Active Fault Tolerant Control for Flexible Spacecraft with Sensor Faults Using Adaptive Integral Sliding Mode

Abstract

In this paper, an active fault tolerant attitude control system is provided for a flexible spacecraft to accommodate the unknown sensor faults. Firstly, a model-based fault estimation technique developed into the process behavior by using a virtual filter and an adaptive observer, and then estimating the amplitude magnitude for the faulty sensor in the presence of system disturbance and parameter uncertainty. Next, reconfigurable attitude controller is developed by combining both integral sliding mode control and linear matrix inequality technique. Meanwhile, the stability of the closed loop attitude systems under the designed active fault tolerant control (FTC) scheme is analyzed by utilizing Lyapunov approach. Finally, the effectiveness of the proposed fault tolerant scheme has justified by simulation result on a flexible spacecraft attitude control systems with a time varying sensor fault.

1 Introduction

The attitude control system is one of the most important problem in spacecraft design. In modern space related missions, there is a high demand on safety, reliability and performance of spacecraft. In the past few years, the attitude control of spacecraft has attracted extensive interests and attentions by many scientists and engineers, many important results have been reported in the published academic journals [1, 2]. For the high-accuracy attitude control performance of flexible spacecraft, all kinds of advanced attitude control schemes are proposed recently, such as an adaptive fuzzy sliding mode controller is designed in [3] for the attitude stabilization of networked flexible spacecraft with parameter uncertainty and time delay, such that the tracking stability of the closed loop attitude system can be guaranteed. In [4], the attitude regulation control problem is studied for the flexible spacecraft. A dynamic disturbance compensator and a novel robust Lyapunov-based controller are designed.

The above mentioned studies focus on modeling uncertainty, disturbance and vibration to flexible appendages and do not consider the unknown faults. However, faults caused by actuators, sensors or other components, frequently occur in practice attitude control systems, which will deteriorate its performance or even shut-down of the system. Therefore, research into fault tolerant control (FTC) for attitude system has been recognized as one of the important aspects in seeking solution to an improved reliability of spacecraft. In the literature, the categories of FTC and be divide into active and passive methods [5]. Passive FTC is just an extension of robust control. Active FTC needs the feedback from the fault information. The designed controller changes in an active way according to the effects of fault. In [6], A novel fault tolerant attitude tracking control scheme is developed for flexible spacecraft with partial loss of actuator effectiveness fault. a modified fault tolerant control law based on adaptive sliding mode and neural network is proposed. In [7], a FTC method is given to the spacecraft attitude maneuvering control systems against actuator complete failure using feedback controller. In, recent years, sliding mode control has been widely applied for design of FTC as an effective control method to compensate for faults in control system. Until now, to the best of our knowledge, the fault tolerant control problem of spacecraft in sensor faulty case has not been fully investigated yet, which remains challenging and motivates us to do this study.

Inspired by the above background, an active FTC scheme is proposed for a flexible spacecraft attitude systems with unknown sensor faults. Compared to the existing literature, the main contributions of this paper are: (i) Sensor fault can be transformed into the actuator fault form without strict conditions and coordinate. (ii) The proposed approach can achieve the FTC task without fault detection module, which brings convenience FTC design in aerospace Engineering.

2 Problem Statement

The dynamics behavior of flexible spacecraft can be described by Zhang et al. [8],

$$ J\ddot{\theta } + \delta^{T} \ddot{\eta } = u $$

(2.1)

$$ \ddot{\eta } + D\dot{\eta } + K\eta + \delta \ddot{\theta } = 0 $$

(2.2)

where $J \in R^{3 \times 3}$ is the total inertia matrix of the spacecraft, $\theta$ is the attitude angle, which includes roll $\theta_{x}$, yaw $\theta_{y}$ and pitch $\theta_{z}$. $\eta \in R^{n \times 3}$ stands for the coupling matrix between elastic and rigid dynamics, $n$ is the number of the elastic modes connected to the rigid body, $u \in R^{3 \times 1}$ is the control torques generating from the flywheels orthogonally mounted on board. $D$ and $K \in R^{n \times n}$ are the damping matrix, stiffness matrix of the appendages and, respectively.

