Adaptive Fault Compensation and Disturbance Suppression Design for Nonlinear Systems with an Aircraft Control Application

A comprehensive adaptive compensation control strategy based on feedback linearization design is proposed for multivariable nonlinear systems with uncertain actuator fault and unknown mismatched disturbances. Firstly, the linear dynamic system is obtained through nonlinear feedback linearization, and the dynamic model of the mismatched disturbances as well as its relevance to the nonlinear system is given. The effect of disturbances on the system output is suppressed with the basic controller of the linearized system. Then, a direct adaptive controller is developed for the multiple uncertain actuator faults. Finally, an integrated algorithm based on adaptive weighted fusion could provide an effective compensation for the effect of multiple uncertain faults and mismatched disturbances. Thus, the stability and asymptotic tracking performance of the closedloop system are ensured. The feasibility and performance of the proposed control strategy are validated by the numerical simulation results.


Introduction
Actuator faults are common in performance-critical systems. The occurrence of faults will cause severe deterioration in performance or even catastrophic problems of system instability. Actuator faults are featured with multiple essential uncertainties, including the fault mode, time, value, and type. Therefore, it is necessary to develop the effective faulttolerant control technology to address the problem associated with the multiple uncertainties of actuator faults, so as to sustain reliability and safety of the closed-loop system.
In recent years, the problem of actuator faults compensation control has attracted more and more attention. A variety of control methods are tested with several profound achievements. Many effective fault-tolerant control methods were reviewed in literatures [1][2][3][4][5]. Multimodel adaptive control methods were employed as a fault compensation in literatures [6][7][8]. Literatures [9][10][11] applied neural network to the design of reconfigurable aircraft control in the case of sensors or actuator faults. The fault recognition and faulttolerant control strategies of the near space vehicle are designed base on the adaptive sliding mode control method in reference [12]. For the spacecraft attitude control system with external disturbances, two kinds of effective faulttolerant control method were proposed in literature [13]. To enhance the overall performance of the multisensor measurement system and reduce the influence of faults of each sensor on the system, a new multisensor information fusion design framework was proposed in reference [14]. Fault detection and diagnosis methods are also widely used to for the problems of component faults in the control system [15]. In literatures [16,17], the adaptive observer design was used to reconstruct actuator faults and a fault-tolerant controller was designed based on estimated information for fault. Besides, adaptive control is also an effective tool with widespread application in fault-tolerant control for both linear and nonlinear systems [18][19][20][21]. Although great practical progress has been made in actuator fault compensation for the nonlinear system, there are still many unresolved problems for control system with uncertain dynamics and actuator faults. For example, the problems of multiple-actuator fault compensation control in the general nonlinear system can be further investigated to improve closed-loop system stability and asymptotic tracking control.
The so-called feedback linearized system refers to a kind of nonlinear system linearized by appropriate nonlinear feedback control [22]. Based on the feedback linearization, the control objectives such as models match, pole assignment, and tracking can be further realized. References [23,24] combined feedback linearization theory with adaptive control and effectively solved the parameter uncertainty and fault-tolerant control problems of nonlinear systems. In addition, the performance of the controlled system suffers from quite different influences due to the various disturbances during the actual operation of the nonlinear system. Therefore, the disturbance suppression problems should be given adequate attention. In literatures [25][26][27], disturbance decoupling for the measurable disturbances in linear systems provides a potential approach for disturbance suppression problems. However, this method is not suitable for nonmeasurable disturbances. The robust control method is proposed for the nonmeasurable disturbances in literatures [28,29], without implementation for control objective of asymptotic tracking. The disturbances suppression method based on adaptive control design can effectively estimate the unknown system parameters and disturbance parameters. In literature [30], the adaptive internal model control method was applied in the spacecraft system to realize the attitude tracking with external disturbances. For general hypersonic vehicles with uncertain system parameters and external disturbances, a new sliding mode control method was proposed in literature [31]. The problem of asymptotic tracking of nonlinear systems under sinusoidal disturbances was investigated in literature [32]. And a disturbance suppression algorithm was proposed for single-input single-output nonlinear systems, but the algorithm is inappropriate for multi-input multioutput nonlinear systems with mismatched disturbances. In addition, the suppression of mismatched disturbances in multi-input and multioutput nonlinear systems were studied in literatures [33][34][35].
Unknown disturbances and uncertain actuator faults may occur simultaneously in the actual operation, which increases the difficulties in asymptotic tracking control for multi-input and multioutput nonlinear systems. Although some theoretical achievements have been made in disturbance suppression and actuator fault compensation for multi-input and multioutput nonlinear systems, some critical problems are left open. The problem of unmatched disturbance suppression in nonlinear systems with uncertain multivariable is solved in literature [35]. On this basis, the problem of multiple uncertain actuator fault compensation and mismatched input disturbance suppression is further studied in this paper for the case of a feedback linearized multivariable nonlinear systems. Compared with some available fault-tolerant control methods, the currently proposed control method presents the following improvement: (1) a new adaptive actuator failure compensation and disturbance rejection scheme with relaxed design conditions is designed for general multivariable nonlinear systems; (2) a new composite fault-tolerant control approach is developed to handle a set of uncertain actuator failures, by using a complete parametrization for estimation of both the failure pattern parameters and the failure value parameters; (3) an adaptive disturbance rejection scheme is developed in details, including error equations, adaptive laws, and stability analysis, for multivariable nonlinear systems with uncertainties from both the actuator failure and unmatched disturbances, such that desired closed-loop performances are ensured including signals boundedness and asymptotic output tracking; and (4) an important aircraft flight control application is conducted.

