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

. A modi ﬁ ed, robust adaptive fault compensation design is proposed for rigid spacecraft systems with uncertain actuator failures and unknown disturbances. The feedback linearization method is ﬁ rst 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, an H ∞ compensator is introduced as an additional item in the basic controller to attenuate the adverse e ﬀ ects 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.


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][2][3][4][5][6][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, simulta-neously [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 performancecritical 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 finitetime 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 fixedtime 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 condi-tions, 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 faulttolerant 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 designed H ∞ 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 well The remaining section is composed as follows.Section 2 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 2 International Journal of Aerospace Engineering 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.

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: where q 0 and q = ½q 1 , q 2 , q 3 T ∈ R 3 denote the scalar and vector parts of the unit-quaternion, respectively.The quaternion also satisfies the constraint equation q T q + q 2 0 = 1, ω ∈ R 3 denotes the inertial angular velocity of the spacecraft expressed in the body frame, and the inertia matrix J ∈ R ð3×3Þ is assumed to be known in this study.The notations ζ × , ∀ζ = ½ζ 1 , ζ 2 , ζ 3 , can be expressed as uðtÞ ∈ R 4 is the control input produced by reaction wheels.As the orientation matrix of the reaction wheel, D ∈ R ð3×4Þ is available for a given spacecraft.In this research, we consider : ð3Þ dðtÞ = ½d 1 , d 2 , d 3 T ∈ R 3 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 dðtÞ is modeled as where c i , a ij , and b ij are unknown amplitudes, and ω ij are known frequencies.
Define x = ½q T , ω T T and y = q as the state and output vector, the spacecraft attitude control system (1) is rewritten as where and 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 where j ∈ f1, 2, 3, 4g, t j > 0, u j0 , and u ji represent unknown failure parameters.f ji ðtÞ, i = 1, 2, ⋯, q j are known.The Equation (9) can also be rewritten into the below-parameterized expression where θ j = ½ u j0 , u j1 , ⋯, u j q j T ∈ R q j +1 , ϖ j ðtÞ = ½1, f j1 ðtÞ, ⋯, f jq j ðtÞ T ∈ R q j +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 input uðtÞ applying to the system is where vðtÞ denotes the control input signal.uðtÞ = ½ u 1 , u 2 , u 3 , u 4 T andσðtÞ = diag fσ 1 , σ 2 , σ 3 , σ 4 g are defined fault pattern matrix, i.e, when the j-th actuator fails σ j ðtÞ = 1, if there is no failure σ j ðtÞ = 0. Substituting Equation 3International Journal of Aerospace Engineering (11) into the second equation of (1), the system model is expressed as For the output yðtÞ = q to track a given command y m ðtÞ, 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, σ ð1Þ = diag f0, 0, 0, 0g (ii) u 1 failure case, σ ð2Þ = diag f1, 0, 0, 0g (iii) u 4 failure case, σ ð3Þ = diag f0, 0, 0, 1g (iv) u 3 failure case, σ ð4Þ = diag f0, 0, 1, 0g (v) u 2 failure case, σ ð5Þ = diag f0, 1, 0, 0g 2.2.1.Control Objective.For system (1), which has one uncertain failure (9) at most, an adaptive controller vðtÞ 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: 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 u 1 failure case can be expended to address the u 2 and u 3 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 of mdimension Definition 2. The system (14) has a vector relative degree f ρ 1 , ρ 2 , ⋯, ρ m g, 1 ≤ ρ i ≤ n, at a point x 0 for ∀x in a neighborhood of if the below two conditions hold mg,  and (ii) the m × m matrix GðxÞ is defined as : ð15Þ Then the system ( 14) has the relative degree ρ = ∑ m i=1 ρ i , with ρ i being the subrelative degree of the i-th output y i = h i ðxÞ.If the equilibrium point of system ( 1) is x 0 = ½0, 0, 0, 0, 0, 0 T , we can obtain that ρ 1 = ρ 2 = ρ 3 = 2 and the relative degree ρ = n = 6.By the twice differentiation to the system output y i , the control input u d in the differential equation is expressed in the form of a nonzero factor.
To be specific, the Equation ( 6) is denoted as where The system ( 14) can be feedback linearized through differential homeomorphic mapping.Supposing there exists a differential homeomorphic mapping with the form International Journal of Aerospace Engineering Under the afore-mentioned coordinate transform, the system (6) can be transformed into three linear subsystems with the following normal form: with (11), the system (18) can be described as where G iσ ðxÞ = G i ðxÞDðI − σðtÞÞ G iσ ðxÞ = G i ðxÞDσðtÞ, and i = 1, 2, 3.

