# A Modified Robust Adaptive Fault Compensation Design for Spacecraft with Guaranteed Transient Performance

Authors: Xuelian Yao; Yi Yang; Mingyu Wu
Journal: International Journal of Aerospace Engineering
 (2023)

---

## Abstract

A modified, robust adaptive fault compensation design is proposed for rigid spacecraft systems with uncertain actuator failures and unknown disturbances. The feedback linearization method is first introduced to linearize the nonlinear dynamics, and a model-reference adaptive controller is designed to suppress the unknown external disturbances and stabilize the linearized system. Then, a composite adaptive controller is developed by integrating multiple controllers designed for the corresponding actuator failure conditions, which can handle the essentially multiple uncertainties (failure time, values, type, and failure pattern) of actuator failures simultaneously. To further improve the transient performance problem in the failure compensation control, anH∞ compensator is introduced as an additional item in the basic controller to attenuate the adverse effects on tracking performance caused by parameter estimation errors. From the theoretical analysis and simulation results, it is obvious that the designed scheme can not only guarantee the stability of the closed-loop system is stable and asymptotical tracking properties for a given reference signal but also greatly improve the transient performance of the spacecraft system during the process of failure compensation.


---

## Body

## 1. Introduction

Component (actuator or sensor) failures and external disturbances are common in performance-critical systems, which can lead to loss of performance and even cause catastrophic accidents. Hence, to maintain an acceptable level of performance and guarantee system stability in the event of uncertain component failures, remarkable progresses have been made in the area of fault-tolerant control (FTC) and disturbance suppression [1–7].Reaction wheels are commonly used in spacecraft as the actuators, which may fail in the course of system operation. Precise attitude control in the case of disturbances and uncertain actuator failures have widely studied in the existing literature. Excellent overviews were provided by the survey papers [8, 9] to make FTC designs for spacecraft control system. In [10], a fault tolerant control scheme was proposed for spacecraft attitude stabilization by integrating learning observer and backstepping control design. A novel adaptive event-triggered controller was designed to handle disturbances, model uncertainties, actuator failures, and limited communication, simultaneously [11]. In this paper, only loss of actuator effectiveness fault was considered in the control system design. Adaptive observer-based fault-tolerant tracking control schemes were widely used to deal with the attitude tracking problem for spacecraft experience disturbances and actuator failures [12, 13]. Fault detection and identification based FTC scheme was designed for spacecraft control system subject to multiple actuator faults, parameter uncertainties, and external disturbances [14]. Nonlinear model predictive control approach was used to control the coupled translational-rotational motion of a spacecraft in the presence of one actuator failure [15]. In [16], a new adaptive attitude tracking control scheme was developed for a flexible spacecraft system subject to external disturbances and uncertain failures. The sliding mode control (SMC) technology is insensitive to some disturbances and uncertainties very much [17]. Numerous works related to SMC-based spacecraft FTC design were reported in [18] and the references therein.For the recently advanced space missions, the system performance either in the stable state and the instantaneous one are equally important. Bad transient performance may exist although the steady performance can be obtained successfully finally, which are potentially dangerous to performance-critical systems. The problem of transient performance improvement has been widely researched based on several inspired control approaches, such as model reference adaptive control (MRAC) [19], H∞ control [20], adaptive control [21, 22] and sliding mode control [23]. To improve the transient dynamics through modifying MRAC design, [24, 25] have used fuzzy logic and genetic algorithm, illustrated some simulations without analytical support to manifest the effectiveness of the designed scheme. Dynamic regressor extension and the technique of mixing parameter estimation were introduced by [26], which removed the assumption of some prior knowledge with high-frequency gain, whereas such a scheme may cause a complicated task for issues of practical interests. Considering that actuator saturation, together with the excessive tracking error of the closed-loop system’s trajectory may also be led by large adaptive rates, with the exception of the plant’s reference model, [27] introduced an auxiliary reference model with the characterization of the classical MRAC framework. However, the simulation results show that the larger values of the user-defined rate parameter will cause larger overshoots. Most of the aforementioned schemes deal only with system parameter uncertainties without considering the uncertainties of actuator failure.Recently, more attention has been paid to the study of FTC designs with guaranteed transient performance [28]. Investigating the prescribe performance fault tolerance control for chaser spacecraft. The uncertainties of model, actuator failure, and external disturbances were summarized as lumped disturbances, which were estimated by a finite-time extended state observer. Based on the estimated information from the observer, an adaptive backstepping controller was designed to achieve the desired trajectory [29]. Addressed the problem of finite-time attitude-tracking control for a rigid spacecraft with inertial uncertainties, external disturbances, actuator saturations, and faults. A fast nonsingular terminal sliding mode manifold integrating with fuzzy approximation technique was constructed to develop an enhanced FTC scheme. It can guarantee the real finite-time stability instead of asymptotical stability. Similar to the study in [29, 30], we proposed a fault-tolerant nonsingular fixed-time control scheme based on neural networks for spacecraft maneuver mission, which can accelerate the convergence rate and improve control accuracy. A robust FTC algorithm was synthesized by employing a low-pass filter and an auxiliary dynamic system along with adaptive backstepping design, which achieved attitude tracking despite the presence of disturbances, actuator faults, and input saturation [31]. In [32], a fault-tolerant controller, based on dynamic surface design and nonlinear extended state observer, was developed for attitude tracking dynamics of the combined spacecraft in the presence of inertia uncertainty, actuator failure, and external disturbance. Such a scheme can drive the attitude tracking error to converge to one small neighborhood of zero. However, the uncertainties of both actuator failure and external disturbance were considered to be lumped disturbances by the previous literatures, and fuzzy logic system, neural networks, or observers were investigated to estimate the lumped disturbance. As [33] pointed out in most conditions, actuator failure and disturbance cannot be handled in the same way due to their different mechanisms.As has been pointed out by [34], the parameter estimation error of the adaptive controller is one of the most significant factors that lead to the undesired transient. It is widely known that unanticipated actuator failures will bring about parametric uncertainties in the system. And the parameter estimation error is inevitable no matter which adaptive control scheme is adopted. Despite recent advances in transient performance improvement design for spacecraft fault-tolerant control system, it is still a challenging problem on how to guarantee transient performance for spacecraft system with uncertain actuator faults and external disturbances. Hence, it is an interesting and meaningful topic to investigate the problem of attitude tracking control with guaranteed transient and steady state performance for spacecraft subject to both actuator failures and external disturbances, which motivates the main results in this paper. The spacecraft attitude control problem under uncertain actuator failures and unknown disturbances is solved by proposed backstepping-based adaptive control scheme in literature [35]. On this basis, the problem of actuator failure compensation design for spacecraft attitude control system with guaranteed transient performance is further studied in this paper. The major contributions and excellence of our proposed methodology are as follows:
(i)
Unlike the works which focus on the design of adaptive FTC for spacecraft systems with the weakness of a bad transient performance under the condition of unanticipated actuator failures, this paper further concerns the transient performance problem for the actuator failure compensation design. A performance index is adopted to assess the degree of the transient performance enhancement and characterize the weight of the designedH∞ compensator in the modified MRAC system(ii)
Compared with the current MRAC based transient performance improvement schemes, this paper first couple the modified MRAC and direct adaptive FTC control techniques to increase the fault tolerance capability of the MRAC scheme(iii)
In contrast to the existing literatures regarding the uncertainties of both actuator failure and external disturbance as lumped disturbances, this paper solve the uncertain actuator failures and disturbances separately according to their different mechanisms(iv)
Detailed analysis of the performance of both transient and system steady state, and the valid proof according to the asymptotic output tracking as wellThe remaining section is composed as follows. Section2 formulates the control problems, describes some preliminaries on the feedback linearization theory, and presents two lemmas, which are important for the compensator design and performance analysis. Section 3 describes the modified MRAC and the adaptive failure compensation design, as well as the closed-loop system performance analysis. Section 4 gives the simulation background and numerical simulation results.


## 2. Spacecraft Model and Problem Description

This chapter first introduces the rigid spacecraft system model and some basic concepts, then describes the actuator fault compensation of the spacecraft system.


### 2.1. Rigid Spacecraft Model

The mathematical model of a rigid spacecraft system is formulated by the following equations:(1)q̇0q̇=12−qTq0I+q×ω,Jω̇=−ω×Jω+Dut+dt,where q0 and q=q1,q2,q3T∈R3 denote the scalar and vector parts of the unit-quaternion, respectively. The quaternion also satisfies the constraint equation qTq+q02=1, ω∈R3 denotes the inertial angular velocity of the spacecraft expressed in the body frame, and the inertia matrix J∈R3×3 is assumed to be known in this study. The notations ζ×, ∀ζ=ζ1,ζ2,ζ3, can be expressed as
(2)ζ×=0−ζ3ζ2ζ30−ζ1−ζ2ζ10.ut∈R4 is the control input produced by reaction wheels. As the orientation matrix of the reaction wheel, D∈R3×4 is available for a given spacecraft. In this research, we consider
(3)D=−100130−101300−113.dt=d1,d2,d3T∈R3 represents disturbance vector, which comes in many forms: gravity gradients, solar pressure, atmospheric drag, pressure forces, and so on. In practice, these forces are bounded. For the major topic of our interest, each component of dt is modeled as
(4)dit=ci+∑j=1niaijsinωijt+∑j=1nibijcosωijt=θdi∗Tϖdit,where ci, aij, and bij are unknown amplitudes, and ωij are known frequencies.
(5)θdi∗=ci0,ai1,⋯,aini,bi1,⋯,biniT∈R2ni+1,ϖdit=1,sinωi1t,⋯,sinωinit,cosωi1t,⋯,cosωinitT∈R2ni+1.Definex=qT,ωTT and y=q as the state and output vector, the spacecraft attitude control system (1) is rewritten as
(6)ẋ=fx+gxudt+gxdt,y=hx,where
(7)fx=1/21−q2ω−1/2ω×q−J−1ω×Jω,gx=g1,g2,g3=0J−1,and
(8)ud=Du.


### 2.2. Control Problem Statement

This research mainly focuses on the issue of the spacecraft attitude control involves uncertain actuator failures. The classical actuator failures is expressed as(9)u¯jt=u¯j0+∑i=1qju¯jifjit,t≥tj,where j∈1,2,3,4, tj>0, u¯j0, and u¯ji represent unknown failure parameters. fjit,i=1,2,⋯,qj are known. The Equation (9) can also be rewritten into the below-parameterized expression
(10)u¯jt=θjTϖjt,where θj=u¯j0,u¯j1,⋯,u¯jqjT∈Rqj+1, ϖjt=1,fj1t,⋯,fjqjtT∈Rqj+1. As specifically pointed out, the failure model (9) can describe stuck-in-place, complete failure, and oscillatory failure which usually occur in the spacecraft system.In the system, if there is any uncertain actuator fault, then the inputut applying to the system is
(11)ut=I−σtvt+σtu¯t,where vt denotes the control input signal. u¯t=u¯1,u¯2,u¯3,u¯4T andσt=diagσ1,σ2,σ3,σ4 are defined fault pattern matrix, i.e, when the j-th actuator fails σjt=1, if there is no failure σjt=0. Substituting Equation (11) into the second equation of (1), the system model is expressed as
(12)Jω̇=−ω×Jω+DI−σtvt+σtu¯t+dt.For the outputyt=q to track a given command ymt, at least three functional inputs at any time are needed, so that the independent control inputs are enough to make sure that the output can track the arbitrary given output signals. Therefore, no more than one actuator failure can be allowed in the system; the compensable failure cases and the corresponding failure patterns are listed as
(i)
no failure case,σ1=diag0,0,0,0(ii)
u1 failure case, σ2=diag1,0,0,0(iii)
u4 failure case, σ3=diag0,0,0,1(iv)
u3 failure case, σ4=diag0,0,1,0(v)
u2 failure case, σ5=diag0,1,0,0


#### 2.2.1. Control Objective

For system (1), which has one uncertain failure (9) at most, an adaptive controller vt is designed to guarantee system stability and output tracking with guaranteed transient performance. To show the design process of failure compensation control accompanied by satisfactory transient performance, we design the adaptive scheme for the three failure patterns as following:
(13)σ1=diag0,0,0,0,σ2=diag1,0,0,0,σ3=diag0,0,0,1.Remark 1.
The studied spacecraft of this paper are actuated by four reaction wheels. We can learn from the orientation matrix of reaction wheel given in (3) that three of them are mounted where their spin axes are parallel to the body frame, respectively, and the other one is mounted where its spin axis points to some fixed direction. According to the configuration of reaction wheels, the control design for the u1 failure case can be expended to address the u2 and u3 failure cases, so in this paper, only the three failure cases (13) are taken into account to demonstrate the detail design process.


