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

Authors: X. Yao; Y. Yang
Journal: International Journal of Aerospace Engineering
 (2020)

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## Abstract

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 closed-loop system are ensured. The feasibility and performance of the proposed control strategy are validated by the numerical simulation results.


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## Body

## 1. 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 fault-tolerant 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 [1–5]. Multimodel adaptive control methods were employed as a fault compensation in literatures [6–8]. Literatures [9–11] applied neural network to the design of reconfigurable aircraft control in the case of sensors or actuator faults. The fault recognition and fault-tolerant 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 fault-tolerant 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 [18–21]. 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 [25–27], 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 [33–35].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.


## 2. 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.


### 2.1. Control Problem Statement

Consider the nonlinear system as below(1)ẋ=fx+gxu+pxdt(2)y=hx,where x∈Rn is state vector, y=y1,y2,⋯,yqT∈Rq is system output, u=u1,u2,⋯,umT∈Rm.is system input, and dt∈Rp is the uncertain external disturbance. fx∈Rn, gx=g1x,g2x,⋯,gmx∈Rn×m,px∈Rn×p, and hx∈Rq are known.


#### 2.1.1. Actuator Fault Model

The classical model of the actuator fault can be represented as [19]
(3)ujt=u¯jt=u¯j0+∑i=1qju¯jifjit,t≥tj,where j∈1,2,⋯,m, tj>0, ¯u¯j0, and u¯ji represent the parameters of the uncertain fault. fjit,i=1,2,⋯,qj are known. The fault model (3) is written in the following parameterized form
(4)u¯jt=θjTϖjt,where θj=u¯j0,u¯j1,⋯,u¯jqjT∈Rqj+1, ϖjt=1,fj1t,⋯,fjqjtT∈Rqj+1.When the uncertain actuator fault occurs in the system, the actual input ut acting on the system can be expressed as
(5)ut=I−σtvt+σtu¯t,where vt is the control input signal to be designed. u¯t=u¯1t,u¯2t,⋯,u¯mtT. σt=diagσ1t,σ2t,⋯,σmt is the corresponding actuator fault mode matrix. If the j actuator fails, then σit=1; otherwise, σit=0. Considering actuator fault (5), the system model can be expressed as
(6)ẋ=fx+gxσtu¯t+gx1−σtvt+pxdty=hx.


#### 2.1.2. External Disturbance Model

The disturbance termpxdt in this paper has the following characteristics: (1) px≠gxα α∈Rm×p indicates that the disturbance signal dt is incompatible with the control signal ut; (2) the component of the disturbance vector dt=d1t,d2t,⋯,dptp∈Rp can be expressed as [36]:
(7)djt=dj0+∑k=1qjdjkΦjkt,j=1,2,⋯,p,and it also can be rewritten in the parameterized form as
(8)djt=θdj∗Tϖdjt,where θdj∗T=dj0,dj1,⋯,djqj∈Rqj+1, ϖdjt=1,Φj1t,⋯,ΦjqjtT∈Rqj+1, j=1,2,⋯,p, dj0,and djk are unknown while Φjkt are known. By selecting appropriate qj and basic function Φjkt, 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.,px=gxα and α∈Rm×p, the control signal can be derived as ut=u1t+u2t, where u1t is the basic control variable that can stabilize the nonlinear multivariable system, and u2t=−αdt is the disturbance suppression component. Without such match, i.e., px≠gxα and α∈Rm×p, the above control method cannot eliminate the influence of disturbances. Therefore, a new control input ut needs to be designed to suppress disturbance.


#### 2.1.3. Control Objective

For system (1) with uncertain actuator faults (3) up to m−qaq≤qa≤m and unmatched external disturbance dt, the number of the faults depends on the actual application. In this paper, qa=m−1. That is, the total actuator faults are no more than m−qa=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
(9)σ1=diag0,0,⋯,0,σ2=diag1,0,⋯,0.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 controllervt 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 closed-loop system and asymptotical tracking performance of system output.


### 2.2. Feedback Linearization

For a multi-input and multioutput nonlinear system(10)ẋ=fx+gxu,y=hxwhere u∈Rm, y∈Rq.Assumption 3.
Supposing the correlation vector asρ1,ρ2,⋯,ρq,1≤ρi≤n in a neighborhood Ω0 at x0∈Rn, if for ∀x∈Ω0, LgjLfkhix=0, 1≤j≤m, 0≤k<ρi−1, 1≤i≤q, and LgjLfρi−1hix0≠0 for j∈1,2,⋯,m.Similarly, for a nonlinear system with input disturbances(11)ẋ=fx+pxdt,y=hx,where dt∈Rp, y∈Rq, and it has a correlation set ν1,ν2,⋯,νq, 1≤νi≤n. The disturbance suppression design of the multivariable nonlinear system in this paper involves the following assumption:Assumption 4.
Ifi=i1,i2,⋯,ip ∈1,2,⋯,m, then ρi=νi; If i≠i1,i2,⋯,ip, then ρi<νi.


#### 2.2.1. Strict Feedback Linearization

Ifρ1+ρ2+⋯+ρq=n, the system (1) can be transformed into a strict feedback subsystem through strict feedback linearization and differential homeomorphic mapping ξ=Tx∈Rn, where
(12)Tx=h1x,Lfh1x,⋯,Lfρ1−1h1x,⋯,hqx,⋯,Lfρq−1hqxTξ=ξ1,1,ξ1,2,⋯,ξ1,ρ1,ξ2,1,⋯,ξq,ρqTThen the dynamic equation becomes(13)ξ̇1,1=ξ1,2ξ̇1,2=ξ1,3⋮ξ̇1,ρ1=b1ξ+A1σxv+A¯1σxu¯t+Δd1x,t⋮ξ̇q,1=ξq,2⋮ξ̇q,ρq=bqξ+Aqσxv+A¯qσxu¯tΔdqx,twhere biξ=Lfρihix, Aiσx, A¯iσx, and Δdi are the ith row of Aσx, A¯σx, and Δd, respectively, Aσx=AxI−σtvt, A¯σx=σtu¯t, with
(14)Ax=Lg1Lfρ1−1h1x⋯LgmLfρ1−1h1xLg1Lfρ2−1h2x⋯LgmLfρ2−1h2x⋯⋯Lg1Lfρq−1hqx⋯LgmLfρq−1hqxΔdx,t=Lp1Lfρ1−1h1x⋯LppLfρ1−1h1xLp1Lfρ2−1h2x⋯LppLfρ2−1h2x⋯⋯Lp1Lfρq−1hqx⋯LppLfρq−1hqxdt=δ1x,δ2x,⋯,δqxT.The system output is expressed as(15)y1=ξ1,1,y2=ξ2,1,⋯,yq=ξq,1.


#### 2.2.2. Partial Feedback Linearization

Ifρ1+ρ2+⋯+ρq<n, only partial feedback linearization can be carried out in system (1) by means of coordinate transformation within the neighborhood of x0. Supposing Tcx is a smooth function with the following form
(16)Tcx=h1x,⋯,Lfρ1−1h1x,⋯,Lfρq−1hqxT.Literature [24] indicates that there is always smooth mapping
(17)Tzx=T1x,⋯,Tn−ρ1+⋯+ρqxT.To form the differential homeomorphism ξT,ηTT=Tx=TcxT,TzxTT, ξ∈Rρ1+ρ2+⋯+ρq, η∈Rn−ρ1+ρ2+⋯+ρq, system (1) is converted into
(18)ξ̇1,1=ξ1,2ξ̇1,2=ξ1,3⋮ξ̇1,ρ1=b1ξ,η+A1σxv+A¯1σxu¯t+Δd1x,t⋮ξ̇q,1=ξq,2⋮ξ̇q,ρq=bqξ,η+Aqσxv+A¯qσxu¯t+Δdqx,t(19)η̇=ψξ,η+Ψgξ,ηu+Ψpξ,ηdt,where biξ,η, Aiσx, A¯iσx, i=1,2,⋯,q have the same definition with the equation (13). Δdix, i=1,2,⋯,q is the unmatched disturbance. ψξ,η=∂Tzx/∂xfxx=T−1ξ,η, Ψgξ,η=∂Tzx/∂xgxx=T−1ξ,η, and Ψpξ,η=∂Tzx/∂xpxx=T−1ξ,η. The system (19) is termed as the zero dynamic form of the multivariable nonlinear system (1).


#### 2.2.3. Nonlinear Feedback Control Law

Based on Assumptions3 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 yi derivatives of ρi in system (1), we can obtain the following equation:
(20)yiρi=Lfρihix+∑j=1mLgjLfρi−1hixuj+δix,where δix=∑j=1pLpjLfρi−1hixdjt. We can further obtain
(21)y1ρ1y2ρ2⋮yqρq=Lfρ1h1xLfρ2h2x⋮Lfρqhqx+Axu+Δdx,t.When m=q and assuming Ax is nonsingular in x0, the control input signal could be rearranged as
(22)ut=−A−1xLfρ1h1xLfρ2h2x⋮Lfρqhqx+A−1xuL.The linearized system can be obtained(23)y1ρ1y2ρ2⋮yqρq=uL1uL2⋮uLq+Δdx,t,where uL is the linear feedback control law to be designed.


#### 2.2.4. Linear Feedback Control

The control law from the linearized system provides the possibility to guarantee the output tracking performance of the system. The control lawuL=uL1,uL2,⋯,uLqT is
(24)uLi=ymiρi+αi1ymiρi−1−yiρi−1+⋯+αiρiymi−yi+δcix,t,where δcix,t=−∑j=1pLpjLfρi−1hixdjt. With substitution of uLi and equation (24) into equation (23), the dynamic equation of the tracking error ei=yi−ymi is obtained as
(25)eiρi+αi1eiρi−1+⋯+αiρiei=0.By selecting appropriate value forαiρi, i=1,2,⋯,q, sρi+αi1sρi−1+⋯+αiρi becomes the Hurwitz polynomial. The output error and its higher derivative ei,ėi,⋯,eiρi−1asymptotically approach to zero as t⟶∞. If ymi,ẏmi,⋯,ymiρi−1 is bounded, then boundedness could be expected for yi,ẏi,⋯,yiρi−1.Remark 5.
If for alli=1,2,⋯,m, ρi<νi, then δix,t=0, in equation (26), i.e., Δdix=0, i=1,2,⋯,m. Thus, equation (21) can be simplified as
(26)y1ρ1y2ρ2⋮yqρq=Lfρ1h1xLfρ2h2x⋮Lfρqhqx+Axu.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 y1ρ1,y1ρ2,⋯,yqρqT=uLt, where uLit can assume the following simplification uLi=ymiρi+αi1ymiρi−1−yiρi−1+⋯+αiρiymi−yi,i=1,2,⋯,m.Remark 6.
Ifρi>νi, i∈1,2,⋯,m, then δix,t in equation (20) is related to the disturbance term dt and its differential term ḋt,⋯,dρi−νit, and could be expressed as a function of the differential term ḋt,⋯,dρi−νitδix,t=−dρi−νi/dt∑j=1pLpjLfρi−1hixdjt. 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.


## 2.1. Control Problem Statement

Consider the nonlinear system as below(1)ẋ=fx+gxu+pxdt(2)y=hx,where x∈Rn is state vector, y=y1,y2,⋯,yqT∈Rq is system output, u=u1,u2,⋯,umT∈Rm.is system input, and dt∈Rp is the uncertain external disturbance. fx∈Rn, gx=g1x,g2x,⋯,gmx∈Rn×m,px∈Rn×p, and hx∈Rq are known.