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: Considering the uncertainty of external disturbance dðtÞ, we set the control signal as where W di denotes the desired control signal generated from a chosen control that is designed for the closed-loop system, u Li is the control law to be proposed, and dðtÞ is the estimator of dðtÞ.Then, the linearized system can be obtained is the regulated output to be controlled, and y ∈ R n y is the measured output.A, B 1 , B 2 , C 1 , and C 2 are matrices of appropriate dimensions and satisfying the assumptions.
If there exists an ε < 0 such that the Riccati equation has a solution P ≥ 0, where R ∈ R n u ×n u and S ∈ R n x ×n x are given positive-definite matrices.
A state feedback controller u s = −ðð1/2εÞR −1 B T 2 PÞx can be designed to stabilize the system (23), and the transfer function G wz ðsÞ between disturbance w and output z satisfy the following condition 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. Let z = HðsÞw, where HðsÞ have all their roots in Re ½s ≥ −δ/2, for any δ ≥ 0 and w ∈ L 2 , we have where kz t k δ 2 is defined as for ∀z ∈ ½0,∞Þ ⟶ R n ' and δ ≥ 0 and t ≥ 0. If HðsÞ is strict, then and if HðsÞ is a stable n-order transfer function, then

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 Figure 1.
To achieve the control objective, we will complete four design steps: Substituting ( 21) into (29), we have and the linearized system € y i = u Li can be rewritten into In this paper, we rewrite the linearized subsystem (31) into the transfer function form as with and its reference model is where Z pi ðsÞ, R pi ðsÞ, Z mi ðsÞ, and R mi ðsÞ are monic Hurwitz polynomials of degree m pi , n pi , , and p mi , respectively, The relative degree of W mi ðsÞ is the same as that of G pi ðsÞ.

Modified MRAC Design.
A model reference adaptive controller is constructed as follows where ðsÞ, and λ i ðsÞ are monic Hurwitz polynomials of degree n pi − q mi − 1; θ i ∈ R 2n i −1 is the vector of controller parameters; u ci 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 be For a given transform function G pi ðsÞ, there is a desired reference value vector θ * i = ½θ * T i1 , θ * i2 T to make the following matching conditions valid Multiplying both sides with y i and using the equation Dividing both sides by k pi Z pi we have The spacecraft system parameters are known; that is, in the MRAC law, θ i is substituted by the desired value θ * i .To be more specific, the control u Li is designed as where K i = ½K i1 , K i2 are the design controller parameters corresponding with θ * i , ξ i = ½ξ i1 , ξ i2 T .Substituting (38) into (30), we have Furthermore, the linearized subsystem ( 18) is described as We then revise the above equation in the form of state space as follows: where , and C ci ∈ R 1×2 are the standard form of integrator chains.We also revise the reference model (33) in the state space as follows: where A mi , B mi , and C mi are the minimal realization of W mi ðsÞ, i.e., and C mi = ½1, 0 ∈ R 1×2 .Denoting e i = ξ i − ξ mi , we can obtain With the system model ( 32), (33) and the controller (34), we obtain 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 G i ðxÞDðI − σÞðv − v * Þ + G i ðxÞðd − dÞ on the closed-loop system is designed by proposing the compensator u ci as an H ∞ optimal controller.Equation ( 45) is considered as a special form of (23).With (45) and Lemma 3, a transient performance compensator u ci is designed for the system (45) by the following steps as shown next.
Step 1: initialize ε i > 0 and 0 < γ i < kW mi ðsÞ/c 0i k ∞ , choose two positive-definite matrices S ci ∈ R 2×2 and R ci ∈ R 1×1 , and solve the following Riccati equation and obtain the state feedback gain If there is no solution, decrease ε i and repeat this step 7 International Journal of Aerospace Engineering again until getting an appropriate positive-definite solution P ci = P T ci > 0 to ensure that the state feedback gain is obtained.The H ∞ compensator which is based on measurement feedback is constructed by state feedback gain K ci , which guarantees that the system (45) with Step 2: Return to step 1 and reduce the transient performance index until getting a gratifying transient or the optimum γ i Remark 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 of H ∞ can be obtained by selecting sufficiently small ε i .However, it may cause the gain K ci 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 < γ i < kW mi ðsÞ/c 0i k ∞ can also achieve a satisfactory transient performance through the following theoretical analysis and simulation.

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 uðtÞ, we write the three subsystems together as where G D ðxÞ = GðxÞD ∈ R 3×4 is regarded as control distribution matrix.
The signal vðtÞ is designed as for a chosen h 21 ðxÞ ∈ R 4×4 , and signal v * 0ð1Þ ðtÞ to be calculated from The solution v * 0ð1Þ ðtÞ may be derived as with matrix function K 21 ðxÞ ∈ R 4×3 .
The signal vðtÞ is proposed as with a chosen matrix h 23 ∈ R 3×3 and a deduced signal v * 0ð3Þ ð tÞ ∈ R 3 from Similarly, we have with matrix function K 23 ∈ R 3×3 and vector K 234 ∈ R 3×1 .