### 2.3. Feedback Linearization

For a kind of nonlinear systems with the input and output ofmdimension
(14)ẋ=fx+gxu,y=hx.Definition 2.
The system (14) has a vector relative degree ρ1,ρ2,⋯,ρm,1≤ρi≤n, at a point x0 for ∀x in a neighborhood of if the below two conditions hold
(i)
LgjLfkhix=0, for all 1≤j≤m,1≤i≤m,0<k<ρi−1 and LgjLfρi−1hix≠0, for some j∈1,2,⋯,m, and(ii)
them×m matrix Gx is defined as(15)Gx=Lg1Lfρ1−1h1x⋯LgmLfρ1−1h1xLg1Lfρ2−1h2x⋯LgmLfρ2−1h2x⋯⋯Lg1Lfρm−1hmx⋯LgmLfρm−1hmx.Then the system (14) has the relative degree ρ=∑i=1mρi, with ρi being the subrelative degree of the i-th output yi=hix. If the equilibrium point of system (1) is x0=0,0,0,0,0,0T, we can obtain that ρ1=ρ2=ρ3=2 and the relative degree ρ=n=6. By the twice differentiation to the system output yi, the control input ud in the differential equation is expressed in the form of a nonzero factor.To be specific, the Equation (6) is denoted as
(16)y¨1y¨2y¨3=F1xF2xF3x+G1xG2xG3xud+dt,where
(17)F1x=−14q1ω12+ω22+ω32+J1−J22J3q2ω1ω2+J1−J32J2q3ω1ω3+J2−J32J1q0ω2ω3,F2x=−14q2ω12+ω22+ω32+J2−J12J3q1ω1ω2+J3−J12J2q0ω1ω3+J2−J32J1q3ω2ω3,F3x=−14q3ω12+ω22+ω32+J1−J22J3q0ω1ω2+J3−J12J2q1ω1ω3+J3−J22J1q2ω2ω3,G1x=q02J1,−q32J2,q22J3,G2x=q32J1,q02J2,−q12J3,G3x=−q22J1,q12J2,q02J3.The system (14) can be feedback linearized through differential homeomorphic mapping. Supposing there exists a differential homeomorphic mapping with the form Tx=ξ=h1x,Lfh1x,h2x,Lfh2x,h3x,Lfh3xT.Under the afore-mentioned coordinate transform, the system (6) can be transformed into three linear subsystems with the following normal form:
(18)ξ̇11=ξ12,ξ̇12=F1x+G1xud+dt,ξ̇21=ξ22,ξ̇22=F2x+G2xud+dt,ξ̇31=ξ32,ξ̇32=F3x+G3xud+dt,y=ξ11,ξ21,ξ31T,with (11), the system (18) can be described as
(19)ξ̇11=ξ12,ξ̇12=F1x+G1σxv+G¯1σxu¯t+G1xdt,ξ̇21=ξ22,ξ̇22=F2x+G2σxv+G¯2σxu¯t+G2xdt,ξ̇31=ξ32,ξ̇32=F3x+G3σxv+G¯3σxu¯t+G3xdt,y=ξ11,ξ21,ξ31T,where Giσx=GixDI−σtG¯iσx=GixDσt, and i=1,2,3.


#### 2.3.1. Nonlinear Feedback Control Law

The feedback linearization design can be applied to generate an ideal controller, on the condition that the system parameters and fault parameters of a nonlinear system (6) are accessible. From Equation (16), we can get the following equation:
(20)y¨i=Fix+Gixud+dt,i=1,2,3.Considering the uncertainty of external disturbancedt, we set the control signal as
(21)Gixud=Wdi≜−Fix+uLi−Gixd^t,where Wdi denotes the desired control signal generated from a chosen control that is designed for the closed-loop system, uLi is the control law to be proposed, and d^t is the estimator of dt. Then, the linearized system can be obtained
(22)y¨i=uLi+Gixdt−d^t,i=1,2,3.Lemma 3. (State-feedbackH∞ optimal control).
Consider a linear time-invariant system(23)ẋ=Ax+B1uw+B2u,z=C1x,y=C2x,where x∈Rnx is the state, u∈Rnu is the control input, uw∈Rnw is the disturbance, z∈Rnz is the regulated output to be controlled, and y∈Rny is the measured output. A, B1, B2, C1, and C2 are matrices of appropriate dimensions and satisfying the assumptions.Assumption 4.
A,B1 is stabilizable; (A, C2) is detectable; B1 is column full rank, and C2 is row full rank; nw≤ny≤nx.
If there exists anε<0 such that the Riccati equation
(24)ATP+PA−ε−1PB2R−1B2TP+γ−1C1TC1+εS=0,has a solution P≥0, where R∈Rnu×nu and S∈Rnx×nx are given positive-definite matrices.
A state feedback controllerus=−1/2εR−1B2TPx can be designed to stabilize the system (23), and the transfer function Gwzs between disturbance w and output z satisfy the following condition
(25)Gwzs∞<γ,for a prespecified constant γ>0. It should be noted that γ can be arbitrarily close to the H∞ optimum by choosing a sufficiently small ε.Lemma 5.
Letz=Hsw, where Hs have all their roots in Res≥−δ/2, for any δ≥0 and w∈L2, we have
(26)zt2δ≤Hs∞δwt2δ,where zt2δ is defined as zt2δ=Δ∫0te−δt−τzTτzτdτ1/2 for ∀z∈0,∞⟶Rn’ and δ≥0 and t≥0. If Hs is strict, then
(27)zt≤Hs2δwt2δ,and if Hs is a stable n-order transfer function, then
(28)zt∞≤2nHs∞w∞.


## 2.1. Rigid Spacecraft Model

The mathematical model of a rigid spacecraft system is formulated by the following equations:(1)q̇0q̇=12−qTq0I+q×ω,Jω̇=−ω×Jω+Dut+dt,where q0 and q=q1,q2,q3T∈R3 denote the scalar and vector parts of the unit-quaternion, respectively. The quaternion also satisfies the constraint equation qTq+q02=1, ω∈R3 denotes the inertial angular velocity of the spacecraft expressed in the body frame, and the inertia matrix J∈R3×3 is assumed to be known in this study. The notations ζ×, ∀ζ=ζ1,ζ2,ζ3, can be expressed as
(2)ζ×=0−ζ3ζ2ζ30−ζ1−ζ2ζ10.ut∈R4 is the control input produced by reaction wheels. As the orientation matrix of the reaction wheel, D∈R3×4 is available for a given spacecraft. In this research, we consider
(3)D=−100130−101300−113.dt=d1,d2,d3T∈R3 represents disturbance vector, which comes in many forms: gravity gradients, solar pressure, atmospheric drag, pressure forces, and so on. In practice, these forces are bounded. For the major topic of our interest, each component of dt is modeled as
(4)dit=ci+∑j=1niaijsinωijt+∑j=1nibijcosωijt=θdi∗Tϖdit,where ci, aij, and bij are unknown amplitudes, and ωij are known frequencies.
(5)θdi∗=ci0,ai1,⋯,aini,bi1,⋯,biniT∈R2ni+1,ϖdit=1,sinωi1t,⋯,sinωinit,cosωi1t,⋯,cosωinitT∈R2ni+1.Definex=qT,ωTT and y=q as the state and output vector, the spacecraft attitude control system (1) is rewritten as
(6)ẋ=fx+gxudt+gxdt,y=hx,where
(7)fx=1/21−q2ω−1/2ω×q−J−1ω×Jω,gx=g1,g2,g3=0J−1,and
(8)ud=Du.


## 2.2. Control Problem Statement

This research mainly focuses on the issue of the spacecraft attitude control involves uncertain actuator failures. The classical actuator failures is expressed as(9)u¯jt=u¯j0+∑i=1qju¯jifjit,t≥tj,where j∈1,2,3,4, tj>0, u¯j0, and u¯ji represent unknown failure parameters. fjit,i=1,2,⋯,qj are known. The Equation (9) can also be rewritten into the below-parameterized expression
(10)u¯jt=θjTϖjt,where θj=u¯j0,u¯j1,⋯,u¯jqjT∈Rqj+1, ϖjt=1,fj1t,⋯,fjqjtT∈Rqj+1. As specifically pointed out, the failure model (9) can describe stuck-in-place, complete failure, and oscillatory failure which usually occur in the spacecraft system.In the system, if there is any uncertain actuator fault, then the inputut applying to the system is
(11)ut=I−σtvt+σtu¯t,where vt denotes the control input signal. u¯t=u¯1,u¯2,u¯3,u¯4T andσt=diagσ1,σ2,σ3,σ4 are defined fault pattern matrix, i.e, when the j-th actuator fails σjt=1, if there is no failure σjt=0. Substituting Equation (11) into the second equation of (1), the system model is expressed as
(12)Jω̇=−ω×Jω+DI−σtvt+σtu¯t+dt.For the outputyt=q to track a given command ymt, at least three functional inputs at any time are needed, so that the independent control inputs are enough to make sure that the output can track the arbitrary given output signals. Therefore, no more than one actuator failure can be allowed in the system; the compensable failure cases and the corresponding failure patterns are listed as
(i)
no failure case,σ1=diag0,0,0,0(ii)
u1 failure case, σ2=diag1,0,0,0(iii)
u4 failure case, σ3=diag0,0,0,1(iv)
u3 failure case, σ4=diag0,0,1,0(v)
u2 failure case, σ5=diag0,1,0,0


### 2.2.1. Control Objective

For system (1), which has one uncertain failure (9) at most, an adaptive controller vt is designed to guarantee system stability and output tracking with guaranteed transient performance. To show the design process of failure compensation control accompanied by satisfactory transient performance, we design the adaptive scheme for the three failure patterns as following:
(13)σ1=diag0,0,0,0,σ2=diag1,0,0,0,σ3=diag0,0,0,1.Remark 1.
The studied spacecraft of this paper are actuated by four reaction wheels. We can learn from the orientation matrix of reaction wheel given in (3) that three of them are mounted where their spin axes are parallel to the body frame, respectively, and the other one is mounted where its spin axis points to some fixed direction. According to the configuration of reaction wheels, the control design for the u1 failure case can be expended to address the u2 and u3 failure cases, so in this paper, only the three failure cases (13) are taken into account to demonstrate the detail design process.


## 2.2.1. Control Objective

For system (1), which has one uncertain failure (9) at most, an adaptive controller vt is designed to guarantee system stability and output tracking with guaranteed transient performance. To show the design process of failure compensation control accompanied by satisfactory transient performance, we design the adaptive scheme for the three failure patterns as following:
(13)σ1=diag0,0,0,0,σ2=diag1,0,0,0,σ3=diag0,0,0,1.Remark 1.
The studied spacecraft of this paper are actuated by four reaction wheels. We can learn from the orientation matrix of reaction wheel given in (3) that three of them are mounted where their spin axes are parallel to the body frame, respectively, and the other one is mounted where its spin axis points to some fixed direction. According to the configuration of reaction wheels, the control design for the u1 failure case can be expended to address the u2 and u3 failure cases, so in this paper, only the three failure cases (13) are taken into account to demonstrate the detail design process.


## 2.3. Feedback Linearization

For a kind of nonlinear systems with the input and output ofmdimension
(14)ẋ=fx+gxu,y=hx.Definition 2.
The system (14) has a vector relative degree ρ1,ρ2,⋯,ρm,1≤ρi≤n, at a point x0 for ∀x in a neighborhood of if the below two conditions hold
(i)
LgjLfkhix=0, for all 1≤j≤m,1≤i≤m,0<k<ρi−1 and LgjLfρi−1hix≠0, for some j∈1,2,⋯,m, and(ii)
them×m matrix Gx is defined as(15)Gx=Lg1Lfρ1−1h1x⋯LgmLfρ1−1h1xLg1Lfρ2−1h2x⋯LgmLfρ2−1h2x⋯⋯Lg1Lfρm−1hmx⋯LgmLfρm−1hmx.Then the system (14) has the relative degree ρ=∑i=1mρi, with ρi being the subrelative degree of the i-th output yi=hix. If the equilibrium point of system (1) is x0=0,0,0,0,0,0T, we can obtain that ρ1=ρ2=ρ3=2 and the relative degree ρ=n=6. By the twice differentiation to the system output yi, the control input ud in the differential equation is expressed in the form of a nonzero factor.To be specific, the Equation (6) is denoted as
(16)y¨1y¨2y¨3=F1xF2xF3x+G1xG2xG3xud+dt,where
(17)F1x=−14q1ω12+ω22+ω32+J1−J22J3q2ω1ω2+J1−J32J2q3ω1ω3+J2−J32J1q0ω2ω3,F2x=−14q2ω12+ω22+ω32+J2−J12J3q1ω1ω2+J3−J12J2q0ω1ω3+J2−J32J1q3ω2ω3,F3x=−14q3ω12+ω22+ω32+J1−J22J3q0ω1ω2+J3−J12J2q1ω1ω3+J3−J22J1q2ω2ω3,G1x=q02J1,−q32J2,q22J3,G2x=q32J1,q02J2,−q12J3,G3x=−q22J1,q12J2,q02J3.The system (14) can be feedback linearized through differential homeomorphic mapping. Supposing there exists a differential homeomorphic mapping with the form Tx=ξ=h1x,Lfh1x,h2x,Lfh2x,h3x,Lfh3xT.Under the afore-mentioned coordinate transform, the system (6) can be transformed into three linear subsystems with the following normal form:
(18)ξ̇11=ξ12,ξ̇12=F1x+G1xud+dt,ξ̇21=ξ22,ξ̇22=F2x+G2xud+dt,ξ̇31=ξ32,ξ̇32=F3x+G3xud+dt,y=ξ11,ξ21,ξ31T,with (11), the system (18) can be described as
(19)ξ̇11=ξ12,ξ̇12=F1x+G1σxv+G¯1σxu¯t+G1xdt,ξ̇21=ξ22,ξ̇22=F2x+G2σxv+G¯2σxu¯t+G2xdt,ξ̇31=ξ32,ξ̇32=F3x+G3σxv+G¯3σxu¯t+G3xdt,y=ξ11,ξ21,ξ31T,where Giσx=GixDI−σtG¯iσx=GixDσt, and i=1,2,3.


