Systems. SIAM J on Control and Optimization, 38(3), pp. 751-766. 90
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IFAC PapersOnLine 52-12 (2019) 85–90
Finite-time Attitude Tracking Control Method of Microsatellite Finite-time Attitude Tracking Control Method of Microsatellite Finite-time Attitude Trackingand Control Method of Microsatellite Based on Adaptive Robustness Neural Network Approximation Finite-time Attitude Trackingand Control Method of Microsatellite Based on Adaptive Robustness Neural Network Approximation Based on Adaptive Robustness and Neural Network Approximation Based on Adaptive Robustness and Neural Network Approximation Zhang Kaicheng*. Wang Feng**
Zhang Kaicheng*. Wang Feng** Zhang Kaicheng*. Wang Feng** Zhang Kaicheng*. Wang Feng** *Research Centre of Satellite Technology, Institute of Technology, Harbin, China Harbin *Research Centre of Satellite Technology, Harbin Institute of Technology, Harbin, China (e-mail: [email protected]) *Research Centre of Satellite Technology, Harbin Institute of Technology, Harbin, China (e-mail: [email protected]) *Research Centre of Satellite Technology, Harbin Institute **Research Centre of Satellite Technology, Harbin Instituteof ofTechnology, Technology,Harbin, Harbin,China China (e-mail: [email protected]) **Research Centre of Satellite Technology, Harbin Institute of Technology, Harbin, China (e-mail: [email protected]) (e-mail: [email protected]) **Research Centre of Satellite(e-mail: Technology, Harbin Institute of Technology, Harbin, China [email protected]) **Research Centre of Satellite(e-mail: Technology, Harbin Institute of Technology, Harbin, China [email protected]) (e-mail: [email protected]) Abstract: Aiming at the attitude tracking control of microsatellites, a finite-time attitude tracking control Abstract:based Aiming at the attitude tracking microsatellites, a finite-time attitude tracking control method on adaptive robustness andcontrol neural of network approximation is proposed. Specifically, based Abstract:based Aiming at the attitude tracking control of microsatellites, a finite-time attitude tracking control method on adaptive robustness and neural network approximation is proposed. Specifically, based Abstract: Aiming at the attitude tracking control of microsatellites, a finite-time attitude tracking control on quaternion and Newton-Euler equation, the model of microsatellite attitude tracking control is method based on adaptive robustness and neural network approximation is proposed. Specifically, based on quaternion and Newton-Euler equation, the model All of terms microsatellite attitude tracking control is method based on adaptive robustness and neural network approximation is proposed. Specifically, based established. Then, a terminal sliding surface is proposed. except control torque are regarded as on quaternion anda terminal Newton-Euler model All of terms microsatellite attitude tracking control as is established. Then, slidingequation, surface isthe proposed. except control torque are on quaternion and Newton-Euler equation, the model of terms microsatellite attitude tracking control as is total disturbance, which is estimated by three groups of adaptive variables and approximated byregarded Chebyshev established. Then, a terminal sliding surface is proposed. All except control torque are regarded total disturbance, which is estimated by threeneural groups of adaptive variables and approximated byaregarded Chebyshev established. Then, a terminal sliding surface is proposed. All terms except control torque are as neural network. Adaptive robustness and network approximation are combined by switching total disturbance, which is estimated byand threeneural groupsnetwork of adaptive variables and neural network. robustness approximation areapproximated combined byby aChebyshev switching total disturbance, which is estimated byand three groups of adaptive variables and function and the Adaptive controller is designed based on it.network After that, the finite-time stability of the is neural network. Adaptive robustness neural approximation areapproximated combined bybycontroller aChebyshev switching function and the controller is designed based on it. After that, the finite-time stability of the controller is neural network. Adaptive robustness and neural network approximation are combined by a switching proved by finite-time Lyapunov stability theorem. Finally, the numerical simulations are carried out and functionbyand the controller is designed based on it. Finally, After that, the finite-time stability are of the controller is proved finite-time Lyapunov stability theorem. the numerical simulations carried out and function the controller isand designed based on After that, the finite-time stability of the controller is have verified correctness effectiveness of it. theFinally, method, indicates that the controller has proved byand finite-time Lyapunov