### 2.1.1. Actuator Fault Model

The classical model of the actuator fault can be represented as [19]
(3)ujt=u¯jt=u¯j0+∑i=1qju¯jifjit,t≥tj,where j∈1,2,⋯,m, tj>0, ¯u¯j0, and u¯ji represent the parameters of the uncertain fault. fjit,i=1,2,⋯,qj are known. The fault model (3) is written in the following parameterized form
(4)u¯jt=θjTϖjt,where θj=u¯j0,u¯j1,⋯,u¯jqjT∈Rqj+1, ϖjt=1,fj1t,⋯,fjqjtT∈Rqj+1.When the uncertain actuator fault occurs in the system, the actual input ut acting on the system can be expressed as
(5)ut=I−σtvt+σtu¯t,where vt is the control input signal to be designed. u¯t=u¯1t,u¯2t,⋯,u¯mtT. σt=diagσ1t,σ2t,⋯,σmt is the corresponding actuator fault mode matrix. If the j actuator fails, then σit=1; otherwise, σit=0. Considering actuator fault (5), the system model can be expressed as
(6)ẋ=fx+gxσtu¯t+gx1−σtvt+pxdty=hx.


### 2.1.2. External Disturbance Model

The disturbance termpxdt in this paper has the following characteristics: (1) px≠gxα α∈Rm×p indicates that the disturbance signal dt is incompatible with the control signal ut; (2) the component of the disturbance vector dt=d1t,d2t,⋯,dptp∈Rp can be expressed as [36]:
(7)djt=dj0+∑k=1qjdjkΦjkt,j=1,2,⋯,p,and it also can be rewritten in the parameterized form as
(8)djt=θdj∗Tϖdjt,where θdj∗T=dj0,dj1,⋯,djqj∈Rqj+1, ϖdjt=1,Φj1t,⋯,ΦjqjtT∈Rqj+1, j=1,2,⋯,p, dj0,and djk are unknown while Φjkt are known. By selecting appropriate qj and basic function Φjkt, 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.,px=gxα and α∈Rm×p, the control signal can be derived as ut=u1t+u2t, where u1t is the basic control variable that can stabilize the nonlinear multivariable system, and u2t=−αdt is the disturbance suppression component. Without such match, i.e., px≠gxα and α∈Rm×p, the above control method cannot eliminate the influence of disturbances. Therefore, a new control input ut needs to be designed to suppress disturbance.


### 2.1.3. Control Objective

For system (1) with uncertain actuator faults (3) up to m−qaq≤qa≤m and unmatched external disturbance dt, the number of the faults depends on the actual application. In this paper, qa=m−1. That is, the total actuator faults are no more than m−qa=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
(9)σ1=diag0,0,⋯,0,σ2=diag1,0,⋯,0.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 controllervt 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 closed-loop system and asymptotical tracking performance of system output.


## 2.1.1. Actuator Fault Model

The classical model of the actuator fault can be represented as [19]
(3)ujt=u¯jt=u¯j0+∑i=1qju¯jifjit,t≥tj,where j∈1,2,⋯,m, tj>0, ¯u¯j0, and u¯ji represent the parameters of the uncertain fault. fjit,i=1,2,⋯,qj are known. The fault model (3) is written in the following parameterized form
(4)u¯jt=θjTϖjt,where θj=u¯j0,u¯j1,⋯,u¯jqjT∈Rqj+1, ϖjt=1,fj1t,⋯,fjqjtT∈Rqj+1.When the uncertain actuator fault occurs in the system, the actual input ut acting on the system can be expressed as
(5)ut=I−σtvt+σtu¯t,where vt is the control input signal to be designed. u¯t=u¯1t,u¯2t,⋯,u¯mtT. σt=diagσ1t,σ2t,⋯,σmt is the corresponding actuator fault mode matrix. If the j actuator fails, then σit=1; otherwise, σit=0. Considering actuator fault (5), the system model can be expressed as
(6)ẋ=fx+gxσtu¯t+gx1−σtvt+pxdty=hx.


## 2.1.2. External Disturbance Model

The disturbance termpxdt in this paper has the following characteristics: (1) px≠gxα α∈Rm×p indicates that the disturbance signal dt is incompatible with the control signal ut; (2) the component of the disturbance vector dt=d1t,d2t,⋯,dptp∈Rp can be expressed as [36]:
(7)djt=dj0+∑k=1qjdjkΦjkt,j=1,2,⋯,p,and it also can be rewritten in the parameterized form as
(8)djt=θdj∗Tϖdjt,where θdj∗T=dj0,dj1,⋯,djqj∈Rqj+1, ϖdjt=1,Φj1t,⋯,ΦjqjtT∈Rqj+1, j=1,2,⋯,p, dj0,and djk are unknown while Φjkt are known. By selecting appropriate qj and basic function Φjkt, 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.,px=gxα and α∈Rm×p, the control signal can be derived as ut=u1t+u2t, where u1t is the basic control variable that can stabilize the nonlinear multivariable system, and u2t=−αdt is the disturbance suppression component. Without such match, i.e., px≠gxα and α∈Rm×p, the above control method cannot eliminate the influence of disturbances. Therefore, a new control input ut needs to be designed to suppress disturbance.


## 2.1.3. Control Objective

For system (1) with uncertain actuator faults (3) up to m−qaq≤qa≤m and unmatched external disturbance dt, the number of the faults depends on the actual application. In this paper, qa=m−1. That is, the total actuator faults are no more than m−qa=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
(9)σ1=diag0,0,⋯,0,σ2=diag1,0,⋯,0.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 controllervt 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 closed-loop system and asymptotical tracking performance of system output.


## 2.2. Feedback Linearization

For a multi-input and multioutput nonlinear system(10)ẋ=fx+gxu,y=hxwhere u∈Rm, y∈Rq.Assumption 3.
Supposing the correlation vector asρ1,ρ2,⋯,ρq,1≤ρi≤n in a neighborhood Ω0 at x0∈Rn, if for ∀x∈Ω0, LgjLfkhix=0, 1≤j≤m, 0≤k<ρi−1, 1≤i≤q, and LgjLfρi−1hix0≠0 for j∈1,2,⋯,m.Similarly, for a nonlinear system with input disturbances(11)ẋ=fx+pxdt,y=hx,where dt∈Rp, y∈Rq, and it has a correlation set ν1,ν2,⋯,νq, 1≤νi≤n. The disturbance suppression design of the multivariable nonlinear system in this paper involves the following assumption:Assumption 4.
Ifi=i1,i2,⋯,ip ∈1,2,⋯,m, then ρi=νi; If i≠i1,i2,⋯,ip, then ρi<νi.


### 2.2.1. Strict Feedback Linearization

Ifρ1+ρ2+⋯+ρq=n, the system (1) can be transformed into a strict feedback subsystem through strict feedback linearization and differential homeomorphic mapping ξ=Tx∈Rn, where
(12)Tx=h1x,Lfh1x,⋯,Lfρ1−1h1x,⋯,hqx,⋯,Lfρq−1hqxTξ=ξ1,1,ξ1,2,⋯,ξ1,ρ1,ξ2,1,⋯,ξq,ρqTThen the dynamic equation becomes(13)ξ̇1,1=ξ1,2ξ̇1,2=ξ1,3⋮ξ̇1,ρ1=b1ξ+A1σxv+A¯1σxu¯t+Δd1x,t⋮ξ̇q,1=ξq,2⋮ξ̇q,ρq=bqξ+Aqσxv+A¯qσxu¯tΔdqx,twhere biξ=Lfρihix, Aiσx, A¯iσx, and Δdi are the ith row of Aσx, A¯σx, and Δd, respectively, Aσx=AxI−σtvt, A¯σx=σtu¯t, with
(14)Ax=Lg1Lfρ1−1h1x⋯LgmLfρ1−1h1xLg1Lfρ2−1h2x⋯LgmLfρ2−1h2x⋯⋯Lg1Lfρq−1hqx⋯LgmLfρq−1hqxΔdx,t=Lp1Lfρ1−1h1x⋯LppLfρ1−1h1xLp1Lfρ2−1h2x⋯LppLfρ2−1h2x⋯⋯Lp1Lfρq−1hqx⋯LppLfρq−1hqxdt=δ1x,δ2x,⋯,δqxT.The system output is expressed as(15)y1=ξ1,1,y2=ξ2,1,⋯,yq=ξq,1.


### 2.2.2. Partial Feedback Linearization

Ifρ1+ρ2+⋯+ρq<n, only partial feedback linearization can be carried out in system (1) by means of coordinate transformation within the neighborhood of x0. Supposing Tcx is a smooth function with the following form
(16)Tcx=h1x,⋯,Lfρ1−1h1x,⋯,Lfρq−1hqxT.Literature [24] indicates that there is always smooth mapping
(17)Tzx=T1x,⋯,Tn−ρ1+⋯+ρqxT.To form the differential homeomorphism ξT,ηTT=Tx=TcxT,TzxTT, ξ∈Rρ1+ρ2+⋯+ρq, η∈Rn−ρ1+ρ2+⋯+ρq, system (1) is converted into
(18)ξ̇1,1=ξ1,2ξ̇1,2=ξ1,3⋮ξ̇1,ρ1=b1ξ,η+A1σxv+A¯1σxu¯t+Δd1x,t⋮ξ̇q,1=ξq,2⋮ξ̇q,ρq=bqξ,η+Aqσxv+A¯qσxu¯t+Δdqx,t(19)η̇=ψξ,η+Ψgξ,ηu+Ψpξ,ηdt,where biξ,η, Aiσx, A¯iσx, i=1,2,⋯,q have the same definition with the equation (13). Δdix, i=1,2,⋯,q is the unmatched disturbance. ψξ,η=∂Tzx/∂xfxx=T−1ξ,η, Ψgξ,η=∂Tzx/∂xgxx=T−1ξ,η, and Ψpξ,η=∂Tzx/∂xpxx=T−1ξ,η. The system (19) is termed as the zero dynamic form of the multivariable nonlinear system (1).


### 2.2.3. Nonlinear Feedback Control Law

Based on Assumptions3 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 yi derivatives of ρi in system (1), we can obtain the following equation:
(20)yiρi=Lfρihix+∑j=1mLgjLfρi−1hixuj+δix,where δix=∑j=1pLpjLfρi−1hixdjt. We can further obtain
(21)y1ρ1y2ρ2⋮yqρq=Lfρ1h1xLfρ2h2x⋮Lfρqhqx+Axu+Δdx,t.When m=q and assuming Ax is nonsingular in x0, the control input signal could be rearranged as
(22)ut=−A−1xLfρ1h1xLfρ2h2x⋮Lfρqhqx+A−1xuL.The linearized system can be obtained(23)y1ρ1y2ρ2⋮yqρq=uL1uL2⋮uLq+Δdx,t,where uL is the linear feedback control law to be designed.


### 2.2.4. Linear Feedback Control

The control law from the linearized system provides the possibility to guarantee the output tracking performance of the system. The control lawuL=uL1,uL2,⋯,uLqT is
(24)uLi=ymiρi+αi1ymiρi−1−yiρi−1+⋯+αiρiymi−yi+δcix,t,where δcix,t=−∑j=1pLpjLfρi−1hixdjt. With substitution of uLi and equation (24) into equation (23), the dynamic equation of the tracking error ei=yi−ymi is obtained as
(25)eiρi+αi1eiρi−1+⋯+αiρiei=0.By selecting appropriate value forαiρi, i=1,2,⋯,q, sρi+αi1sρi−1+⋯+αiρi becomes the Hurwitz polynomial. The output error and its higher derivative ei,ėi,⋯,eiρi−1asymptotically approach to zero as t⟶∞. If ymi,ẏmi,⋯,ymiρi−1 is bounded, then boundedness could be expected for yi,ẏi,⋯,yiρi−1.Remark 5.
If for alli=1,2,⋯,m, ρi<νi, then δix,t=0, in equation (26), i.e., Δdix=0, i=1,2,⋯,m. Thus, equation (21) can be simplified as
(26)y1ρ1y2ρ2⋮yqρq=Lfρ1h1xLfρ2h2x⋮Lfρqhqx+Axu.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 y1ρ1,y1ρ2,⋯,yqρqT=uLt, where uLit can assume the following simplification uLi=ymiρi+αi1ymiρi−1−yiρi−1+⋯+αiρiymi−yi,i=1,2,⋯,m.Remark 6.
Ifρi>νi, i∈1,2,⋯,m, then δix,t in equation (20) is related to the disturbance term dt and its differential term ḋt,⋯,dρi−νit, and could be expressed as a function of the differential term ḋt,⋯,dρi−νitδix,t=−dρi−νi/dt∑j=1pLpjLfρi−1hixdjt. 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.