### 2.3.1. Nonlinear Feedback Control Law

The feedback linearization design can be applied to generate an ideal controller, on the condition that the system parameters and fault parameters of a nonlinear system (6) are accessible. From Equation (16), we can get the following equation:
(20)y¨i=Fix+Gixud+dt,i=1,2,3.Considering the uncertainty of external disturbancedt, we set the control signal as
(21)Gixud=Wdi≜−Fix+uLi−Gixd^t,where Wdi denotes the desired control signal generated from a chosen control that is designed for the closed-loop system, uLi is the control law to be proposed, and d^t is the estimator of dt. Then, the linearized system can be obtained
(22)y¨i=uLi+Gixdt−d^t,i=1,2,3.Lemma 3. (State-feedbackH∞ optimal control).
Consider a linear time-invariant system(23)ẋ=Ax+B1uw+B2u,z=C1x,y=C2x,where x∈Rnx is the state, u∈Rnu is the control input, uw∈Rnw is the disturbance, z∈Rnz is the regulated output to be controlled, and y∈Rny is the measured output. A, B1, B2, C1, and C2 are matrices of appropriate dimensions and satisfying the assumptions.Assumption 4.
A,B1 is stabilizable; (A, C2) is detectable; B1 is column full rank, and C2 is row full rank; nw≤ny≤nx.
If there exists anε<0 such that the Riccati equation
(24)ATP+PA−ε−1PB2R−1B2TP+γ−1C1TC1+εS=0,has a solution P≥0, where R∈Rnu×nu and S∈Rnx×nx are given positive-definite matrices.
A state feedback controllerus=−1/2εR−1B2TPx can be designed to stabilize the system (23), and the transfer function Gwzs between disturbance w and output z satisfy the following condition
(25)Gwzs∞<γ,for a prespecified constant γ>0. It should be noted that γ can be arbitrarily close to the H∞ optimum by choosing a sufficiently small ε.Lemma 5.
Letz=Hsw, where Hs have all their roots in Res≥−δ/2, for any δ≥0 and w∈L2, we have
(26)zt2δ≤Hs∞δwt2δ,where zt2δ is defined as zt2δ=Δ∫0te−δt−τzTτzτdτ1/2 for ∀z∈0,∞⟶Rn’ and δ≥0 and t≥0. If Hs is strict, then
(27)zt≤Hs2δwt2δ,and if Hs is a stable n-order transfer function, then
(28)zt∞≤2nHs∞w∞.


## 2.3.1. Nonlinear Feedback Control Law

The feedback linearization design can be applied to generate an ideal controller, on the condition that the system parameters and fault parameters of a nonlinear system (6) are accessible. From Equation (16), we can get the following equation:
(20)y¨i=Fix+Gixud+dt,i=1,2,3.Considering the uncertainty of external disturbancedt, we set the control signal as
(21)Gixud=Wdi≜−Fix+uLi−Gixd^t,where Wdi denotes the desired control signal generated from a chosen control that is designed for the closed-loop system, uLi is the control law to be proposed, and d^t is the estimator of dt. Then, the linearized system can be obtained
(22)y¨i=uLi+Gixdt−d^t,i=1,2,3.Lemma 3. (State-feedbackH∞ optimal control).
Consider a linear time-invariant system(23)ẋ=Ax+B1uw+B2u,z=C1x,y=C2x,where x∈Rnx is the state, u∈Rnu is the control input, uw∈Rnw is the disturbance, z∈Rnz is the regulated output to be controlled, and y∈Rny is the measured output. A, B1, B2, C1, and C2 are matrices of appropriate dimensions and satisfying the assumptions.Assumption 4.
A,B1 is stabilizable; (A, C2) is detectable; B1 is column full rank, and C2 is row full rank; nw≤ny≤nx.
If there exists anε<0 such that the Riccati equation
(24)ATP+PA−ε−1PB2R−1B2TP+γ−1C1TC1+εS=0,has a solution P≥0, where R∈Rnu×nu and S∈Rnx×nx are given positive-definite matrices.
A state feedback controllerus=−1/2εR−1B2TPx can be designed to stabilize the system (23), and the transfer function Gwzs between disturbance w and output z satisfy the following condition
(25)Gwzs∞<γ,for a prespecified constant γ>0. It should be noted that γ can be arbitrarily close to the H∞ optimum by choosing a sufficiently small ε.Lemma 5.
Letz=Hsw, where Hs have all their roots in Res≥−δ/2, for any δ≥0 and w∈L2, we have
(26)zt2δ≤Hs∞δwt2δ,where zt2δ is defined as zt2δ=Δ∫0te−δt−τzTτzτdτ1/2 for ∀z∈0,∞⟶Rn’ and δ≥0 and t≥0. If Hs is strict, then
(27)zt≤Hs2δwt2δ,and if Hs is a stable n-order transfer function, then
(28)zt∞≤2nHs∞w∞.


## 3. Actuator Failure Compensation Design

In this section, a direct adaptive control scheme is proposed to be combined with a basic control law derived from the modified MRAC design, which is not only adaptive to unknown disturbances but also able to handle uncertain patterns, values, and times of actuator failures. The simplified block diagram is shown in Figure1.Figure 1
Block diagram of control system.To achieve the control objective, we will complete four design steps:(1)
Derive a desired control signal from modified MRAC technique for the closed-loop system to achieve the desired system performance(2)
Design a composite nominal controller to handle all possible failure patterns simultaneously, with the known parameters of actuator failures(3)
Develop an adaptive control scheme with the estimation of failure parameters updated based on system performance errors, and(4)
Analyze transient and steady-state performance for failure accommodationIn order to obtain an appropriate adaptive lawvt, the error of the control signal equation which is led by the actuator uncertainties is examined. A desired control equation Gixudt=Wdi (with udt=Dut) is defined to be satisfied by the nominal control signal v∗t to be designed in the next section. In the light of (11), we define u∗t as u∗t≜I−σv∗t+σu¯t and obtain
(29)Gixudt=Wdi+GixDI−σvt−v∗t.Substituting (21) into (29), we have
(30)Gixudt=−Fix+uLi−Gixd^t+GixDI−σvt−v∗t,and the linearized system y¨i=uLi can be rewritten into
(31)y¨i=uLi+GixDI−σvt−v∗t+Gixdt−d^t.In this paper, we rewrite the linearized subsystem (31) into the transfer function form as
(32)yi=Gpisuri=kpiZpisRpisuLi+GixI−σv−v∗+Gixd−d^,with uri=uLi+GixDI−σv−v∗+Gid−d^, and its reference model is
(33)ymi=Wmisri=kmiZmisRmisri,where Zpis, Rpis, Zmis, and Rmis are monic Hurwitz polynomials of degree mpi, npi, , and pmi, respectively, ρi=npi−mpi=pmi−qmi=2. The relative degree of Wmis is the same as that of Gpis.


### 3.1. Modified MRAC Design

A model reference adaptive controller is constructed as follows(34)uLi=θiTωi+c0iri+uci,c0i=kmikpi,where ωi=ωi1T,yiT, θi=θi1T,θi2T, ωi1=αis/Λisy, αis=snpi−2,snpi−3,⋯,s,1, Λis=Zmiλis, and λis are monic Hurwitz polynomials of degree npi−qmi−1; θi∈R2ni−1 is the vector of controller parameters; uci is a proper H∞ compensator to be designed later.Remark 6.
To derive the relation between the parameter estimation error of actuator failure and transient performance, we just talk about the uncertainties of actuator failure and disturbance assuming the system parameters are known. If all the system parameters are known, the nominal controller of MRAC can beuLi=θi∗Tωi+c0iri+uci, with known θi∗=θi1∗T,θi2∗T.For a given transform functionGpis, there is a desired reference value vector θi∗=θi1∗T,θi2∗T to make the following matching conditions valid
(35)c0ikpiZpiRpi−kpiZpiαiTsθi1∗/Λis+θi2∗=Wmis.Multiplying both sides withyi and using the equation Rpisyi=kpiZpisurikpiZpi, we have
(36)coikpiZpiyi=WmiskpiZpiuri−kpiθi1∗TαisΛisyi+θi2∗Zpiyi.Dividing both sides bykpiZpi we have
(37)yi=Wmisc0iuri−θi1∗TαisΛisyi−θi2∗yi.The spacecraft system parameters are known; that is, in the MRAC law,θi is substituted by the desired value θi∗. To be more specific, the control uLi is designed as
(38)uLi=θi1∗TαisΛisyi+θi2∗yi+c0iri+uci=−Kiξi+c0iri+uci,where Ki=Ki1,Ki2 are the design controller parameters corresponding with θi∗, ξi=ξi1,ξi2T. Substituting (38) into (30), we have
(39)Gixudt=−Fix−Gixd^t+GixDI−σvt−v∗t−Kiξi+c0iri+uci.Furthermore, the linearized subsystem (18) is described as
(40)ξ̇i1=ξi2,ξ̇i2=−Kiξi+c0iri+uci+GixDI−σvt−v∗t+Gixdt−d^t.We then revise the above equation in the form of state space as follows:(41)ξ̇i=Aci−BciKiξi+BciGixDI−σv−v∗+Gixd−d^+uci+c0iri,yi=Cciξi,whereAci∈R2×2, Bci∈R2×1, and Cci∈R1×2 are the standard form of integrator chains. We also revise the reference model (33) in the state space as follows:
(42)ξ̇mi=Amiξmi+Bmiri,ymi=Cmiξmi,where Ami, Bmi, and Cmi are the minimal realization of Wmis, i.e.,
(43)Ami=01−ai1−ai2∈R2×2,Bmi=0,1T∈R2×1,and Cmi=1,0∈R1×2. Denoting ei=ξi−ξmi, we can obtain
(44)ėi=Aci−BciKiξi+BciGixDI−σv−v∗+Gixd−d^+uci+c0iri−Amiξmi−Bmiri.If the controller parametersKi is set to satisfy Aci−BciKi=Ami and Bcic0i=Bmi, then
(45)ėi=Amiei+Bmic0iGixDI−σv−v∗+Gixd−d^+uci,ei1=Cmiei.With the system model (32), (33) and the controller (34), we obtain
(46)yi=Wmisc0iGixDI−σv−v∗+Gixd−d^+c0iri+uci.Remark 7.
For the tracking error dynamics (45) and output dynamic (46), one may discover that the parameter estimation error items d−d^ and the controller parameter errors v−v∗ can be regarded as the disturbance input and plays an important role in system tracking performance. Hence, the attenuation of the disturbance made by GixDI−σv−v∗+Gixd−d^ on the closed-loop system is designed by proposing the compensator uci as an H∞ optimal controller.Equation (45) is considered as a special form of (23). With (45) and Lemma 3, a transient performance compensator uci is designed for the system (45) by the following steps as shown next.Step 1: initializeεi>0 and 0<γi<Wmis/c0i∞, choose two positive-definite matrices Sci∈R2×2 and Rci∈R1×1, and solve the following Riccati equation
(47)yi=Wmisc0iGixDI−σv−v∗+Gixd−d^+c0iri+uci.and obtain the state feedback gain
(48)yi=Wmisc0iGixDI−σv−v∗+Gixd−d^+c0iri+uci.If there is no solution, decreaseεi and repeat this step again until getting an appropriate positive-definite solution Pci=PciT>0 to ensure that the state feedback gain is obtained. The H∞ compensator which is based on measurement feedback
(49)uci=Kciei,is constructed by state feedback gain Kci, which guarantees that the system (45) with GixDI−σv−v∗+Gixd−d^ disturbance attenuation γi is stable.Step 2: Return to step 1 and reduce the transient performance index until getting a gratifying transient or the optimumγiRemark 8.
The standard algorithm can be used to confirm whether the positive-definite solution of a Riccati equation exists. According to the above processes, the optimal compensator ofH∞ can be obtained by selecting sufficiently small εi. However, it may cause the gain Kci too large. For the whole control system, the large gain will reduce the stability margin and increase the influence of measurement noise. In practical application, a compromise is usually adopted in the H∞ compensator design. It turns out that the suboptimum H∞ compensator for a given 0<γi<Wmis/c0i∞ can also achieve a satisfactory transient performance through the following theoretical analysis and simulation.


### 3.2. Nominal Compensation Design

As mentioned previously, the spacecraft system is turned into three-linear subsystem (18) derived from the feedback linearization technique. To derive the failure compensation control law ut, we write the three subsystems together as
(50)y¨1,y¨2,y¨3T=Fx+GDxI−σv+σu¯+Gxdt,where GDx=GxD∈R3×4 is regarded as control distribution matrix.


#### 3.2.1. Design for no Failure Case

On this condition,σt=σ1=diag0,0,0,0, ut=vt, and GDxvt=Wdt. The signal vt is designed as
(51)vt=v1∗t=h21xv01∗tfor a chosen h21x∈R4×4, and signal v01∗t to be calculated from
(52)GDxh21xv01∗t=Wdt.The solutionv01∗t may be derived as
(53)v01∗t=K21xWdt,with matrix function K21x∈R4×3.


#### 3.2.2. Design for theu1 Failure Case

In this situation,σt=σ2=diag1,0,0,0, u1=u¯1, and ui=vi for i=2,3,4, with GDx=GD1,GD2,GD3,GD4=GD1,GD2∈R3×4 for GD2=GD2,GD3,GD4∈R3×3, v=v1,v2,v3,v4T=v1,va2TT∈R4 for va2=v2,v3,v4T∈R3, equation GDxut=Wdt becomes
(54)GD1u¯1t+GD2va2t=Wdt.In this situation, the signalv1 is set to be v1=0. A nonsingular matrix function h22x∈R3×3 is chosen to set
(55)vt=v1t,va2TtT=v2∗t=0,va2∗TtT,va2∗t=h22xv02∗t,with v02∗t∈R3 to be deduced from
(56)GD1u¯1t+GD2h22xv02∗t=Wdt.We can obtain(57)v02∗t=K22xWdt+K221xu¯1t,with matrix function K22x∈R3×3 and vector K221x∈R3×1.


