Fuzzy logic based fault-tolerant attitude control for nonlinear flexible spacecraft with sampled-data input

Abstract

This paper deals with the problem of fault-tolerant control (FTC) for nonlinear flexible spacecraft subject to actuator faults and input sampling. Different from the existing approaches, the considered nonlinear flexible spacecraft is approximated by Takagi–Sugeno (T–S) Fuzzy techniques with connecting a set of linear models. To tackle the loss effectiveness of actuator, a general model of actuator failure is developed first. Then, the sampled-data input is converted into a time-delay term by input delay method, and the time delay caused by signal transmission is also taken care of here. By constructing a mixed linear-quadratic and $H_{\infty }$ performance, a robust FTC algorithm with less conservatism is proposed to ensure the required attitude maneuvering of flexible spacecraft in the presence of external disturbance and elastic vibration. Finally, a practical example with simulation results are carried out to show the effectiveness of the proposed FTC strategy.

Introduction

Over the past half-century, lots of nations, such as Europe, USA, and China, focus on developing the programs of deep space explorations to serve people in the fields of navigation, communication, and so on. The growing requirements of various exploration missions have considerably promoted the development of spacecrafts [1]. From the point of reducing launch cost and increasing service life, most of advanced spacecrafts often employ large-scale and lightly damping structures, such as antenna and solar arrays. The difficulties always occur in the attitude control of such flexible spacecrafts because of the complicated and serious coupling effects between the elastic and rigid modes. The environmental disturbances acted on these weakly damping structures, such as aerodynamic and gravitational torques, would induce continuous elastic vibration which could badly degrade the precision of attitude control. Clearly, it is really a difficult and challenging work to control the attitude of flexible spacecraft with high precision [2].

Indeed, how to make system robust to elastic vibration is not the only problem we have to face. As known to all, the reliability of attitude control system in flexible spacecraft determines whether the required exploration missions can be carried out successfully. After launching a spacecraft, it is impossible to replace the hardware in which some faults occur. Hence, how to improve the fault tolerance capability of spacecraft is another one of the crucial problems which deserves to be investigated. Actuator is a significant cog in the machinery of attitude control system, which transforms the command into physical control torque [3], [4], [5]. Due to component ageing or damage, actuator often suffers from the failures or faults which prevent it from performing command completely. Obviously, this would lead to the errors between the required and real flight attitude since the main control surfaces cannot be moved to the right positions and become a potential source of instability and control performance reduction [6]. The growing demands for solving such problems have resulted in a broad and deep investigation of FTC strategy in control systems [7]. The primary focuses of FTC scheme are to ensure the system stability and to hold the control performances of acceptable level with the occurrence of component faults. According to whether the fault sets can be predefined in the process of controller design or not, the development of FTC is separated into two directions, i.e., active FTC method and passive FTC method [8]. Active FTC method employs a fault detection and diagnosis (FDD) part to obtain the information of failure (like size or shape) firstly, and then reconfigure the controller based on these information. Because of this real-time property, active FTC has the ability to handle a lot of faults [9], [24], [25]. However, the time consumption on detecting and processing failure is a potential threat to system stability. By contrast, the passive one has the merits in handling actuator faults without considering their information, therefore, it is easy to carry out [29], [30].