## 2.2.1. Strict Feedback Linearization

Ifρ1+ρ2+⋯+ρq=n, the system (1) can be transformed into a strict feedback subsystem through strict feedback linearization and differential homeomorphic mapping ξ=Tx∈Rn, where
(12)Tx=h1x,Lfh1x,⋯,Lfρ1−1h1x,⋯,hqx,⋯,Lfρq−1hqxTξ=ξ1,1,ξ1,2,⋯,ξ1,ρ1,ξ2,1,⋯,ξq,ρqTThen the dynamic equation becomes(13)ξ̇1,1=ξ1,2ξ̇1,2=ξ1,3⋮ξ̇1,ρ1=b1ξ+A1σxv+A¯1σxu¯t+Δd1x,t⋮ξ̇q,1=ξq,2⋮ξ̇q,ρq=bqξ+Aqσxv+A¯qσxu¯tΔdqx,twhere biξ=Lfρihix, Aiσx, A¯iσx, and Δdi are the ith row of Aσx, A¯σx, and Δd, respectively, Aσx=AxI−σtvt, A¯σx=σtu¯t, with
(14)Ax=Lg1Lfρ1−1h1x⋯LgmLfρ1−1h1xLg1Lfρ2−1h2x⋯LgmLfρ2−1h2x⋯⋯Lg1Lfρq−1hqx⋯LgmLfρq−1hqxΔdx,t=Lp1Lfρ1−1h1x⋯LppLfρ1−1h1xLp1Lfρ2−1h2x⋯LppLfρ2−1h2x⋯⋯Lp1Lfρq−1hqx⋯LppLfρq−1hqxdt=δ1x,δ2x,⋯,δqxT.The system output is expressed as(15)y1=ξ1,1,y2=ξ2,1,⋯,yq=ξq,1.


## 2.2.2. Partial Feedback Linearization

Ifρ1+ρ2+⋯+ρq<n, only partial feedback linearization can be carried out in system (1) by means of coordinate transformation within the neighborhood of x0. Supposing Tcx is a smooth function with the following form
(16)Tcx=h1x,⋯,Lfρ1−1h1x,⋯,Lfρq−1hqxT.Literature [24] indicates that there is always smooth mapping
(17)Tzx=T1x,⋯,Tn−ρ1+⋯+ρqxT.To form the differential homeomorphism ξT,ηTT=Tx=TcxT,TzxTT, ξ∈Rρ1+ρ2+⋯+ρq, η∈Rn−ρ1+ρ2+⋯+ρq, system (1) is converted into
(18)ξ̇1,1=ξ1,2ξ̇1,2=ξ1,3⋮ξ̇1,ρ1=b1ξ,η+A1σxv+A¯1σxu¯t+Δd1x,t⋮ξ̇q,1=ξq,2⋮ξ̇q,ρq=bqξ,η+Aqσxv+A¯qσxu¯t+Δdqx,t(19)η̇=ψξ,η+Ψgξ,ηu+Ψpξ,ηdt,where biξ,η, Aiσx, A¯iσx, i=1,2,⋯,q have the same definition with the equation (13). Δdix, i=1,2,⋯,q is the unmatched disturbance. ψξ,η=∂Tzx/∂xfxx=T−1ξ,η, Ψgξ,η=∂Tzx/∂xgxx=T−1ξ,η, and Ψpξ,η=∂Tzx/∂xpxx=T−1ξ,η. The system (19) is termed as the zero dynamic form of the multivariable nonlinear system (1).


## 2.2.3. Nonlinear Feedback Control Law

Based on Assumptions3 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 yi derivatives of ρi in system (1), we can obtain the following equation:
(20)yiρi=Lfρihix+∑j=1mLgjLfρi−1hixuj+δix,where δix=∑j=1pLpjLfρi−1hixdjt. We can further obtain
(21)y1ρ1y2ρ2⋮yqρq=Lfρ1h1xLfρ2h2x⋮Lfρqhqx+Axu+Δdx,t.When m=q and assuming Ax is nonsingular in x0, the control input signal could be rearranged as
(22)ut=−A−1xLfρ1h1xLfρ2h2x⋮Lfρqhqx+A−1xuL.The linearized system can be obtained(23)y1ρ1y2ρ2⋮yqρq=uL1uL2⋮uLq+Δdx,t,where uL is the linear feedback control law to be designed.


## 2.2.4. Linear Feedback Control

The control law from the linearized system provides the possibility to guarantee the output tracking performance of the system. The control lawuL=uL1,uL2,⋯,uLqT is
(24)uLi=ymiρi+αi1ymiρi−1−yiρi−1+⋯+αiρiymi−yi+δcix,t,where δcix,t=−∑j=1pLpjLfρi−1hixdjt. With substitution of uLi and equation (24) into equation (23), the dynamic equation of the tracking error ei=yi−ymi is obtained as
(25)eiρi+αi1eiρi−1+⋯+αiρiei=0.By selecting appropriate value forαiρi, i=1,2,⋯,q, sρi+αi1sρi−1+⋯+αiρi becomes the Hurwitz polynomial. The output error and its higher derivative ei,ėi,⋯,eiρi−1asymptotically approach to zero as t⟶∞. If ymi,ẏmi,⋯,ymiρi−1 is bounded, then boundedness could be expected for yi,ẏi,⋯,yiρi−1.Remark 5.
If for alli=1,2,⋯,m, ρi<νi, then δix,t=0, in equation (26), i.e., Δdix=0, i=1,2,⋯,m. Thus, equation (21) can be simplified as
(26)y1ρ1y2ρ2⋮yqρq=Lfρ1h1xLfρ2h2x⋮Lfρqhqx+Axu.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 y1ρ1,y1ρ2,⋯,yqρqT=uLt, where uLit can assume the following simplification uLi=ymiρi+αi1ymiρi−1−yiρi−1+⋯+αiρiymi−yi,i=1,2,⋯,m.Remark 6.
Ifρi>νi, i∈1,2,⋯,m, then δix,t in equation (20) is related to the disturbance term dt and its differential term ḋt,⋯,dρi−νit, and could be expressed as a function of the differential term ḋt,⋯,dρi−νitδix,t=−dρi−νi/dt∑j=1pLpjLfρi−1hixdjt. 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.


## 3. Actuator Fault Compensation and Disturbance Suppression Design

If the relevance of the systemρ1,ρ2,⋯,ρq satisfies ρ1+ρ2+⋯+ρq=n, the system with uncertain actuator fault can be linearized by strict feedback and converted into
(27)ξ̇1,1=ξ1,2,ξ̇1,2=ξ1,3,⋮ξ̇1,ρ1=b1ξ+A1σxv+A¯1σxu¯t+Δd1x,t⋮ξ̇q,1=ξq,2,⋮ξ̇q,ρq=bqξ+Aqσxv+A¯qσxu¯t+Δdqx,ty1=ξ1,1,y2=ξ2,1,⋯,yq=ξq,1.Based onuLi, i=1,2,⋯,q in equation (24), the control signal wdt∈Rqof the system (27) could be determined through nonlinear feedback, if
(28)Aσxvt+A¯σxu¯t=wdt.The control signal can guarantee asymptotic output tracking, i.e.,limt→∞yt−ymt=0. With occurrence of uncertain actuator fault, the control input signal vt could be calculated according to equation (28).


### 3.1. Adaptive Disturbance Suppression Design

The control signalwdt and the feedback linearization are determined in this chapter. The detailed derivation includes adaptive controller, error equation, parameter adaptive updating law, and stability analysis.


#### 3.1.1. Adaptive Feedback Linearization Design

In the disturbance modeldjt=θdj∗Tϖdjt, ϖdjtj=1,2,⋯,p are known functions while θdj∗ are unknown parameters. The unknown parameters could be estimated with d^jt=θ^djTϖdjt in the disturbance suppression design, where θ^dj is the estimate of the disturbance parameter θdj∗, j=1,2,⋯,p. Based on the estimation, the adaptive linear control law is obtained as
(29)ut=−A−1xLfρ1h1xLfρ2h2x⋮Lfρqhqx+A−1xu^L,where u^L is the estimated value of uL, and its estimated component is
(30)u^Li=ymiρi+αi1ymiρi−1−yiρi−1+αiρiymi−yi+δ^cix,t,δ^cix,t=−∑j=1pLpjLfρi−1hixd^jt.


#### 3.1.2. Error Model

Letz1,1=ξ1,1−ym1, z1,2=ξ1,2−ẏm1, ⋯, z1,ρ1=ξ1,ρ1−ym1ρ1−1, ⋯zi,ρi=ξi,ρi−ymiρi−1, ⋯, zq,ρq=ξq,ρq−ymqρq−1, z∈Rρ1+ρ2+⋯+ρq=z1T,z2T,⋯,zqTT=z1,1,⋯,z1,ρ1,⋯,zi,ρi,⋯,zq,ρqT. Combining the system output in equation (15): y1=ξ1,1, y2=ξ2,1, ⋯, yq=ξq,1 and ei=yi−ymi, one can obtain z1,1=e1, z1,2=ė1, ⋯, z1,ρ1=e1ρ1−1, ⋯, zi,ρi=eiρi−1, ⋯, zq,ρq=eqρq−1. And the state error equation of the multi-input multioutput system is calculated by
(31)ż=Azz+BzE~d,where Az=diagAz1,Az2,⋯,Azq, BzE~d=Bz1TE~d1,Bz2TE~d2,⋯,BzqTE~dqT, and E~di=∑j=1pLpjLfρi−1hixθ~djTtϖdj,
(32)Azi=010⋯00001⋯00⋯⋯−αi1−αi2⋯−αiρi−2−αiρi−1−αiρi,Bzi=0,0,⋯,1T∈Rρi,i=1,2,⋯,q.


#### 3.1.3. Adaptive Laws

Based on error system (31), an adaptive law is incorporated to update unknown disturbance parameters θ^dj,j=1,2,⋯,p. Lyapunov function is designed in following form
(33)Vd=12zTPz+12∑j=1pθ~djTΓdj−1θ~dj,where adaptive gain matrix Γdj=ΓdjT>0, P∈Rn×n is positive definite symmetric matrix and satisfies the following equation
(34)PAz+AzTP=−Q,where Q=QT>0. Taking the derivative with respect to Vd gives
(35)V̇d=12żTPz+12zTPż+∑j=1pθ~djTΓdj−1θ~̇dj=12zTPAz+AzTPz+zTPBzE~d+∑j=1pθ~djTΓdj−1θ~̇dj=−12zTQz+ZPE~d1,⋯,E~dqT+∑j=1pθ~djTΓdj−1θ~̇dj=−12zTQz+∑j=1p∑i=1qZPiθ~djTϖρi,j+∑j=1pθ~djTΓdj−1θ~̇dj,where ZP=ZP1,ZP2,⋯,ZPqT, ZPi, i=1,2,⋯,q are the components of zTP∈R1×ρ1+ρ2+⋯+ρq, ϖρi,j=LpjLfρi−1hixϖdj. Design control equation is given by
(36)wdt=Δ−Lfρ1h1xLfρ2h2x⋮Lfρqhqx+u^L,and the adaptive law of the parameter θ^dj is
(37)θ^̇dj=∑i=1qΓdjZPiϖρi,j,j=1,2,⋯,p.With a substitution into equation (33), the following could be obtained
(38)V̇d=−12zTQz≤0.So it is ensured thatθ^dj∈L∞, θ^̇dj∈L2∩L∞. The stability of the closed-loop system can be determined from the negative definition of V̇d and limt→∞zi,1t=limt→∞yit−ymt=0, i=1,2,⋯,q. It indicates that a desired performance is achieved with the control system.