#### 3.2.3. Design for theu4 Failure Case

Similarly,σt=σ3=diag0,0,0,1, u4=u¯4, and v4t are chosen as v4t=0 and vit=uit for i=1,2,3. With GDx=GD1,GD2,GD3,GD4=GD1,GD4∈R3×4 for GD1=GD1,GD2,GD3∈R3×3 and vt=va3Tt,v4tT for va3t=v1t,v2t,v3tT∈R3, and GDxut=Wdt becomes
(58)GD1va3t+GD4u¯4t=Wdt.The signalvt is proposed as
(59)vt=va3Tt,v4tT=v3∗t=va3∗Tt,0T,va3∗t=h23v03∗t,with a chosen matrix h23∈R3×3 and a deduced signal v03∗t∈R3 from
(60)GD1h23v03∗t+GD4u¯4t=Wdt.Similarly, we have(61)v03∗t=K23Wdt+K234u¯4t,with matrix functionK23∈R3×3 and vector K234∈R3×1.


#### 3.2.4. Composite Control Law

Define three indicator functionsχj∗, j=1,2,3, which are corresponding to the considered three failure models σj and j=1,2,3, respectively. That is, if σ=σj and χj∗=1; otherwise, χj∗=0 and j=1,2,3. Then, a synthetic control law is obtained by integrating the three individual controllers
(62)v∗t=∑j=13χj∗tvj∗t,to handle the three failure cases.Forva2∗t=h22v02∗t, with v02∗t in (57), signal va2∗t can be further expressed as
(63)va2∗t=h22K22Wdt+h22K221u¯1t∈R3.Foru¯1t in (10), we express
(64)u¯1t=θ1∗Tϖ1t=ϖ1Ttθ1∗,where θ1∗=u¯10,u¯11,⋯,u¯1q1T∈Rq1+1, and ϖ1t=1,f11t,⋯,f1q1tT∈Rq1+1.Therefore, we have(65)χ2∗tva2∗t=χ21∗t,χ22∗t,χ23∗th22K22Wdt+θ11∗Ttϖ1tϕ2,1,θ12∗Ttϖ1tϕ22,θ13∗Ttϖ1tϕ23T,where χ2i∗t=χ2∗t, θ1i∗t=χ2∗tθ1∗, i=1,2,3, and ϕ2=h22K221=ϕ21,ϕ22,ϕ23T.In the same way, to designatevχ11∗t and vχ33∗t, we express
(66)χ1∗tv1∗t=diagχ11∗t,χ12∗t,χ13∗t,χ14∗th21K21Wdt(67)χ3∗tva3∗t=diagχ31∗t,χ32∗t,χ33∗th23K23Wdt+θ41∗Ttϖ4tϕ31,θ42∗Ttϖ4tϕ32,θ43∗Ttϖ4tϕ33T,where χ1i∗t=χ1∗t, i=1,2,3,4, χ3i∗t=χ3∗t, and θ4i∗t=χ3∗tθ4∗, i=1,2,3.


### 3.3. Adaptive Fault Tolerant Control Design

The adaptive version of the nominal control law is as follows (62):
(68)vt=∑j=13χjtvjt=∑j=13vχjjt=vχ11t+0,vχ2a2TtT+vχ3a3Tt,0T.In view of (65)–(67), we derive
(69)vχ11t=Δdiagχ11t,χ12t,χ13t,χ14th21K21Wdt,(70)vχ2a2t=Δdiagχ21t,χ22t,χ23th22K22Wdt+θ11Ttϖ1tϕ21,θ12Ttϖ1tϕ22,θ13Ttϖ1tϕ23T,(71)vχ3a3t=Δdiagχ31t,χ32t,χ33th23K23Wdt+θ41Ttϖ4tϕ31,θ42Ttϖ4tϕ32,θ43Ttϖ4tϕ33T,where χjit, θ1it, and θ4it are the estimates of χji∗t, θ1i∗t, and θ4i∗t, respectively.From (65)–(71), we obtain
(72)vt−v∗t=v~χ11t+0,v~χ2a2TtT+v~χ3a3Tt,0T,where
(73)v~χ11t=χ~11t,χ~12t,χ~13t,χ~14th21K21Wdt,v~χ2a2t=diagχ~21t,χ~22t,χ~23th22K22Wdt+θ~11Ttϖ1tϕ21,θ~12Ttϖ1tϕ22,θ~13Ttϖ1tϕ23T,v~χ3a3t=Δdiagχ~31t,χ~32t,χ~33th23K23Wdt+θ~41Ttϖ4tϕ31,θ~42Ttϖ4tϕ32,θ~43Ttϖ4tϕ33T.


#### 3.3.1. State Error Equations

Introducinge∈R6=e1T,e2T,e3TT=e11,e12,e21,e22,e31,e32T, ei1=yi−ymi=ξi1−ξmi1, ei2=ξ̇i1−ξ̇mi1=ėi1, and i=1,2,3. With the three subsystems
(74)ėi=Amiei+Bmic0iGixDI−σv−v∗−1/2εiRci−1BciTPciei+Gixd−d^,i=1,2,3,we have
(75)e¨11+α12ė11+α11e11e¨21+α22ė21+α21e21e¨31+α32ė31+α31e31=1c0GxDI−σv−v∗+Gxd−d^−1/2ε1c0Rc1−1Bc1TPc1e11/2ε2c0Rc2−1Bc2TPc2e21/2ε3c0Rc3−1Bc3TPc3e3.Ifσ=σ1=diag0,0,0,0, with (72) and (73), we rewrite (75) as
(76)e¨11+α12ė11+α11e11e¨21+α22ė21+α21e21e¨31+α32ė31+α31e31=1c0∑i=14GDiχ~1iν1i+∑i=13GDi+1χ~2iν2i+∑i=13GDiχ~3iν3i+∑i=13GDi+1θ~1iTϖ1ϕ2i+∑i=13GDiθ~4iTϖ4ϕ3i+1c0∑j=13G1jθ~djTtϖdj∑j=13G2jθ~djTtϖdj∑j=13G3jθ~djTtϖdj−12ε1c0Rc1−1Bc1TPc1e112ε2c0Rc2−1Bc2TPc2e212ε3c0Rc3−1Bc3TPc3e3≜E~1.Ifσ=σ2=diag1,0,0,0, with (72) and (73), we rewrite (75) as
(77)e¨11+α12ė11+α11e11e¨21+α22ė21+α21e21e¨31+α32ė31+α31e31=1c0∑i=24GDiχ~1iν1i+∑i=13GDi+1χ~2iν2i+∑i=23GDiχ~3iν3i+∑i=13GDi+1θ~1iTϖ1ϕ2i+∑i=23GDiθ~4iTϖ4ϕ3i+1c0∑j=13G1jθ~djTtϖdj∑j=13G2jθ~djTtϖdj∑j=13G3jθ~djTtϖdj−12ε1c0Rc1−1Bc1TPc1e112ε2c0Rc2−1Bc2TPc2e212ε3c0Rc3−1Bc3TPc3e3≜E~2Ifσ=σ3=diag0,0,0,1, with (72) and (73), we rewrite (75) as
(78)e¨11+α12ė11+α11e11e¨21+α22ė21+α21e21e¨31+α32ė31+α31e31=1c0∑i=13GDiχ~1iν1i+∑i=12GDi+1χ~2iν2i+∑i=13GDiχ~3iν3i+∑i=12GDi+1θ~1iTϖ1ϕ2i+∑i=13GDiθ~4iTϖ4ϕ3i+1c0∑j=13G1jθ~djTtϖdj∑j=13G2jθ~djTtϖdj∑j=13G3jθ~djTtϖdj−12ε1c0Rc1−1Bc1TPc1e112ε2c0Rc2−1Bc2TPc2e212ε3c0Rc3−1Bc3TPc3e3≜E~3where GD=GD1,GD2,GD3,GD4, ν1=h21K21Wd=ν11,ν12,ν13,ν14T, ν2=h22K22Wd=ν21,ν22,ν23T, and ν3=h23K23Wd=ν31,ν32,ν33T, θ~dj=θdj−θ^dj.From Equations (76)–(78), we can obtain the state error equation
(79)ė=Ame+BmE~k=Ame+BmE~k,where Am=diagAm1,Am2,Am3∈R6×6 and Ekj is the j th component of E~k, k=1,2,3, BmE~k=Bm1TE~k1,Bm2TE~k2,Bm3TE~k3T∈R6, and Bm=diagBm1,Bm2,Bm3∈R6×3.


#### 3.3.2. Adaptive Laws

According to the state error Equation (79), adaptive laws are chosen for the parameter θ^di and the parameters χ1it, χ2it, χ3it, θ1it, and θ2it of the failure compensator as
(80)θ^̇di=1c0∑j=13ΓdiϖdiePjGji,(81)χ̇1it=−γ1ic0∑j=13ePjGDiν1ii=2,3,−γ1ic0∑j=13ePjGDiν1i+fχ1ii=1,4,(82)χ̇2it=−γ2ic0∑j=13ePjGDi+1ν2ii=1,2,γ2ic0∑j=13ePjGDi+1ν2i+fχ2ii=3,(83)χ̇3it=−γ3ic0∑j=13ePjGDiν3ii=2,3,−γ3ic0∑j=13ePjGDiν3i+fχ3ii=1,(84)θ̇1tt=−1c0∑j=13Γ1iePjGDi+1ϖ1ϕ2ii=1,2,−1c0∑j=13Γ1iePjGDi+1ϖ1ϕ2i+fθ1ii=3,(85)θ̇4tt=−1c0∑j=13Γ4iePjGDiϖ1ϕ2ii=2,3,−1c0∑j=13Γ4iePjGDiϖ1ϕ2i+fθ4ii=1,where Γdi=ΓdiT>0, Γ1i=Γ1iT>0, Γ4i=Γ4iT>0, γ1i>0, γ2i>0, and γ3i>0 are the adaptive gains, and fχ1i is the projection algorithm. Consequently, based on adaptive laws χ̇11=−γ11/c0∑j=13ePjGD1ν11+fχ11, we can derive that 0≤χ11≤1 and χ11−χ11∗fχ11≤0. fχ2i, fχ3i, fθ1i, and fθ4i have the same characteristics with fχ1i.


#### 3.3.3. Stability Performance Analysis

(i)
For periodt∈T0,∞, σ=σ1. Lyapunov function is defined as(86)V0=12eTPe+12∑i=13θ~diTΓdi−1θ~di+12∑i=14χ~1i2γ1i−1+∑i=13χ~2i2γ2i−1+∑i=13χ~3i2γ3i−1+∑i=13θ~1iTΓ1i−1θ~1i+∑i=13θ~4iTΓ4i−1θ~4i.By differentiatingV0 in the interval T0,∞, we can obtain
(87)V̇0=12eTAmTPe+12eTPAme+1c0∑j=13∑i=14ePjGDiχ~1iν1i+∑j=13∑i=12ePjGDi+1χ~2iν2i+∑i=14γ1i−1χ~1iχ~̇1i+∑i=13γ2i−1χ~2iχ~̇2i+∑i=13γ3i−1χ~3iχ~̇3i+∑i=13θ~1iTΓ1i−1θ~̇1i+∑i=13θ~4iTΓ4i−1θ~̇4i,where eP=eP1,eP2,eP3∈R1×3 and ePi are the i+1-th column components of eTP∈R1×6, i=1,2,3, and P=diagPc1,Pc2,Pc3.Substituting Equations (80)–(85) into (87), one would have
(88)V̇0=12∑i=13eiTAmiTPci+PciAmi−1εiPciBmiRci−1BmiTPciei,t∈T0,T1.Based on of Lemma3, we can obtain
(89)V̇0=−12∑i=13eiTγi−1CmiTCmi+εiSiei≤0.(ii)
If actuatoru1 fails over the period T1,∞, i.e., σ=σ2, we define(90)V1=12eTPe+12∑i=13θ~diTΓdi−1θ~di+12∑i=24χ~1i2γ1i−1+∑i=13χ~2i2γ2i−1+∑i=23χ~3i2γ3i−1+∑i=13θ~1iTΓ1i−1θ~1i+∑i=23θ~4iTΓ4i−1θ~4i.By differentiatingV1 in the interval T1,∞ and combining Equations (80)–(84), we can obtain
(91)V̇1=−12∑i=13eiTγi−1CmiTCmi+εiSiei≤0.(iii)
Assume thatu1 is normal, and only u4 fails at T1 and remains failed on the interval T1,∞; that is, σ=σ3. We define(92)V2=12eTPe+12∑i=13θ~diTΓdi−1θ~di+12∑i=13χ~1i2γ1i−1+∑i=12χ~2i2γ2i−1+∑i=13χ~3i2γ3i−1+∑i=12θ~1iTΓ1i−1θ~1i+∑i=13θ~4iTΓ4i−1θ~4i.The time derivative ofV2 is
(93)V̇2=−12∑i=13eiTγi−1CmiTCmi+εmiSiei≤0,t∈T1,T2.WithV̇k,k=0,1,2≤0 for three different failure scenarios and the adopted projection scheme of adaptive laws, we can conclude that all the signals in the close-loop system are bounded, and the output error gradually decreases to zero over time.To sum up, the below theorem is obtained.Theorem 9.
For the spacecraft system (1) with potential uncertain actuator faults (9) and unknown disturbances (4), controller (68) designed based on an H∞ transient performance compensator (49), and its parameter adaptive laws (80)–(85) can ensure that the system is stable and perform the given maneuvers asymptotically, if for any failure pattern σt belongs to failure pattern set Σ=σj,j=1,2,3. The following condition holds the following equivalent actuation matrix. Gσx=GxDI−σt is a full rank in the domain U (definition is U⊂R6⟶V⊂R3).