By introducing state variable $x = \left[ {\begin{array}{*{20}c} \theta & {\dot{\theta }} \\ \end{array} } \right]^{T}$, the state space model of flexible spacecraft attitude dynamics with sensor faults is described by,

$$ \dot{x} = (A + \Delta A )x + Bu + Bd $$

(2.3)

$$ y = Cx + f_{s} $$

(2.4)

where $A = \left[ {\begin{array}{*{20}c} 0 & {I_{3 \times 3} } \\ 0 & 0 \\ \end{array} } \right]$, $B = \left[ {\begin{array}{*{20}c} 0 \\ {(J - \delta^{T} \delta )^{ - 1} } \\ \end{array} } \right]$, $C = \left[ {\begin{array}{*{20}c} {I_{4} } \\ 0 \\ \end{array} } \right]^{T}$, $\Delta A = MFE$ is a real time-varying matrices representing parameter uncertainty with $M$, $E$ being known real constant matrices of appropriate dimensions, and $F$ is an unknown real time-varying matrix satisfying the following inequality $F^{T} F < I_{4}$, $d = \delta^{T} (D\dot{\eta } + K\eta )$ is the disturbance vector caused by elastic vibration of flexible appendages, which satisfies $\left\| d \right\| \le d_{0}$ and $d_{0}$ is a unknown positive scalar. $f_{s} (t) \in R^{4 \times 4}$ represents the unknown sensor faults.

Before ending this section, we recall the following assumptions and lemmas, which will be essential to realize the active FTC design later.

Assumption 1

The sensor fault satisfies the following norm bounded constraint: $\left\| {f_{s} } \right\| \le l$ and $\left\| {f_{s} } \right\| \le l_{d}$, $l$ and $l_{d}$ are two unknown positive scalars.

Lemma 1

Hao et al. [9] consider the following linear system,

$$ x = Ax + B\omega $$

(2.5)

$$ y = Cx + D\omega $$

(2.6)

If there exist a positive scalar $\gamma$ and a symmetric positive define matrix $P$, such that the following inequality holds,

$$ \left[ {\begin{array}{*{20}l} {A^{T} P + PA} \hfill & {PB} \hfill & {C^{T} } \hfill \\ {B^{T} P} \hfill & { - \gamma^{2} I} \hfill & {D^{T} } \hfill \\ C \hfill & D \hfill & { - I} \hfill \\ \end{array} } \right] < 0 $$

(2.7)

then above system is stable with a given disturbance attenuation level $\gamma$.

3 Sensor Fault Estimation Scheme

In this section, a sensor fault estimation approach will be developed. For this purpose, define a new state variable $z{ \in }R^{4 \times 4}$, which can be achieved by passing the faulty system output (2.4) through the following virtual filter.

$$ \dot{z} = - A_{f} z + A_{f} y $$

(3.1)

where $A_{f} { \in }R^{4 \times 4}$ is the parameter matrix and it is Hurwitz.

Let $\bar{x} = \left[ {\begin{array}{*{20}c} x & z \\ \end{array} } \right]^{T}$, the original faulty system (2.3) and (2.4) can be transformed into the following augment system,

$$ \dot{\bar{x}} = (A_{a} + \Delta A_{a} )x + B_{a} u + B_{a} d + B_{a} f_{s} $$

(3.2)

$$ z = C_{a} \bar{x} $$

(3.3)

where

$A_{a} = \left[ {\begin{array}{*{20}c} A & {0_{6 \times 4} } \\ {A_{f} C} & { - A_{f} } \\ \end{array} } \right]$ $\Delta A_{a} = \left[ {\begin{array}{*{20}c} {\Delta A} & {0_{6 \times 4} } \\ {0_{4 \times 6} } & {0_{4 \times 4} } \\ \end{array} } \right]$ $B_{a} = \left[ {\begin{array}{*{20}c} B \\ {0_{4 \times 3} } \\ \end{array} } \right]$ $D_{a} = \left[ {\begin{array}{*{20}c} {0_{6 \times 4} } \\ {A_{f} } \\ \end{array} } \right]$ $C_{a} = \left[ {\begin{array}{*{20}c} {0_{6 \times 4} } \\ {I_{4} } \\ \end{array} } \right]^{T}$. After transformation by this way, the original sensor fault can be treated as actuator fault in system (3.2) and (3.3). The estimated value of sensor fault can be obtained by the above designed adaptive fault estimation observer,

$$ \dot{\hat{\bar{x}}} = A_{a} \hat{\bar{x}} + B_{a} u + D_{a} \hat{f}_{s} + L (z - \hat{z} ) $$

(3.4)

$$ \dot{\hat{f}}_{s} = KR (z - \hat{z} )- \sigma K\hat{f}_{s} $$

(3.5)

where $\hat{\bar{x}}$, $\hat{f}_{s}$ and $\hat{z}$ are the estimation of $\bar{x}$, $f_{s}$ and $z$ respectively. $L$ is the designed observer gain matrix. $K$ is a positive define weighting matrix, $\sigma$ is a positive constant scalar and satisfies $\sigma > \lambda_{ \hbox{max} } (K^{ - 1} )$.

Denoting $e_{x} = \bar{x} - \hat{\bar{x}}$, $e_{f} = f_{s} - \hat{f}_{s}$, in the following, the fist main result is given in the form of Theorem 1.