stability theorem. thewhich numerical simulations are carried outquick and have verified the correctness and effectiveness of the method, which indicates that the controller has quick proved by finite-time Lyapunov stability theorem. Finally, the numerical simulations are carried out and convergence speed and excellent control precision. have verified speed the correctness and effectiveness of the method, which indicates that the controller has quick convergence and excellent control precision. have verified speed the correctness and effectiveness of the method, which indicates that the controller has quick convergence and excellent control precision. Copyright © attitude 2019. Thecontrol, Authors. Published by Elsevier Ltd. All rights reserved. Keywords: sliding mode, adaptive control, sliding surfaces, neural network convergence speed and excellent control precision. Keywords: attitude control, sliding mode, adaptive control, sliding surfaces, neural network Keywords: attitude control, sliding mode, adaptive control, sliding surfaces, neural network Keywords: attitude control, sliding mode, adaptive control, sliding surfaces, neural network disturbance. In this paper, three groups of adaptive variables 1. INTRODUCTION disturbance. Intothis paper,the three groups variables are introduced estimate upper boundof ofadaptive total disturbance, 1. INTRODUCTION disturbance. Intothis paper,the three groups ofof adaptive variables are introduced estimate upper bound total disturbance, 1. INTRODUCTION disturbance. Intothis paper, three variables and Chebyshev neural network isgroups introduced to approximate With the development of aerospace technology and the and are introduced estimate the upper boundofofadaptive total disturbance, 1. INTRODUCTION Chebyshev neural network is introduced to approximate With the development of aerospace technology and the are introduced to estimate the upper bound of total disturbance, total disturbance function. Control based on adaptive expansion of satellite applications, and Chebyshev neural networkControl is introduced to on approximate With the development of aerospace high-precision technology andspace the total disturbance function. basedprecision adaptive expansion of satellite applications, high-precision space and Chebyshev neural network is introduced to approximate robustness converges quickly, but control after With theand development of payloads aerospacerequire technology andspace the robustness missions higher demands total disturbance function. Control basedprecision on adaptive expansion ofhigh-precision satellite applications, high-precision converges quickly, butcontrol control after missions and high-precision payloads require higher demands total disturbance function. Control based on adaptive convergence is poor. However, based on neural expansion ofhigh-precision satellite applications, high-precision space convergence on satellite attitude control systems. order to meetdemands mission robustness converges butcontrol controlbased precision after missions and payloadsIn higher is poor. quickly, However, on neural on satellite attitude control systems. Inrequire order to meet mission robustness converges quickly, butcontrol controlbased precision after network approximation has slow convergence speed and missions and high-precision payloads require higher demands requirements, high-precision attitude control is required. In convergence is poor. However, on neural on satellite attitude control systems. Incontrol order toismeet mission approximation has slow convergence speed and requirements, high-precision attitude required. In network convergence is poor. However, control based on neural excellent control precision after convergence, which is on satellite attitude control systems. In order to meet mission recent years, a series of researches have been made on satellite network approximation has slow convergence speed and requirements, attitude is required. In excellent control precision afterTherefore, convergence, which isa recent years, a high-precision series of researches havecontrol been made on satellite network approximation has slow convergence speed and opposite of adaptive robustness. in this paper, requirements, high-precision attitude control is required. In attitude control by scholars, including optimal control (Wang excellent ofcontrol precision afterTherefore, convergence, recent years, a series of researches haveoptimal been made on satellite adaptive robustness. in thiswhich paper, is a attitude control by scholars, including control (Wang opposite excellent precision afterTherefore, convergence, switching function is introduced to combine adaptive recent years, a adaptive series of researches haveoptimal been onfeedback satellite & Ge 2002)., control (Wang et al. made 2015), opposite ofcontrol adaptive robustness. in thiswhich paper, isa attitude control by scholars, including control (Wang switching function is introduced to combine adaptive & Ge 2002)., control (Wang et al. 2015), feedback opposite adaptive Therefore, in and this aadaptive paper, robustness and neuralrobustness. network approximation controla attitude control by 1994), scholars, including control control (Jing etadaptive al. sliding modeoptimal control (Deng &(Wang Song robustness switchingofand function isnetwork introduced to combine & Ge 2002)., adaptive control (Wang et al. 2015), feedback neural approximation andhas aadaptive control control (Jing et al. 1994), sliding mode control (Deng & Song switching function is introduced to combine method based on that is developed, which quick & Ge 2002)., control al. 2015), H2 / H (Wang(Wang et al.et 2009), andfeedback various 2013), robustnessbased and neural network approximation a control al.control control (Jing etadaptive 1994), sliding mode control (Deng & Song method on and that is developed, whichand has quick robustness and neural network approximation and a control convergence speed excellent control precision. H / H control (Wang et al. 2009), and various 2013), 2 control (Jing et al. methods 1994), sliding (Deng &2015). Song convergence method based on and thatexcellent is developed, which has quick integrated (Zou mode etet al.al.control 2010, Li et speed control precision. H 22control / H control (Wang 2009), andal.various 2013), based on and thatexcellent is developed, which has quick integrated methods (Zou etet al.al. 2010, Li et 2015). method H 2control / methods H control (Wang 2009), andal.various 2013), convergence speed control precision. The control above can make the closed-loop system sectionsspeed in this paper are arranged as follows. In Section integrated control methods (Zou et al.the 2010, Li et al. system 2015). Other convergence and excellent control precision. The control methods above can make closed-loop in this are arranged In attitude Section integrated (Zou et al.the 2010, Li et al. system 2015). stable, but control most ofmethods them merely ensure asymptotically stable, Other 2, thesections kinematic andpaper dynamic model as forfollows. satellite The control methods above can make closed-loop Other sections in this paper are arranged as follows. In attitude Section stable, but most oftheoretically them merely ensure asymptotically stable, 2, the kinematic and dynamic model for satellite The control methods above can make the closed-loop system which means that it takes an infinitely long time Other sections in this paper are arranged as follows. In attitude Section tracking is established. Then some definitions and lemmas are stable, but most oftheoretically them merelyitensure asymptotically stable, 2, the kinematic and dynamic model for satellite which means that takes an infinitely long time tracking is established. Then someterminal definitions and lemmas are stable, but most oftheoretically them merelyitensure asymptotically stable, for the means control error to converge to zeroan and as a result, some 2, the kinematic and dynamic model for satellite attitude introduced. In Section 3, the sliding surface is which that takes infinitely long time tracking is established. Then someterminal definitions and lemmas are for the means control error to zeroanand as aemergence result, In Section 3, the sliding surface is which theoretically it to takes infinitely longsome time on-orbit tasks that cannot beconverge well accomplished. The of introduced. tracking is established. Then some definitions and lemmas are designed. In Section 4, adaptive robustness and neural network for the control error to converge to zero and as a result, some introduced. In Section 3, therobustness terminal and sliding surface is on-orbit tasks cannot bemethod well accomplished. of designed. In Section 4, adaptive neural network for the control error to converge to zero andThe as aemergence result, some the finite-time control makes it possible to overcome In are Section 3, theand terminal sliding is approximation introduced combined by a surface switching on-orbit tasks cannot bemethod well accomplished. The emergence of introduced. designed. In Section 4, adaptive robustness and neural network the finite-time control makes it possible to overcome approximation are introduced and combined by a switching on-orbit tasks cannot be well accomplished. The emergence of the disadvantage that the convergence time is too long and designed. In Section 4, adaptive robustness and neural network function, then a finite-time controller based on that is designed the finite-time control method makes it possible to overcome approximation are introduced and combined by aisswitching the disadvantage that the convergence time is too long and function, thenInaSection finite-time controller based on that designed4 the finite-time method makes it possible to overcome therefore ensurecontrol the timeliness of the task. are introduced and combined by in aisswitching and proved. 