Adaptive Finite-Time Command Filtered Fault-Tolerant Control for Uncertain Spacecraft with Prescribed Performance

Chen, Zhongtian; Chen, Qiang; He, Xiongxiong; Sun, Mingxuan

doi:https://doi.org/10.1155/2018/4912483

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Control Design for Systems Operating in Complex Environments

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Research Article | Open Access

Volume 2018 | Article ID 4912483 | https://doi.org/10.1155/2018/4912483

Adaptive Finite-Time Command Filtered Fault-Tolerant Control for Uncertain Spacecraft with Prescribed Performance

Zhongtian Chen,¹Qiang Chen,¹Xiongxiong He,¹and Mingxuan Sun¹

Guest Editor: Junpei Zhong

Received17 Aug 2018

Accepted23 Oct 2018

Published06 Nov 2018

Abstract

In this paper, an adaptive finite-time fault-tolerant control scheme is proposed for the attitude stabilization of rigid spacecrafts. A first-order command filter is presented at the second step of the backstepping design to approximate the derivative of the virtual control, such that the singularity problem caused by the differentiation of the virtual control is avoided. Then, an adaptive fuzzy finite-time backstepping controller is developed to achieve the finite-time attitude stabilization subject to inertia uncertainty, external disturbance, actuator saturation, and faults. Through using an error transformation, the prescribed performance boundary is incorporated into the controller design to guarantee the prescribed performance of the system output. Numerical simulations demonstrate the effectiveness of the proposed scheme.

1. Introduction

Due to the significant role in guaranteeing the success of any spacecraft related mission, the attitude control of the spacecraft has obtained much attention, and numerous control schemes are proposed, such as adaptive control, sliding mode control, backstepping control, control, finite-time control, and so on [1–8]. Considering the specificity of the spacecraft’s working environment, the hardware of the spacecraft is unlikely repairable, and the faults or failures cannot be fixed with replacement parts after the spacecraft is launched. The existence of faults or failures can potentially cause all kinds of safety, economic, and environment problems, and they should be considered in the attitude control design.

Different from the conventional control system without considering the possibility of fault occurrence, the fault-tolerant control (FTC) can guarantee desirable performance properties even the actuators are not healthy. In general, the existing FTC technique can be roughly classified into two categories: active FTC and passive FTC. The active FTC relies on the fault detection and diagnosis (FDD) algorithm to provide the real-time information of the system status and then reconfigure the controller to achieve the control objective. For instance, an active fault-tolerant control was proposed for the flexible spacecraft in [9], such that the attitude stabilization was achieved. Unlike active FTC, by using a single fixed controller, passive FTC dose not require any online fault information and reconfiguration mechanism. Therefore, the passive FTC is more succinct, easy to compute and suitable to the actual application. Most of FTC controllers designed for the spacecraft attitude control are passive [10–12]. In [10], an adaptive fuzzy fault-tolerant controller was proposed to achieve the spacecraft attitude tracking. A sliding mode control based attitude fault-tolerant controller was presented in [11], and the attitude was stabilized for the satellite with solar flaps. For those spacecrafts with redundancy actuators, the common method is using the actuator distribution matrix. In [12], a fault-tolerant control method based on distribution matrix was presented for the spacecraft attitude tracking, such that the finite-time convergence of the tracking error was achieved.

Actuator saturation is another issue worthy of study, the actual actuators in the spacecraft have the nominal limit of the output, and saturated output definitely effaces the system performance. Although there exist works concerning the attitude control design with actuator saturation [13–15], the possibility of the emerging of actuator failure at the same time is ignored. When the actuator fault happens, in order to maintain the performance of the system, the need for large control torque leads to severe actuator saturation. Until now, numerous works have considered both issues and designed controllers for the spacecraft attitude control [16–18].

Convergence speed is always significant in practically spacecraft system, and the finite-time control is able to provide faster convergence performance and higher tracking precision. Most of the finite-time control methods applied to the spacecraft have two kinds: homogeneous method [19, 20] and Lyapunov method [21–24]. In [19], a local continuous finite-time control scheme was proposed for the spacecraft system with an unknown inertia matrix and the homogeneous method based controller achieved attitude stabilization within a finite time. Homogeneous system theory was used to design a finite-time controller in [20], and the spacecraft attitude was stabilized within a finite time even with actuator saturation. In [5, 21], two adaptive terminal sliding mode controllers were proposed based on the Lyapunov method, such that the spacecraft attitude and angular velocity could converge to a small region of the origin within a finite time, respectively. Furthermore, the recent works [12, 16, 22] have achieved the finite-time spacecraft attitude stabilization or tracking in the presence of the actuator saturation and faults.

