Adaptive Fault-tolerant Control with Control Allocation for Flight Systems with Severe Actuator Failures and Input Saturation

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1 213 American Control Conference (ACC) Washington, DC, USA, Jne 17-19, 213 Adaptive Falt-tolerant Control with Control Allocation for Flight Systems with Severe Actator Failres and Inpt Satration Man Wang, Jianying Yang, Gozheng Qin and Yingxin Yan Abstract This paper proposes a novel adaptive falt tolerant control strategy for flight systems which have no sfficient actator redndancy after severe actator failres. In addition to distribting the control signals to the remaining actators based on the effectiveness of actators, this strategy tilizes model reference adaptive control (MRAC) to compensate for the tracking error cased by severe failres. Frthermore, an improved weighting algorithm and an anti-satration controller are developed to compensate for the satration error. Finally, a simlation of the satellite lanch vehicle is condcted to demonstrate the effectiveness of the proposed strategy. Compared to the traditional falt tolerant control with control allocation method, the proposed strategy gives better performance. I. INTRODUCTION Actator failres of aircraft significantly redce reliability performance and even can case catastrophic accidents. Based on this consideration, designers often add redndant actators to aircraft, which can provide degrees of freedom to design falt tolerant control () systems [1]. Zhang presented an overview of historical, crrent, and ftre developments on systems in [2]. Control allocation (CA) is an effective method that can manage actator redndancy, i.e., distribte the virtal control law reqirements to the redndant actators in the best manner while acconting for their constraints [3]. Particlarly, when some of the actators lose effectiveness or become damaged, CA is able to tilize the remaining healthy actators to ensre that the aircraft performs well. Several poplar approaches and applications have been developed for CA. Oppenheimer gave a srvey of linear control allocation techniqes, and presented the methods in detail in [4]. Bodson evalated the performance and comptational reqirements of optimization methods for control allocation in []. Härkegård compared control allocation with optimal control for solving actator redndancy in [6]. Arn Kishore considered CA along with distrbance rejection and gave an algorithm that pdated the weighting matrix to deal with actator limits in [7]. CA is widely sed to realize, especially in aerospace systems. Zho introdced two reconfigrable control allocation schemes and illstrated them with an nmanned *This work is spported by the National Natral Science Key Fondation of China nder Grant No , No and No.99163; Fondation of National Defence B321123; Fondation of national astronatica key lab ( ). M. Wang, J. Yang and G. Qin are with the State Key Laboratory for Trblence and Complex Systems, Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 1871, P. R. Chinamairo26@gmail.com;yjy@pk.ed.cn; gzhqin@gmail.com Y. Yan is with the BeiJing Institte of Nearspace Vehicle s System Engineering, Beijing 176, P. R. China yanyingxin@yahoo.cn r(t) Fig. 1. x(t) Baseline Adaptive Falttolerant v(t) Control Allocation Control Reallocation (t) a (t) system The overall strctre of (solid line) and A (dotted line). aerial vehicle nder stck actator failres in [9]. Alwi combined sliding modes control with control allocation for in [1]. Casavola sed an ad hoc online parameter estimator and a control allocation algorithm to realize control reconfigration adaptively [11]. The control strctre of the traditional methods can be described by Fig. 1 (solid line). However, the existing literatres only deal with the sitation that actator redndancy still exists after a failre. If most actators of the aircraft experience failres, control allocation alone will not be able to realize. In this paper, a novel method combining CA with model reference adaptive control (MRAC) will be explored to achieve, and the control strctre is described by Fig. 1 (dotted line). Besides achieving CA, MRAC can garantee the closed-loop stability of the system, redce the error cased by the severe failre, and track the reference model well. What s more, de to the severe failre, the weighting algorithm (WA) which can handle the satration error nder the normal sitation in [7] is no longer effective, and the satration error may reslt in a bad effect on the stability of the system. To solve this problem, we compensate for the satration error sing a feedback method. The rest of this paper is organized as