Problem Description and Knowledge Preparation
This chapter first describes the problem of actuator fault compensation and disturbance suppression of the systems with redundant actuators and then introduces some basic concepts involved in this paper.

Control Problem Statement.
Consider the nonlinear system as below where x ∈ R n is state vector, y = ½y 1 , y 2 , ⋯, y q T ∈ R q is system output, u = ½u 1 , u 2 , ⋯, u m T ∈ R m: is system input, and dðtÞ ∈ R p is the uncertain external disturbance. f ðxÞ ∈ R n , gðxÞ = ½g 1 ðxÞ, g 2 ðxÞ, ⋯, g m ðxÞ ∈ R n×m , pðxÞ ∈ R n×p , and hðxÞ ∈ R q are known.
where j ∈ f1, 2, ⋯, mg, t j > 0,¯ u j0 , and u ji represent the parameters of the uncertain fault. f ji ðtÞ, i = 1, 2, ⋯, q j are known. The fault model (3) is written in the following parameterized form 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 :When the uncertain actuator fault occurs in the system, the actual input uðtÞ acting on the system can be expressed as International Journal of Aerospace Engineering where vðtÞ is the control input signal to be designed. uðtÞ = ½ u 1 ðtÞ, u 2 ðtÞ, ⋯, u m ðtÞ T . σðtÞ = diagfσ 1 ðtÞ, σ 2 ðtÞ, ⋯, σ m ðtÞg is the corresponding actuator fault mode matrix. If the j actuator fails, then σ i ðtÞ = 1; otherwise, σ i ðtÞ = 0. Considering actuator fault (5), the system model can be expressed as 2.1.2. External Disturbance Model. The disturbance term pðxÞdðtÞ in this paper has the following characteristics: (1) pðxÞ ≠ gðxÞα α ∈ R m×p indicates that the disturbance signal dðtÞ is incompatible with the control signal uðtÞ; (2) the component of the disturbance vector dðtÞ = ½d 1 ðtÞ, d 2 ðtÞ, ⋯, d p ðtÞ p ∈ R p can be expressed as [36]: and it also can be rewritten in the parameterized form as where θ * T dj = ½d j0 , d j1 , ⋯, d j q j ∈ R q j +1 , ϖ dj ðtÞ = ½1, Φ j1 ðtÞ, ⋯, Φ jq j ðtÞ T ∈ R q j +1 , j = 1, 2, ⋯, p, d j0 ,and d jk are unknown while Φ jk ðtÞ are known. By selecting appropriate q j and basic function Φ jk ðtÞ, the disturbance model (7) can offer an approximate description for many practical disturbance signals, such as constant value, sinusoidal signal, and nonsinusoidal time-varying disturbance. Remark 1. When the disturbance is consistent with the control input, i.e., pðxÞ = gðxÞα and α ∈ R m×p , the control signal can be derived as uðtÞ = u 1 ðtÞ + u 2 ðtÞ, where u 1 ðtÞ is the basic control variable that can stabilize the nonlinear multivariable system, and u 2 ðtÞ = −αdðtÞ is the disturbance suppression component. Without such match, i.e., pðxÞ ≠ gðxÞα and α ∈ R m×p , the above control method cannot eliminate the influence of disturbances. Therefore, a new control input uðtÞ needs to be designed to suppress disturbance.