#### 3.3.4. Transient Performance Analysis

Then, the transient performance is analyzed by the criteria of the bound of bothL∞ and mean square tracking error at any time.With (45) and (49), we can get the following output tracking error dynamic equation
(94)ei1=Wmisc0iGixDI−σv−v∗+Gixd−d^.Since the order of the stable reference modelWmis is pmi, it can be derived from Lemma 5 that
(95)ei1t∞≤2pmiWmis∞GixDI−σv−v∗+Gixd−d^.From Theorem9, we have θ~di∈L∞ and v−v∗∈L∞. With Lemma 3, one can prove that Wmis is stable and
(96)Wmis∞<γi,and thereby, we have
(97)ei1t∞=Δsupt≥0ei1t≤γici,where ci>0. Then, according to
(98)∫t1t2Hsx2dt≤Hs∞2∫t1t2x2dt,we have
(99)1t∫t0t0+tei12dτ≤Wmis∞21t∫t0t0+tGixDI−σv−v∗+Gixd−d∧2dτ≤γi2ci.Theorem 10.
For the improved controller (34), the performance index of H∞ compensator is γi, and the output tracking error ei1=yi−ymi and i=1,2,3 of the system (32) satisfies the following inequality condition:
(100)1t∫t0t0+tei12dτ≤Wmis∞21t∫t0t0+tGixDI−σv−v∗+Gixd−d∧2dτ≤γi2ci.where constant ci>0.According to Theorem10, the transient characteristics rely on the performance level of the H∞ compensator. Both Theorem 9 and Theorem 10 show that the control objective is reached by our proposed control scheme.


## 3.1. Modified MRAC Design

A model reference adaptive controller is constructed as follows(34)uLi=θiTωi+c0iri+uci,c0i=kmikpi,where ωi=ωi1T,yiT, θi=θi1T,θi2T, ωi1=αis/Λisy, αis=snpi−2,snpi−3,⋯,s,1, Λis=Zmiλis, and λis are monic Hurwitz polynomials of degree npi−qmi−1; θi∈R2ni−1 is the vector of controller parameters; uci is a proper H∞ compensator to be designed later.Remark 6.
To derive the relation between the parameter estimation error of actuator failure and transient performance, we just talk about the uncertainties of actuator failure and disturbance assuming the system parameters are known. If all the system parameters are known, the nominal controller of MRAC can beuLi=θi∗Tωi+c0iri+uci, with known θi∗=θi1∗T,θi2∗T.For a given transform functionGpis, there is a desired reference value vector θi∗=θi1∗T,θi2∗T to make the following matching conditions valid
(35)c0ikpiZpiRpi−kpiZpiαiTsθi1∗/Λis+θi2∗=Wmis.Multiplying both sides withyi and using the equation Rpisyi=kpiZpisurikpiZpi, we have
(36)coikpiZpiyi=WmiskpiZpiuri−kpiθi1∗TαisΛisyi+θi2∗Zpiyi.Dividing both sides bykpiZpi we have
(37)yi=Wmisc0iuri−θi1∗TαisΛisyi−θi2∗yi.The spacecraft system parameters are known; that is, in the MRAC law,θi is substituted by the desired value θi∗. To be more specific, the control uLi is designed as
(38)uLi=θi1∗TαisΛisyi+θi2∗yi+c0iri+uci=−Kiξi+c0iri+uci,where Ki=Ki1,Ki2 are the design controller parameters corresponding with θi∗, ξi=ξi1,ξi2T. Substituting (38) into (30), we have
(39)Gixudt=−Fix−Gixd^t+GixDI−σvt−v∗t−Kiξi+c0iri+uci.Furthermore, the linearized subsystem (18) is described as
(40)ξ̇i1=ξi2,ξ̇i2=−Kiξi+c0iri+uci+GixDI−σvt−v∗t+Gixdt−d^t.We then revise the above equation in the form of state space as follows:(41)ξ̇i=Aci−BciKiξi+BciGixDI−σv−v∗+Gixd−d^+uci+c0iri,yi=Cciξi,whereAci∈R2×2, Bci∈R2×1, and Cci∈R1×2 are the standard form of integrator chains. We also revise the reference model (33) in the state space as follows:
(42)ξ̇mi=Amiξmi+Bmiri,ymi=Cmiξmi,where Ami, Bmi, and Cmi are the minimal realization of Wmis, i.e.,
(43)Ami=01−ai1−ai2∈R2×2,Bmi=0,1T∈R2×1,and Cmi=1,0∈R1×2. Denoting ei=ξi−ξmi, we can obtain
(44)ėi=Aci−BciKiξi+BciGixDI−σv−v∗+Gixd−d^+uci+c0iri−Amiξmi−Bmiri.If the controller parametersKi is set to satisfy Aci−BciKi=Ami and Bcic0i=Bmi, then
(45)ėi=Amiei+Bmic0iGixDI−σv−v∗+Gixd−d^+uci,ei1=Cmiei.With the system model (32), (33) and the controller (34), we obtain
(46)yi=Wmisc0iGixDI−σv−v∗+Gixd−d^+c0iri+uci.Remark 7.
For the tracking error dynamics (45) and output dynamic (46), one may discover that the parameter estimation error items d−d^ and the controller parameter errors v−v∗ can be regarded as the disturbance input and plays an important role in system tracking performance. Hence, the attenuation of the disturbance made by GixDI−σv−v∗+Gixd−d^ on the closed-loop system is designed by proposing the compensator uci as an H∞ optimal controller.Equation (45) is considered as a special form of (23). With (45) and Lemma 3, a transient performance compensator uci is designed for the system (45) by the following steps as shown next.Step 1: initializeεi>0 and 0<γi<Wmis/c0i∞, choose two positive-definite matrices Sci∈R2×2 and Rci∈R1×1, and solve the following Riccati equation
(47)yi=Wmisc0iGixDI−σv−v∗+Gixd−d^+c0iri+uci.and obtain the state feedback gain
(48)yi=Wmisc0iGixDI−σv−v∗+Gixd−d^+c0iri+uci.If there is no solution, decreaseεi and repeat this step again until getting an appropriate positive-definite solution Pci=PciT>0 to ensure that the state feedback gain is obtained. The H∞ compensator which is based on measurement feedback
(49)uci=Kciei,is constructed by state feedback gain Kci, which guarantees that the system (45) with GixDI−σv−v∗+Gixd−d^ disturbance attenuation γi is stable.Step 2: Return to step 1 and reduce the transient performance index until getting a gratifying transient or the optimumγiRemark 8.
The standard algorithm can be used to confirm whether the positive-definite solution of a Riccati equation exists. According to the above processes, the optimal compensator ofH∞ can be obtained by selecting sufficiently small εi. However, it may cause the gain Kci too large. For the whole control system, the large gain will reduce the stability margin and increase the influence of measurement noise. In practical application, a compromise is usually adopted in the H∞ compensator design. It turns out that the suboptimum H∞ compensator for a given 0<γi<Wmis/c0i∞ can also achieve a satisfactory transient performance through the following theoretical analysis and simulation.


## 3.2. Nominal Compensation Design

As mentioned previously, the spacecraft system is turned into three-linear subsystem (18) derived from the feedback linearization technique. To derive the failure compensation control law ut, we write the three subsystems together as
(50)y¨1,y¨2,y¨3T=Fx+GDxI−σv+σu¯+Gxdt,where GDx=GxD∈R3×4 is regarded as control distribution matrix.


### 3.2.1. Design for no Failure Case

On this condition,σt=σ1=diag0,0,0,0, ut=vt, and GDxvt=Wdt. The signal vt is designed as
(51)vt=v1∗t=h21xv01∗tfor a chosen h21x∈R4×4, and signal v01∗t to be calculated from
(52)GDxh21xv01∗t=Wdt.The solutionv01∗t may be derived as
(53)v01∗t=K21xWdt,with matrix function K21x∈R4×3.


### 3.2.2. Design for theu1 Failure Case

In this situation,σt=σ2=diag1,0,0,0, u1=u¯1, and ui=vi for i=2,3,4, with GDx=GD1,GD2,GD3,GD4=GD1,GD2∈R3×4 for GD2=GD2,GD3,GD4∈R3×3, v=v1,v2,v3,v4T=v1,va2TT∈R4 for va2=v2,v3,v4T∈R3, equation GDxut=Wdt becomes
(54)GD1u¯1t+GD2va2t=Wdt.In this situation, the signalv1 is set to be v1=0. A nonsingular matrix function h22x∈R3×3 is chosen to set
(55)vt=v1t,va2TtT=v2∗t=0,va2∗TtT,va2∗t=h22xv02∗t,with v02∗t∈R3 to be deduced from
(56)GD1u¯1t+GD2h22xv02∗t=Wdt.We can obtain(57)v02∗t=K22xWdt+K221xu¯1t,with matrix function K22x∈R3×3 and vector K221x∈R3×1.


### 3.2.3. Design for theu4 Failure Case

Similarly,σt=σ3=diag0,0,0,1, u4=u¯4, and v4t are chosen as v4t=0 and vit=uit for i=1,2,3. With GDx=GD1,GD2,GD3,GD4=GD1,GD4∈R3×4 for GD1=GD1,GD2,GD3∈R3×3 and vt=va3Tt,v4tT for va3t=v1t,v2t,v3tT∈R3, and GDxut=Wdt becomes
(58)GD1va3t+GD4u¯4t=Wdt.The signalvt is proposed as
(59)vt=va3Tt,v4tT=v3∗t=va3∗Tt,0T,va3∗t=h23v03∗t,with a chosen matrix h23∈R3×3 and a deduced signal v03∗t∈R3 from
(60)GD1h23v03∗t+GD4u¯4t=Wdt.Similarly, we have(61)v03∗t=K23Wdt+K234u¯4t,with matrix functionK23∈R3×3 and vector K234∈R3×1.


### 3.2.4. Composite Control Law

Define three indicator functionsχj∗, j=1,2,3, which are corresponding to the considered three failure models σj and j=1,2,3, respectively. That is, if σ=σj and χj∗=1; otherwise, χj∗=0 and j=1,2,3. Then, a synthetic control law is obtained by integrating the three individual controllers
(62)v∗t=∑j=13χj∗tvj∗t,to handle the three failure cases.Forva2∗t=h22v02∗t, with v02∗t in (57), signal va2∗t can be further expressed as
(63)va2∗t=h22K22Wdt+h22K221u¯1t∈R3.Foru¯1t in (10), we express
(64)u¯1t=θ1∗Tϖ1t=ϖ1Ttθ1∗,where θ1∗=u¯10,u¯11,⋯,u¯1q1T∈Rq1+1, and ϖ1t=1,f11t,⋯,f1q1tT∈Rq1+1.Therefore, we have(65)χ2∗tva2∗t=χ21∗t,χ22∗t,χ23∗th22K22Wdt+θ11∗Ttϖ1tϕ2,1,θ12∗Ttϖ1tϕ22,θ13∗Ttϖ1tϕ23T,where χ2i∗t=χ2∗t, θ1i∗t=χ2∗tθ1∗, i=1,2,3, and ϕ2=h22K221=ϕ21,ϕ22,ϕ23T.In the same way, to designatevχ11∗t and vχ33∗t, we express
(66)χ1∗tv1∗t=diagχ11∗t,χ12∗t,χ13∗t,χ14∗th21K21Wdt(67)χ3∗tva3∗t=diagχ31∗t,χ32∗t,χ33∗th23K23Wdt+θ41∗Ttϖ4tϕ31,θ42∗Ttϖ4tϕ32,θ43∗Ttϖ4tϕ33T,where χ1i∗t=χ1∗t, i=1,2,3,4, χ3i∗t=χ3∗t, and θ4i∗t=χ3∗tθ4∗, i=1,2,3.


## 3.2.1. Design for no Failure Case

On this condition,σt=σ1=diag0,0,0,0, ut=vt, and GDxvt=Wdt. The signal vt is designed as
(51)vt=v1∗t=h21xv01∗tfor a chosen h21x∈R4×4, and signal v01∗t to be calculated from
(52)GDxh21xv01∗t=Wdt.The solutionv01∗t may be derived as
(53)v01∗t=K21xWdt,with matrix function K21x∈R4×3.


## 3.2.2. Design for theu1 Failure Case

In this situation,σt=σ2=diag1,0,0,0, u1=u¯1, and ui=vi for i=2,3,4, with GDx=GD1,GD2,GD3,GD4=GD1,GD2∈R3×4 for GD2=GD2,GD3,GD4∈R3×3, v=v1,v2,v3,v4T=v1,va2TT∈R4 for va2=v2,v3,v4T∈R3, equation GDxut=Wdt becomes
(54)GD1u¯1t+GD2va2t=Wdt.In this situation, the signalv1 is set to be v1=0. A nonsingular matrix function h22x∈R3×3 is chosen to set
(55)vt=v1t,va2TtT=v2∗t=0,va2∗TtT,va2∗t=h22xv02∗t,with v02∗t∈R3 to be deduced from
(56)GD1u¯1t+GD2h22xv02∗t=Wdt.We can obtain(57)v02∗t=K22xWdt+K221xu¯1t,with matrix function K22x∈R3×3 and vector K221x∈R3×1.