Theorem 1

Based on the proposed observer (3.4) and (3.5), the estimation error dynamics are uniformly ultimately bounded, if there exists asymmetric positive definite matrix $P_{1}$ and a matrix $Q$ with appropriate dimensions such that

$$ P_{1} D_{a} = C_{a}^{T} R $$

(3.6)

$$ \left[ {\begin{array}{*{20}l} {P_{1} A + A^{T} P - QC_{a} - C_{a}^{T} Q^{T} } \hfill & {P_{1} M} \hfill & {P_{1} B_{a} } \hfill \\ {M^{T} P_{1} } \hfill & { - \varepsilon I} \hfill & 0 \hfill \\ {B_{a}^{T} P_{1} } \hfill & 0 \hfill & { - \varepsilon I} \hfill \\ \end{array} } \right] > 0 $$

(3.7)

The observer gain can calculated by $L = P_{1}^{ - 1} Q$. The estimation errors are bounded and converge with a exponential rate greater than ${\text{e}}^{{ - \frac{k}{h}}}$ to the residual set $\Omega :\left\{ {e_{x} ,e_{f} \left| {\lambda_{ \hbox{min} } (P_{1} )} \right.\left\| {e_{x} } \right\|^{2} + \lambda_{ \hbox{min} } (K^{ - 1} )\left\| {e_{f} } \right\|^{2} \le \frac{\delta h}{k}} \right\}$.

Proof

Choosing a Lyapunov candidate function as follows,

$$ V_{1} = e_{x}^{T} P_{1} e_{x} + e_{{_{f} }}^{T} P_{1} e_{f} $$

(3.8)

Under the Assumption 1 and some constraints mentioned above, it is easily obtained that, $2e_{f} K^{ - 1} \dot{f}_{s} \le \lambda_{ \hbox{max} } (K^{ - 1} )(\left\| {e_{f} } \right\|^{2} + l_{{_{d} }}^{2} )$, $2\sigma e_{f}^{T} f_{s} \le \sigma (\left\| {e_{f} } \right\|^{2} + l^{2} )$,$2e_{x}^{T} P_{1} \varDelta Ax \le \frac{1}{\varepsilon }e_{x}^{T} P_{1} B_{a} B_{{_{a} }}^{T} P_{{_{1} }}^{T} e_{x} + \varepsilon d^{T} d$.

Then, the derivative of (3.8) satisfies the following inequality relationship,

$$ \dot{V}_{1} \le - k(\left\| {e_{x} } \right\|^{2} + \left\| {e_{f} } \right\|^{2} ) + \delta $$

(3.9)

where $k = { \hbox{min} }(\lambda_{\hbox{max} } (\varGamma ),\sigma - \lambda_{ \hbox{max} } (K^{ - 1} ))$, $\delta = \lambda_{ \hbox{max} } (K^{ - 1} )l_{d}^{2} + \sigma l^{2} + \varepsilon \left\| x \right\|^{2} + \varepsilon d_{0}^{2}$, $\varGamma = - (A_{a} - LC_{a} )^{T} P_{1} - P_{1} (A_{a} - LC_{a} )- \frac{1}{\varepsilon }P_{1} MM^{T} P_{{_{1} }}^{T} - \frac{1}{\varepsilon }P_{1} B_{a} B_{{_{a} }}^{T} P_{{_{1} }}^{T}$.

Noting that $V_{1} \le h (\left\| {e_{x} } \right\|^{2} + \left\| {e_{f} } \right\|^{2} )$, $h = {\text{max(}}\lambda_{ \hbox{max} } (P_{1} ),\lambda_{ \hbox{max} } (K^{ - 1}_{1} ) )$, and substituting it into (3.9), it yields $\dot{V}_{1} \le - \frac{k}{h}V_{1} + \delta$, which implies that Theorem 1 holds. This completes the proof.

4 Fault Tolerant Controller Design and Stability Analysis

In this section, the compensated output signal $y_{c} = y - \hat{f}_{s}$ is used to construct a integral sliding surface as follow,

$$ s = G(y_{c} - y_{c} (t_{0} ) + \int\limits_{{t_{0} }}^{t} {y_{c} (\tau {)}} d\tau ) $$

(4.1)

where $G = (CB)^{ + } - Y(I - (CB)(CB)^{ + } )$ and $Y$ is an arbitrary matrix. $t_{0}$ represents the initial time of the attitude system.

The designed controller is composed of a linear and a nonlinear part as follows,

$$ u = u_{l} + u_{n} $$

(4.2)

where $u_{l} = Ny_{c}$, $u_{n} = - (\hat{\rho } + \lambda )\frac{s}{\left\| s \right\|}$, $\dot{\hat{\rho }} = \frac{1}{\nu }\left\| s \right\|$. $N$ is the designed matrix, $\lambda$ is a constant scalar and $\hat{\rho }$ is the estimation of the unknown term $\rho = l_{d} + \left\| G \right\|\left\| {\dot{e}_{f} } \right\| + \left\| {G + N} \right\|\left\| {e_{f} } \right\|$. $\nu > 0$ is a constant adaptation gain.