Control Objective.
For system (1) with uncertain actuator faults (3) up to m − q a q ≤ q a ≤ m and unmatched external disturbance dðtÞ, the number of the faults depends on the actual application. In this paper, q a = m − 1. That is, the total actuator faults are no more than m − q a = 1, but it is impossible to identify in advance the exact amount of faults. The actuator fault compensation method designed in this case can be applied to the problem of simultaneous or alternating faults of multiple actuators. The mathematical expressions of the corresponding fault modes are In this paper, a fault compensation control algorithm is developed based on the following assumptions to achieve the above control objectives. Assumption 2. When at most one actuator of system (1) fails and the fault information is available, it is still possible to design effective control methods to adjust the residual actuators adaptively so that the system still fulfills the desired control objective.
The goal of this paper is to design an adaptive controller vðtÞ to solve the issue due to multiple uncertainties of faults and disturbances, especially the uncertain fault mode, in order to guarantee the stability of the closedloop system and asymptotical tracking performance of system output.

Feedback Linearization.
For a multi-input and multioutput nonlinear system where u ∈ R m , y ∈ R q .

Nonlinear Feedback Control Law.
Based on Assumptions 3 and 4, if the system parameters and fault parameters of nonlinear system (1) are accessible, feedback linearization design can be used to design an ideal controller. By taking y i derivatives of ρ i in system (1), we can obtain the following equation: where We can further obtain When m = q and assuming AðxÞ is nonsingular in x 0 , the control input signal could be rearranged as International Journal of Aerospace Engineering The linearized system can be obtained where u L is the linear feedback control law to be designed.

Linear Feedback
Control. The control law from the linearized system provides the possibility to guarantee the output tracking performance of the system. The control law where With substitution of u Li and equation (24) into equation (23), the dynamic equation of the tracking error e i = y i − y mi is obtained as By selecting appropriate value for α iρ i , i = 1, 2, ⋯, q, s ρ i + α i1 s ρ i −1 +⋯+α iρ i becomes the Hurwitz polynomial.
The output error and its higher derivative e i , _ e i , ⋯, e ðρ i −1Þ i asymptotically approach to zero as t ⟶ ∞. If y mi , _ y mi , ⋯, y ðρ i −1Þ mi is bounded, then boundedness could be expected for (26), i.e., Δ di ðxÞ = 0, i = 1, 2, ⋯, m. Thus, equation (21) can be simplified as Linearized system becomes disturbance-free, and disturbance suppression is unnecessary in the basic feedback control system. In addition, in combination with equation (24), equation (21) can be further expressed as , where u Li ðtÞ can assume the following simplification u Li = y (20) is related to the disturbance term dðtÞ and its differential term _ dðtÞ, ⋯, d ρ i −ν i ðtÞ, and could be expressed as a function of In this case, in order to achieve disturbance suppression and asymptotic tracking control, it is necessary to acquire the differential information of the disturbance in advance. However, the derivation process is rather complicated. Therefore, such situation is not considered in this design.

Actuator Fault Compensation and Disturbance Suppression Design
If the relevance of the system fρ 1 , ρ 2 , ⋯, ρ q g satisfies ρ 1 + ρ 2 +⋯+ρ q = n, the system with uncertain actuator fault can be linearized by strict feedback and converted into Based on u Li , i = 1, 2, ⋯, q in equation (24), the control signal w d ðtÞ ∈ R q of the system (27) could be determined through nonlinear feedback, if The control signal can guarantee asymptotic output tracking, i.e., lim t→∞ ðyðtÞ − y m ðtÞÞ = 0. With occurrence of uncertain actuator fault, the control input signal vðtÞ could be calculated according to equation (28).
whereû L is the estimated value of u L , and its estimated component iŝ Combining the system output in equation (15): y 1 = ξ 1,1 , y 2 = ξ 2,1 , ⋯, y q = ξ q,1 and e i = y i − y mi , one can obtain z 1,1 = e 1 , z 1,2 = _ e 1 , ⋯, z 1,ρ 1 = e . And the state error equation of the multi-input multioutput system is calculated by where , 3.1.3. Adaptive Laws. Based on error system (31), an adaptive law is incorporated to update unknown disturbance parameters b θ dj , j = 1, 2, ⋯, p. Lyapunov function is designed in following form where adaptive gain matrix Γ dj = Γ T dj > 0, P ∈ R n×n is positive definite symmetric matrix and satisfies the following equation where Q = Q T > 0. Taking the derivative with respect to V d gives where Design control equation is given by and the adaptive law of the parameter b θ dj is With a substitution into equation (33), the following could be obtained The stability of the closed-loop system can be determined from the negative definition of _ V d and lim t→∞ z i,1 ðtÞ = lim t→∞ ð y i ðtÞ − y m ðtÞÞ = 0, i = 1, 2, ⋯, q. It indicates that a desired performance is achieved with the control system.