## 3.2.3. Design for theu4 Failure Case

Similarly,σt=σ3=diag0,0,0,1, u4=u¯4, and v4t are chosen as v4t=0 and vit=uit for i=1,2,3. With GDx=GD1,GD2,GD3,GD4=GD1,GD4∈R3×4 for GD1=GD1,GD2,GD3∈R3×3 and vt=va3Tt,v4tT for va3t=v1t,v2t,v3tT∈R3, and GDxut=Wdt becomes
(58)GD1va3t+GD4u¯4t=Wdt.The signalvt is proposed as
(59)vt=va3Tt,v4tT=v3∗t=va3∗Tt,0T,va3∗t=h23v03∗t,with a chosen matrix h23∈R3×3 and a deduced signal v03∗t∈R3 from
(60)GD1h23v03∗t+GD4u¯4t=Wdt.Similarly, we have(61)v03∗t=K23Wdt+K234u¯4t,with matrix functionK23∈R3×3 and vector K234∈R3×1.


## 3.2.4. Composite Control Law

Define three indicator functionsχj∗, j=1,2,3, which are corresponding to the considered three failure models σj and j=1,2,3, respectively. That is, if σ=σj and χj∗=1; otherwise, χj∗=0 and j=1,2,3. Then, a synthetic control law is obtained by integrating the three individual controllers
(62)v∗t=∑j=13χj∗tvj∗t,to handle the three failure cases.Forva2∗t=h22v02∗t, with v02∗t in (57), signal va2∗t can be further expressed as
(63)va2∗t=h22K22Wdt+h22K221u¯1t∈R3.Foru¯1t in (10), we express
(64)u¯1t=θ1∗Tϖ1t=ϖ1Ttθ1∗,where θ1∗=u¯10,u¯11,⋯,u¯1q1T∈Rq1+1, and ϖ1t=1,f11t,⋯,f1q1tT∈Rq1+1.Therefore, we have(65)χ2∗tva2∗t=χ21∗t,χ22∗t,χ23∗th22K22Wdt+θ11∗Ttϖ1tϕ2,1,θ12∗Ttϖ1tϕ22,θ13∗Ttϖ1tϕ23T,where χ2i∗t=χ2∗t, θ1i∗t=χ2∗tθ1∗, i=1,2,3, and ϕ2=h22K221=ϕ21,ϕ22,ϕ23T.In the same way, to designatevχ11∗t and vχ33∗t, we express
(66)χ1∗tv1∗t=diagχ11∗t,χ12∗t,χ13∗t,χ14∗th21K21Wdt(67)χ3∗tva3∗t=diagχ31∗t,χ32∗t,χ33∗th23K23Wdt+θ41∗Ttϖ4tϕ31,θ42∗Ttϖ4tϕ32,θ43∗Ttϖ4tϕ33T,where χ1i∗t=χ1∗t, i=1,2,3,4, χ3i∗t=χ3∗t, and θ4i∗t=χ3∗tθ4∗, i=1,2,3.


## 3.3. Adaptive Fault Tolerant Control Design

The adaptive version of the nominal control law is as follows (62):
(68)vt=∑j=13χjtvjt=∑j=13vχjjt=vχ11t+0,vχ2a2TtT+vχ3a3Tt,0T.In view of (65)–(67), we derive
(69)vχ11t=Δdiagχ11t,χ12t,χ13t,χ14th21K21Wdt,(70)vχ2a2t=Δdiagχ21t,χ22t,χ23th22K22Wdt+θ11Ttϖ1tϕ21,θ12Ttϖ1tϕ22,θ13Ttϖ1tϕ23T,(71)vχ3a3t=Δdiagχ31t,χ32t,χ33th23K23Wdt+θ41Ttϖ4tϕ31,θ42Ttϖ4tϕ32,θ43Ttϖ4tϕ33T,where χjit, θ1it, and θ4it are the estimates of χji∗t, θ1i∗t, and θ4i∗t, respectively.From (65)–(71), we obtain
(72)vt−v∗t=v~χ11t+0,v~χ2a2TtT+v~χ3a3Tt,0T,where
(73)v~χ11t=χ~11t,χ~12t,χ~13t,χ~14th21K21Wdt,v~χ2a2t=diagχ~21t,χ~22t,χ~23th22K22Wdt+θ~11Ttϖ1tϕ21,θ~12Ttϖ1tϕ22,θ~13Ttϖ1tϕ23T,v~χ3a3t=Δdiagχ~31t,χ~32t,χ~33th23K23Wdt+θ~41Ttϖ4tϕ31,θ~42Ttϖ4tϕ32,θ~43Ttϖ4tϕ33T.


### 3.3.1. State Error Equations

Introducinge∈R6=e1T,e2T,e3TT=e11,e12,e21,e22,e31,e32T, ei1=yi−ymi=ξi1−ξmi1, ei2=ξ̇i1−ξ̇mi1=ėi1, and i=1,2,3. With the three subsystems
(74)ėi=Amiei+Bmic0iGixDI−σv−v∗−1/2εiRci−1BciTPciei+Gixd−d^,i=1,2,3,we have
(75)e¨11+α12ė11+α11e11e¨21+α22ė21+α21e21e¨31+α32ė31+α31e31=1c0GxDI−σv−v∗+Gxd−d^−1/2ε1c0Rc1−1Bc1TPc1e11/2ε2c0Rc2−1Bc2TPc2e21/2ε3c0Rc3−1Bc3TPc3e3.Ifσ=σ1=diag0,0,0,0, with (72) and (73), we rewrite (75) as
(76)e¨11+α12ė11+α11e11e¨21+α22ė21+α21e21e¨31+α32ė31+α31e31=1c0∑i=14GDiχ~1iν1i+∑i=13GDi+1χ~2iν2i+∑i=13GDiχ~3iν3i+∑i=13GDi+1θ~1iTϖ1ϕ2i+∑i=13GDiθ~4iTϖ4ϕ3i+1c0∑j=13G1jθ~djTtϖdj∑j=13G2jθ~djTtϖdj∑j=13G3jθ~djTtϖdj−12ε1c0Rc1−1Bc1TPc1e112ε2c0Rc2−1Bc2TPc2e212ε3c0Rc3−1Bc3TPc3e3≜E~1.Ifσ=σ2=diag1,0,0,0, with (72) and (73), we rewrite (75) as
(77)e¨11+α12ė11+α11e11e¨21+α22ė21+α21e21e¨31+α32ė31+α31e31=1c0∑i=24GDiχ~1iν1i+∑i=13GDi+1χ~2iν2i+∑i=23GDiχ~3iν3i+∑i=13GDi+1θ~1iTϖ1ϕ2i+∑i=23GDiθ~4iTϖ4ϕ3i+1c0∑j=13G1jθ~djTtϖdj∑j=13G2jθ~djTtϖdj∑j=13G3jθ~djTtϖdj−12ε1c0Rc1−1Bc1TPc1e112ε2c0Rc2−1Bc2TPc2e212ε3c0Rc3−1Bc3TPc3e3≜E~2Ifσ=σ3=diag0,0,0,1, with (72) and (73), we rewrite (75) as
(78)e¨11+α12ė11+α11e11e¨21+α22ė21+α21e21e¨31+α32ė31+α31e31=1c0∑i=13GDiχ~1iν1i+∑i=12GDi+1χ~2iν2i+∑i=13GDiχ~3iν3i+∑i=12GDi+1θ~1iTϖ1ϕ2i+∑i=13GDiθ~4iTϖ4ϕ3i+1c0∑j=13G1jθ~djTtϖdj∑j=13G2jθ~djTtϖdj∑j=13G3jθ~djTtϖdj−12ε1c0Rc1−1Bc1TPc1e112ε2c0Rc2−1Bc2TPc2e212ε3c0Rc3−1Bc3TPc3e3≜E~3where GD=GD1,GD2,GD3,GD4, ν1=h21K21Wd=ν11,ν12,ν13,ν14T, ν2=h22K22Wd=ν21,ν22,ν23T, and ν3=h23K23Wd=ν31,ν32,ν33T, θ~dj=θdj−θ^dj.From Equations (76)–(78), we can obtain the state error equation
(79)ė=Ame+BmE~k=Ame+BmE~k,where Am=diagAm1,Am2,Am3∈R6×6 and Ekj is the j th component of E~k, k=1,2,3, BmE~k=Bm1TE~k1,Bm2TE~k2,Bm3TE~k3T∈R6, and Bm=diagBm1,Bm2,Bm3∈R6×3.


### 3.3.2. Adaptive Laws

According to the state error Equation (79), adaptive laws are chosen for the parameter θ^di and the parameters χ1it, χ2it, χ3it, θ1it, and θ2it of the failure compensator as
(80)θ^̇di=1c0∑j=13ΓdiϖdiePjGji,(81)χ̇1it=−γ1ic0∑j=13ePjGDiν1ii=2,3,−γ1ic0∑j=13ePjGDiν1i+fχ1ii=1,4,(82)χ̇2it=−γ2ic0∑j=13ePjGDi+1ν2ii=1,2,γ2ic0∑j=13ePjGDi+1ν2i+fχ2ii=3,(83)χ̇3it=−γ3ic0∑j=13ePjGDiν3ii=2,3,−γ3ic0∑j=13ePjGDiν3i+fχ3ii=1,(84)θ̇1tt=−1c0∑j=13Γ1iePjGDi+1ϖ1ϕ2ii=1,2,−1c0∑j=13Γ1iePjGDi+1ϖ1ϕ2i+fθ1ii=3,(85)θ̇4tt=−1c0∑j=13Γ4iePjGDiϖ1ϕ2ii=2,3,−1c0∑j=13Γ4iePjGDiϖ1ϕ2i+fθ4ii=1,where Γdi=ΓdiT>0, Γ1i=Γ1iT>0, Γ4i=Γ4iT>0, γ1i>0, γ2i>0, and γ3i>0 are the adaptive gains, and fχ1i is the projection algorithm. Consequently, based on adaptive laws χ̇11=−γ11/c0∑j=13ePjGD1ν11+fχ11, we can derive that 0≤χ11≤1 and χ11−χ11∗fχ11≤0. fχ2i, fχ3i, fθ1i, and fθ4i have the same characteristics with fχ1i.


### 3.3.3. Stability Performance Analysis

(i)
For periodt∈T0,∞, σ=σ1. Lyapunov function is defined as(86)V0=12eTPe+12∑i=13θ~diTΓdi−1θ~di+12∑i=14χ~1i2γ1i−1+∑i=13χ~2i2γ2i−1+∑i=13χ~3i2γ3i−1+∑i=13θ~1iTΓ1i−1θ~1i+∑i=13θ~4iTΓ4i−1θ~4i.By differentiatingV0 in the interval T0,∞, we can obtain
(87)V̇0=12eTAmTPe+12eTPAme+1c0∑j=13∑i=14ePjGDiχ~1iν1i+∑j=13∑i=12ePjGDi+1χ~2iν2i+∑i=14γ1i−1χ~1iχ~̇1i+∑i=13γ2i−1χ~2iχ~̇2i+∑i=13γ3i−1χ~3iχ~̇3i+∑i=13θ~1iTΓ1i−1θ~̇1i+∑i=13θ~4iTΓ4i−1θ~̇4i,where eP=eP1,eP2,eP3∈R1×3 and ePi are the i+1-th column components of eTP∈R1×6, i=1,2,3, and P=diagPc1,Pc2,Pc3.Substituting Equations (80)–(85) into (87), one would have
(88)V̇0=12∑i=13eiTAmiTPci+PciAmi−1εiPciBmiRci−1BmiTPciei,t∈T0,T1.Based on of Lemma3, we can obtain
(89)V̇0=−12∑i=13eiTγi−1CmiTCmi+εiSiei≤0.(ii)
If actuatoru1 fails over the period T1,∞, i.e., σ=σ2, we define(90)V1=12eTPe+12∑i=13θ~diTΓdi−1θ~di+12∑i=24χ~1i2γ1i−1+∑i=13χ~2i2γ2i−1+∑i=23χ~3i2γ3i−1+∑i=13θ~1iTΓ1i−1θ~1i+∑i=23θ~4iTΓ4i−1θ~4i.By differentiatingV1 in the interval T1,∞ and combining Equations (80)–(84), we can obtain
(91)V̇1=−12∑i=13eiTγi−1CmiTCmi+εiSiei≤0.(iii)
Assume thatu1 is normal, and only u4 fails at T1 and remains failed on the interval T1,∞; that is, σ=σ3. We define(92)V2=12eTPe+12∑i=13θ~diTΓdi−1θ~di+12∑i=13χ~1i2γ1i−1+∑i=12χ~2i2γ2i−1+∑i=13χ~3i2γ3i−1+∑i=12θ~1iTΓ1i−1θ~1i+∑i=13θ~4iTΓ4i−1θ~4i.The time derivative ofV2 is
(93)V̇2=−12∑i=13eiTγi−1CmiTCmi+εmiSiei≤0,t∈T1,T2.WithV̇k,k=0,1,2≤0 for three different failure scenarios and the adopted projection scheme of adaptive laws, we can conclude that all the signals in the close-loop system are bounded, and the output error gradually decreases to zero over time.To sum up, the below theorem is obtained.Theorem 9.
For the spacecraft system (1) with potential uncertain actuator faults (9) and unknown disturbances (4), controller (68) designed based on an H∞ transient performance compensator (49), and its parameter adaptive laws (80)–(85) can ensure that the system is stable and perform the given maneuvers asymptotically, if for any failure pattern σt belongs to failure pattern set Σ=σj,j=1,2,3. The following condition holds the following equivalent actuation matrix. Gσx=GxDI−σt is a full rank in the domain U (definition is U⊂R6⟶V⊂R3).


