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

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.


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][2][3][4][5][6][7][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 faulttolerant 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][11][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][14][15], the possibility of the emerging of actuator failure at the same time is ignored.When the actuator fault happens, in order 2 Complexity 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][17][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][22][23][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 finitetime 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][31][32][33], funnel control [34][35][36], prescribed performance control (PPC) [37][38][39][40][41][42], and so on.In order to guarantee the prescribed performance imposed on the transient and steadystate 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 faulttolerant 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 steadystate 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 faulttolerant control scheme is proposed and followed by stability analysis.Simulation results are provided in Section 5, and the conclusion is summarized in Section 6.

Problem Formulation
Considering the attitude stabilization problem for a rigid spacecraft, the modified Rodrigues parameter (MRP) based spacecraft system is described as [44] where  = [ 1 ,  2 ,  3 ]  is the spacecraft attitude in body frame with respect to the inertial frame presented by MPRs,  3 is the identity matrix, and the operational symbol  × denotes the following skew-symmetric matrix for any vector = [ 1 ,  2 ,  3 ]  is the body frame angular velocity with respect to inertial frame. =  0 +Δ is the inertia matrix of the spacecraft, where  0 denotes the nonsingular known nominal value of the inertia matrix and Δ 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 as where the commanded control which needs to be designed later, sat(V  ) = sgn(V  ) × min{|V  |, V  } with sgn(⋅) denoting the sign function and V  being the th axis maximum torque.
The nonlinear saturation function sat(V  ) can be approximated by the following smooth function (V  ): Then, sat(V) is rewritten as where According to the mean value theorem, there exist constants   (0 <   < 1),  = 1, 2, 3, such that the following inequality holds: where where The control torque ( 4) is rewritten as where Γ = , and there exists an unknown positive constant  satisfying 0 <  ≤ ‖Γ‖ < 1.
From ( 1), (2), and ( 9), the spacecraft system is further formulated as where () =  −1 0 (−Δ ω− × ) is the uncertainty and () =  −1 0 (  (V)+).Because   (V) 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 1/4 ≤ ‖‖ ≤ 1/2.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.

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

Finite-Time Differentiator.
The first-order Levant differentiator [46] is formulated as where   is the input signal,  1 ,  2 > 0 are the parameters, and  1 and  are the estimations of   and α  , respectively.Following lemma holds if the parameters  1 and  2 are chosen properly.
Lemma 1 (see [46]).If the input   is not affected by the noise, the following equalities are true within a finite time If the input is affected by noise and satisfying |  −    | ≤  1 , where    is the original signal without interference, the following inequalities hold in a finite time: where  1 and  2 are positive constants depended on the design parameters of the differentiator.

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: Rule : where  = [ 1 ,  2 , . . .,   ]  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.

Complexity
Combining the singleton fuzzifier, center average defuzzification, and product inference, the output of the FLS is obtained as where   = max ∈    () and     (  ) is the membership function value of the fuzzy variable.The fuzzy basis function is defined as Define  = [ 1 ,  2 , . . .,   ]  = [ 1 ,  2 , . . .,  3 ]  as the ideal constant weight vector, and the ( 17) is expressed as where Φ() = [Φ 1 (), Φ 2 (), . . ., Φ  ()]  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 Ω, then for any constant  > 0, there exists an FLS   Φ() such that

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  0 ,  ∞ , 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].
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  = [ 1 ,  2 ,  3 ]  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).

Main Results
4.1.Control Design.Define virtual states  1 and  2 as where   = [ 1 ,  2 ,  3 ]  is the output of the following finite-time command filter where the input  = [ 1 ,  2 ,  3 ]  is the virtual control to be designed later and the output   =  1 ,  = 1, 2, 3.
Based on the command filtered backstepping control approach, the compensated stabilization errors are given by  1 =  1 −  1 ,  2 =  2 −  2 , where  1 and  2 are the error compensating signals to reduce the influence of   − .
Choose the Lyapunov function as The virtual control  and error compensation  1 are designed as and where  1 ,  ), sgn( 13 )]  with  1 (0) = 0,  = 1, 2, 3. Substituting ( 32) and ( 33) into (31) yields Construct the second Lyapunov function as Taking the derivative of  2 along with ( 10) and ( 29) yields The error compensation is designed as 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  > 0, there always exists an FLS such that where where The commanded controller is designed as where  2 ,  2 , ℎ are positive design parameters, It is obtained for the commanded controller (42) that Using ( 39)-( 43), the derivative of the  2 is simplified as The update law of θ is designed as where  and  1 are positive design parameters.

Stability Analysis.
Before providing the stability analysis, the following two lemmas are given.
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.
According to (44), the time derivative of  is Substituting update law (45) into (49) yields According to Young's inequality, the following inequality is hold: Considering 0 <  < 1, it is concluded that Using (51), (52), and Lemma 4, the time derivative of  is expressed as which leads to where }, and According to Lemma 5, it is concluded that  1 and  2 converge to the small region ‖  ‖ ≤ if the finite-time convergence of the   is guaranteed, then it is confirmed that the states  1 and  2 can converge into a small neighbourhood of the origin within a finite time.
Since  2 = ξ 2 = [0, 0, 0]  is given in the control design, in order to show that  1 is bounded in a finite time, the following Lyapunov function is chosen: Differentiating  3 along with (33) yields According to the Lemma 1, ‖  − ‖ ≤  can be achieved in a finite time  2 , and combining ‖‖ ≤ 1/2, the following inequality is obtained: Substituting (57) and  2 = [0, 0, 0]  into (56) leads to According to Lemma 5, it is illustrated that  1 can converge to the origin in the finite time  3 by choosing suitable parameter satisfying  1 > (1/2).Since  1 =  1 +  1 and  2 =  2 +  2 , the transformed error  =  1 converges to a Complexity 7 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  1 ,  2 ,  1 ,  2 ,  1 and decrease of  lead to better convergence speed, but large  1 ,  1 and small  will result in chattering problem.Consequently, the choice of the parameters  1 ,  1 , and  should be considered with a trade between the convergence speed and chattering reduction.
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.
M2: the conventional backstepping control scheme [26].The virtual control , commanded controller V, and fuzzy adaptive update law are given by where  1 = ,  2 =  − , and the control parameters  1 ,  2 , , ℎ are set the same as M1 scheme.M3: the adaptive fast terminal sliding mode control scheme proposed in [22].The sliding function  is given by where  1 ,  2 are design parameters and where  > 0 is a small constant scalar; 0 <  < 1 is a design parameter.
The control law is where ,  are design parameters, sig 0.   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 θ is shown in Figure 5, and it is clear that the parameter θ converges to a positive constant.From Figures

Figure 1 :
Figure 1: Attitude  along with corresponding performance bounds.

Table 1 :
Parameters of the control scheme.the origin within the finite time   =  1 + 2 +  3 , and from the definition  =  2 +   ,  is also bounded.