Robust fault-tolerant control for flexible spacecraft against partial actuator failures

In this paper, we present a robust fault-tolerant control scheme to achieve attitude control of flexible spacecraft with disturbances and actuator failures. It is shown that the control algorithms are not only attenuate exogenous bounded disturbances with attenuation level, but also able to tolerate partial loss of actuator effectiveness. The proposed controller design is simple and can guarantee the faulty closed-loop system to be quadratically stable with a prescribed upper bound of the cost function. The design algorithms are obtained by combining free weighting matrices method with linear matrix inequality technique. The effectiveness of the proposed design method is demonstrated in a spacecraft attitude control system subject to loss of actuator effectiveness.


Introduction
High precision attitude control has been a difficult and important problem for flexible spacecraft in commu-R. Zhang · J. Qiao · L. Guo School of Instrumentation Science and Opto-Electronics Engineering, Beihang University, Beijing 100191, China T. Li (B) Institute of Advanced Technology, Nanjing University of Posts and Telecommunications, Nanjing 210046, China e-mail: litaojia@163.com nication, navigation, remote sensing, and other spacerelated missions. It is because modern spacecraft often employ large, deployed and light damping structures (such as solar paddles and antenna reflectors) to provide sufficient power supply and reduce launch costs [1][2][3][4][5][6]. During the control of the rigid body attitude, actuators play an important role of linking control commands to physical actions [7,8]. Normally, the actuators should execute commands demanded by the controller faithfully and completely. In this condition, the actuators need to be 100 % effective. However, when a fault occurs in the actuator, the handicapped actuator may not complete the control command fully. Naturally, the control channel effectiveness (or lack of it) becomes an appropriate measure of the severity of the actuator fault [9]. In an spacecraft, actuator faults may cause discrepancies between the desired and the actual movements of these control surfaces due to incorrect supply pressure in the hydraulic lines, change in hydraulic compliance, and line leakage [10]. Any of these problems can prevent the primary control surfaces such as elevators, ailerons, or rudder from moving to the positions demanded by the controller [9]. On the other hand, the complex space structure may lead to the decreased rigidity and low-frequency elastic modes. However, elastic vibration of the flexible appendages may cause degradation of the performance of attitude control [7,11]. Thus, the desired control scheme should tolerate partial loss of actuator effectiveness and be robust enough to overcome various disturbances from structural vibrations of the flexible appendages.
Due to the increasing demands for high reliability and survivability of the complex control systems, the fault-tolerant control (FTC) has attracted extensive interests and attention [12][13][14][15][16][17][18][19][20][21]. FTC can be divided into passive FTC [12,13] and active ones [14,15]. An active FTC uses the diagnosis results provided by the fault detection and diagnosis to actively adjust the control efforts, thus is potentially capable of dealing with a larger number of faults [14,15]. Compared with the active FTC, the passive one has the advantage of not requiring the exact actuator fault information; thus, it is simple to implement. The passive FTC can also ensure system stability and desired performance after the actuator fault occurs and before the fault detection and diagnosis phase finishes [12,13].
Motivated by the preceding discussion, in this paper, a passive FTC scheme for flexible spacecraft with disturbances and partial loss of actuator effectiveness is studied. First, the partial loss of actuator effectiveness problem is transformed into uncertain parameters problem. Second, the fault tolerant control is designed by combining H ∞ control technique and robust control method. The proposed control algorithms are not only attenuate disturbances from structural vibrations of the flexible appendages with H ∞ attenuation level, but also able to robust to partial loss of actuator effectiveness. Meanwhile, the resultant FT controller may be simply designed and can guarantee the faulty closed-loop system to be quadratically stable with a prescribed upper bound of the cost function. Finally, a numerical example is shown to demonstrate the good performance of our method.
The rest of this paper is organized as follows. The single-axis model of flexible spacecraft model and partial loss of actuator effectiveness are described in Sect. 2. The passive FT controller is designed and analyzed in Sect. 3. Numerical simulations on different control effectiveness factor situations are presented in Sect. 4 to demonstrate the performance of the proposed control method. Finally, we conclude the paper in Sect. 5.
Notation: Throughout this paper, R n denotes the ndimensional Euclidean space; the space of squareintegrable vector functions over [0, ∞) is denoted by l 2 [0, ∞); the superscripts " " and "−1" stand for matrix transposition and matrix inverse, respectively; P > (≥ 0) means that P is real symmetric and positive definite (semidefinite). The identity andzero matrices are denoted by I and 0, respectively, with appropriate dimensions. In symmetric block matrices or complex matrix expressions, diag{. . .} stands for a blockdiagonal matrix, and * represents a term that is induced by symmetry. For a vector ν(t), its norm is given by Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for related algebraic operations.