### 3.3.4. Transient Performance Analysis

Then, the transient performance is analyzed by the criteria of the bound of bothL∞ and mean square tracking error at any time.With (45) and (49), we can get the following output tracking error dynamic equation
(94)ei1=Wmisc0iGixDI−σv−v∗+Gixd−d^.Since the order of the stable reference modelWmis is pmi, it can be derived from Lemma 5 that
(95)ei1t∞≤2pmiWmis∞GixDI−σv−v∗+Gixd−d^.From Theorem9, we have θ~di∈L∞ and v−v∗∈L∞. With Lemma 3, one can prove that Wmis is stable and
(96)Wmis∞<γi,and thereby, we have
(97)ei1t∞=Δsupt≥0ei1t≤γici,where ci>0. Then, according to
(98)∫t1t2Hsx2dt≤Hs∞2∫t1t2x2dt,we have
(99)1t∫t0t0+tei12dτ≤Wmis∞21t∫t0t0+tGixDI−σv−v∗+Gixd−d∧2dτ≤γi2ci.Theorem 10.
For the improved controller (34), the performance index of H∞ compensator is γi, and the output tracking error ei1=yi−ymi and i=1,2,3 of the system (32) satisfies the following inequality condition:
(100)1t∫t0t0+tei12dτ≤Wmis∞21t∫t0t0+tGixDI−σv−v∗+Gixd−d∧2dτ≤γi2ci.where constant ci>0.According to Theorem10, the transient characteristics rely on the performance level of the H∞ compensator. Both Theorem 9 and Theorem 10 show that the control objective is reached by our proposed control scheme.


## 3.3.1. State Error Equations

Introducinge∈R6=e1T,e2T,e3TT=e11,e12,e21,e22,e31,e32T, ei1=yi−ymi=ξi1−ξmi1, ei2=ξ̇i1−ξ̇mi1=ėi1, and i=1,2,3. With the three subsystems
(74)ėi=Amiei+Bmic0iGixDI−σv−v∗−1/2εiRci−1BciTPciei+Gixd−d^,i=1,2,3,we have
(75)e¨11+α12ė11+α11e11e¨21+α22ė21+α21e21e¨31+α32ė31+α31e31=1c0GxDI−σv−v∗+Gxd−d^−1/2ε1c0Rc1−1Bc1TPc1e11/2ε2c0Rc2−1Bc2TPc2e21/2ε3c0Rc3−1Bc3TPc3e3.Ifσ=σ1=diag0,0,0,0, with (72) and (73), we rewrite (75) as
(76)e¨11+α12ė11+α11e11e¨21+α22ė21+α21e21e¨31+α32ė31+α31e31=1c0∑i=14GDiχ~1iν1i+∑i=13GDi+1χ~2iν2i+∑i=13GDiχ~3iν3i+∑i=13GDi+1θ~1iTϖ1ϕ2i+∑i=13GDiθ~4iTϖ4ϕ3i+1c0∑j=13G1jθ~djTtϖdj∑j=13G2jθ~djTtϖdj∑j=13G3jθ~djTtϖdj−12ε1c0Rc1−1Bc1TPc1e112ε2c0Rc2−1Bc2TPc2e212ε3c0Rc3−1Bc3TPc3e3≜E~1.Ifσ=σ2=diag1,0,0,0, with (72) and (73), we rewrite (75) as
(77)e¨11+α12ė11+α11e11e¨21+α22ė21+α21e21e¨31+α32ė31+α31e31=1c0∑i=24GDiχ~1iν1i+∑i=13GDi+1χ~2iν2i+∑i=23GDiχ~3iν3i+∑i=13GDi+1θ~1iTϖ1ϕ2i+∑i=23GDiθ~4iTϖ4ϕ3i+1c0∑j=13G1jθ~djTtϖdj∑j=13G2jθ~djTtϖdj∑j=13G3jθ~djTtϖdj−12ε1c0Rc1−1Bc1TPc1e112ε2c0Rc2−1Bc2TPc2e212ε3c0Rc3−1Bc3TPc3e3≜E~2Ifσ=σ3=diag0,0,0,1, with (72) and (73), we rewrite (75) as
(78)e¨11+α12ė11+α11e11e¨21+α22ė21+α21e21e¨31+α32ė31+α31e31=1c0∑i=13GDiχ~1iν1i+∑i=12GDi+1χ~2iν2i+∑i=13GDiχ~3iν3i+∑i=12GDi+1θ~1iTϖ1ϕ2i+∑i=13GDiθ~4iTϖ4ϕ3i+1c0∑j=13G1jθ~djTtϖdj∑j=13G2jθ~djTtϖdj∑j=13G3jθ~djTtϖdj−12ε1c0Rc1−1Bc1TPc1e112ε2c0Rc2−1Bc2TPc2e212ε3c0Rc3−1Bc3TPc3e3≜E~3where GD=GD1,GD2,GD3,GD4, ν1=h21K21Wd=ν11,ν12,ν13,ν14T, ν2=h22K22Wd=ν21,ν22,ν23T, and ν3=h23K23Wd=ν31,ν32,ν33T, θ~dj=θdj−θ^dj.From Equations (76)–(78), we can obtain the state error equation
(79)ė=Ame+BmE~k=Ame+BmE~k,where Am=diagAm1,Am2,Am3∈R6×6 and Ekj is the j th component of E~k, k=1,2,3, BmE~k=Bm1TE~k1,Bm2TE~k2,Bm3TE~k3T∈R6, and Bm=diagBm1,Bm2,Bm3∈R6×3.


## 3.3.2. Adaptive Laws

According to the state error Equation (79), adaptive laws are chosen for the parameter θ^di and the parameters χ1it, χ2it, χ3it, θ1it, and θ2it of the failure compensator as
(80)θ^̇di=1c0∑j=13ΓdiϖdiePjGji,(81)χ̇1it=−γ1ic0∑j=13ePjGDiν1ii=2,3,−γ1ic0∑j=13ePjGDiν1i+fχ1ii=1,4,(82)χ̇2it=−γ2ic0∑j=13ePjGDi+1ν2ii=1,2,γ2ic0∑j=13ePjGDi+1ν2i+fχ2ii=3,(83)χ̇3it=−γ3ic0∑j=13ePjGDiν3ii=2,3,−γ3ic0∑j=13ePjGDiν3i+fχ3ii=1,(84)θ̇1tt=−1c0∑j=13Γ1iePjGDi+1ϖ1ϕ2ii=1,2,−1c0∑j=13Γ1iePjGDi+1ϖ1ϕ2i+fθ1ii=3,(85)θ̇4tt=−1c0∑j=13Γ4iePjGDiϖ1ϕ2ii=2,3,−1c0∑j=13Γ4iePjGDiϖ1ϕ2i+fθ4ii=1,where Γdi=ΓdiT>0, Γ1i=Γ1iT>0, Γ4i=Γ4iT>0, γ1i>0, γ2i>0, and γ3i>0 are the adaptive gains, and fχ1i is the projection algorithm. Consequently, based on adaptive laws χ̇11=−γ11/c0∑j=13ePjGD1ν11+fχ11, we can derive that 0≤χ11≤1 and χ11−χ11∗fχ11≤0. fχ2i, fχ3i, fθ1i, and fθ4i have the same characteristics with fχ1i.


## 3.3.3. Stability Performance Analysis

(i)
For periodt∈T0,∞, σ=σ1. Lyapunov function is defined as(86)V0=12eTPe+12∑i=13θ~diTΓdi−1θ~di+12∑i=14χ~1i2γ1i−1+∑i=13χ~2i2γ2i−1+∑i=13χ~3i2γ3i−1+∑i=13θ~1iTΓ1i−1θ~1i+∑i=13θ~4iTΓ4i−1θ~4i.By differentiatingV0 in the interval T0,∞, we can obtain
(87)V̇0=12eTAmTPe+12eTPAme+1c0∑j=13∑i=14ePjGDiχ~1iν1i+∑j=13∑i=12ePjGDi+1χ~2iν2i+∑i=14γ1i−1χ~1iχ~̇1i+∑i=13γ2i−1χ~2iχ~̇2i+∑i=13γ3i−1χ~3iχ~̇3i+∑i=13θ~1iTΓ1i−1θ~̇1i+∑i=13θ~4iTΓ4i−1θ~̇4i,where eP=eP1,eP2,eP3∈R1×3 and ePi are the i+1-th column components of eTP∈R1×6, i=1,2,3, and P=diagPc1,Pc2,Pc3.Substituting Equations (80)–(85) into (87), one would have
(88)V̇0=12∑i=13eiTAmiTPci+PciAmi−1εiPciBmiRci−1BmiTPciei,t∈T0,T1.Based on of Lemma3, we can obtain
(89)V̇0=−12∑i=13eiTγi−1CmiTCmi+εiSiei≤0.(ii)
If actuatoru1 fails over the period T1,∞, i.e., σ=σ2, we define(90)V1=12eTPe+12∑i=13θ~diTΓdi−1θ~di+12∑i=24χ~1i2γ1i−1+∑i=13χ~2i2γ2i−1+∑i=23χ~3i2γ3i−1+∑i=13θ~1iTΓ1i−1θ~1i+∑i=23θ~4iTΓ4i−1θ~4i.By differentiatingV1 in the interval T1,∞ and combining Equations (80)–(84), we can obtain
(91)V̇1=−12∑i=13eiTγi−1CmiTCmi+εiSiei≤0.(iii)
Assume thatu1 is normal, and only u4 fails at T1 and remains failed on the interval T1,∞; that is, σ=σ3. We define(92)V2=12eTPe+12∑i=13θ~diTΓdi−1θ~di+12∑i=13χ~1i2γ1i−1+∑i=12χ~2i2γ2i−1+∑i=13χ~3i2γ3i−1+∑i=12θ~1iTΓ1i−1θ~1i+∑i=13θ~4iTΓ4i−1θ~4i.The time derivative ofV2 is
(93)V̇2=−12∑i=13eiTγi−1CmiTCmi+εmiSiei≤0,t∈T1,T2.WithV̇k,k=0,1,2≤0 for three different failure scenarios and the adopted projection scheme of adaptive laws, we can conclude that all the signals in the close-loop system are bounded, and the output error gradually decreases to zero over time.To sum up, the below theorem is obtained.Theorem 9.
For the spacecraft system (1) with potential uncertain actuator faults (9) and unknown disturbances (4), controller (68) designed based on an H∞ transient performance compensator (49), and its parameter adaptive laws (80)–(85) can ensure that the system is stable and perform the given maneuvers asymptotically, if for any failure pattern σt belongs to failure pattern set Σ=σj,j=1,2,3. The following condition holds the following equivalent actuation matrix. Gσx=GxDI−σt is a full rank in the domain U (definition is U⊂R6⟶V⊂R3).


## 3.3.4. Transient Performance Analysis

Then, the transient performance is analyzed by the criteria of the bound of bothL∞ and mean square tracking error at any time.With (45) and (49), we can get the following output tracking error dynamic equation
(94)ei1=Wmisc0iGixDI−σv−v∗+Gixd−d^.Since the order of the stable reference modelWmis is pmi, it can be derived from Lemma 5 that
(95)ei1t∞≤2pmiWmis∞GixDI−σv−v∗+Gixd−d^.From Theorem9, we have θ~di∈L∞ and v−v∗∈L∞. With Lemma 3, one can prove that Wmis is stable and
(96)Wmis∞<γi,and thereby, we have
(97)ei1t∞=Δsupt≥0ei1t≤γici,where ci>0. Then, according to
(98)∫t1t2Hsx2dt≤Hs∞2∫t1t2x2dt,we have
(99)1t∫t0t0+tei12dτ≤Wmis∞21t∫t0t0+tGixDI−σv−v∗+Gixd−d∧2dτ≤γi2ci.Theorem 10.
For the improved controller (34), the performance index of H∞ compensator is γi, and the output tracking error ei1=yi−ymi and i=1,2,3 of the system (32) satisfies the following inequality condition:
(100)1t∫t0t0+tei12dτ≤Wmis∞21t∫t0t0+tGixDI−σv−v∗+Gixd−d∧2dτ≤γi2ci.where constant ci>0.According to Theorem10, the transient characteristics rely on the performance level of the H∞ compensator. Both Theorem 9 and Theorem 10 show that the control objective is reached by our proposed control scheme.


## 4. Simulations

MATLAB/SIMULINK software has been used to carry out numerical simulations to verify the effectiveness and performance of the proposed control scheme. The nominal moments of inertia parametersJ=diag40.45,42.09,42.36kg⋅m2, orientation matrix of the reaction wheel, and the external disturbances dt=sin0.01t+1,1.5cos0.01t−1,2sin0.01t+1×10−3N⋅m are taken from [36].