5, the controller designed Section the disadvantage that the convergence time is too long and approximation function, then a finite-time controller based on that designed4 therefore ensure the timeliness of the task. and proved. In Section 5, the controller designed in Section the disadvantage that the convergence time is too long and function, then finite-time controller based on thatinisSection designed4 is applied to In a amicrosatellite. therefore ensure the timeliness of the task. and proved. Section 5, the controller designed Different from previous literature, in this paper, a finite-time is applied to In a microsatellite. therefore ensure the timeliness of the task. and proved. Section 5, the controller designed in Section 4 Different from previous literature, in this paper, a finite-time attitude tracking control literature, method using terminal mode is applied to a microsatellite. Different from previous in this paper, sliding a finite-time is applied to a microsatellite. 2. SATELLITE MODEL AND CONTROL THEORY attitude tracking control method using terminal sliding mode Different fromTerminal previous literature, in this a finite-time is developed. isterminal anpaper, important part of 2. SATELLITE MODEL AND CONTROL THEORY attitude tracking controlsliding methodmode using sliding mode is developed. Terminal sliding mode isterminal an important part of 2. SATELLITE MODEL AND CONTROL THEORY attitude tracking control method using sliding mode finite-time control, which contains a fractional power term and is developed. Terminal sliding mode is an important part of 2. SATELLITE finite-time control, which contains a fractional power term and 2.1 Attitude Model ofMODEL SatelliteAND CONTROL THEORY is developed. Terminal sliding mode is an important part of can converge in finite time. finite-time control, which contains a fractional power term and 2.1 Attitude Model of Satellite can converge in finite time. finite-time control, which contains a fractional power term and Attitude Model of Satellite can converge in finite time. In all terms except control torque are 2.1 2.1 Attitude of kinematic Satellite equations based on quaternion In this paper,Model attitude cansatellite convergedynamics, in finite time. In satellite dynamics, all terms except control torque are In this paper, attitude kinematic equations based on quaternion regarded as total disturbance, which includes external T T T In satelliteasdynamics, all terms except control torque are In this paper, attitude kinematic equations T based Q q q q0 on q1 quaternion q2 q3 TT are used. Quaternion regarded total disturbance, which includes external T T 0 In satellite dynamics, all terms except control torque are disturbance and parameter uncertainties and many coupling T In this paper, attitude kinematic equations based on quaternion regarded asand total disturbance, which and includes external are used. Quaternion Q q00 q T T q00 q11 q22 q33 T disturbance parameter uncertainties many coupling regarded asand total disturbance, which includes external describes terms. Adaptive robustness and neural network approximation q0 q T Tand q0 isq1 theq2angular q3 T are used. the Quaternion attitude ofQthe satellite, disturbance parameter uncertainties and many coupling terms. Adaptive robustness and neural network approximation Q q0 q and q0 isq1 theq2angular q3 are used. Quaternion describesofthe attitude ofThe theattitude disturbance andthat parameter uncertainties and many coupling velocity are two ways are often used to deal with unknown satellite, the satellite. kinematic equations are terms. Adaptive robustness and neural network approximation describes the attitude of the satellite, and is the angular are two ways that are often used to deal with unknown theattitude satellite.ofThe kinematic are terms. Adaptive robustness and neural network approximation is the angular describesofthe theattitude satellite, and equations are two ways that are often used to deal with unknown velocity velocity of the satellite. The attitude kinematic equations are are two ways that are often used to deal with unknown CopyrightCopyright © 2019 IFAC 85rights velocity of the satellite. The attitude kinematic equations are 2405-8963 © 2019. The Authors. Published by Elsevier Ltd. All reserved. Copyright 2019responsibility IFAC 85 Control. Peer review©under of International Federation of Automatic Copyright © 2019 IFAC 85 10.1016/j.ifacol.2019.11.074 Copyright © 2019 IFAC 85
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1 q 2 q q0 E3 q 1 q T 0 2 0 q3 where q 0 q3 q2 q1 matrix of vector q .
satisfy which make the system x f x V x V x 0 , the system will be finite-time stable.
(1)
Definition 2 For the dynamic system x f x , x R n ,if
q2 q1 is the cross-multiplication 0
there is 0 and T 0 , which make x t for all t T , the system will be actual finite-time stable.