Stability Performance Analysis
(i) For period t ∈ ½T 0 ,∞Þ, σ = σ ð1Þ .Lyapunov function is defined as By differentiating V 0 in the interval ½T 0 , ∞Þ, we can obtain where e P = ½e P1 , e P2 , e P3 ∈ R 1×3 and e Pi are the ði + 1Þ-th column components of e T P ∈ R 1×6 , i = 1, 2, 3, and P = diag f P c1 , P c2 , P c3 g.Substituting Equations ( 80)-( 85) into (87), one would have Based on of Lemma 3, we can obtain (ii) If actuator u 1 fails over the period ðT 1 , ∞Þ, i.e., σ = σ ð2Þ , we define By differentiating V 1 in the interval ðT 1 , ∞Þ and International Journal of Aerospace Engineering combining Equations ( 80)-(84), we can obtain (iii) Assume that u 1 is normal, and only u 4 fails at T 1 and remains failed on the interval ðT 1 , ∞Þ; that is, σ = σ ð3Þ .We define The time derivative of V 2 is With _ V k , ðk = 0, 1, 2Þ ≤ 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 σðtÞ belongs to failure pattern set Σ = fσ ðjÞ , j = 1, 2, 3g.The following condition holds the following equivalent actuation matrix.G σ ðxÞ = GðxÞDðI − σðtÞÞ is a full rank in the domain Since the order of the stable reference model W mi ðsÞ is p mi , it can be derived from Lemma 5 that From Theorem 9, we have e θ di ∈ L ∞ and ðv − v * Þ ∈ L ∞ .With Lemma 3, one can prove that W mi ðsÞ is stable and and thereby, we have where c i > 0.Then, according to we have Theorem 10.For the improved controller (34), the performance index of H ∞ compensator is γ i , and the output tracking error e i1 = y i − y mi and i = 1, 2, 3 of the system (32) satisfies the following inequality condition: where constant c i > 0.
According to Theorem 10, 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.
The design parameters are chosen as A second-order reference model W mi ðsÞ = 1/ðs 2 + 2s + 1Þ and the reference input signal r i ðtÞ = 0, i = 1, 2, 3, are chosen to generate the given command y m ðtÞ to be tracked by system output yðtÞ.
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.

Simulation Results.
To demonstrate the superior performance of the proposed control scheme, two actuator failure conditions are simulated: Case 1-intermittent fault occurring in actuator u 1 -and Case 2-Alternate faults occurring in actuators u 1 and u 4 .
Case 1. Intermittent fault occurring in actuator u 1 .In this case, the following failure conditions are considered as follows: (i) When 0 ≤ t < 100s, all the reaction wheels function healthily, u i ðtÞ = v i ðtÞ, i = 1, 2, 3, 4 (ii) When 100s ≤ t < 200s, actuator u 1 failed, u 1 ðtÞ = 2 Nm and u i ðtÞ = v i ðtÞ, i = 2, 3, 4 (iv) When t ≥ 300s, actuator u 1 is out of control, u 1 ðtÞ = 0:75 sin ð0:25tÞNm, u i ðtÞ = v i ðtÞ, i = 2, 3, 4 Figure 2(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 faulttolerant 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 u 1 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 Actuator u 1 and u 4 .In this case, the following failure conditions are considered as follows: (i) When 0 ≤ t < 100s, all the reaction wheels function healthily, u i ðtÞ = v i ðtÞ, i = 1, 2, 3, 4  This example represents the severe case in which both two actuators experience failure at the moments of t = 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 u 1 is stuck at the instant t = 100s and becomes normal at t = 200s, that is an intermittent failure occurred in actuator u 1 , 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 u 4 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   In 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 different γ i (i = 1, 2), it can be found from Figures 2(d) and 3(d) that with smaller γ 1 = 0:1, the compensator takes more effects on transient oscillations inhibition with a smaller following error of trajectory than that with γ 2 = 3, which keeps an agreement with the performance analysis by the criterion of L 1 bound or mean squared value.

Conclusions
With multiple uncertain actuator faults, the control system of rigid spacecraft attitude is 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, an H ∞ 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.

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International Journal of Aerospace Engineering (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 accommodation In order to obtain an appropriate adaptive law vðtÞ, the error of the control signal equation which is led by the actuator uncertainties is examined.A desired control equation G i ðxÞu d ðtÞ = W di (with u d ðtÞ = DuðtÞ) is defined to be satisfied by the nominal control signal v * ðtÞ to be designed in the next section.In the light of (11), we define u * ðtÞ as u * ðtÞ ≜ ðI − σÞv * ðtÞ + σ uðtÞ and obtain

3. 3 . 4 .
Transient Performance Analysis.Then, the transient performance is analyzed by the criteria of the bound of both L ∞ and mean square tracking error at any time.With (45) and (49), we can get the following output tracking error dynamic equation Control input signals for different fault scenarios
System outputs for different γ in the MMAC design
System outputs for different γ in the MMAC design