The adaptive backstepping approach [25, 26], as a recursive Lyapunov-based scheme, has emerged as a powerful method to construct controllers for nonlinear systems since early 1990s. There are several works using the backstepping method to design controllers of the spacecraft [3, 27]; however, the closed-loop stability is achieved when time goes to infinity. When designing a finite-time backstepping controller, the differentiation of the virtual controller in recursive steps may lead to the singularity problem. Recently, the finite-time command filtered backstepping approach was proposed in [28, 29], where the first-order Levant differentiator was used to approximate the derivative of the virtual controller, such that the finite-time convergence can be achieved when the system model was known or partially known.

All of the aforementioned works mainly focus on the steady-state behavior but ignore the transient performance such as convergence rate and overshot. To achieve the specific goals of spacecraft missions which need a specific rotating speed or a limitation of angular velocity, prescribed transient performance of the system output is very important. The widely used techniques to improve transient performance mainly include barrier Lyapunov function (BLF) [30–33], funnel control [34–36], prescribed performance control (PPC) [37–42], and so on. In order to guarantee the prescribed performance imposed on the transient and steady-state output error, the prescribed performance control (PPC) method was proposed by Bechlioulis and Rovithakis for uncertain nonlinear systems [37]. For the spacecraft attitude system with input saturation, a PPC based adaptive fault-tolerant control was presented in [43], such that the output of the system was constrained by the prescribed performance. Nevertheless, the designed controller in [43] could only guarantee the asymptotic uniform ultimate boundedness of the spacecraft system as the time goes to infinity.

Motivated by the aforementioned discussions, the fuzzy finite-time attitude stabilization problem for spacecraft systems under the actuator saturation and faults is studied in this paper, and a fuzzy finite-time fault-tolerant controller is proposed to achieve prescribed transient and steady-state performance of the system output. However, it is a challenging work to design a controller to guarantee the prescribed transient performance when considering the input constraints including actuator saturation and faults. On the one hand, when the initial state is far away from the equilibrium state, the required control input is usually set relatively large to guarantee the fast transient response. But on the other hand, due to the effect of the input saturation and actuator fault, it is a hard work to keep the satisfactory transient response as usual. The main contributions of this paper are listed as follows.

(1) A first-order command filter is presented at the second step of the backstepping design to approximate the derivative of the virtual control, such that the singularity problem caused by the differentiation of the virtual control is avoided.

(2) An adaptive fuzzy finite-time backstepping controller is developed to achieve the finite-time attitude stabilization subject to inertia uncertainty, external disturbance, actuator saturation, and faults.

(3) Through using an error transformation, the prescribed performance boundary is incorporated into the controller design to guarantee the prescribed performance of the system output.

The rest of this paper is organized as follows. Section 2 states the formulation of the spacecraft attitude stabilization problem. In Section 3, some preliminary knowledge and lemmas are given. In Section 4, the fuzzy finite-time fault-tolerant control scheme is proposed and followed by stability analysis. Simulation results are provided in Section 5, and the conclusion is summarized in Section 6.

2. Problem Formulation

Considering the attitude stabilization problem for a rigid spacecraft, the modified Rodrigues parameter (MRP) based spacecraft system is described as [44]where is the spacecraft attitude in body frame with respect to the inertial frame presented by MPRs, is the identity matrix, and the operational symbol denotes the following skew-symmetric matrix for any vector is the body frame angular velocity with respect to inertial frame. is the inertia matrix of the spacecraft, where denotes the nonsingular known nominal value of the inertia matrix and is the bounded uncertainty. and in (2) denote the control torque and the bounded external disturbance torque, respectively. Considering the input saturation and actuator faults in the rigid spacecraft, the actual control torque is further formulated aswhere with is the fault matrix and is the saturated control input satisfying , where is the commanded control which needs to be designed later, with denoting the sign function and being the th axis maximum torque.

The nonlinear saturation function can be approximated by the following smooth function :

Then, is rewritten aswhere and stands for the approximation error. It is noted that is a bounded function and its bound satisfying .

According to the mean value theorem, there exist constants , , such that the following inequality holds:where . By setting , (6) is rewritten aswhere .

The control torque (4) is rewritten aswhere , and there exists an unknown positive constant satisfying .

From (1), (2), and (9), the spacecraft system is further formulated aswhere is the uncertainty and . Because and are bounded, there exists an unknown positive constant satisfying .

The matrix has the following properties [45]:

From (11) and (12), it is obtained that .