Lemma 2 If there is a continuous differentiable positive definite function V x , real number 0 , 0 1 and
0 , which make the system x f x satisfy V x V x , the system will be actual finite-time
In this paper, we use the Newton-Euler equation to establish the attitude dynamic equation
J J u d
(2)
stable.
where J 33 is the inertia matrix of the satellite, and u31 , d 31 , 31 are the control torque, the external disturbance torque and the disturbance torque caused by actuator saturation respectively. We assume that disturbance d and are both bounded. Further we assume that there are m0i 0 , m2i 0 and i max 0 , so that their components satisfy 2
d i m0i m2i , i 1, 2,3
(3)
i i max , i 1, 2,3
(4)
In addition, the following theorem is used in the following proof process. Lemma 3 For any real number xi , if 0 w 1 , the following inequality holds.
3 2 xi xi i 1 i 1 3
1 w
1 w 2
3. SLIDING SURFACE DESIGN
For the attitude tracking problem, the target attitude is represented by quaternion Qd , then attitude tracking error
The sliding surface is designed as follows.
quaternion Qe =Qd1 Q represents the error between the satellite attitude quaternion and the target quaternion. The Qe is direction cosine matrix decided by
where
s e kf qe ,
T e
T e e
e0 e
and s e ksig qe , 1
Assume the target angular velocity is d , the error angular velocity and its derivative can be described as
e C d e eCd C d
1 2 , 2 1 , 2 T
f qe , f qe1 , f qe 2 , f qe3 , . is a small
(5)
positive number.
(6)
Since
Then, the attitude model based on tracking error quaternion is 1 qe 2 qe qe 0 E3 e q 1 q T e e e 0 2 J e J J e C d C d u d
(10)
si 0 or qei qei sgn q ei , (11) f qei , 2 1 sgn qei qei 2 qei , si 0 and qei
C q q qe E3 2q q 2q q . 2 e0
(9)
sliding
e kdiag qei sgn qei
surface
(10)
satisfies
when s 0 , we select the
Lyapunov function V qeT qe 1 qe 0 and its derivative 1 2
(7)
V1 2qeT qe 2qe 0 1 qe 0 qeT e . According to Lemma 3, 1
(8)
1 3 2 2 V1 k qei k qeT qe 2 . Since V1 2qeT qe , we i 1 can finally get the inequality
2.2 Finite-time Control Theory Definition 1 For the dynamic system x f x , x R n , if
1
1
1 2 V1 kV1 2 0 (12) 2 According to Lemma 1, we can get the following conclusion.
there is a constant T 0 , which makes lim x t 0 and t T
x t 0 for all t T , the system will be finite-time stable.
Theorem 1 For the microsatellite attitude tracking control system described by (7) and (8), if the system state maintains on the sliding surface (10), the system state can converge to the equilibrium point along the sliding surface in finite time.
Lemma 1 If there is a continuous differentiable positive definite function V x , real number 0 and 0 1 ,
86
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control method. In order to realize smooth switching between adaptive control method and NN control method, define the switching function (Wu et al., 2013)
4. CONTROL LAW DESIGN 4.1 Proposal of the Control Law
0, e a e 2 a 2 n n sin h e 1 cos 2 b2 a 2 2 1, e b
Substituting the derivative of (10) into (8) (13)
J s D u
where
1 1 si 0 or qei qe qe0 E 3 , 2 Jkdiag qei R q 1 J 2 diag sgn qei qei 2 qe qe 0 E3 , 2 1 si 0 and qei
and
D J J C d C d R e d
Theorem 2 For the microsatellite attitude tracking control system described by (7) and (8), the sliding surface is selected as (10). If we use the control law
total disturbance. In the perspective of adaptive control, based on the assumption provided by (3) and (4), the components of D satisfy
JC
3
ij
d CQ ij
J ij C d Qij j 1
ai bi ci
m2i
u k s sig s 1 h ωe uN h ωe ua
(18)
uN Wˆ Φ(ω) εˆ sgn s
(19)
2 ua aˆ bˆ ω cˆ ω sgn s
(20)
m0i i max
J
2
Wˆ W 1 1 h ωe sΦT ω W 2Wˆ εˆ 1 1 h ωe s 2 εˆ
(14)
(22) (23) (24) (25)
the satellite attitude will converge to a small neighbourhood near zero in finite time, where k s 0 , 0 1 , a1 0 , a 2 0 , b1 0 , b 2 0 , c1 0 , c 2 0 , W 1 0 ,
In the perspective of neural network control, the unknown function D can be approximated by neural network. In this paper, Chebyshev neural network is used, which does not require extra parameters and is faster to calculate. Chebyshev neural network is based on Chebyshev polynomials, which can be obtained by using the two-term recursive formula (Zou et al. 2010) Ti 1 x 2 xTi x Ti 1 x
(21)
aˆ a1 h ωe s a 2 aˆ bˆ b1 h ωe s ω b 2 bˆ 2 cˆ c1 h ωe s ω c 2 cˆ
Therefore, three groups of adaptive variables ( a31 , b31 and c31 ) are introduced to estimate the upper bound of the total disturbance. aˆ , bˆ , cˆ are their estimations and a a aˆ , b b bˆ , c c cˆ are their estimation errors respectively.