### 3.2. Adaptive Fault Compensation Control Design

Supposing the fault information (fault mode, fault value, and fault time) is known. Two ideal controllersv1∗t and v2∗t are designed for the two cases (without fault and actuator u1 fault). Through weighted fusion design, an integrated controller v∗t is obtained, which can deal with the simultaneous coexistence of two fault modes mentioned above.


#### 3.2.1. Fault Free Condition

In this case,σ=diag0,0,⋯,0, the control equation (28) is Axvt=wdt. By selecting an appropriate h21x∈Rm×m−1, the following equation could be satisfied
(39)vt=v1∗t=h21xv01∗tBy solving the equationAxh21xv01∗t=wdt, we could obtain the following equation
(40)v01∗t=K21xwdt,where K21x∈Rm−1×q.


#### 3.2.2.u1 Fault Condition

In case ofu1 fault condition, σ=diag1,0,⋯,0, u1=u¯1, ui=vi, i=2,⋯,m, Ax=A1,A2,⋯,Am=A1,A2∈Rq×m, where A2=A2,⋯,Am∈Rq×m−1, v=v1,va2TT∈Rm, where va2=v2,⋯,vmT∈Rm−1, by selecting the appropriate matrix equation h22x∈Rm−1×m−1, one could have va2∗t=h22xv02∗t. And the equation could be solved
(41)A1u¯1t+A2h22xv02∗t=wdt.The ideal controller under this condition is(42)v02∗t=K22xwdt+K221xu¯1t.


#### 3.2.3. Integrated Control Law

The fault index function is defined as(43)χ1∗=1faultfreecondition0others,χ2∗=1u1faultcondition0others.With a weighted fusion of controllerv1∗t and v2∗t, an ideal integrated controller structure is achieved.
(44)v∗t=χ1∗tv1∗t+χ2∗tv2∗t=vχ11∗t+0,vχ2a2∗TtT,where vχ2a2∗t=χ2∗th22K22xwdt+χ2∗th22K221xu¯1t.


#### 3.2.4. Adaptive Controller Structure

From equation (44), the structure of adaptive controller can be deduced as
(45)vt=vχ11t+vχ22t=vχ11t+0,vχ2a2TtT,where
(46)vχ11=Δdiagχ1,1,⋯,χ1,mh21K21wd(47)vχ2a2=Δdiagχ2,1,⋯,χ2,m−1h22K22wd+θ11Tϖ1ϕ2,1θ12Tϖ1ϕ2,2⋮θ1m−1Tϖ1ϕ2,m−1.χj,i and θ1i are the estimated value of χj,i∗ and θ1i∗, χ1,i∗=χ1∗, i=1,⋯,m, χ2,i∗=χ2∗, θ1i∗=χ2∗θ1∗, i=1,2,⋯,m−1.Remark 7.
As the number offjit increase, the parameters of the actuator failure (including the parameters of failure indicator function χi∗ and χ2∗, failure model θ1∗ also increases. In our proposed actuator failure compensation design, all the unknown parameters will be estimated multiple (m or m−1) times based on χ1,i∗=χ1∗, i=1,⋯,m, χ2,i∗=χ2∗, θ1i∗=χ2∗θ1∗, 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.


#### 3.2.5. Error Equations

Equation (21) could be rewritten as
(48)y1ρ1y1ρ2⋮yqρq=Lfρ1h1xLfρ2h2x⋮Lfρqhqx+Axu−wdt+wdt+δ1xδ2x⋮δqx=ym1ρ1+α11ym1ρ1−1−y1ρ1−1+α1ρ1ym1−y1ym2ρ2+α21ym2ρ2−1−y2ρ2−1+α2ρ2ym2−y2⋮ymqρq+αq1ymqρq−1−yqρq−1+αqρqymq−yq+AxI−σtvt−v∗t+δ1xδ2x⋮δqx−δ^1xδ^2x⋮δ^qx.To obtain the output erroret=yt−ymt and the parameter estimation error χ~1,it, χ~2,it, the dynamic formula between θ~1it and θ~dj is reformulated as
(49)e1ρ1+⋯+α1ρ1e1e2ρ2+⋯+α2ρ2e2⋮eqρq+⋯+αqρqeq=AxI−σtv−v∗+E~d1,E~d2,⋯,E~dqT.Ifσ=σ1=diag0,⋯,0
(50)e1ρ1+⋯+α1ρ1e1e2ρ2+⋯+α2ρ2e2⋮eqρq+⋯+αqρqeq=∑i=1mAiχ~1,iv1,i+∑i=1m−1Ai+1χ~2,iv2,i+∑i=1m−1Ai+1θ~1iTϖ1ϕ2,i+E~d1,E~d2,⋯,E~dqT=∑i=1mAiχ~1,iv1,i+∑i=1m−1Ai+1χ~2,iv2,i+∑i=1m−1Ai+1θ~1iTϖ1ϕ2,i+∑j=1pθ~djTϖρ1,j∑j=1pθ~djTϖρ2,j⋮∑j=1pθ~djTϖρq,j=ΔE~1,where h21K21wd=v1,1,⋯,v1,mT, h22K22wd=v2,1,⋯,v2,m−1T, ϖρi,j=LpjLfρi−1hixϖdj.Ifσ=σ2=diag1,⋯,0, equation (49) can be expressed as
(51)e1ρ1+⋯+α1ρ1e1e2ρ2+⋯+α2ρ2e2⋮eqρq+⋯+αqρqeq=∑i=2mAiχ~1,iv1,i+∑i=1m−1Ai+1χ~2,iv2,i+∑i=1m−1Ai+1θ~1iTϖ1ϕ2,i+∑j=1pθ~djTϖρ1,j,∑j=1pθ~djTϖρ2,j,⋯,∑j=1pθ~djTϖρq,jT=ΔE~2.The state error equation can be obtained from equations ((50), (51)).
(52)ż=Azz+BzE~k,where Az=diagAz1,⋯,Azq∈Rρ1+⋯+ρq×ρ1+⋯+ρq, E~kj is the jth component of E~k, k=1,2, BzE~k=Bz1TE~k1,Bz2TE~k2,⋯,BzqTE~kqT∈Rρ1+⋯+ρq.


#### 3.2.6. Adaptive Laws

Based on state error equation (52), the adaptive laws could be derived with projection algorithm, parameters χ1,i, i=1,⋯,m, χ2,i and θ1i, i=1,2,⋯,m−1, are estimated as
(53)χ̇1,it=−γ1izTPBZAiv1,ii=2,⋯,m−γ1izTPBZAiv1,i+fχ1,ii=1(54)χ̇2,it=−γ2izTPBzAi+1v2,i,i=1,⋯,m−1(55)θ̇1it=−zTPBzAi+1Γ1iϖ1ϕ2,i,i=1,⋯,m−1where Γ1i=Γ1iT>0, γ1i>0 and γ2i>0 are the adaptive gains, fχ1,i is the projection algorithm. Consequently, according to adaptive laws χ̇1,1=−γ11zTPBzA1v1,1+fχ1,1, we can derive that 0≤χ1,1≤1 and
(56)χ1,1−χ1,1∗fχ1,1≤0.


#### 3.2.7. Performance Analysis

(I) For time periodt∈T0,T1, T1=∞, σ=σ1=diag0,⋯,0. The Lyapunov function is defined as
(57)V0=12zTPz+12∑i=1mχ~1,i2γ1i−1+∑i=1m−1χ~2,i2γ2i−1+∑i=1m−1θ~1iTΓ1i−1θ~1i+12∑j=1pθ~djTΓdj−1θ~dj.Combining equations ((37), (50), (51), (52), (53), (54), (55)) one would have the derivative of V0
(58)V̇0=−zTQz≤0,t∈T0,T1.Thus, it can be proved that the designed adaptive controller and its parameter adaptive laws could ensure the desired system performance under the free fault condition, i.e.,ξ, χ~1,it, χ~2,it, θ~1it, and θ~dj are all bounded, and the output error asymptotes to zero as time going on.(II) If actuatoru1 has faults in time period T1,T2 (T2=∞), i.e., σ=σ2=diag1,⋯,0, the Lyapunov function is defined as
(59)V1=12zTPz+12∑i=2mχ~1,i2γ1i−1+∑i=1m−1χ~2,i2γ2i−1+∑i=1m−1θ~1iTΓ1i−1θ~1i+12∑j=1pθ~djTΓdj−1θ~dj.Combining equations ((37), (50), (51), (52), (53), (54), (55)) gives the derivative of V1,
(60)V̇1=−zTQz≤0,t∈T1,T2.The above equation indicates thatξ, χ~1,it, i=2,⋯,m, χ~2,it, θ~1it, i=1,⋯,m−1 and θ~dj, j=1,2,⋯,p are bounded when actuator u1 is failed. In addition, the adaptive projection algorithm χ̇1,1t=−γ1izTPBzAiv1,i+fχ1,i can ensure 0≤χ1,1t≤1. Thus, it can be verified that with increase in t, the closed-loop system is stable and the output asymptotically approaches zero: limt→∞yt−ymt=0. In conclusion, the following theorem can be obtained.Theorem 8.
For the multivariable nonlinear system (1) with potential uncertain actuator fault (3) and mismatched disturbance dt, controller (45) and its parameter adaptive laws can ensure the closed-loop system stability and asymptotic tracking output: limt→∞y−ym=0, ifρ1+ρ2+⋯+ρq=nand the equivalent control matrix Aσx=AxI−σtvt in uncertain fault condition has full rank in the domain U(definition is U⊂Rn⟶V⊂Rq).


### 3.3. Fault Compensation Design of Zero Dynamic System

Ifρ1+ρ2+⋯+ρq<n, there is differential homeomorphism
(61)ξT,ηTT=Tx=TcxT,TzxTT.The nonlinear system with uncertain actuator faultẋt=fx+gxσtu¯t+gx1−σtvt+pxdt, y=hx is converted into
(62)ξ̇1,1=ξ1,2ξ̇1,2=ξ1,3⋯ξ̇1,ρ1=b1ξ,η+A1xI−σv+A1xσu¯+Δd1x,t⋯ξ̇i,ρi=ξi,ρi+1⋯ξ̇i,ρi=biξ,η+AixI−σv+Aixσu¯+Δdix,tξ̇q,1=ξq,2⋯ξ̇q,ρq=bqξ,η+AqxI−σv+Aqxσu¯+Δdqx,t,and zero dynamic subsystem
(63)η̇=ψξ,η+Ψσξ,ηu¯+Ψ¯σξ,ηv+Ψpξ,ηdt,where Tcx=h1x,⋯,Lfρ1−1h1x,⋯,Lfρq−1hqxT, Tzx definitely exists and is nonunique. Ψσξ,η=∂Tzx/∂xgxσ, Ψ¯σξ,η=∂Tzx/∂xgxI−σ is related to the fault mode σ.


#### 3.3.1. Stable Zero Dynamic Assumption

To ensure the stability of the closed-loop system and outputyit asymptotic tracking reference signal ymit, the differentials of ρi, i=1,2,⋯,q of ymit 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 ut, dt, and ξ, the zero dynamic subsystem given by
(64)η̇=ψξ,η+Ψσξ,ηu¯+Ψpξ,ηd+Ψ¯σξ,ηvξ,η,χ~1,i,χ~2,i,θ~1icould guarantee input state stability.Remark 10.
Based on Assumption9, if σ∈Σ in any fault case, the state ξ, fault signal u¯, and the designed feedback control signal vξ,η,χ~1,i,χ~2,i,θ~1i are all bounded while dt 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 vξ,η,χ~1,i,χ~2,i,θ~1i is bounded.Combined with Assumption3, the signal vt 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 (Assumption9) and the equivalent control matrix Aσx 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: limt→∞yt−ymt=0 in the case of multiple uncertain actuator faults (3) and unknown disturbances.Proof.
Assuming one of the actuators failed at timeT1, and the system has no fault during time period T1,T2, it can be derived according to the performance analysis in Section 3.2 that the estimated parameters ξ, χ~1,it, χ~2,it, and θ~1it are bounded and the state error z asymptotically approaches to zero as t tends towards infinity. Boundedness of vt 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.