Adaptive Fault Compensation Control
Design. Supposing the fault information (fault mode, fault value, and fault time) is known. Two ideal controllers v * ð1Þ ðtÞ and v * ð2Þ ðtÞ are designed for the two cases (without fault and actuator u 1 fault). Through weighted fusion design, an integrated controller v * ðtÞ is obtained, which can deal with the simultaneous coexistence of two fault modes mentioned above. 6 International Journal of Aerospace Engineering
The ideal controller under this condition is With a weighted fusion of controller v * ð1Þ ðtÞ and v * ð2Þ ðtÞ, an ideal integrated controller structure is achieved.

Remark 7.
As the number of f ji ðtÞ increase, the parameters of the actuator failure (including the parameters of failure indicator function χ * i and χ * 2 , failure model θ * 1 also increases. In our proposed actuator failure compensation design, all the unknown parameters will be estimated multiple (m or m − 1) times based on χ * 1,i = χ * 1 , i = 1, ⋯, m, χ * 2,i = χ * 2 , θ * 1ðiÞ = χ * 2 θ * 1 , i = 1, 2, ⋯, m − 1. With the development of science and technology, the computers have become more advanced, the computation complexity can be solved effectively.

Theorem 8.
For the multivariable nonlinear system (1) with potential uncertain actuator fault (3) and mismatched disturbance dðtÞ, controller (45) and its parameter adaptive laws can ensure the closed-loop system stability and asymptotic tracking output: lim t→∞ ðy − y m Þ = 0, ifρ 1 + ρ 2 +⋯+ρ q = nand the equivalent control matrix A σ ðxÞ = AðxÞðI − σðtÞÞvðtÞ in uncertain fault condition has full rank in the domain U (definition is U ⊂ R n ⟶ V ⊂ R q ).

Stable Zero Dynamic Assumption.
To ensure the stability of the closed-loop system and output y i ðtÞ asymptotic tracking reference signal y mi ðtÞ, the differentials of ρ i , i = 1, 2, ⋯, q of y mi ðtÞ are bounded and piecewise continuous. In this paper, the controller is developed based on the following assumption: Assumption 9. The nonlinear system (1) still belongs to the minimum phase system under condition of centralized arbitrary fault, which is considered as the fault mode of this paper. That is, with input of uðtÞ, dðtÞ, and ξ, the zero dynamic subsystem given by could guarantee input state stability.
Remark 10. Based on Assumption 9, if σ ∈ Σ in any fault case, the state ξ, fault signal u, and the designed feedback control signal vðξ, η,χ 1,i ,χ 2,i ,θ 1ðiÞ Þ are all bounded while dðtÞ is bounded disturbance. According to the input state stability condition of the zero dynamic system, η is bounded. Combined with the performance analysis results in Section 3.2, it can be inferred that the nonlinear feedback control signal designed in this paper vðξ, η,χ 1,i ,χ 2,i ,θ 1ðiÞ Þ is bounded.
Combined with Assumption 3, the signal vðtÞ of adaptive fault compensation designed for the partial feedback linearization system (18) is similar to that for full feedback linearization system in Section 3.2. The detailed derivation is not rendered. The closed-loop system has the following desired control performance. Theorem 11. Based on the input state stability condition of zero dynamic (Assumption 9) and the equivalent control matrix A σ ðxÞ in uncertain fault with row full ranks in domain of U, the adaptive controller (45) and its parameter adaptive law can achieve desired stability for closed-loop system (3) and asymptotic tracking output: lim t→∞ ðyðtÞ − y m ðtÞÞ = 0 in the case of multiple uncertain actuator faults (3) and unknown disturbances.
Proof. Assuming one of the actuators failed at time T 1 , and the system has no fault during time period ðT 1 , T 2 Þ, it can be derived according to the performance analysis in Section 3.2 that the estimated parameters ξ,χ 1,i ðtÞ,χ 2,i ðtÞ, andθ 1ðiÞ 9 International Journal of Aerospace Engineering ðtÞ are bounded and the state error z asymptotically approaches to zero as t tends towards infinity. Boundedness of vðtÞ could be further confirmed from equations ((45), (46), (47)). Input state stability and row full rank constitute an estimation criterion for performance of a closed-loop system in terms of stability and asymptotical tracking capability.