W 2 0 ,
1 0 ,
(15)
s s1 s2 s3
2 0 ,
sig s s1 sgn s1
s2 sgn s2
T
,
T
s3 sgn s3 .
The block diagram of the controller is illustrated in Figure 1.
where T0 x 1 , T1 x x . An enhanced pattern using
Negative Feedback
Chebyshev polynomials for a vector X x1 ,, xm R m is T
TSM
given by qe
T
X 1, T1 x1 ,, Tn x1 ,, T1 xm ,, Tn xm (16) Thus, total disturbance can be approximated by D ω W ω
and the update law
2
*
, a e b
Combining two control methods with the switching function, the control law is proposed below.
is the
e
3 Di j 1
87
(17)
Kinematics
Qd
s h e
Adaptive Control NN Control
ks sig s
h ωe ua
+
1 h ω u
+
e
+ N
u
Switching Function
Dynamics
ω
ωe
+
ωd -
where W * is the optimal weight matrix and is the approximation error. Wˆ , ˆ are their estimations and W * Wˆ , ˆ are their estimation errors W respectively.
Figure 1 Controller block diagram 4.2 Proof of the Control Law Due to the switching function, (13) is rewritten as
The control method to be proposed in this paper is a combination of adaptive control method and neural network
Js 1 h ωe W *Φ(ω) ε h ωe D u 87
(26)
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The Lyapunov function is selected as
If
1 T 1 1 T tr W TW V2 s Js ε ε 2 2W 1 2 1
1 T 1 T a T a b b c c 2a1 2b1 2c1
Its derivative is
2
1
a1
W 1W 2
1
1 T ε T εˆ 1 h ωe 1 s a aˆ h ωe a1 s a1
1 T ˆ 1 T b b h ωe b1 s c cˆ h ωe c1 s b1 c1
1
1 2
1 s T Js 2
1 2
(30)
, and J max is the maximum eigenvalue
is satisfied.
1 T 1 T b b c c 2b1 2c1
2
2kW
2
W 2 2 kW tr W TW 2
(32)
2
Therefore, the following inequality holds. 2
(33)
According to Lemma 2, the system is finite-time stable and we have completed the proof of Theorem 2.
(31)
5. NUMERICAL SIMULATION AND DISCUSSIONS
1 2
i 0
1
1
2
V2 V2
2 kW 2 W2 tr W TW W W 2 tr W TWˆ
1
where W 0 0 a 0 b 0 c 0 .
1
where W
2 kW
1 1 1 T 1 T tr W TW s T Js ε ε a a 2 2 2 2 1 W1 a1
Rewrite it as
1
2 kW W 2 tr W TWˆ W 2 tr W TW W 2 tr W TW
and
1 1 1 2 2 2 1 1 1 T T T ε ε tr W W V2 s Js 2 2W 1 2 1 1 1 1 1 T 2 1 T 2 1 T 2 a a b b c c 2a1 2b1 2c1 W 0 0 a 0 b 0 c 0
For any 0 kW 2 ,
W 0
2
The first item in the right side satisfies
1 2 J max 2 of the inertia matrix J .
2
1 where i 0 max i 2 i T i ,1 i 2 i T iˆ , i1i 2 , and i 2 ki 2 ki refers to , a , b and c .