## 3.1. Adaptive Disturbance Suppression Design

The control signalwdt and the feedback linearization are determined in this chapter. The detailed derivation includes adaptive controller, error equation, parameter adaptive updating law, and stability analysis.


### 3.1.1. Adaptive Feedback Linearization Design

In the disturbance modeldjt=θdj∗Tϖdjt, ϖdjtj=1,2,⋯,p are known functions while θdj∗ are unknown parameters. The unknown parameters could be estimated with d^jt=θ^djTϖdjt in the disturbance suppression design, where θ^dj is the estimate of the disturbance parameter θdj∗, j=1,2,⋯,p. Based on the estimation, the adaptive linear control law is obtained as
(29)ut=−A−1xLfρ1h1xLfρ2h2x⋮Lfρqhqx+A−1xu^L,where u^L is the estimated value of uL, and its estimated component is
(30)u^Li=ymiρi+αi1ymiρi−1−yiρi−1+αiρiymi−yi+δ^cix,t,δ^cix,t=−∑j=1pLpjLfρi−1hixd^jt.


### 3.1.2. Error Model

Letz1,1=ξ1,1−ym1, z1,2=ξ1,2−ẏm1, ⋯, z1,ρ1=ξ1,ρ1−ym1ρ1−1, ⋯zi,ρi=ξi,ρi−ymiρi−1, ⋯, zq,ρq=ξq,ρq−ymqρq−1, z∈Rρ1+ρ2+⋯+ρq=z1T,z2T,⋯,zqTT=z1,1,⋯,z1,ρ1,⋯,zi,ρi,⋯,zq,ρqT. Combining the system output in equation (15): y1=ξ1,1, y2=ξ2,1, ⋯, yq=ξq,1 and ei=yi−ymi, one can obtain z1,1=e1, z1,2=ė1, ⋯, z1,ρ1=e1ρ1−1, ⋯, zi,ρi=eiρi−1, ⋯, zq,ρq=eqρq−1. And the state error equation of the multi-input multioutput system is calculated by
(31)ż=Azz+BzE~d,where Az=diagAz1,Az2,⋯,Azq, BzE~d=Bz1TE~d1,Bz2TE~d2,⋯,BzqTE~dqT, and E~di=∑j=1pLpjLfρi−1hixθ~djTtϖdj,
(32)Azi=010⋯00001⋯00⋯⋯−αi1−αi2⋯−αiρi−2−αiρi−1−αiρi,Bzi=0,0,⋯,1T∈Rρi,i=1,2,⋯,q.


### 3.1.3. Adaptive Laws

Based on error system (31), an adaptive law is incorporated to update unknown disturbance parameters θ^dj,j=1,2,⋯,p. Lyapunov function is designed in following form
(33)Vd=12zTPz+12∑j=1pθ~djTΓdj−1θ~dj,where adaptive gain matrix Γdj=ΓdjT>0, P∈Rn×n is positive definite symmetric matrix and satisfies the following equation
(34)PAz+AzTP=−Q,where Q=QT>0. Taking the derivative with respect to Vd gives
(35)V̇d=12żTPz+12zTPż+∑j=1pθ~djTΓdj−1θ~̇dj=12zTPAz+AzTPz+zTPBzE~d+∑j=1pθ~djTΓdj−1θ~̇dj=−12zTQz+ZPE~d1,⋯,E~dqT+∑j=1pθ~djTΓdj−1θ~̇dj=−12zTQz+∑j=1p∑i=1qZPiθ~djTϖρi,j+∑j=1pθ~djTΓdj−1θ~̇dj,where ZP=ZP1,ZP2,⋯,ZPqT, ZPi, i=1,2,⋯,q are the components of zTP∈R1×ρ1+ρ2+⋯+ρq, ϖρi,j=LpjLfρi−1hixϖdj. Design control equation is given by
(36)wdt=Δ−Lfρ1h1xLfρ2h2x⋮Lfρqhqx+u^L,and the adaptive law of the parameter θ^dj is
(37)θ^̇dj=∑i=1qΓdjZPiϖρi,j,j=1,2,⋯,p.With a substitution into equation (33), the following could be obtained
(38)V̇d=−12zTQz≤0.So it is ensured thatθ^dj∈L∞, θ^̇dj∈L2∩L∞. The stability of the closed-loop system can be determined from the negative definition of V̇d and limt→∞zi,1t=limt→∞yit−ymt=0, i=1,2,⋯,q. It indicates that a desired performance is achieved with the control system.


## 3.1.1. Adaptive Feedback Linearization Design

In the disturbance modeldjt=θdj∗Tϖdjt, ϖdjtj=1,2,⋯,p are known functions while θdj∗ are unknown parameters. The unknown parameters could be estimated with d^jt=θ^djTϖdjt in the disturbance suppression design, where θ^dj is the estimate of the disturbance parameter θdj∗, j=1,2,⋯,p. Based on the estimation, the adaptive linear control law is obtained as
(29)ut=−A−1xLfρ1h1xLfρ2h2x⋮Lfρqhqx+A−1xu^L,where u^L is the estimated value of uL, and its estimated component is
(30)u^Li=ymiρi+αi1ymiρi−1−yiρi−1+αiρiymi−yi+δ^cix,t,δ^cix,t=−∑j=1pLpjLfρi−1hixd^jt.


## 3.1.2. Error Model

Letz1,1=ξ1,1−ym1, z1,2=ξ1,2−ẏm1, ⋯, z1,ρ1=ξ1,ρ1−ym1ρ1−1, ⋯zi,ρi=ξi,ρi−ymiρi−1, ⋯, zq,ρq=ξq,ρq−ymqρq−1, z∈Rρ1+ρ2+⋯+ρq=z1T,z2T,⋯,zqTT=z1,1,⋯,z1,ρ1,⋯,zi,ρi,⋯,zq,ρqT. Combining the system output in equation (15): y1=ξ1,1, y2=ξ2,1, ⋯, yq=ξq,1 and ei=yi−ymi, one can obtain z1,1=e1, z1,2=ė1, ⋯, z1,ρ1=e1ρ1−1, ⋯, zi,ρi=eiρi−1, ⋯, zq,ρq=eqρq−1. And the state error equation of the multi-input multioutput system is calculated by
(31)ż=Azz+BzE~d,where Az=diagAz1,Az2,⋯,Azq, BzE~d=Bz1TE~d1,Bz2TE~d2,⋯,BzqTE~dqT, and E~di=∑j=1pLpjLfρi−1hixθ~djTtϖdj,
(32)Azi=010⋯00001⋯00⋯⋯−αi1−αi2⋯−αiρi−2−αiρi−1−αiρi,Bzi=0,0,⋯,1T∈Rρi,i=1,2,⋯,q.


## 3.1.3. Adaptive Laws

Based on error system (31), an adaptive law is incorporated to update unknown disturbance parameters θ^dj,j=1,2,⋯,p. Lyapunov function is designed in following form
(33)Vd=12zTPz+12∑j=1pθ~djTΓdj−1θ~dj,where adaptive gain matrix Γdj=ΓdjT>0, P∈Rn×n is positive definite symmetric matrix and satisfies the following equation
(34)PAz+AzTP=−Q,where Q=QT>0. Taking the derivative with respect to Vd gives
(35)V̇d=12żTPz+12zTPż+∑j=1pθ~djTΓdj−1θ~̇dj=12zTPAz+AzTPz+zTPBzE~d+∑j=1pθ~djTΓdj−1θ~̇dj=−12zTQz+ZPE~d1,⋯,E~dqT+∑j=1pθ~djTΓdj−1θ~̇dj=−12zTQz+∑j=1p∑i=1qZPiθ~djTϖρi,j+∑j=1pθ~djTΓdj−1θ~̇dj,where ZP=ZP1,ZP2,⋯,ZPqT, ZPi, i=1,2,⋯,q are the components of zTP∈R1×ρ1+ρ2+⋯+ρq, ϖρi,j=LpjLfρi−1hixϖdj. Design control equation is given by
(36)wdt=Δ−Lfρ1h1xLfρ2h2x⋮Lfρqhqx+u^L,and the adaptive law of the parameter θ^dj is
(37)θ^̇dj=∑i=1qΓdjZPiϖρi,j,j=1,2,⋯,p.With a substitution into equation (33), the following could be obtained
(38)V̇d=−12zTQz≤0.So it is ensured thatθ^dj∈L∞, θ^̇dj∈L2∩L∞. The stability of the closed-loop system can be determined from the negative definition of V̇d and limt→∞zi,1t=limt→∞yit−ymt=0, i=1,2,⋯,q. It indicates that a desired performance is achieved with the control system.


## 3.2. Adaptive Fault Compensation Control Design

Supposing the fault information (fault mode, fault value, and fault time) is known. Two ideal controllersv1∗t and v2∗t are designed for the two cases (without fault and actuator u1 fault). Through weighted fusion design, an integrated controller v∗t is obtained, which can deal with the simultaneous coexistence of two fault modes mentioned above.


### 3.2.1. Fault Free Condition

In this case,σ=diag0,0,⋯,0, the control equation (28) is Axvt=wdt. By selecting an appropriate h21x∈Rm×m−1, the following equation could be satisfied
(39)vt=v1∗t=h21xv01∗tBy solving the equationAxh21xv01∗t=wdt, we could obtain the following equation
(40)v01∗t=K21xwdt,where K21x∈Rm−1×q.


### 3.2.2.u1 Fault Condition

In case ofu1 fault condition, σ=diag1,0,⋯,0, u1=u¯1, ui=vi, i=2,⋯,m, Ax=A1,A2,⋯,Am=A1,A2∈Rq×m, where A2=A2,⋯,Am∈Rq×m−1, v=v1,va2TT∈Rm, where va2=v2,⋯,vmT∈Rm−1, by selecting the appropriate matrix equation h22x∈Rm−1×m−1, one could have va2∗t=h22xv02∗t. And the equation could be solved
(41)A1u¯1t+A2h22xv02∗t=wdt.The ideal controller under this condition is(42)v02∗t=K22xwdt+K221xu¯1t.


### 3.2.3. Integrated Control Law

The fault index function is defined as(43)χ1∗=1faultfreecondition0others,χ2∗=1u1faultcondition0others.With a weighted fusion of controllerv1∗t and v2∗t, an ideal integrated controller structure is achieved.
(44)v∗t=χ1∗tv1∗t+χ2∗tv2∗t=vχ11∗t+0,vχ2a2∗TtT,where vχ2a2∗t=χ2∗th22K22xwdt+χ2∗th22K221xu¯1t.


### 3.2.4. Adaptive Controller Structure

From equation (44), the structure of adaptive controller can be deduced as
(45)vt=vχ11t+vχ22t=vχ11t+0,vχ2a2TtT,where
(46)vχ11=Δdiagχ1,1,⋯,χ1,mh21K21wd(47)vχ2a2=Δdiagχ2,1,⋯,χ2,m−1h22K22wd+θ11Tϖ1ϕ2,1θ12Tϖ1ϕ2,2⋮θ1m−1Tϖ1ϕ2,m−1.χj,i and θ1i are the estimated value of χj,i∗ and θ1i∗, χ1,i∗=χ1∗, i=1,⋯,m, χ2,i∗=χ2∗, θ1i∗=χ2∗θ1∗, i=1,2,⋯,m−1.Remark 7.
As the number offjit increase, the parameters of the actuator failure (including the parameters of failure indicator function χi∗ and χ2∗, failure model θ1∗ also increases. In our proposed actuator failure compensation design, all the unknown parameters will be estimated multiple (m or m−1) times based on χ1,i∗=χ1∗, i=1,⋯,m, χ2,i∗=χ2∗, θ1i∗=χ2∗θ1∗, 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.