It needs to mention that the aforementioned FTC approaches are all constructed on the basis of mathematical model. In reality, the dynamics of flexible spacecraft are always highly nonlinear. However, most established control algorithms are based on an approximate linear model which does not take care of the nonlinear characteristics of physical spacecraft [10], [11]. In spite of the fact that linear model can simplify the analysis and synthesis of system, it also restricts the designer to procure the controllers which are applicable to realistic systems. To tackle this problem, a lot of attention have been paid to the nonlinear control methods for flight systems [12], [14], [15]. In addition to these developments, nowadays, there is a novel trend in using T–S fuzzy technique to approximate the dynamics of nonlinear systems by blending a series of local linear models [16]. Due to the powerful advantages of approximating any smooth nonlinear dynamics with any accuracy, T–S fuzzy method has been investigated extensively and deeply in recent years. Such as, [17] addressed the stability analysis, [18], [19], [20], [21], [22], [23] reported the $H_{\infty }$ control and filtering problems, [24], [25] performed the studies of fault detection, and [26] developed a new solution framework for adaptive output feedback control of T–S fuzzy systems with multiple input–output delays and under dynamics and actuator failure uncertainties. When it comes to the field of aerospace engineering, Qi et al. [13] presented an adaptive FTC scheme for hypersonic vehicles with unknown parameters and uncertain actuator faults based on a combination of adaptive backstepping control and dynamic surface control technique. Huo et al. [14] discussed the topic of adaptive fuzzy control for rigid spacecraft where T–S fuzzy was taken to estimate nonlinear disturbances. Park et al. [27] established a nonlinear model of rigid spacecraft by T–S fuzzy approach and employed an inverse optimal controller to realize attitude maneuvering. Jiang et al. [28] studied the problem of T–S fuzzy modeling and adaptive reliable control for near space spacecraft. Unfortunately, so far few efforts are made to investigate the problem of attitude control for nonlinear flexible spacecraft by T–S fuzzy approach.

From the above discussions, two questions emerge: first, is it possible to use T–S fuzzy method to model the nonlinear flexible spacecraft with sampled-data input and actuator failures? And second, under the occurrence of component faults, can we construct a passive FTC law for nonlinear flexible spacecraft to achieve rest-to-rest attitude maneuvering and simultaneously reject external disturbance and elastic vibration? Motivated by answering these questions, we start this study.

In this paper, the FTC problem is investigated for the nonlinear flexible spacecraft with actuator faults and sampled-data input. Different from most existing methods, the nonlinear dynamics of flexible spacecraft are established by T–S fuzzy strategy. A more reasonable communication model for sampled-data system arises by considering the time delay of signal transmission, and a general actuator fault model is taken care of in the design process. Based on the mixed linear-quadratic and $H_{\infty }$ performance, a sampled-data FTC law is formulated to implement the required attitude for flexible spacecraft and to suppress the disturbances which come from the environmental torques and elastic vibration at the same time. Moreover, the closed-loop system is robust to the situations with parameter uncertainties. Finally, a practical and illustrative design example is exhibited to prove the effectiveness of the theoretical developments of this paper.

The remainder contents of this work will proceed as follows: In Section 2, the T–S fuzzy model of nonlinear flexible spacecraft with actuator failure and sampled-data input is derived firstly, and then we formulate the control objectives of this paper. The main results of FTC strategy with known and unknown actuator faults are exhibited in Section 3. On account of proving the merits and potential of proposed algorithm, a practical example is presented in Section 4. Finally, Section 5 gives the conclusion of this paper.

Notations: For a general matrix X, X^T and X⁻¹ refer to its transpose and inverse, respectively. ${[X]}_s$ is the simplified notation of $X+X^T$. $\rho (X)$ gives the maximum eigenvalue of X. $\mathrm{diag}\{\cdots \}$ denotes a diagonal matrix. $X>0$ $(X<0)$ indicates X is a positive (negative) definite matrix. I $(0)$ represents a compatibly dimensioned identity (zero) matrix. ${\mathbb{R}}^n$ and ${\mathbb{R}}^{n\times m}$ describe n-dimensional vector space and $n\times m\mathrm{-}\mathrm{dimensional}$ matrix space, respectively. For an arbitrary vector $a={[a_1{a}_2{a}_3]}^T$, $\parallel a\parallel$ denotes its Euclidean norm and $\mathcal{S}(a)$ represents its skew-symmetric matrix which is stated as $\mathcal{S}(a)=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill -a_3\hfill & \hfill a_2\hfill \\ \hfill a_3\hfill & \hfill 0\hfill & \hfill -a_1\hfill \\ \hfill -a_2\hfill & \hfill a_1\hfill & \hfill 0\hfill \end{array}\right].$ $L_2[0,t^{*})$ denotes the linear space of square integrable vectors over the time interval $[0,t^{*})$, $\forall t^{*}\ge 0$. Any vector in this space has the property of $\sqrt{{\int }_0^{t^{*}}\parallel ·{\parallel }^2\mathit{dt}}<\infty$. The symbol “⋆” denotes the symmetric term in a matrix.