(29)
a 2 a T aˆ b 2 b T bˆ c 2 c T cˆ
ks
Then (29) can be rewritten as
V2 k s s T sig s W 2 tr W TWˆ 2 ε T εˆ
where
then
1
1 T i i i 2 i T iˆ 2i1
Then considering the update law (21)~(25), we get
3 2 ks s T sig s ks si i 1
,
Define
Follow the similar process, we can get
1 k s s T sig s tr W T Wˆ 1 h ωe W 1 sΦ T (ω) V2 =
1
T
.
2
Taking control law (18) into (28)
tr W TW 1 , W 1 W 2 tr W TWˆ
1 tr W TW W 2 tr W Wˆ 2W 1
c1
W 1
1 T ˆ 1 T b b c cˆ
b1
W 0 max W 2 tr W TW ,1 W 2 tr W TWˆ 2kW
s T 1 h ω W *Φ(ω) ε h ω D u V e e 2 1 1 1 T T T (28) tr W Wˆ ε εˆ a aˆ
W 1
tr W TW 1 , W W 2 tr W TW ; and if 2kW
2
W 2 2 kW
(27)
1
W 2 2 kW
W 2 tr W TWˆ .
In order to illustrate the performance of the controller designed in Section 4, a model was built in Simulink, and the controller was applied to the microsatellite attitude model. According to the microsatellite parameters provided in the literature (Lu & Xia 2013), the parameters used in the simulations are shown in Table 1. 88
2019 IFAC ACA August 27-30, 2019. Cranfield, UK
Inertia matrix Initial attitude
Target attitude
Zhang Kaicheng et al. / IFAC PapersOnLine 52-12 (2019) 85–90
89
Table 1 Simulation Parameters 20 1.2 0.9 J 1.2 17 1.4 0.9 1.4 15 0 Q0
0.06
0.8832
0.04 0.05 , T
0.3 0.2 0.3
T
sin 0.01t 4 rad / s , d 0.05 sin 0.03t 6 sin 0.02t
(b) Angular velocity error
Qd 1 0 0 0
T
Sliding surface parameters
k 1 , 0.5 , 0.1
6 cos 10x t 8sin 30 t 20 Disturbance d 3sin 2 y t 6 cos 50 t 30 104 torque ( Nm ) 6sin 10z t 16sin 40 t 20 Controller k s 20 , 0.1 parameters W 1 20 , W 2 0.2 , 10 , 1 a1 b1 c1 ˆ Update law 0.1 , W 0 03301 , 2 a2 b2 c2 parameters T ˆ 0 0 0 0 , aˆ 0 bˆ 0 cˆ 0
0.01
(c) Sliding variable
0.01 0.01
T
(d) Control torque Figure 2 Simulation results According to (18), if we set h ωe 1 , the controller (18) will
ua 0.5N m ， ub 0.5N m Torque limit Measure Power 4 10 14 s 2 , bias 10 6 rad / s noise Switching a 0.01 ， b 0.02 function The simulation result is presented in Figure 2.
become an “adaptive-only” controller. Similarly, if h ωe 0 , the controller (18) will become an “NN-only” controller. In order to further illustrate the performance of the proposed controller (18), it is compared with “adaptive-only” and “NNonly” controllers.
It can be observed from Figure 2 that both the attitude error and angular velocity error are bounded and will converge to sets around the origin in finite time, which indicates that the tracking objective is accomplished with the proposed controller based on adaptive robustness and neural network.
Two more simulations are made using “adaptive-only” controller and “NN-only” controller. Simulation parameters used in each simulation are the same of those shown in Table 1. Comparisons between controllers are shown in Figure 3.