### 3.2.5. Error Equations

Equation (21) could be rewritten as
(48)y1ρ1y1ρ2⋮yqρq=Lfρ1h1xLfρ2h2x⋮Lfρqhqx+Axu−wdt+wdt+δ1xδ2x⋮δqx=ym1ρ1+α11ym1ρ1−1−y1ρ1−1+α1ρ1ym1−y1ym2ρ2+α21ym2ρ2−1−y2ρ2−1+α2ρ2ym2−y2⋮ymqρq+αq1ymqρq−1−yqρq−1+αqρqymq−yq+AxI−σtvt−v∗t+δ1xδ2x⋮δqx−δ^1xδ^2x⋮δ^qx.To obtain the output erroret=yt−ymt and the parameter estimation error χ~1,it, χ~2,it, the dynamic formula between θ~1it and θ~dj is reformulated as
(49)e1ρ1+⋯+α1ρ1e1e2ρ2+⋯+α2ρ2e2⋮eqρq+⋯+αqρqeq=AxI−σtv−v∗+E~d1,E~d2,⋯,E~dqT.Ifσ=σ1=diag0,⋯,0
(50)e1ρ1+⋯+α1ρ1e1e2ρ2+⋯+α2ρ2e2⋮eqρq+⋯+αqρqeq=∑i=1mAiχ~1,iv1,i+∑i=1m−1Ai+1χ~2,iv2,i+∑i=1m−1Ai+1θ~1iTϖ1ϕ2,i+E~d1,E~d2,⋯,E~dqT=∑i=1mAiχ~1,iv1,i+∑i=1m−1Ai+1χ~2,iv2,i+∑i=1m−1Ai+1θ~1iTϖ1ϕ2,i+∑j=1pθ~djTϖρ1,j∑j=1pθ~djTϖρ2,j⋮∑j=1pθ~djTϖρq,j=ΔE~1,where h21K21wd=v1,1,⋯,v1,mT, h22K22wd=v2,1,⋯,v2,m−1T, ϖρi,j=LpjLfρi−1hixϖdj.Ifσ=σ2=diag1,⋯,0, equation (49) can be expressed as
(51)e1ρ1+⋯+α1ρ1e1e2ρ2+⋯+α2ρ2e2⋮eqρq+⋯+αqρqeq=∑i=2mAiχ~1,iv1,i+∑i=1m−1Ai+1χ~2,iv2,i+∑i=1m−1Ai+1θ~1iTϖ1ϕ2,i+∑j=1pθ~djTϖρ1,j,∑j=1pθ~djTϖρ2,j,⋯,∑j=1pθ~djTϖρq,jT=ΔE~2.The state error equation can be obtained from equations ((50), (51)).
(52)ż=Azz+BzE~k,where Az=diagAz1,⋯,Azq∈Rρ1+⋯+ρq×ρ1+⋯+ρq, E~kj is the jth component of E~k, k=1,2, BzE~k=Bz1TE~k1,Bz2TE~k2,⋯,BzqTE~kqT∈Rρ1+⋯+ρq.


### 3.2.6. Adaptive Laws

Based on state error equation (52), the adaptive laws could be derived with projection algorithm, parameters χ1,i, i=1,⋯,m, χ2,i and θ1i, i=1,2,⋯,m−1, are estimated as
(53)χ̇1,it=−γ1izTPBZAiv1,ii=2,⋯,m−γ1izTPBZAiv1,i+fχ1,ii=1(54)χ̇2,it=−γ2izTPBzAi+1v2,i,i=1,⋯,m−1(55)θ̇1it=−zTPBzAi+1Γ1iϖ1ϕ2,i,i=1,⋯,m−1where Γ1i=Γ1iT>0, γ1i>0 and γ2i>0 are the adaptive gains, fχ1,i is the projection algorithm. Consequently, according to adaptive laws χ̇1,1=−γ11zTPBzA1v1,1+fχ1,1, we can derive that 0≤χ1,1≤1 and
(56)χ1,1−χ1,1∗fχ1,1≤0.


### 3.2.7. Performance Analysis

(I) For time periodt∈T0,T1, T1=∞, σ=σ1=diag0,⋯,0. The Lyapunov function is defined as
(57)V0=12zTPz+12∑i=1mχ~1,i2γ1i−1+∑i=1m−1χ~2,i2γ2i−1+∑i=1m−1θ~1iTΓ1i−1θ~1i+12∑j=1pθ~djTΓdj−1θ~dj.Combining equations ((37), (50), (51), (52), (53), (54), (55)) one would have the derivative of V0
(58)V̇0=−zTQz≤0,t∈T0,T1.Thus, it can be proved that the designed adaptive controller and its parameter adaptive laws could ensure the desired system performance under the free fault condition, i.e.,ξ, χ~1,it, χ~2,it, θ~1it, and θ~dj are all bounded, and the output error asymptotes to zero as time going on.(II) If actuatoru1 has faults in time period T1,T2 (T2=∞), i.e., σ=σ2=diag1,⋯,0, the Lyapunov function is defined as
(59)V1=12zTPz+12∑i=2mχ~1,i2γ1i−1+∑i=1m−1χ~2,i2γ2i−1+∑i=1m−1θ~1iTΓ1i−1θ~1i+12∑j=1pθ~djTΓdj−1θ~dj.Combining equations ((37), (50), (51), (52), (53), (54), (55)) gives the derivative of V1,
(60)V̇1=−zTQz≤0,t∈T1,T2.The above equation indicates thatξ, χ~1,it, i=2,⋯,m, χ~2,it, θ~1it, i=1,⋯,m−1 and θ~dj, j=1,2,⋯,p are bounded when actuator u1 is failed. In addition, the adaptive projection algorithm χ̇1,1t=−γ1izTPBzAiv1,i+fχ1,i can ensure 0≤χ1,1t≤1. Thus, it can be verified that with increase in t, the closed-loop system is stable and the output asymptotically approaches zero: limt→∞yt−ymt=0. In conclusion, the following theorem can be obtained.Theorem 8.
For the multivariable nonlinear system (1) with potential uncertain actuator fault (3) and mismatched disturbance dt, controller (45) and its parameter adaptive laws can ensure the closed-loop system stability and asymptotic tracking output: limt→∞y−ym=0, ifρ1+ρ2+⋯+ρq=nand the equivalent control matrix Aσx=AxI−σtvt in uncertain fault condition has full rank in the domain U(definition is U⊂Rn⟶V⊂Rq).


## 3.2.1. Fault Free Condition

In this case,σ=diag0,0,⋯,0, the control equation (28) is Axvt=wdt. By selecting an appropriate h21x∈Rm×m−1, the following equation could be satisfied
(39)vt=v1∗t=h21xv01∗tBy solving the equationAxh21xv01∗t=wdt, we could obtain the following equation
(40)v01∗t=K21xwdt,where K21x∈Rm−1×q.


## 3.2.2.u1 Fault Condition

In case ofu1 fault condition, σ=diag1,0,⋯,0, u1=u¯1, ui=vi, i=2,⋯,m, Ax=A1,A2,⋯,Am=A1,A2∈Rq×m, where A2=A2,⋯,Am∈Rq×m−1, v=v1,va2TT∈Rm, where va2=v2,⋯,vmT∈Rm−1, by selecting the appropriate matrix equation h22x∈Rm−1×m−1, one could have va2∗t=h22xv02∗t. And the equation could be solved
(41)A1u¯1t+A2h22xv02∗t=wdt.The ideal controller under this condition is(42)v02∗t=K22xwdt+K221xu¯1t.


## 3.2.3. Integrated Control Law

The fault index function is defined as(43)χ1∗=1faultfreecondition0others,χ2∗=1u1faultcondition0others.With a weighted fusion of controllerv1∗t and v2∗t, an ideal integrated controller structure is achieved.
(44)v∗t=χ1∗tv1∗t+χ2∗tv2∗t=vχ11∗t+0,vχ2a2∗TtT,where vχ2a2∗t=χ2∗th22K22xwdt+χ2∗th22K221xu¯1t.


## 3.2.4. Adaptive Controller Structure

From equation (44), the structure of adaptive controller can be deduced as
(45)vt=vχ11t+vχ22t=vχ11t+0,vχ2a2TtT,where
(46)vχ11=Δdiagχ1,1,⋯,χ1,mh21K21wd(47)vχ2a2=Δdiagχ2,1,⋯,χ2,m−1h22K22wd+θ11Tϖ1ϕ2,1θ12Tϖ1ϕ2,2⋮θ1m−1Tϖ1ϕ2,m−1.χj,i and θ1i are the estimated value of χj,i∗ and θ1i∗, χ1,i∗=χ1∗, i=1,⋯,m, χ2,i∗=χ2∗, θ1i∗=χ2∗θ1∗, i=1,2,⋯,m−1.Remark 7.
As the number offjit increase, the parameters of the actuator failure (including the parameters of failure indicator function χi∗ and χ2∗, failure model θ1∗ also increases. In our proposed actuator failure compensation design, all the unknown parameters will be estimated multiple (m or m−1) times based on χ1,i∗=χ1∗, i=1,⋯,m, χ2,i∗=χ2∗, θ1i∗=χ2∗θ1∗, 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.


## 3.2.5. Error Equations

Equation (21) could be rewritten as
(48)y1ρ1y1ρ2⋮yqρq=Lfρ1h1xLfρ2h2x⋮Lfρqhqx+Axu−wdt+wdt+δ1xδ2x⋮δqx=ym1ρ1+α11ym1ρ1−1−y1ρ1−1+α1ρ1ym1−y1ym2ρ2+α21ym2ρ2−1−y2ρ2−1+α2ρ2ym2−y2⋮ymqρq+αq1ymqρq−1−yqρq−1+αqρqymq−yq+AxI−σtvt−v∗t+δ1xδ2x⋮δqx−δ^1xδ^2x⋮δ^qx.To obtain the output erroret=yt−ymt and the parameter estimation error χ~1,it, χ~2,it, the dynamic formula between θ~1it and θ~dj is reformulated as
(49)e1ρ1+⋯+α1ρ1e1e2ρ2+⋯+α2ρ2e2⋮eqρq+⋯+αqρqeq=AxI−σtv−v∗+E~d1,E~d2,⋯,E~dqT.Ifσ=σ1=diag0,⋯,0
(50)e1ρ1+⋯+α1ρ1e1e2ρ2+⋯+α2ρ2e2⋮eqρq+⋯+αqρqeq=∑i=1mAiχ~1,iv1,i+∑i=1m−1Ai+1χ~2,iv2,i+∑i=1m−1Ai+1θ~1iTϖ1ϕ2,i+E~d1,E~d2,⋯,E~dqT=∑i=1mAiχ~1,iv1,i+∑i=1m−1Ai+1χ~2,iv2,i+∑i=1m−1Ai+1θ~1iTϖ1ϕ2,i+∑j=1pθ~djTϖρ1,j∑j=1pθ~djTϖρ2,j⋮∑j=1pθ~djTϖρq,j=ΔE~1,where h21K21wd=v1,1,⋯,v1,mT, h22K22wd=v2,1,⋯,v2,m−1T, ϖρi,j=LpjLfρi−1hixϖdj.Ifσ=σ2=diag1,⋯,0, equation (49) can be expressed as
(51)e1ρ1+⋯+α1ρ1e1e2ρ2+⋯+α2ρ2e2⋮eqρq+⋯+αqρqeq=∑i=2mAiχ~1,iv1,i+∑i=1m−1Ai+1χ~2,iv2,i+∑i=1m−1Ai+1θ~1iTϖ1ϕ2,i+∑j=1pθ~djTϖρ1,j,∑j=1pθ~djTϖρ2,j,⋯,∑j=1pθ~djTϖρq,jT=ΔE~2.The state error equation can be obtained from equations ((50), (51)).
(52)ż=Azz+BzE~k,where Az=diagAz1,⋯,Azq∈Rρ1+⋯+ρq×ρ1+⋯+ρq, E~kj is the jth component of E~k, k=1,2, BzE~k=Bz1TE~k1,Bz2TE~k2,⋯,BzqTE~kqT∈Rρ1+⋯+ρq.