follows. Section II describes the system with actator failres and introdces the control objective. Section III gives a detailed description of falt tolerant control allocation sbject to actator satration. The adaptive falt tolerant control (A) method is proposed in Section IV. A satellite lanch vehicle model is presented and simlated in section V. Some conclding remarks are given in Section VI. II. PROBLEM STATEMENT Fig. 2 describes the A strategy. The baseline controller is designed according to [7], and it offers the initial vale for the adaptive falt tolerant controller. The objective of the adaptive falt tolerant controller is to track the reference model with desired performance when the system experiences a severe failre. The control reallocation modle is y(t) /$ AACC 141

2 r(t) x(t) Baseline v Adaptive Falttolerant v c(t) v s(t) Control Reallocation Antisatration (t) Weighting Algorithm (t) a (t) system y(t) B. Control Objective The control objective is to design an adaptive falt-tolerant feedback controller ˆK(t) R n m and tilize control reallocation strategy to achieve desired performance when the aircraft experiences a severe failre, and then find a control inpt (t) satisfying (3) and (4). Fig. 2. The detailed description of A. III. CONTROL REALLOCATION According to (4), the virtal inpt is defined as able to allocate the virtal inpt based on the actator effectiveness matrix. In addition to improving the weighting algorithm, an anti-satration feedback controller is proposed to stabilize the system when it experiences a severe failre. A. Plant Description This paper considers a linear time-invariant system sbject to actator failres described by ẋ(t) = Ax(t) + B a (t) y(t) = Cx(t) where x(t) R m, a (t) R q and y(t) R n are the state vector, actal control inpt entering the system and controlled otpt, respectively. A,B,C are constant matrices of appropriate dimensions. R q q is a piecewise constant ncertain control effectiveness matrix that can be expressed as = diag(δ 1,δ 2,...,δ q ), where δ i 1,i = 1,2,...,q indicates the effectiveness of the ith actator. For example, δ i = 1 indicates the ith actator is working well; δ i = means the ith actator has damaged completely; < δ i < 1 implies the ith actator has a failre. Denoting the satration error as ũ(t) = a (t) (t) and B = B f, the state eqation can be modified as (1) ẋ(t) = Ax(t) + B f (t) + B f ũ(t) (2) where (t) is the control inpt. The following assmptions on the system are reqired: (a) The nmber of control inpts is greater than the nmber of control variables, that is, q > m. (b) The matrix B f can be factorized into a matrix denoted as B v R m n with fll colmn rank and a matrix denoted as B R n q with fll row rank. (c) The control inpt (t) satisfies min (t) max (3) where max := [ max1, max2,..., maxq ] T and min := [ min1, min2,..., minq ] T are the pper bond and lower bond of the actators respectively. (d) (A,B f ) is controllable. Assmption (a) means the aircraft has redndant actators. Assmption (b) implies the following relations: B f (t) = B v B(t) = B v v(t) (4) where v(t) R n is defined as the virtal inpt. Assmption (c) presents the actator limits of the system. v(t) = B(t) () Different from the traditional control allocation problem, the matrix B incorporates the falt information. Generally, the control reallocation problem can be formlated as the optimization problem [1]: min (t) T W, sbject to (4) (6) where W = diag(w 1,w 2,...,w q ) is a diagonal positive definite weighting matrix, and the scalar w i is the weight of the ith actator. The soltion of problem (6) is [1]: (t) = W 1 B T (BW 1 B T ) 1 v(t) (7) Eqation (7) can be sed to distribte the virtal inpt to the still operating actators. Unfortnately, the optimal soltion is sometimes infeasible de to the actator limits. To solve this problem, a weighting algorithm is provided in [7]. The scalar w i is pdated when the ith actator exceeds the actator limits. In order to improve this algorithm, a small positive scalar < η 1 is mltiplied by the actator limits in this paper. With the η-weighting algorithm, more parameters can be sed to pdate the weighting matrix. Denoting Ŵ = ΘWΘ, where Θ = diag(θ 1,θ 2,...,θ q ) is a positive definite matrix and (1 + ε) i η i max, i (t) > η i max θ i = 1, η i min i (t) η i max (8) (1 + ε) i η i min, i (t) < η i min where ε is a small positive scalar, (7) can be modified as (t) = Ŵ 1 B T (BŴ 1 B T ) 1 v(t) (9) Unfortnately, this algorithm may become invalid when the aircraft experiences a severe failre. As most actators experience failres, the satration errors will be cased in the remaining operating actators, which may not be eliminated by the η-weighting algorithm. The larger satration errors will indce performance degradation and closed-loop instability. To solve this problem, an anti-satration controller is designed to compensate for the errors when the η-weighting algorithm is invalid [12]. Theorem 1. When a severe failre occrs, the satration error ũ can be compensated for by the following antisatration controller v s : v s = Bũ (1) 142