Problem formulation and preliminaries
Similarly to the references [11,22], the single-axis model can be derived from the nonlinear attitude dynamics of the flexible spacecraft. In this paper, the problem is simplified and only considers the singleaxis rotational maneuver. It is assumed that this model includes one rigid body and one flexible appendage, and the relative elastic spacecraft model is described as: where θ(t) is the attitude angle, J is the spacecraft inertia about the pitch axis, η(t) is the flexible modal coordinate, F is the rigid-elastic coupling matrix, u F (t) is the control torque generated by the reaction wheels that are installed in the flexible space- is the damping ratio, and i , (i = 1, . . . n) is the modal frequency. Λ = diag{ 2 1 , . . . 2 n } is stiffness matrix. n is their dimensions. Since vibration energy is concentrated in low-frequency modes in a flexible structure, its reduced order model can be obtained by modal truncation. In this paper, only the first two bending modes are taken into account. Then we can get To formulate the FTC problem, the fault model must be established. According to the fault type for flightcontrol system established in [15,18], the fault type considered in this study is the loss of actuator effectiveness. We use u F (t) to describe the control signal as follows: where ω is the control effectiveness factor and satisfied by the following form: where ω i is an unknown constant, ω i and ω i represent the known lower and upper bounds of ω i , respectively. For simplicity, we introduce the following notation: wherê Then, we have Remark 1 When ω i = ω i = 1, then the ith actuator is considered to be fault-free. Nevertheless, when 0 ≤ ω i < 1, the considered fault is a partial loss of control effectiveness. Specially, when ω i = 0, the ith actuator is considered to be failure and the actuator is out of order. On the other hand, it is noted that loss of actuator effectiveness problem may be transformed into uncertain parameters problem by using fault description method in (7), which will also make the partial actuator failures problem to easily solve in subsequent section. Letting , then, the system (2) with actuator faults (7), can be transformed into the state-space form where is as the disturbance due to elastic vibration of the flexible appendages, which is supposed w(t) belongs to l 2 [0, ∞) and satisfied ||w(t)|| ≤ δ and For system (8), the following FT controller is employed via state feedback where K is the gain of controller and need to design. Then, with the control law (9), the system (8) can be expressed as follows The objective of this paper is to design FT controller such that • The closed-loop system (10) with w(t) = 0 is asymptotically stable; • In the case when w(t) = 0, the following cost function associated with closed-loop system (10) satisfies where J * > 0 is a specified constant, Z and R are given positive definite matrices; • Under the zero initial condition, the closed-loop system (10) satisfies ||y|| 2 < γ ||w(t)||, for any nonzero w(t) that belongs to l 2 [0, 1), where γ > 0 is a prescribed scalar. Now, we give the following lemma which are needed in the proof of main results.

Fault-tolerant control design
In this section, we will give design method based on LMI to compute the FT controller gain.

