Advances in Computer Science Research (ACSR), volume 76 7th International Conference on Education, Management, Information and Mechanical Engineering (EMIM 2017) Satellite Attitude Control with Actuator Failure Haitao Meng1,a, Aihua Zhang1,b, Xing Huo1,c and Zhiyong She2,d 1College of Engineering, Bohai University, Jinzhou 121013, China 2 China Beijing Aerospace Technology Institute [email protected], [email protected],[email protected], [email protected] Keywords: Observer; Backstepping; Actuator fault; Attitude stabilization control Abstract. A method of attitude stabilization control, based on observer design is proposed for an on-orbiting spacecraft in the presence of partial loss of actuator effectiveness, external disturbance and actuator control input saturation problem. In this approach, observer is employed to estimate the value of actuator fault, and a Backstepping attitude controller is then designed to achieve fault tolerant control and external disturbance rejection. The Lyapunov stability analysis shows that the closed-loop attitude system is guaranteed to be almost asymptotically stable. The simulation results show the effectiveness of the derived control law. Introduction It is inevitable that the satellite components is in trouble, so it is one of the hot spots in the research of attitude control to design the control algorithm to achieve the fault tolerant control. A spacecraft attitude compensation controller is designed using adaptive control method in literature [13]. In literature [14], a fault tolerant attitude control strategy is proposed for the non singular terminal sliding mode control method. In literature [9], a passive fault tolerant controller is designed based on time delay method. The attitude tracking control can be realized in the case of 4 flywheels, but the controller must obtain the fault information of the actuators. Application of two stage Kalman filtering algorithm in literature [11]. The Backstepping control method has many advantages such as the decomposition of the complex nonlinear system into many subsystems and the simple design procedure of the controller. Therefore the backstepping can be used to design the attitude controller. Considering the problem of actuator failure, external disturbance and actuator control input constraint in this paper, a method of attitude stabilization control is proposed to achieve high precision and high stability attitude stabilization control. Mathematical Model of the Satellite For description method using Euler angles of satellite attitude exists the singular problems, this paper adopts MRPs describe the satellite attitude, and then the rigid satellite attitude mathematical model can be described by: 1 Τ × Τ σɺ =()1 − σ σI + 2 σ + 2 σ σ ω 4 3 = F ()σ ω (1) × Jωɺ + ω J ω = τ + d (2) Τ ω=[ ω ω ω ] ∈ R3 where 1 2 3 is that coordinate system relative to the inertial coordinate system of Τ τ=[ τ τ τ ] ∈ R3 angular velocity to project on the body coordinate system, 1 2 3 is the total control Τ d=[ d d d] ∈ R3 moment that is actually applied to satellite coordinate system, 1 2 3 is external disturbs 3× 3 matrix of the satellite. Matrix JR∈ (positive definite and symmetric) is the moment of inertia of Copyright © 2017, the Authors. Published by Atlantis Press. 323 This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/). Advances in Computer Science Research (ACSR), volume 76 the satellite array. In particular, Ep.(1) is the satellite attitude kinematics, and Ep. (2) is a rigid satellite attitude dynamics, and all of them is satisfied with Assumption 1. Assumption 1: Assumes that the external disturbance d is bounded, and there is constant d > 0 d≤ d max to make max to be established. Considering actuator partial failure and the fault is modeled as a multiplicative factor, so the actuator failure acting on the total control torque of the satellite ontology is as follows: τ= ρ τ ()t c (3) Τ τ= [ τ τ τ ] ∈ R3 where c c1 c 2 c 3 is actuator command control torque, ρ= [ ρ ρ ρ ]Τ ∈ 3× 3 ()()()()t diag( 1 t 2 t 3 t) R 0<ρ (), t ≤ 1 i = 1,2,3 is actuator fault degree, i , and where ρ ()t τ= τ i =1 and i ci , actuator normal operation and control torque is consistent with the actual torque; 0<ρ (t ) < 1 τ= ρ()t ρ < ρ If i , i i ci i , actuator partial failure. Controller Design In this approach, observer is employed to estimate the value of actuator faults, and a Backstepping attitude controller is then designed. ρ Observer Design. Because the fault factor ()t is a diagonal matrix, Eq. (3) can be rewritten by ρ τ= ρ ()()tc U t (4) Τ = [τ τ τ ] ∈ 3× 3 Τ U diag( c1()()() t c 2 t c 3 t) R ρ()()()()t= [ ρ t ρ t ρ t ] where , 1 2 3 , the satellite state dynamics Eq. (2) can be rewritten in the form of partial failure Eq.