3 Proof: Combining (2) with (4), the closed-loop system can be written as ẋ = Ax + B v v + B f ũ = Ax + B v (v c + v s ) + B f ũ = Ax + B v v c + B v (v s + Bũ) (11) where v c is the otpt of the adaptive falt-tolerant controller, as is shown in Fig. 2. It is clear that sbstitting (1) into (11) can compensate for the satration error. The proof is completed. A. Baseline IV. CONTROLLER DESIGN The baseline controller v c R n is designed as v c = K x + K ref r (12) where K R n m and K ref R n n denote the state feedback controller and feedforward controller respectively. The state feedback controller is designed sing the H method. In order to track the reference inpt r(t) R n, the feedforward controller for the nominal system is designed as K ref = K C B v AC (13) where B v and C are the psedo inverse of B v and C. The detailed description can be fond in [14]. Note that the baseline controller can stabilize the system and track the reference signal in a falt free condition. The controller need not be reconfigred when an actator failre occrs dring the operation if there is sfficient actator redndancy. Under this circmstance, combining the baseline controller with CA strategy can achieve satisfactory performance easily. However, when most actators lose effectiveness or become damaged, i.e., redndant actators no longer exist, the traditional CA strategy will fail. Ths, an adaptive falt-tolerant controller is proposed in this paper to solve this problem. B. Adaptive Falt-tolerant The ideal system with no actator failres (the effectiveness matrix = I) is chosen as the reference system. Denoting A + B v K = A m B v K ref = B m (14) the reference system can be defined as ẋ m = A m x m + B m r () where A m R m m is a Hrwitz matix; x m R m is the desired state. Given a symmetric positive definite matrix Q R m m, the following Lyapnov fnction has a niqe soltion P R m m [16]: A T mp + PA m = Q, P = P T > (16) The baseline controller garantees that the reference system has satisfactory stability and dynamic performance, bt it may be invalid when a severe failre occrs. De to the ncertainty of the failre, an adaptive control signal can be sed to compensate for the failre. The adaptive falt-tolerant controller is designed as ˆv c = ˆK ref r + ˆKx (17) where ˆK ref and ˆK are the estimates of K ref and K respectively. Denoting the error signals as and with (13), we can get K ref = ˆK ref K ref K = ˆK K (18) ˆK ref = ˆKC B v AC (19) Combining with (2), (4), (1) and (19), the closed-loop system can be rewritten as ẋ = Ax + B v ( ˆK ref r + ˆKx) = Ax + B v ( ˆKC B v AC )r + ˆKx) = Ax + B v ( (K + K)C B v AC )r + ˆKx) = Ax + B v (K ref KC )r + B v K x + B v Kx = A m x + B m r + B v K(x C r) (2) Denoting e = x x m, combining with (17) and (18) yields the error dynamics ė = A m e + B v K(x C r) (21) The error dynamic (21) associates the parameter error K with the tracking error e. Ths, the following theorem can be sed to generate the estimated parameter ˆK. Theorem 2. For the control system (1), the adaptive controller (17) with the gain adaption law ˆK = Γ k B v T Pe(x C r) T (22) where Γ k R n n is a positive definite matrix, garantee that all closed-loop signals remain bonded and that the tracking error e(t) = x(t) x m (t) converges to as t. Proof: In order to analyze the closed-loop stability and tracking performance, define the following candidate Lyapnov fnction: V = e T Pe + trace( K T Γ k 1 K) (23) where P is given by (16), and trace( ) is the trace of a matrix. Then, V = (ė T Pe + e T Pė) + 2trace( K T Γ k 1 K) = e T (A T mp + PA m )e +2trace[ K T B v T Pe(x C r) T + K T Γ k 1 ˆK] (24) As K = ˆK, sbstitting the adaptive law (22) into (24) gives V = e T Qe (2) Since V (t) is a positive definite fnction and V (t), it is easy to obtain V (t) L, which indicates that e(t), K(t) L. Therefore, all the closed-loop signals are 143