Stability of closed-loop system
Theorem 1 Given a scalar γ > 0 and matrices Z > 0, R > 0, the system (10) is asymptotically stable and satisfies y(t) 2 < γ w(t) 2 for any non-zero w(t) ∈ l 2 [0, ∞) under the zero initial condition if there exist a matrix P > 0, invertible matrices Γ 1 and Γ 2 such that the following inequality holds: Furthermore, an upper bound of performance index (11) is given by Proof The first step is to analyze the asymptotic stability of system (10). Consider system (10) in the absence of w(t), we choose a Lyapunov function as V (t) = x (t)Px(t). For any matrix Γ 1 and invertible matrix Γ 2 of appropriate dimensions, we have Differentiating V (t) along the trajectory of the system (10) and adding (14) to it giveṡ According to (12) and Lemma 2, it impliesV (t) < 0. Hence, the system (10) is asymptotically stable. Next, we consider the upper bound of cost function (11). It is noted thaṫ When (12) holds, we havė Integrating both sides of (17) from t = 0 to t = ∞, we obtain At last, we shall establish the H ∞ performance of the system (10) under zero initial condition. Let It can be shown that for any non-zero w(t) ∈ l 2 [0, ∞) and t > 0, It is clear that for any matrix Γ 1 and invertible matrix Γ 2 of appropriate dimensions, the following equality always is true Then, we have According to (21) and (20), J h < 0 follows from (12), which implies that y(t) 2 < γ w(t) 2 holds for any non-zero w(t) ∈ l 2 [0, ∞). This complete the proof.
Remark 2 In Theorem 1, slack variables Γ 1 and Γ 2 are introduced. It is noted that Γ 1 and Γ 2 are useless for reducing the conservatism of stability conditions in [24,25]. However, they can more relaxed design of fault tolerant controller later on since they need only be invertible matrices rather than positive definite matrices. Meanwhile, it is noted that Γ 1 and Γ 2 are not invertible matrices, equality (14) also is true. Here, in order to design fault tolerant controller gain, we assume them to invertible matrices.

Controller design
On the basis of Theorem 1, we will present a design method of robust FT controller in the following.
Theorem 2 For given scalars γ > 0, κ and matrices Z > 0, R > 0, if there exist a scalar δ > 0, a matrix P > 0, invertible matrix V 1 and any matrix W such that the following inequality holds: the closed-loop system (10) is asymptotically stable under the FT controller u(t) = W V −1 1 x(t) and satisfies y(t) 2 < γ w(t) 2 for any non-zero w(t) ∈ l 2 [0, ∞) with the zero initial condition. Moreover, an upper bound of performance index (11) is given by where Proof Suppose there exists δ > 0 such that the inequality (22) holds and Let (7), we can obtain L L ≤ I . According to Lemma 1, there are M + SL N + N L (t)S < 0 holds, that is From Schur Complement theory, we have Then, pre-multiplying (25) by diag{V − 1 , V − 1 , I, I } and post-multiplying by diag{V −1 1 , V −1 1 , I, I } and defining some matrices as follows: Thus, (25) became to (12). From Theorem 1, it is clear that the closed-loop system (10) is asymptotically stable and satisfies y(t) 2 < γ w(t) 2 . The proof is completed.
Remark 3 By employing Lemma 1 and the fault description (7), the partial loss of actuator effectiveness problem is transformed into uncertain parameters problem. Thus, a feasible robust FTC scheme is obtained in Theorem 2, which only depends on the control effectiveness factorω. On the other hand, the designed controller is described in form of LMI, which is easy to solve by using Matlab LMI toolbox.

Numerical examples
In this section, the faut tolerant control scheme will be applied to a spacecraft with one flexible appendage. Since low-frequency modes are generally dominant in a flexible system, only the lowest two bending modes have been considered for the implemented spacecraft model. Thus, we suppose that ω 1 = 3.17 rad/s, ω 2 = 7.38 rad/s with damping ξ 1 = 0.001, ξ 2 = 0.015, We where the coupling coefficients of the first two bending modes are F 1 = 1.27814, F 2 =   Angle rate (deg/s) angle rate (effectiveness factor is 0.2) angle rate (effectiveness factor is 0.5) angle rate (effectiveness factor is 1) Fig. 1 The responses of pitch attitude angle and angle rate under partial actuator failure The results we obtained by using Theorem 2 are given in Tables 1 and 2. Figure 1 shows the responses of pitch attitude angle and angle rate under partial actuator failure, which the control effectiveness factors are chosen as when ω = 0.2,ω = 0.5ω = 1 respectively. Figure 2 shows the responses of control effect whenω = 0.2,ω = 0.5ω = 1. From these figures, it is clear that the response performance can be guaranteed under the robust FT controller.

Conclusion
In this paper, a robust FTC scheme has been investigated for flexible spacecraft. The LMI-based conditions are formulated for the existence of the admissible controller, which ensures that the faulty closed-loop system is asymptotically stable with a H ∞ disturbance attenuation level and partial loss of actuator effectiveness. Numerical simulations have shown the performance of the attitude control system. Further improvement in composite disturbance-observer with feedback control for flexible spacecraft will be considered in our future work.