## 3.2.6. Adaptive Laws

Based on state error equation (52), the adaptive laws could be derived with projection algorithm, parameters χ1,i, i=1,⋯,m, χ2,i and θ1i, i=1,2,⋯,m−1, are estimated as
(53)χ̇1,it=−γ1izTPBZAiv1,ii=2,⋯,m−γ1izTPBZAiv1,i+fχ1,ii=1(54)χ̇2,it=−γ2izTPBzAi+1v2,i,i=1,⋯,m−1(55)θ̇1it=−zTPBzAi+1Γ1iϖ1ϕ2,i,i=1,⋯,m−1where Γ1i=Γ1iT>0, γ1i>0 and γ2i>0 are the adaptive gains, fχ1,i is the projection algorithm. Consequently, according to adaptive laws χ̇1,1=−γ11zTPBzA1v1,1+fχ1,1, we can derive that 0≤χ1,1≤1 and
(56)χ1,1−χ1,1∗fχ1,1≤0.


## 3.2.7. Performance Analysis

(I) For time periodt∈T0,T1, T1=∞, σ=σ1=diag0,⋯,0. The Lyapunov function is defined as
(57)V0=12zTPz+12∑i=1mχ~1,i2γ1i−1+∑i=1m−1χ~2,i2γ2i−1+∑i=1m−1θ~1iTΓ1i−1θ~1i+12∑j=1pθ~djTΓdj−1θ~dj.Combining equations ((37), (50), (51), (52), (53), (54), (55)) one would have the derivative of V0
(58)V̇0=−zTQz≤0,t∈T0,T1.Thus, it can be proved that the designed adaptive controller and its parameter adaptive laws could ensure the desired system performance under the free fault condition, i.e.,ξ, χ~1,it, χ~2,it, θ~1it, and θ~dj are all bounded, and the output error asymptotes to zero as time going on.(II) If actuatoru1 has faults in time period T1,T2 (T2=∞), i.e., σ=σ2=diag1,⋯,0, the Lyapunov function is defined as
(59)V1=12zTPz+12∑i=2mχ~1,i2γ1i−1+∑i=1m−1χ~2,i2γ2i−1+∑i=1m−1θ~1iTΓ1i−1θ~1i+12∑j=1pθ~djTΓdj−1θ~dj.Combining equations ((37), (50), (51), (52), (53), (54), (55)) gives the derivative of V1,
(60)V̇1=−zTQz≤0,t∈T1,T2.The above equation indicates thatξ, χ~1,it, i=2,⋯,m, χ~2,it, θ~1it, i=1,⋯,m−1 and θ~dj, j=1,2,⋯,p are bounded when actuator u1 is failed. In addition, the adaptive projection algorithm χ̇1,1t=−γ1izTPBzAiv1,i+fχ1,i can ensure 0≤χ1,1t≤1. Thus, it can be verified that with increase in t, the closed-loop system is stable and the output asymptotically approaches zero: limt→∞yt−ymt=0. In conclusion, the following theorem can be obtained.Theorem 8.
For the multivariable nonlinear system (1) with potential uncertain actuator fault (3) and mismatched disturbance dt, controller (45) and its parameter adaptive laws can ensure the closed-loop system stability and asymptotic tracking output: limt→∞y−ym=0, ifρ1+ρ2+⋯+ρq=nand the equivalent control matrix Aσx=AxI−σtvt in uncertain fault condition has full rank in the domain U(definition is U⊂Rn⟶V⊂Rq).


## 3.3. Fault Compensation Design of Zero Dynamic System

Ifρ1+ρ2+⋯+ρq<n, there is differential homeomorphism
(61)ξT,ηTT=Tx=TcxT,TzxTT.The nonlinear system with uncertain actuator faultẋt=fx+gxσtu¯t+gx1−σtvt+pxdt, y=hx is converted into
(62)ξ̇1,1=ξ1,2ξ̇1,2=ξ1,3⋯ξ̇1,ρ1=b1ξ,η+A1xI−σv+A1xσu¯+Δd1x,t⋯ξ̇i,ρi=ξi,ρi+1⋯ξ̇i,ρi=biξ,η+AixI−σv+Aixσu¯+Δdix,tξ̇q,1=ξq,2⋯ξ̇q,ρq=bqξ,η+AqxI−σv+Aqxσu¯+Δdqx,t,and zero dynamic subsystem
(63)η̇=ψξ,η+Ψσξ,ηu¯+Ψ¯σξ,ηv+Ψpξ,ηdt,where Tcx=h1x,⋯,Lfρ1−1h1x,⋯,Lfρq−1hqxT, Tzx definitely exists and is nonunique. Ψσξ,η=∂Tzx/∂xgxσ, Ψ¯σξ,η=∂Tzx/∂xgxI−σ is related to the fault mode σ.


### 3.3.1. Stable Zero Dynamic Assumption

To ensure the stability of the closed-loop system and outputyit asymptotic tracking reference signal ymit, the differentials of ρi, i=1,2,⋯,q of ymit 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 ut, dt, and ξ, the zero dynamic subsystem given by
(64)η̇=ψξ,η+Ψσξ,ηu¯+Ψpξ,ηd+Ψ¯σξ,ηvξ,η,χ~1,i,χ~2,i,θ~1icould guarantee input state stability.Remark 10.
Based on Assumption9, if σ∈Σ in any fault case, the state ξ, fault signal u¯, and the designed feedback control signal vξ,η,χ~1,i,χ~2,i,θ~1i are all bounded while dt 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 vξ,η,χ~1,i,χ~2,i,θ~1i is bounded.Combined with Assumption3, the signal vt 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 (Assumption9) and the equivalent control matrix Aσx 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: limt→∞yt−ymt=0 in the case of multiple uncertain actuator faults (3) and unknown disturbances.Proof.
Assuming one of the actuators failed at timeT1, and the system has no fault during time period T1,T2, it can be derived according to the performance analysis in Section 3.2 that the estimated parameters ξ, χ~1,it, χ~2,it, and θ~1it are bounded and the state error z asymptotically approaches to zero as t tends towards infinity. Boundedness of vt 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.


## 3.3.1. Stable Zero Dynamic Assumption

To ensure the stability of the closed-loop system and outputyit asymptotic tracking reference signal ymit, the differentials of ρi, i=1,2,⋯,q of ymit 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 ut, dt, and ξ, the zero dynamic subsystem given by
(64)η̇=ψξ,η+Ψσξ,ηu¯+Ψpξ,ηd+Ψ¯σξ,ηvξ,η,χ~1,i,χ~2,i,θ~1icould guarantee input state stability.Remark 10.
Based on Assumption9, if σ∈Σ in any fault case, the state ξ, fault signal u¯, and the designed feedback control signal vξ,η,χ~1,i,χ~2,i,θ~1i are all bounded while dt 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 vξ,η,χ~1,i,χ~2,i,θ~1i is bounded.Combined with Assumption3, the signal vt 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 (Assumption9) and the equivalent control matrix Aσx 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: limt→∞yt−ymt=0 in the case of multiple uncertain actuator faults (3) and unknown disturbances.Proof.
Assuming one of the actuators failed at timeT1, and the system has no fault during time period T1,T2, it can be derived according to the performance analysis in Section 3.2 that the estimated parameters ξ, χ~1,it, χ~2,it, and θ~1it are bounded and the state error z asymptotically approaches to zero as t tends towards infinity. Boundedness of vt 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.


## 4. 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.


### 4.1. 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]
(65)V̇=Fxcosα+Fzsinαmr+d1,α̇=qr+−Fxsinα+Fzcosαm+d2,θ̇=qr,q̇=MIy+d3,where V is the aircraft speed, α is the attack angle, θ is the angle of pitch, qr is the pitch rate, mr is the mass, Iy is the rotational inertia, M is the pitch moment, and d1, d2, and d3 are turbulence disturbance signals
(66)Fx=qrSCxα,qr,δe1,δe2+T1cosγ1+T2cosγ2−mrgsinθ,Fz=q¯SCzα,qr,δe1,δe2+T1sinγ1+T2sinγ2+mrgcosθ,M=q¯cSCmα,qr,δe1,δe2,q¯=1/2ρV2 is the dynamic pressure, ρ is the air density, S is the density of the wing, c is the average chord, and T1 and T2 are thrusters. Cx, Cz, and Cm are given by
(67)Cx=Cx1α+Cx2α2+Cx3+Cx4k1δe1+k2δe2,Cz=Cz1α+Cz2α2+Cz3+Cz4k1δe1+k2δe2+Cz5q,Cm=Cm1α+Cm2α2+Cm3+Cm4k1δe1+k2δe2+Cm5q,where δe1 and δe2 are the two actuators that require fault compensation.


#### 4.1.1. State Space Representation

The state variablesx1, x2, x3, and x4 are represented by V, α, θ, and q, respectively. The input variables δe1, δe2, T1, and T2 are represented by u1, u2, u3, and u4. Nonlinear system (1) can be expressed as
(68)ẋ1=c1Tφ0x2x12+φ1xcosx2+c2Tφ0x2x12+φ2xsinx2+k1g1xu1+k2g1u2+g31xu3+g41xu4+d1,ẋ2=x4−c1Tφ0x2x1+φ1x1x1sinx2+c2Tφ0x2x1+φ2x1x1cosx2+k1g2xu1+k2g2xu2+g32xu3+g42xu4+d2,ẋ3=x4,ẋ4=ϕx+b1x12u1+b2x22u2+d3,where φ0x2=x2,x22,1T, φ1x=p0sinx3, φ2x=p1x4x12+p0cosx3, g1x=a1x12cosx2+a2x12sinx2, g2x=−a1x1sinx2+a2x1cosx2, g31x=cosr1cosx2+sinr2sinx2, g32x=−cosr1sinx2/x1+sinr1cosx2/x1, g41x=cosr1cosx2+sinr2sinx2, g42x=−cosr2sinx2/x1+sinr2cosx2/x1, and ϕx=x12x2,x12x22,x12,x12x4T. B, k1, k2, c1, c2, p1, a1, a2, r1, r2, b1, and b2 are known constants.


#### 4.1.2. 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 yt=x1,x2,x3T could track the desired control instruction ymt=ym1,ym2,ym3T=3sin0.1t+88,1.2sin0.1t,3sin0.1tT. According to Theorem 11, ρ1=v1=1, ρ2=v2=1, ρ3=v3=2, and ρ1+ρ2+ρ3=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:
(69)diag1,0,0,0,diag0,1,0,0,diag0,0,1,0,diag0,0,0,1,anddiag0,0,0,0.


#### 4.1.3. Numerical Simulation Conditions

The aircraft parameters in reference [36] are as follows: mr=4600kg, g=9.80665m/s2, S=39.02m2, c=1.98m, r1=arctan53/1216, r2=arctan2/45, ρ=0.7377kg/m3, Iy=31027kg·m2, Cx1=0.39, Cx2=2.9099, Cx3=−0.0758, Cx4=0.0961, Cz1=−7.0186, Cz2=4.1109, Cz3=−0.3112, Cz4=−0.2340, Cz5=−0.1023, Cm1=−0.8789, Cm2=−3.8520, Cm3=−0.0108, Cm4=−1.8987, and Cm5=−0.6266. The disturbances are given by d1t=30cos5t+50N, d2t=15sin8t+30cos4tN, and d3t=20sin10t+10N.m.During the simulation verification, the following fault conditions are incorporated: (i) Whent<150s, the system is in the absence of faults: uit=vit, i=1,2,3,4; (ii) When 150s≤t≤300s, actuator u1 is stuck: u1t=0 deg, uit = vit, i=2,3,4; (iii) When 300s≤t≤400s, actuator u1 returns to normal: uit=vit, i=1,2,3,4; (iv) When t≥400s, actuator u4 is stuck: u4t=300 N, uit=vit, i=1,2,3.