4 bonded, and e(t) L 2. According to (21) and the bondedness of the closed-loop signal, we can get ė(t) L, and ths lim t e(t) =. De to e(t) L 2 L, ˆK(t) L 2 L and x, r L, all signals and estimated parameters are bonded, realizing lim t [x(t) x m (t)] =. 2 1 Weighting Algorithm 2 1 η Weighting Algorithm V. APPLICATION EXAMPLE Satellite lanch vehicles (SLVs) play an important role in placing artificial satellites and space stations into earth orbit, and they reqire accrate positioning of the payload in the desired orbit, even in adverse conditions []. Ths, it is necessary to design a control system for SLVs to meet the desired reqirements. In this paper, a SLV model is simlated to demonstrate the effectiveness of the proposed algorithm. The physical system of SLVs chosen in this paper has eight actators to control for thrsters, ths it has enogh redndant actators. This paper focses on garanteeing the performance of SLVs withot any significant degradation when a severe failre occrs, i.e., when most actators lose effectiveness or become damaged. Compared to the traditional CA strategy, the proposed MRAC scheme associated with CA has a significant advantage. The state space model of SLV is given by (1). The state incldes the pitch θ, pitch rate θ, yaw ψ, yaw rate ψ, roll φ and roll rate φ, and they are controlled by eight actators. x = [θ θ ψ ψ φ T φ] y = [θ ψ φ] T = [ ] T The control signals i [,] 18,i = 1,...,4 are mainly sed to control the actators of strapons, and the control inpts i [,] π 18,i =,...,8 are main engine trsters. The system matrices are given in the Appendix, and the reference inpts are described as: π 7 1 WA bond η WA bond Fig. 3. Case 1: control inpt 7 of the weighting algorithm (WA) and the η-weighting algorithm (η-wa). B. Case 2 The simlation reslts of the novel A method and the traditional method as shown in Fig. 1 are compared in this case. The otpt tracking errors and control inpts are shown in Fig. and Fig. 6. Assme that 1, 2, 3 lose 3% effectiveness and 4,, 6 are totally damaged at the beginning of the flight. Under this circmstance, the η- weighting algorithm becomes invalid since 7 has exceeded the actator limits. The anti-satration controller is designed according to (1). As shown in Fig., the traditional method exhibits large tracking error in pitch angle. By contrast, the tracking errors of the A method are significantly redced by sing the gain adaption law (22). r i (t) = { 4deg, s t < 2s, else, i = 1,2, Weighting Algorithm A. Case 1 The ideal system in (1) with = I is sed to verify the advantages of the η-weighting algorithm. The simlation reslts of the weighting algorithm with diag(η 1,η 2,...η 8 ) = I and the η-weighting algorithm with diag(η 1,η 2,...η 8 ) = diag( 1 3, 1 3, 1 3, 1 3, 1 3, 1 3, 1 3, 1 3 ) are shown in Fig. 3 and Fig. 4. The two algorithms have the same ε and the same initial vale of Ŵ. The η-weighting algorithm can enforce all control inpts into actator limits, whereas the original weighting algorithm can not de to 7. The simlation reslts of 7 are depicted in Fig. 3. The variation of Ŵ indicates the points where there was a chance of satration with prior vale of Ŵ, as shown in Fig. 4. Obviosly, Ŵ in the η- weighting algorithm pdates more freqently than that in the weighting algorithm, and ths the η-weighting algorithm improves the efficiency. Trace(W) Trace(W) η Weighting Algorithm Fig. 4. Case 1: trace(w) of the weighting algorithm (WA) and the η- weighting algorithm (η-wa). 144

5 θ φ Pitch otpt x Roll otpt A 4 A Fig.. ψ x 1 Yaw otpt A Case 2: tracking errors of and A. VI. CONCLUSIONS In this paper we addressed some open problems in falt tolerant control () of systems with severe actator failres. We presented a soltion to sch a problem that there is no sfficient actator redndancy after the failres. A modified method, which combines control allocation with model reference adaptive control, was developed and shown to garantee the stability and tracking performance of the system. The practical limitations sch as actator satration was also taken into consideration in this paper. An improved weighting algorithm and an anti-satration controller were developed to compensate for the satration error. Finally, simlation of the SLV model demonstrated that the proposed method gave better tracking error performance. Ftre work will focs on extension to the case with timevarying actator failres A A A A 2 bond Fig A A A A Case 2: control inpts of and A. APPENDIX The system matrices of the SLV model are A = C = B v = e e e 4.468e ] 4 [ e e e e 3 4.6e e B = B = B vb REFERENCES [1] S. Isik, O. Tekinalp and I. Yavrck. Falt tolerant control of an over actated UAV. AIAA Gidance, Navigation, and Control Conference, 211 [2] Y. M. Zhang and J. Jiang. Bibliographical review on reconfigrable falt-tolerant control systems. IFAC Annal Review in Control, 28. [3] J. Peterson and M. Bodson. Interior-point algorithms for control allocation. Jornal of Gidance, Control, and Dynamics, 28(3): , 2. 14

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