Theorem 2

If there exist a symmetric matrix $P_{2}$ and output feedback gain $N$ satisfy the following linear matrix inequality,

$$ \left[ \varXi \right]_{34 \times 34} < 0 $$

(4.3)

where $\varXi_{11} = (\varphi A - BGC )P_{2} + P_{2} (\varphi A - BGC )^{T}$, $\varphi = I - BGC$, $\varXi_{12} = - BG + BN$, $\varXi_{13} = - BG$, $\varXi_{14} = P_{2}$, $\varXi_{15} = \varphi M$, $\varXi_{16} = \varXi_{18} = P_{2} C^{T}$, $\varXi_{17} = BN$, $\varXi_{22} = \varXi_{33} = - \gamma_{{_{2} }}^{2} I$, $\varXi_{28} = I$, $\varXi_{44} = \varXi_{55} = - \mu_{{_{2} }}^{ - 1} I$, $\varXi_{66} = \varXi_{77} = \varXi_{88} = - I$, $\varXi_{i,j}$ are null for others. The asymptotical stability of the system can be guaranteed.

Proof

When the system has the ideal sliding mode motion on the sliding surface, i.e. $\dot{s} = 0$, the equivalent control can be designed as,

$$ u_{eq} = - G (C (A + \Delta A + I )x + CBd + \dot{e}_{f} + e_{f} )+ u_{n} $$

(4.4)

Substituting (4.4) into (2.3), the sliding dynamics can be obtained,

$$ \dot{x} = ( (I - BGC ) (A + \Delta A )- BGC + BNC )x + \bar{B}\omega $$

(4.5)

$$ y_{c} = Cx + \bar{D}\omega $$

(4.6)

where $\omega = \left[ {\begin{array}{*{20}c} {e_{f} } & {\dot{e}_{f} } \\ \end{array} } \right]^{T}$, $\bar{B} = \left[ {\begin{array}{*{20}c} {B(N - G)} & { - BG} \\ \end{array} } \right]^{T}$ and $\bar{D} = \left[ {\begin{array}{*{20}c} I & 0 \\ \end{array} } \right]^{T}$.

By applying Lemma 1 to (4.5) and (4.6), Schur complement and some calculations, the linear matrix inequality condition (4.3) can be obtained.

Consider the following Lyapunov function candidate,

$$ V_{2} = \frac{1}{2} (s^{T} s + \nu \tilde{\rho }^{2} ) $$

(4.7)

where $\tilde{\rho } = \rho - \hat{\rho }$ is the adaptive estimation error.

Its derivative follows from (4.2) that,

$$ \dot{V}_{2} \le (\varepsilon \left\| x \right\| - \lambda )\left\| s \right\| $$

(4.8)

where $\varepsilon = \left\| {GCA} \right\| + \left\| {GC} \right\| + \left\| {NC} \right\| + \left\| {GCM} \right\|\left\| E \right\|$. Choosing $\lambda > \varepsilon \beta$ with a given scalar $\beta$, then $\dot{V}_{2} < 0$. Which means that the system will be stably even in faulty conditions. This completes the proof.

5 Simulation Example

In this section, the good performance of the proposed active fault tolerant control approach is verified by considering a flexible spacecraft attitude control system model which is given by Zhang et al. [8]. Suppose that the sensor fault occurs on the first measurement channel at 10 s and the fault is a time-varying signal. Namely, $f_{s} = \left[ {\begin{array}{*{20}c} {f_{1} } & {0_{1 \times 3} } \\ \end{array} } \right]^{T}$. The expressions of fault is described as $f_{s} = 0.8\sin (0.9t)$, when $t > 10s$.

Firstly, Fig. 1 provides evidence that our dynamic sensor estimation scheme has excellent fault estimation capability. Furthermore, the simulation is performed with classical output feedback control strategy. Figure 2a shows the bad effect of sensor fault on the flexible spacecraft attitude system. From Fig. 2b, with the implement of the active FTC scheme designed in this paper, it is not difficult to find that the effect of sensor fault to the closed-loop attitude systems is removed.

Fig. 1

Sensor fault estimation result (a) and its error (b)

Fig. 2

Attitude angle responses in sensor faulty case using classical output feedback control (a) and using active FTC approach designed in this paper (b)

6 Conclusion

In this paper, an active FTC strategy for flexible spacecraft under sensor fault case is designed. The sensor fault is transformed into actuator fault form of an augment system and fault estimation is achieve based on an adaptive observer. By applying adaptive integral sliding mode, a FTC controller is proposed, which can ensure the attitude system normal operation although sensor fault occurs. In the finally, simulation results show the excellent performance of our approach.