### 4.2. Simulation Results

In the simulation, the parameters of the adaptive controlleru^L are α11=α21=α31=α32=0.75, and the other design parameters are as follows:
(1)
Initial state:x0=0.008,0.72,0.008,0.008T
(2)
The base function in disturbance model (8), initial disturbance parameter, and the adaptive gain are ϖd1=ϖd2=ϖd3=1,sin8t,cos4t,sin10t,cos5tT, θ^d10=θ^d20=θ^d20=0,0,0,0,0, and Γd1=Γd2=Γd3=10I5, respectively
(3)
The base function in actuator failure model (3), initial failure parameter, and the adaptive gain are ϖ1t=ϖ4t=1,sintT∈R2, χ110,χ120,χ130,χ140=1,1,1,1, χ210,χ220,χ230=0,0,0, χ310,χ320,χ330=0,0,0, θ110=θ120=θ130=0,0T, θ410=θ420=θ430=0,0T, γ11=γ12=γ13=γ14=1, γ21=γ22=γ23=1,γ31=γ32=γ33=1, and Γ1i=Γ4i=5I2Simulation results are shown in Figures1–3, including a comparison between the actual output of the system and 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.Figure 1
System outputs and reference outputs.Figure 2
Tracking errors.Figure 3
Control inputs.It can be seen from Figures1 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∈0,150s. 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 u1 fails at t=150s and actuator u4 fails at t=400s (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 χ1,i, χ2,i, χ3,i, θ1i, and θ4i of χ1,i∗χ2,i∗, χ3,i∗, θ1i∗, and θ4i∗ for ymt are shown in Figures 4–8, which indicate that all signals in the adaptive control system are bounded, and the desired performance is met.Figure 4
Parameter estimatesχ1,i of χ1,i∗.Figure 5
Parameter estimatesχ2,i of χ2,i∗.Figure 6
Parameter estimatesχ3,i of χ3,i∗.Figure 7
Parameter estimatesθ1i of θ1i∗.Figure 8
Parameter estimatesθ4i of θ4i∗.


## 4.1. 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]
(65)V̇=Fxcosα+Fzsinαmr+d1,α̇=qr+−Fxsinα+Fzcosαm+d2,θ̇=qr,q̇=MIy+d3,where V is the aircraft speed, α is the attack angle, θ is the angle of pitch, qr is the pitch rate, mr is the mass, Iy is the rotational inertia, M is the pitch moment, and d1, d2, and d3 are turbulence disturbance signals
(66)Fx=qrSCxα,qr,δe1,δe2+T1cosγ1+T2cosγ2−mrgsinθ,Fz=q¯SCzα,qr,δe1,δe2+T1sinγ1+T2sinγ2+mrgcosθ,M=q¯cSCmα,qr,δe1,δe2,q¯=1/2ρV2 is the dynamic pressure, ρ is the air density, S is the density of the wing, c is the average chord, and T1 and T2 are thrusters. Cx, Cz, and Cm are given by
(67)Cx=Cx1α+Cx2α2+Cx3+Cx4k1δe1+k2δe2,Cz=Cz1α+Cz2α2+Cz3+Cz4k1δe1+k2δe2+Cz5q,Cm=Cm1α+Cm2α2+Cm3+Cm4k1δe1+k2δe2+Cm5q,where δe1 and δe2 are the two actuators that require fault compensation.


### 4.1.1. State Space Representation

The state variablesx1, x2, x3, and x4 are represented by V, α, θ, and q, respectively. The input variables δe1, δe2, T1, and T2 are represented by u1, u2, u3, and u4. Nonlinear system (1) can be expressed as
(68)ẋ1=c1Tφ0x2x12+φ1xcosx2+c2Tφ0x2x12+φ2xsinx2+k1g1xu1+k2g1u2+g31xu3+g41xu4+d1,ẋ2=x4−c1Tφ0x2x1+φ1x1x1sinx2+c2Tφ0x2x1+φ2x1x1cosx2+k1g2xu1+k2g2xu2+g32xu3+g42xu4+d2,ẋ3=x4,ẋ4=ϕx+b1x12u1+b2x22u2+d3,where φ0x2=x2,x22,1T, φ1x=p0sinx3, φ2x=p1x4x12+p0cosx3, g1x=a1x12cosx2+a2x12sinx2, g2x=−a1x1sinx2+a2x1cosx2, g31x=cosr1cosx2+sinr2sinx2, g32x=−cosr1sinx2/x1+sinr1cosx2/x1, g41x=cosr1cosx2+sinr2sinx2, g42x=−cosr2sinx2/x1+sinr2cosx2/x1, and ϕx=x12x2,x12x22,x12,x12x4T. B, k1, k2, c1, c2, p1, a1, a2, r1, r2, b1, and b2 are known constants.


### 4.1.2. 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 yt=x1,x2,x3T could track the desired control instruction ymt=ym1,ym2,ym3T=3sin0.1t+88,1.2sin0.1t,3sin0.1tT. According to Theorem 11, ρ1=v1=1, ρ2=v2=1, ρ3=v3=2, and ρ1+ρ2+ρ3=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:
(69)diag1,0,0,0,diag0,1,0,0,diag0,0,1,0,diag0,0,0,1,anddiag0,0,0,0.


### 4.1.3. Numerical Simulation Conditions

The aircraft parameters in reference [36] are as follows: mr=4600kg, g=9.80665m/s2, S=39.02m2, c=1.98m, r1=arctan53/1216, r2=arctan2/45, ρ=0.7377kg/m3, Iy=31027kg·m2, Cx1=0.39, Cx2=2.9099, Cx3=−0.0758, Cx4=0.0961, Cz1=−7.0186, Cz2=4.1109, Cz3=−0.3112, Cz4=−0.2340, Cz5=−0.1023, Cm1=−0.8789, Cm2=−3.8520, Cm3=−0.0108, Cm4=−1.8987, and Cm5=−0.6266. The disturbances are given by d1t=30cos5t+50N, d2t=15sin8t+30cos4tN, and d3t=20sin10t+10N.m.During the simulation verification, the following fault conditions are incorporated: (i) Whent<150s, the system is in the absence of faults: uit=vit, i=1,2,3,4; (ii) When 150s≤t≤300s, actuator u1 is stuck: u1t=0 deg, uit = vit, i=2,3,4; (iii) When 300s≤t≤400s, actuator u1 returns to normal: uit=vit, i=1,2,3,4; (iv) When t≥400s, actuator u4 is stuck: u4t=300 N, uit=vit, i=1,2,3.


## 4.1.1. State Space Representation

The state variablesx1, x2, x3, and x4 are represented by V, α, θ, and q, respectively. The input variables δe1, δe2, T1, and T2 are represented by u1, u2, u3, and u4. Nonlinear system (1) can be expressed as
(68)ẋ1=c1Tφ0x2x12+φ1xcosx2+c2Tφ0x2x12+φ2xsinx2+k1g1xu1+k2g1u2+g31xu3+g41xu4+d1,ẋ2=x4−c1Tφ0x2x1+φ1x1x1sinx2+c2Tφ0x2x1+φ2x1x1cosx2+k1g2xu1+k2g2xu2+g32xu3+g42xu4+d2,ẋ3=x4,ẋ4=ϕx+b1x12u1+b2x22u2+d3,where φ0x2=x2,x22,1T, φ1x=p0sinx3, φ2x=p1x4x12+p0cosx3, g1x=a1x12cosx2+a2x12sinx2, g2x=−a1x1sinx2+a2x1cosx2, g31x=cosr1cosx2+sinr2sinx2, g32x=−cosr1sinx2/x1+sinr1cosx2/x1, g41x=cosr1cosx2+sinr2sinx2, g42x=−cosr2sinx2/x1+sinr2cosx2/x1, and ϕx=x12x2,x12x22,x12,x12x4T. B, k1, k2, c1, c2, p1, a1, a2, r1, r2, b1, and b2 are known constants.


## 4.1.2. 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 yt=x1,x2,x3T could track the desired control instruction ymt=ym1,ym2,ym3T=3sin0.1t+88,1.2sin0.1t,3sin0.1tT. According to Theorem 11, ρ1=v1=1, ρ2=v2=1, ρ3=v3=2, and ρ1+ρ2+ρ3=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:
(69)diag1,0,0,0,diag0,1,0,0,diag0,0,1,0,diag0,0,0,1,anddiag0,0,0,0.


## 4.1.3. Numerical Simulation Conditions

The aircraft parameters in reference [36] are as follows: mr=4600kg, g=9.80665m/s2, S=39.02m2, c=1.98m, r1=arctan53/1216, r2=arctan2/45, ρ=0.7377kg/m3, Iy=31027kg·m2, Cx1=0.39, Cx2=2.9099, Cx3=−0.0758, Cx4=0.0961, Cz1=−7.0186, Cz2=4.1109, Cz3=−0.3112, Cz4=−0.2340, Cz5=−0.1023, Cm1=−0.8789, Cm2=−3.8520, Cm3=−0.0108, Cm4=−1.8987, and Cm5=−0.6266. The disturbances are given by d1t=30cos5t+50N, d2t=15sin8t+30cos4tN, and d3t=20sin10t+10N.m.During the simulation verification, the following fault conditions are incorporated: (i) Whent<150s, the system is in the absence of faults: uit=vit, i=1,2,3,4; (ii) When 150s≤t≤300s, actuator u1 is stuck: u1t=0 deg, uit = vit, i=2,3,4; (iii) When 300s≤t≤400s, actuator u1 returns to normal: uit=vit, i=1,2,3,4; (iv) When t≥400s, actuator u4 is stuck: u4t=300 N, uit=vit, i=1,2,3.


## 4.2. Simulation Results

In the simulation, the parameters of the adaptive controlleru^L are α11=α21=α31=α32=0.75, and the other design parameters are as follows:
(1)
Initial state:x0=0.008,0.72,0.008,0.008T
(2)
The base function in disturbance model (8), initial disturbance parameter, and the adaptive gain are ϖd1=ϖd2=ϖd3=1,sin8t,cos4t,sin10t,cos5tT, θ^d10=θ^d20=θ^d20=0,0,0,0,0, and Γd1=Γd2=Γd3=10I5, respectively
(3)
The base function in actuator failure model (3), initial failure parameter, and the adaptive gain are ϖ1t=ϖ4t=1,sintT∈R2, χ110,χ120,χ130,χ140=1,1,1,1, χ210,χ220,χ230=0,0,0, χ310,χ320,χ330=0,0,0, θ110=θ120=θ130=0,0T, θ410=θ420=θ430=0,0T, γ11=γ12=γ13=γ14=1, γ21=γ22=γ23=1,γ31=γ32=γ33=1, and Γ1i=Γ4i=5I2Simulation results are shown in Figures1–3, including a comparison between the actual output of the system and 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.Figure 1
System outputs and reference outputs.Figure 2
Tracking errors.Figure 3
Control inputs.It can be seen from Figures1 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∈0,150s. 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 u1 fails at t=150s and actuator u4 fails at t=400s (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 χ1,i, χ2,i, χ3,i, θ1i, and θ4i of χ1,i∗χ2,i∗, χ3,i∗, θ1i∗, and θ4i∗ for ymt are shown in Figures 4–8, which indicate that all signals in the adaptive control system are bounded, and the desired performance is met.Figure 4
Parameter estimatesχ1,i of χ1,i∗.Figure 5
Parameter estimatesχ2,i of χ2,i∗.Figure 6
Parameter estimatesχ3,i of χ3,i∗.Figure 7
Parameter estimatesθ1i of θ1i∗.Figure 8
Parameter estimatesθ4i of θ4i∗.


## 5. Conclusions

For multivariable nonlinear systems with multiple uncertain actuator faults and mismatched input disturbances, a control method of adaptive fault and disturbance compensation 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 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.



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*Source: 4531302-2020-03-25.xml*