The control objective in this paper is to develop a fuzzy fault-tolerant finite-time control scheme for the spacecraft with inertia uncertainty, extra disturbance, input saturation, and actuator faults, such that the system output converges into a small region of the origin within the prescribed bounds in a finite time.

3. Preliminaries

In this section, some preliminary knowledge critical for control design and satiability analysis is presented.

3.1. Finite-Time Differentiator

The first-order Levant differentiator [46] is formulated aswhere is the input signal, are the parameters, and and are the estimations of and , respectively. Following lemma holds if the parameters and are chosen properly.

Lemma 1 (see [46]). If the input is not affected by the noise, the following equalities are true within a finite timeIf the input is affected by noise and satisfying , where is the original signal without interference, the following inequalities hold in a finite time:where and are positive constants depended on the design parameters of the differentiator.

3.2. Fuzzy Logic System

A typical fuzzy logic system (FLS) consists of four parts: the fuzzy rules, the fuzzifier, the fuzzy inference engine, and the defuzzifier. The foundation of the FLS is a group of fuzzy If-Then rules as follows: ,

where is the input of the FLS, is the FLS output, and denote the fuzzy sets relating to the membership functions and , respectively, and is the rules number.

Combining the singleton fuzzifier, center average defuzzification, and product inference, the output of the FLS is obtained as

where and is the membership function value of the fuzzy variable. The fuzzy basis function is defined as

Define as the ideal constant weight vector, and the (17) is expressed aswhere is the basis function vector. The relationship between the FLS and the unknown nonlinear function in the system is given in the following lemma.

Lemma 2. For any continuous function defined on a compact , then for any constant , there exists an FLS such that

3.3. Prescribed Performance Function

As a priori guaranteeing prescribed behavioral bounds on the output of the system, the prescribed performance function (PPF) is designed as follows:where , , and are positive parameters, where is the prescribed minimum exponential convergence rate and stands for the maximum steady-state error, respectively, and it guarantees the following inequality:

In order to relax the assumption that the initial condition should be precisely known to guarantee the prescribed transient in the classic PPF, e.g., [37], the functions and are satisfying the following properties [47].

(1) and are positive and strictly decreasing; (2) , , , , and , ,

An example of such and is given bywhere are positive constants.

For the purpose of designing the control law to guarantee the prescribed performance bounds (22), the error transformations are presented as

Through employing the error transformation (24), the output of the original system can be guaranteed within the prescribed bound provided that the transformed error is stable.

Lemma 3 (see [37, 48]). System (10) is invariant under the error transformation (24), and the stabilization of the transformed error can guarantee the output converge with the prescribed performance described by (22).

The derivative of the is given aswhere , isand the , is

Substituting (1) into (25) yieldswhere and .

4. Main Results

4.1. Control Design

Define virtual states and aswhere is the output of the following finite-time command filterwhere the input is the virtual control to be designed later and the output .

Based on the command filtered backstepping control approach, the compensated stabilization errors are given by , , where and are the error compensating signals to reduce the influence of .

Choose the Lyapunov function as . Using (28), the time derivative of is

The virtual control and error compensation are designed asandwhere , are design parameters, , , and with , .

Substituting (32) and (33) into (31) yields

Construct the second Lyapunov function as

Taking the derivative of along with (10) and (29) yields

The error compensation is designed as . Let . Substituting (34) into (36) and using the fact that is symmetric lead to

The fuzzy logic systems (19) are utilized to approximate the unknown nonlinear . From Lemma 2, for any given constant , there always exists an FLS such thatwhere is the basis function vector, the approximation error satisfied , and .

By Young’s inequality and (38), the following inequalities hold:where .

The commanded controller is designed aswhere are positive design parameters, , adn is the estimation of .

It is obtained for the commanded controller (42) that

Using (39)-(43), the derivative of the is simplified as

The update law of is designed aswhere and are positive design parameters.

4.2. Stability Analysis

Before providing the stability analysis, the following two lemmas are given.

Lemma 4 (see [49]). For and , , the following inequality holds:

Lemma 5 (see [50]). For any real number , an continuous positive-definite Lyapunov function satisfied the from as , then can be achieved in a finite time and the setting time can be estimated bywhere is the initial value of .

Theorem 6. Consider the spacecraft stabilization system described in (1) and (2) subject to input saturation and actuator fault (4) with finite-time command filter (30), the virtual control (32), the error compensation (33), the controller (42), and the update law (45), and then
(i) the transformed error converges into a small region of the origin in a finite time, and is bounded in a finite time;
(ii) the prescribed control performance of (22) is preserved.