(3) ω= − ω× ω + ρ + Jɺ J U() t d (5) According to the attitude dynamics of spacecraft actuator fault Eq.(5), the design follows the ρ observer to estimate the fault ()t . ⌢ ⌢× ⌢ ⌢ Jωɺ = − ω J ω + U ρ() t ⌢ ⌢ −Γ()()ω − ω −l sng ω − ω 1 (6) ⌢ ⌢ ⌢ ρ()()t= l ρ t − Τ + l ( ω − ω) 2 3 (7) ⌢ ρ⌢ ρ whereω and ()t are the respectively estimated value of ω and ()t .Τ ∈ R is the update time of 3× 3 l∈ R the observer. Γ ∈ R , i + is Observer gain. In order to evaluate the performance of the observer I Eq.(6) - Eq.(7) fault estimation, the energy index p is defined. 1 t 2 2 I= e + σ dt p ∫ t 0 (8) τ≤ τ Attitude Stabilization Controller Design. When i max (i=1, 2, 3), application of observer ρ Eq.(6) - Eq.(7) can realize the accurate estimation of the failurepart ()t . = σ x1 dt x = σ x = ω Variable ∫ , 2 and 3 . According to Eq.(1) and Eq.(5), there are 324 Advances in Computer Science Research (ACSR), volume 76 xɺ = x 1 2 (9) xɺ = F() x x 2 2 3 (10) × ⌢ ⌢ Jxɺ = − x Jx +ρ()() t τ − δ t τ + d 3 3 3 c c (11) ⌢ ρ⌢= [] ρ ⌢ ρ ⌢ ρ ⌢ Τ δ= [] δ δ δ Τ where ()t diag ( 1 2 3 ) ， ()t diag ( 1 2 3 ). From the Eq.(9) to Eq.(11) shown in the form of the system structure, the standard Backstepping method can be used to design the controller. To carry out the following transformation z= x z = x −α z = x − α 1 1 2 2 1 3 3 2 (12) α ∈ R3 α ∈ R3 Where 1 and 2 are the virtual control inputs for backstepping control. The following steps can be divided into the controller design Step 1: According to the formula Eq.(12), available zɺ = xɺ = x = z + α 1 1 2 2 1 (13) V= 0.5 zΤ z Selected Lyapunov candidate function 1 1 1 and design the virtual control Variable .where α = −c x c 1 1 1 and where 1 is constant. Vzzzzczɺ =ΤΤɺ =() − = − cz2 + zzΤ 1111211 11 12 (14) 2 z = 0 Vɺ = − c z z So when 2 , 1 1 1 , 1 will be at the asymptotic convergence. z Step 2: according to the type of 2 derivative Eq.(16) available. zɺ = xɺ −αɺ = F() x x + c x 2 2 1 2 3 1 2 (15) V= V + 0.5 zΤ z α Select another Lyapunov function 2 1 2 2 and 2 for the design of virtual control input. α =F−1() x( − z − c x − c z ) 2 2 1 1 2 2 2 (16) c Where 2 is constant. Apply (16), there are ɺ = −2 +ΤΤ + [ + ] V2 cz 1 1 zzzFxxcx 1 2 2() 2 3 1 2 = −c z2 − c z2 + zΤ F() x x 1 1 2 2 2 2 3 (17) 2 2 z = 0 Vɺ = − c z − c z The same type from Eq.(17) to prove that when 3 , 2 1 1 1 1 . At this z z time 1 and 2 will converge. Aiming at the fault spacecraft attitude control system Eq.(1), Eq.(5), the application of observer Eq.(6) design command control: τ= Sat( v, τ ) c max (18) τ ∈ 3 The controller c input v R is designed as 325 Advances in Computer Science Research (ACSR), volume 76 −1 ⌢ − dF() x = ρ 1 2 ( + + ) v() t()[ t J z1 c 1 x 2 c 2 z 2 dt +−1 ( + + ) F() x2 zɺ 1 c 1 xɺ 2 c 2 zɺ 2 ] +× − − − Τ x3 Jx 3 c 3 z 3 k 1 xa F() x2 z 2 χ 2 zɺ − Τd 3 } χz + εexp( − βt ) Τd 3 (19) χ= τ + d ε ∈ R k∈ R β c Where Τd max max , + is an arbitrary small constant. 1 + , and 3 for control gain, x∈ R a + is the state variable of the following auxiliary system. ρ⌢ 2∆ 2 = − − ()t u −ρ⌢ ∆ xɺa k2 xa xa () t u x 2 a (20) ∆u =τ − v k∈ R c , 2 + . If 2 − > −k1 −1 > c31 0, k 2 0 2 2 (21) The whole closed-loop attitude control system is stable. Prove: By Eq.(11) and Eq.(16) available =−1 − × +ρ τ + zɺ3 J x 3 Jx 3 () tc d dF−1() x + 2 (z+ c x + c z ) dt 1 1 2 2 2 +F−1() x( zɺ + c xɺ + cz ɺ ) 2 1 1 2 2 2 (22) If a Lyapunov candidate function is defined = +1Τ + 1 Τ V3 V 2 z 3 Jz 3 xa x a 2 2 (23) The function of the time derivative of V3, and put Eq.(22) into ɺ = −2 −2 −2 − 2 V3 c 1 z 1 c 2 z 2 c 3 z 3 k 2 xa −ρ⌢ 2 ∆ 2 −Τρ⌢ ∆ ()t u xa () t u ⌢ +Τ ρ⌢ ∆ − − δ τ z3{() t u k 1 xa () t c z χ 2 +d − 3 Τd } zχ+ εexp( − β t ) 3 Τd (24) ⌢ −δ()t τ + d ≤ τ + d According to the characteristic of saturation function, c max max .