Applications in Aircraft Control System
In this section, the proposed control method is applied to the aircraft control system, so that the developed control algorithm could be comprehensively validated. The numerical simulation results show that this method can offer an effective compensation for uncertain actuator fault in the case of gust disturbance.

Aircraft Dynamics in Turbulent
Flow. The research of aircraft dynamic model under turbulence conditions in reference [37] shows that the longitudinal nonlinear dynamic model of the aircraft can be expressed as [38,39] where V is the aircraft speed, α is the attack angle, θ is the angle of pitch, q r is the pitch rate, m r is the mass, I y is the rotational inertia, M is the pitch moment, and d 1 , d 2 , and d 3 are turbulence disturbance signals F x = q r SC x α, q r , δ e1 , δ e2 ð Þ + T 1 cos γ 1 + T 2 cos γ 2 − m r g sin θ ð Þ, F z = qSC z α, q r , δ e1 , δ e2 ð Þ + T 1 sin γ 1 + T 2 sin γ 2 + m r g cos θ ð Þ, q = 1/2ρV 2 is the dynamic pressure, ρ is the air density, S is the density of the wing, c is the average chord, and T 1 and T 2 are thrusters. C x , C z , and C m are given by where δ e1 and δ e2 are the two actuators that require fault compensation.

Control Objectives.
For the aircraft control system (68) with uncertain turbulent disturbance and actuator faults, an adaptive fault compensation controller is designed to ensure that the stability of the closed-loop system is satisfied and that the system output yðtÞ = ½x 1 , x 2 , x 3 T could track the desired control instruction y m ðtÞ = ½y m1 , y m2 , y m3 T = ½3 sin ð0:1tÞ + 88,1:2 sin ð0:1tÞ, 3 sin ð0:1tÞ T . According to Theorem 11, ρ 1 = v 1 = 1, ρ 2 = v 2 = 1, ρ 3 = v 3 = 2, and ρ 1 + ρ 2 + ρ 3 = 4. The system satisfies Assumption 3 without zero dynamic subsystem after feedback linearization. The following fault modes corresponding to the requirements of fault compensation can be obtained: Simulation results are shown in Figures 1-3, including a comparison between the actual output of the system and 11 International Journal of Aerospace Engineering the corresponding reference signal, the tracking error of the system, and the control input signals of four actuators acting on the system in the aircraft.
It can be seen from Figures 1 and 2 that during the actual operation, the designed control algorithm can always fulfill the control objective of system stability and asymptotic tracking, irrespective of normal operation or uncertainties in time, value, or fault model. The results in Figure 3 show that the system has external disturbance and no actuator fault during the period t ∈ ½0,150sÞ. In the process of the asymptotic tracking of a given instruction, a transient response appears and decreases with time. The robustness of the proposed control method is verified through the results. When the actuator u 1 fails at t = 150s and actuator u 4 fails at t = 400 s (shown in Figure 3), the simulation results demonstrate the effectiveness of the proposed adaptive compensation algorithm for both actuator fault and the disturbance. Moreover, the estimates of the adaptive controller parameters χ 1,i , χ 2,i , χ 3,i ,

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International Journal of Aerospace Engineering θ 1ðiÞ , and θ 4ðiÞ of χ * 1,i χ * 2,i , χ * 3,i , θ * 1ðiÞ , and θ * 4ðiÞ for y m ðtÞ are shown in Figures 4-8, which indicate that all signals in the adaptive control system are bounded, and the desired performance is met.

Conclusions
For multivariable nonlinear systems with multiple uncertain actuator faults and mismatched input disturbances, a control method of adaptive fault and disturbance com-pensation is proposed in this paper, with the following main conclusions. (1) An adaptive algorithm is adopted to establish the relation, and a set of adaptive fault compensation controllers is constructed based on parameter estimation. Then, a weighted algorithm is used to fuse multiple controllers into a comprehensive controller, so as to solve multiple uncertain actuator faults. (2) Under the condition of uncertain fault, a new parametric design method is adopted to obtain the parameter adaptive law of the fault compensation controller, so that the desired   13 International Journal of Aerospace Engineering performance of the closed-loop system can be guaranteed.
(3) The effectiveness of the proposed theoretical method is verified by the simulation results of aircraft control under fault and disturbance conditions. The problem of fault compensation control for multivariable nonlinear system with known parameters is studied in this paper. (4) The proposed method can be further extended to solve the problem of fault compensation of the system with unknown parameters.

Data Availability
The data (System parameters and Simulation parameters) used to support the findings of this study are included within the article.

Conflicts of Interest
The authors declare that they have no conflicts of interest.  14 International Journal of Aerospace Engineering