(a) Attitude error comparison ( qe1 )
(a) Attitude error
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2019 IFAC ACA 90 August 27-30, 2019. Cranfield, UK
Zhang Kaicheng et al. / IFAC PapersOnLine 52-12 (2019) 85–90
Deng, L., and Song, S. (2013). Flexible spacecraft attitude robust tracking control based on fractional order sliding mode. Acta Aeronautica, 34(08), pp. 1915-1923. (in Chinese) Du, H., and Li, S. (2012). Finite-time attitude stabilization for a spacecraft using homogeneous method. Journal of Guidance, Control, and Dynamics, 35(3), pp. 740-748. Jing, W., Huang, W., Wu, Y., and Yang, D. (1994). QuasiEuler-angles feedback control of spacecraft attitude maneuver. Journal of Astronautics, (02), pp. 41-47+107. (in Chinese)
(b) Angular velocity error comparison ( x ) Figure 3 Control comparison Figure 3 shows the variation of some classical variables ( qe1 and x ) during the simulation. It can be concluded the controller proposed in this paper is a perfect combination of “adaptive-only” controller and “NN-only” controller. In the early stage of control process, the convergence speed of the proposed controller is almost the same as that of “adaptiveonly” controller; in the later stage of control process, the control precision of the proposed controller after convergence is almost the same as that of “NN-only” controller. It shows that the controller proposed in this paper has quicker convergence speed and higher control precision. The performance comparison of controllers is shown in Table 2.
Li, L., Hou, J., Shi, X., and Yang, J. (2015). Adaptive sliding mode control for spacecraft attitude tracking system. Electric Machines and Control, 19(02), pp. 96-100+108. (in Chinese) Lu, K., and Xia, Y. (2013). Adaptive attitude tracking control for rigid spacecraft with finite-time convergence. Oxford: Pergamon Press, Inc. Ran, D., Ni, Q., Sheng, T., and Chen, X. (2017). Spacecraft attitude maneuver algorithm design based on adaptive secondorder terminal sliding mode. Journal of National University of Defense Technology, 39(01), pp. 6-10. (in Chinese) Wang, H. (2015). Novel terminal sliding mode based finite time control research for spacecraft application. Master Degree. Civil Aviation University of China. (in Chinese)
Table 2 Control performance comparison Proposed Adaptive NN only controller only qe precision
1 10 5
2 104
1 10 5
e precision ( rad / s )
1 10 5
3 10 5
1 10 5
s precision
5 10 5
5 10 3
5 10 5
Convergence time ( s )
50
50
75
Wang, J., Zeng, M., and Xu, W. (2009). Nonlinear H control of large angle attitude maneuvers for satellite using sum of squares. 2009 International Conference on Mechatronics and Automation. Wang, W., Ren, Y., Li Y., and Luo Y. (2015). Nonlinear robust adaptive attitude maneuver control law for spacecraft. Systems Engineering and Electronics, 37(01), pp. 135-140. (in Chinese) Wang, Z., and Ge, X. (2002). Attitude control research of rigid spacecraft with two momentum wheels. Journal of Beijing Institute of Machinery, 17(03), pp. 1-5. (in Chinese) Wu, J., Chen, W., Zhao, D. and Li, J. (2013). Globally stable direct adaptive backstepping NN control for uncertain nonlinear strict-feedback systems. Neurocomputing, 122, pp.134-147.
6. CONCLUSION In this paper, a finite-time controller based on adaptive robustness and neural network approximation is designed for the problem of microsatellite attitude tracking control. A switching function combines adaptive robustness and neural network approximation. In the controller, the terminal sliding mode is adopted and a fractional power term is introduced in the sliding surface to ensure finite-time convergence. Lyapunov stability theorem is used to prove the stability of the controller. Numerical simulations have shown that the proposed controller has as fast convergence speed as controller based on adaptive robustness and as excellent precision as controller based on neural network approximation.
Zhang, A., Zhang, Z., and Huo, X. (2015). Finite-time Spacecraft Attitude Stabilization Control Subject to Parameter Uncertainties. Information and Control, 44(03), pp. 303308+315. (in Chinese) Zhou, Y., Zhu, W., and Du, H. (2017). Global finite-time attitude regulation using bounded feedback for a rigid spacecraft. Control Theory and Technology, 15(1), pp. 26-33. Zou, A., Kumar, K., and Hou, Z. (2010). Quaternion-Based Adaptive Output Feedback Attitude Control of Spacecraft Using Chebyshev Neural Networks. IEEE Transactions on Neural Networks, 21(9), pp.1457-1471.
REFERENCES Bhat, S., Bernstein D. (2000). Finite-time Stability of Continuous Autonomous Systems. SIAM J on Control and Optimization, 38(3), pp. 751-766. 90