Proof. Construct the Lyapunov function aswhere .
According to (44), the time derivative of isSubstituting update law (45) into (49) yieldsAccording to Young’s inequality, the following inequality is hold:Considering , it is concluded thatUsing (51), (52), and Lemma 4, the time derivative of is expressed aswhich leads towhere , , and . According to Lemma 5, it is concluded that and converge to the small region , , in a finite time . From the definition , , if the finite-time convergence of the is guaranteed, then it is confirmed that the states and can converge into a small neighbourhood of the origin within a finite time.
Since is given in the control design, in order to show that is bounded in a finite time, the following Lyapunov function is chosen:Differentiating along with (33) yieldsAccording to the Lemma 1, can be achieved in a finite time , and combining , the following inequality is obtained:Substituting (57) and into (56) leads toAccording to Lemma 5, it is illustrated that can converge to the origin in the finite time by choosing suitable parameter satisfying . Since and , the transformed error converges to a small region of the origin within the finite time , and from the definition , is also bounded. According to Lemma 3, the stabilization of transformed error is sufficient to guarantee converge with prescribed performance described by (22). This completes the proof.

Remark 7. From (54) and (58), it is seen that the increase of and decrease of lead to better convergence speed, but large and small will result in chattering problem. Consequently, the choice of the parameters , and should be considered with a trade between the convergence speed and chattering reduction.

5. Simulation

In order to illustrate the effectiveness of the proposed control scheme, the simulation results and discussions are presented in this section. The spacecraft model is expressed as (1) and (2) where the initial values of the state parameters are set as

The nominal inertia matrix is and the uncertainty is

The external disturbance is

The maximum torque of the actuators is , . In order to reflect the fault condition of real actuators such as flywheels, a time-varying loss of actuator effectiveness fault is given as [9]

In practice, to perform a high-precision stabilization and safety during the maneuvers, the state should be stabilize to a small region, i.e., , , and no overshoot is allowed.

The comparative simulations are given to verify the effectiveness of the proposed control scheme. For the notation convenience, the three compared control schemes are given as follows.

M1: the proposed control scheme including the finite-time command filter (30), virtual control (32), error compensation (33), commanded controller (42), and the update law (45). The control parameter settings are shown in Table 1.

M2: the conventional backstepping control scheme [26]. The virtual control , commanded controller , and fuzzy adaptive update law are given bywhere , , and the control parameters are set the same as M1 scheme.

M3: the adaptive fast terminal sliding mode control scheme proposed in [22]. The sliding function is given bywhere , are design parameters and iswhere is a small constant scalar; is a design parameter.

The control law iswhere are design parameters, , , and iswhere is a small constant scalar, , and the update laws are given bywhere , are design parameters. The parameters are chosen as , , , , , , , , , and

The simulation results are shown in Figures 1–5. The attitude described by MRPs , , along with the corresponding performance bounds is depicted in Figure 1. As shown in Figure 1, although all the three schemes could achieve the attitude stabilization, the M2 and M3 fail to meet the prescribed transient and steady-state error bound, while the proposed M1 scheme remains in the prescribed bound all the time. The angular velocities of the three schemes are depicted in Figure 2, which shows that M1 can provide better angular velocity performance than M2 and M3. From Figures 1 and 2, it is concluded that the attitude stabilization with prescribed performance and high precision is achieved in a finite time with the proposed control scheme. Figure 3 shows the actual control signals , which are quite similar for different three schemes. The three-dimensional trajectory of the transformed error is shown in Figure 4; it can be seen that the transformed error can converge to the neighbourhood of the origin in a finite time (around 40 s) subject to the actuator fault and input saturation. The convergence performance of the estimated parameter is shown in Figure 5, and it is clear that the parameter converges to a positive constant. From Figures 1–5, it is concluded that the proposed control scheme can achieve the prescribed performance within a finite time in the presence of the inertia uncertainty, external disturbance, actuator saturation, and faults.

6. Conclusion

The problem of attitude stabilization with guaranteed transient and steady-state performance has been investigated in this paper for the spacecraft systems with inertia uncertainty, external disturbance, actuator saturation, and faults. The singularity problem caused by the differentiation of the virtual control is avoided by the proposed first-order command filter. Then, incorporate the prescribed performance boundary into the controller design by using the error transformation to guarantee the prescribed performance of the system output. The control scheme stabilizes the system within a finite time with the proposed adaptive finite-time fault-tolerant control scheme. Finally, simulation results have been provided to verify the effectiveness of the proposed control algorithm.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by National Natural Science Foundation of China (Projects Nos. 61473262, 61573320, and 61403343) and Zhejiang Provincial Natural Science Foundation (No. LY17F030018).

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Copyright © 2018 Zhongtian Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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