### 4.1. Simulation Conditions

The attitude values att=0 are given by q00=0.8834, q0=0.03,−0.02,−0.03T, and ω0=0,0,0T. For simulation, the initial values of indicator function and failure parameter estimates are chosen as: χ1i0=1,i=1,2,3,4, χ2i0=0, χ3i0=0,i=1,2,3; θ1i0=0,0T, andθ4i0=0,0T. Basis functions of failure model (10) and disturbance model (4) are ϖ1t=ϖ4t=1,sin0.25tT∈R2 and ϖd1=ϖd2=ϖd3=1,sin0.01t,cos0.01tT∈R3.The design parameters are chosen asγ1i=1, γ2i=1, γ3i=1, c0i=c0=1, γi=1, Ri=2, εi=4, i=1,2,3,
(101)K=12.23.98.52.79.12.3,Γdi=3.303.303.33.33.33.33.3,Γ1i=16.300016.300016.3,Γ4i=31.400031.400031.4,Pc1=3.536.321.054.39,Pc2=3.5310.511.056.58,Pc3=3.534.231.053.79.A second-order reference modelWmis=1/s2+2s+1 and the reference input signal rit=0,i=1,2,3, are chosen to generate the given command ymt to be tracked by system output yt.In the numerical simulation, for comparison, three cases are conducted: (1) attitude tracking control using our proposed modified MRAC based adaptive failure compensation controller (68) (denoted as “MMRAC based FTC”); (2) attitude tracking control using the standard MRAC-based adaptive failure compensation controller (denoted as “SMRAC based FTC”) without transient performance compensator; and (3) attitude tracking control using the direct adaptive failure compensation controller (denoted as “DAC-based FTC”) in [35]. The attitude tracking responses are analyzed to study the performances of the controllers.


### 4.2. Simulation Results

To demonstrate the superior performance of the proposed control scheme, two actuator failure conditions are simulated: Case1—intermittent fault occurring in actuator u1—and Case 2—Alternate faults occurring in actuators u1 and u4.Case 1. Intermittent fault occurring in actuatoru1.
In this case, the following failure conditions are considered as follows:(i)
When0≤t<100s, all the reaction wheels function healthily, uit=vit,i=1,2,3,4(ii)
When100s≤t<200s, actuator u1 failed, u1t=2Nm and uit=vit,i=2,3,4(iii)
Whent≥200s, actuator u1 returns to normal, uit=vit,i=1,2,3,4(iv)
Whent≥300s, actuator u1 is out of control, u1t=0.75sin0.25tNm, uit=vit,i=2,3,4Figure2(a) shows the simulated results obtained by including the faulty actuators for three controllers, namely the designed MMRAC based fault tolerant controller marked with solid line, the SMRAC-based fault-tolerant controller marked with dashed line, and the DAC-based fault-tolerant controller marked with dotted line. All the three fault-tolerant controllers can compensate for both the constraint and time varying faults, although the system performance degrades to some degree, the overshoot a setting time increase significantly once the failure is introduced. However, the system ultimately regulates the attitude to zero asymptotically; that is, the attitude stabilization maneuver is still performed successfully due to the fault-tolerant performance of the three controllers to uncertain actuator failures. It can be found from Figure 2(c) at the moments when actuator u1 failed at t=100s and t=300s (shown in Figure 2(b), which is corresponding with the simulation conditions), the overshoot of the system implemented by our designed MMRAC-based fault-tolerant controller are smaller than the SMRAC- and DAC-based fault-tolerant controller even before the actuators have failed, and this is because the effect caused by parameter estimation errors of both failure and disturbances on the transient performance has been reduced by the H∞ compensator. These simulations demonstrate the theoretical result that the desired performance of the system can be achieved by the proposed fault tolerant control even if the faults are unknown in advance.Case 2. Alternate Faults Occurring in Actuatoru1 and u4.
In this case, the following failure conditions are considered as follows:(i)
When0≤t<100s, all the reaction wheels function healthily, uit=vit,i=1,2,3,4(ii)
When100s≤t<200s, actuator u1 failed, u1t=3.5Nm and uit=vit,i=2,3,4(iii)
Whent≥200s, actuator u1 returns to normal, uit=vit,i=2,3,4(iv)
Whent≥300s, actuator u4 is out of control, u4t=1.75sin0.25tNm,uit=vit,i=1,2,3Figure 2
Attitude tracking control using MMRAC-, SMRAC-, and DAC-based fault-tolerant control schemes for Case1.
(a)
Response of system output(b)
Control input signals for different fault scenarios(c)
Responses of tracking errors(d)
System outputs for differentγ in the MMAC designThis example represents the severe case in which both two actuators experience failure at the moments oft=100s and t=300s, respectively. Actually, the failure conditions in Case 2 can be regarded as a mixed pattern of intermittent failure and permanent failure. As shown in Figure 3(b), when u1 is stuck at the instant t=100s and becomes normal at t=200s, that is an intermittent failure occurred in actuator u1, which is activated and inactivated by itself. The system produces an erroneous result when such fault is active during the period of 100s≤t<200s and produces a correct result when it is inactive for t≥200s. At t=300s, actuator u4 undergoes time-varying failure and never come back as normal. Figure 3(a) shows the results using the three different control laws based on the same simulation conditions. Clearly, compared with attitude tracking response for Case 1, the attitude control performance deteriorates severely due to the multiple uncertainties of actuator failure, with severe overshoots in the attitude orientation, although the objective of asymptotic attitude tracking can be achieved finally, as clearly indicated from the output tracking errors present in Figure 3(c).Figure 3
Attitude tracking control using MMRAC-, SMRAC-, and DAC-based fault-tolerant control schemes for Case2.
(a)
Response of system output(b)
Control input signals for different fault scenarios(c)
Responses of tracking errors(d)
System outputs for differentγ in the MMAC designIn summary, for both the normal, intermittent failure and permanent failure cases, the proposed controller significantly improves the normal control performance of the closed-loop attitude system compared to the SMRAC- and DAC-based fault-tolerant control approaches. For the cases with actuator faults, the proposed method gives better transient performance than those controllers without including the transient performance compensator. As the faults become more severe, the proposed controller still guarantees system stability and asymptotic output tracking of a given command.Furthermore, comparing the modified control scheme with differentγi (i=1,2), it can be found from Figures 2(d) and 3(d) that with smaller γ1=0.1, the compensator takes more effects on transient oscillations inhibition with a smaller following error of trajectory than that with γ2=3, which keeps an agreement with the performance analysis by the criterion of L1 bound or mean squared value.


## 4.1. Simulation Conditions

The attitude values att=0 are given by q00=0.8834, q0=0.03,−0.02,−0.03T, and ω0=0,0,0T. For simulation, the initial values of indicator function and failure parameter estimates are chosen as: χ1i0=1,i=1,2,3,4, χ2i0=0, χ3i0=0,i=1,2,3; θ1i0=0,0T, andθ4i0=0,0T. Basis functions of failure model (10) and disturbance model (4) are ϖ1t=ϖ4t=1,sin0.25tT∈R2 and ϖd1=ϖd2=ϖd3=1,sin0.01t,cos0.01tT∈R3.The design parameters are chosen asγ1i=1, γ2i=1, γ3i=1, c0i=c0=1, γi=1, Ri=2, εi=4, i=1,2,3,
(101)K=12.23.98.52.79.12.3,Γdi=3.303.303.33.33.33.33.3,Γ1i=16.300016.300016.3,Γ4i=31.400031.400031.4,Pc1=3.536.321.054.39,Pc2=3.5310.511.056.58,Pc3=3.534.231.053.79.A second-order reference modelWmis=1/s2+2s+1 and the reference input signal rit=0,i=1,2,3, are chosen to generate the given command ymt to be tracked by system output yt.In the numerical simulation, for comparison, three cases are conducted: (1) attitude tracking control using our proposed modified MRAC based adaptive failure compensation controller (68) (denoted as “MMRAC based FTC”); (2) attitude tracking control using the standard MRAC-based adaptive failure compensation controller (denoted as “SMRAC based FTC”) without transient performance compensator; and (3) attitude tracking control using the direct adaptive failure compensation controller (denoted as “DAC-based FTC”) in [35]. The attitude tracking responses are analyzed to study the performances of the controllers.


## 4.2. Simulation Results

To demonstrate the superior performance of the proposed control scheme, two actuator failure conditions are simulated: Case1—intermittent fault occurring in actuator u1—and Case 2—Alternate faults occurring in actuators u1 and u4.Case 1. Intermittent fault occurring in actuatoru1.
In this case, the following failure conditions are considered as follows:(i)
When0≤t<100s, all the reaction wheels function healthily, uit=vit,i=1,2,3,4(ii)
When100s≤t<200s, actuator u1 failed, u1t=2Nm and uit=vit,i=2,3,4(iii)
Whent≥200s, actuator u1 returns to normal, uit=vit,i=1,2,3,4(iv)
Whent≥300s, actuator u1 is out of control, u1t=0.75sin0.25tNm, uit=vit,i=2,3,4Figure2(a) shows the simulated results obtained by including the faulty actuators for three controllers, namely the designed MMRAC based fault tolerant controller marked with solid line, the SMRAC-based fault-tolerant controller marked with dashed line, and the DAC-based fault-tolerant controller marked with dotted line. All the three fault-tolerant controllers can compensate for both the constraint and time varying faults, although the system performance degrades to some degree, the overshoot a setting time increase significantly once the failure is introduced. However, the system ultimately regulates the attitude to zero asymptotically; that is, the attitude stabilization maneuver is still performed successfully due to the fault-tolerant performance of the three controllers to uncertain actuator failures. It can be found from Figure 2(c) at the moments when actuator u1 failed at t=100s and t=300s (shown in Figure 2(b), which is corresponding with the simulation conditions), the overshoot of the system implemented by our designed MMRAC-based fault-tolerant controller are smaller than the SMRAC- and DAC-based fault-tolerant controller even before the actuators have failed, and this is because the effect caused by parameter estimation errors of both failure and disturbances on the transient performance has been reduced by the H∞ compensator. These simulations demonstrate the theoretical result that the desired performance of the system can be achieved by the proposed fault tolerant control even if the faults are unknown in advance.Case 2. Alternate Faults Occurring in Actuatoru1 and u4.
In this case, the following failure conditions are considered as follows:(i)
When0≤t<100s, all the reaction wheels function healthily, uit=vit,i=1,2,3,4(ii)
When100s≤t<200s, actuator u1 failed, u1t=3.5Nm and uit=vit,i=2,3,4(iii)
Whent≥200s, actuator u1 returns to normal, uit=vit,i=2,3,4(iv)
Whent≥300s, actuator u4 is out of control, u4t=1.75sin0.25tNm,uit=vit,i=1,2,3Figure 2
Attitude tracking control using MMRAC-, SMRAC-, and DAC-based fault-tolerant control schemes for Case1.
(a)
Response of system output(b)
Control input signals for different fault scenarios(c)
Responses of tracking errors(d)
System outputs for differentγ in the MMAC designThis example represents the severe case in which both two actuators experience failure at the moments oft=100s and t=300s, respectively. Actually, the failure conditions in Case 2 can be regarded as a mixed pattern of intermittent failure and permanent failure. As shown in Figure 3(b), when u1 is stuck at the instant t=100s and becomes normal at t=200s, that is an intermittent failure occurred in actuator u1, which is activated and inactivated by itself. The system produces an erroneous result when such fault is active during the period of 100s≤t<200s and produces a correct result when it is inactive for t≥200s. At t=300s, actuator u4 undergoes time-varying failure and never come back as normal. Figure 3(a) shows the results using the three different control laws based on the same simulation conditions. Clearly, compared with attitude tracking response for Case 1, the attitude control performance deteriorates severely due to the multiple uncertainties of actuator failure, with severe overshoots in the attitude orientation, although the objective of asymptotic attitude tracking can be achieved finally, as clearly indicated from the output tracking errors present in Figure 3(c).Figure 3
Attitude tracking control using MMRAC-, SMRAC-, and DAC-based fault-tolerant control schemes for Case2.
(a)
Response of system output(b)
Control input signals for different fault scenarios(c)
Responses of tracking errors(d)
System outputs for differentγ in the MMAC designIn summary, for both the normal, intermittent failure and permanent failure cases, the proposed controller significantly improves the normal control performance of the closed-loop attitude system compared to the SMRAC- and DAC-based fault-tolerant control approaches. For the cases with actuator faults, the proposed method gives better transient performance than those controllers without including the transient performance compensator. As the faults become more severe, the proposed controller still guarantees system stability and asymptotic output tracking of a given command.Furthermore, comparing the modified control scheme with differentγi (i=1,2), it can be found from Figures 2(d) and 3(d) that with smaller γ1=0.1, the compensator takes more effects on transient oscillations inhibition with a smaller following error of trajectory than that with γ2=3, which keeps an agreement with the performance analysis by the criterion of L1 bound or mean squared value.


## 5. Conclusions

With multiple uncertain actuator faults, the control system of rigid spacecraft attitude is optimized by designing an improved adaptive FTC method in this study. And the major conclusions are as follows. (1) A model reference adaptive control algorithm combing with feedback linearization technology is adopted to propose a basic control law for achieving the desired closed-loop system performance. Then, as an additional item of robust adaptive control, anH∞ compensator is introduced to optimize the transient performance. (2) Based on the modified basic control design, multiple targeted adaptive failure compensators are designed to handle the corresponding failure patterns. Multiple controllers are fused into a comprehensive controller by using a weighted algorithm, thus multiple uncertain actuator faults are solved. (3) Under different fault conditions with and without additional transient performance improvers, contrastive simulation analysis of output tracking control is adopted to proof the significance of the designed theoretical method. (4) The fault compensation control for spacecraft whose parameters are known, is researched in this paper. The way to overcoming the fault compensation difficulty on the spacecraft with unknown system parameters can be found by the further extended method which is described in this research.



---

*Source: 6413085-2023-03-07.xml*