# Multiplicative Fault Estimation-Based Adaptive Sliding Mode Fault-Tolerant Control Design for Nonlinear Systems

Authors: Ali Ben Brahim; Slim Dhahri; Fayçal Ben Hmida; Anis Sellami
Journal: Complexity
 (2018)

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## Abstract

This article deals with the sliding mode fault-tolerant control (FTC) problem for a nonlinear system described under Takagi-Sugeno (T-S) fuzzy representation. In particular, the nonlinear system is corrupted with multiplicative actuator faults, process faults, and uncertainties. We start by constructing the separated FTC design to ensure robust stability of the closed-loop nonlinear system. First, we propose to conceive an adaptive observer in order to estimate nonlinear system states, as well as robust multiplicative fault estimation. The novelty of the proposed approach is that the observer gains are obtained by solving the multiobjective linear matrix inequality (LMI) optimization problem. Second, an adaptive sliding mode controller is suggested to offer a solution to stabilize the closed-loop system despite the occurrence of real fault effects. Compared with the separated FTC, this paper provides an integrated sliding mode FTC in order to achieve an optimal robustness interaction between observer and controller models. Thus, in a single-step LMI formulation, sufficient conditions are developed with multiobjective optimization performances to guarantee the stability of the closed-loop system. At last, nonlinear simulation results are given to illustrate the effectiveness of the proposed FTC to treat multiplicative faults.


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## Body

## 1. Introduction

In the last few years, there has been a growing interest in fault-tolerant control (FTC) design based on a fault estimation (FE) technique (location, occurrence time, and magnitude). Active FTC takes a primordial place in modern control application, system reliability, and supervision of industrial technology.The main challenge of active FTC is to conceive a robust controller such that the closed-loop system is stable with acceptable performances even with the presence simultaneously of faults and uncertainties. In the literature, several approaches have been proposed to explore this powerful issue (see for instance [1–6] and the references herein). In practical applications, most of the systems are complex and usually having hard nonlinearities, so it is significant to study FE and FTC for nonlinear systems. Since their excellent ability to describe a nonlinear system, very interesting approaches have represented nonlinear systems under the T-S fuzzy form [7]. Actually, in the presence of system uncertainties, several attempts have been oriented to the fault diagnosis and FTC of nonlinear systems (see for instance [8–13] and the references herein.Popular FE approaches have been developed in a precise and effective way for nonlinear systems, where fault is modelled as additive changes appearing in actuators or sensors [14–18]. The major drawback of the preceding approaches resides primarily on treating actuator and sensor faults with additive terms. However, in practical engineering, it is often the case when some actuator faults and component faults occur in a multiplicative form. Thus, multiplicative faults are mixed with the inputs and outputs of the system. In this way, estimation of the characteristic and magnitude of unknown multiplicative faults has been a growing interest in modern control theory in recent years. It is practically important to decouple their eventual parameter or structure effects in the system or in the process model subject to ameliorate FTC design for a large class of nonlinear systems. Special attentions have already been made in the application of observer design to achieve multiplicative FE for linear and nonlinear systems [19, 20]. In [21–24], robust observers have been used to estimate unknown fault for linear systems. The developed schemes are based on treating multiplicative faults with additive terms. Nevertheless, this approach may not be practical in all situations. For this purpose, it is not an easy task; real effect multiplicative fault detection, location, and estimation especially for nonlinear systems have been a significant research activity in the past decade. Zhang et al. [25] considers the problem of fault diagnosis for a class of nonlinear systems with unstructured modeling uncertainty. The proposed approach addresses the detection and isolation of nonlinear fault function that are modeled as measurable signals. More recently, in [26], a robust adaptive observer FE approach is discussed in order to extract the real component fault effects for Lipschitz nonlinear systems subject to unknown disturbances.


### 1.1. Contributions

Regarding the fact that multiplicative faults have not yet been fully tackled, an FE-based adaptive sliding mode FTC scheme for a class of uncertain nonlinear systems, approximated by the T-S representation, is of great interest in this paper. Thus, sliding mode control has been widely studied and employed in industrial applications based on its computational simplicity and in particular strong robustness against uncertainties or disturbances. The main contributions of the present paper are divided as follows:(1)
In the first scheme, we propose a separated sliding mode FTC for the closed-loop nonlinear system subject to both multiplicative faults and uncertainties. More precisely, it should be pointing that we consider multiplicative faults a partial loss of actuator effectiveness and parameter changes in the nonlinear system state matrix.
(2)
In the second scheme, this paper provides an integrated sliding mode FTC in order to achieve an optimal robustness interaction between observer and controller models. Thus, in a single-step LMI formulation, sufficient conditions are developed withH∞ optimization performances to guarantee the stability of the closed-loop system. In particular, the fault nonlinear function satisfies a Lipschitz condition. In this study, we use a multiobjective LMI optimization approach in which the Lipschitz constant and uncertainty attenuation level are maximized simultaneously.The remainder of this paper is organized as follows: Section2 gives the description of the nonlinear system. In Section 3, we describe the proposed T-S adaptive observer design. Section 4 presents the sliding mode controller structure. Sections 5 and 6 propose, respectively, the design of separated and integrated sliding mode FTC schemes to stabilize the closed-loop system. The simulation example is given in Section 7 based on nonlinear simulation illustrating the effectiveness of the proposed schemes. Finally, Section 8 presents some concluding remarks.


## 1.1. Contributions

Regarding the fact that multiplicative faults have not yet been fully tackled, an FE-based adaptive sliding mode FTC scheme for a class of uncertain nonlinear systems, approximated by the T-S representation, is of great interest in this paper. Thus, sliding mode control has been widely studied and employed in industrial applications based on its computational simplicity and in particular strong robustness against uncertainties or disturbances. The main contributions of the present paper are divided as follows:(1)
In the first scheme, we propose a separated sliding mode FTC for the closed-loop nonlinear system subject to both multiplicative faults and uncertainties. More precisely, it should be pointing that we consider multiplicative faults a partial loss of actuator effectiveness and parameter changes in the nonlinear system state matrix.
(2)
In the second scheme, this paper provides an integrated sliding mode FTC in order to achieve an optimal robustness interaction between observer and controller models. Thus, in a single-step LMI formulation, sufficient conditions are developed withH∞ optimization performances to guarantee the stability of the closed-loop system. In particular, the fault nonlinear function satisfies a Lipschitz condition. In this study, we use a multiobjective LMI optimization approach in which the Lipschitz constant and uncertainty attenuation level are maximized simultaneously.The remainder of this paper is organized as follows: Section2 gives the description of the nonlinear system. In Section 3, we describe the proposed T-S adaptive observer design. Section 4 presents the sliding mode controller structure. Sections 5 and 6 propose, respectively, the design of separated and integrated sliding mode FTC schemes to stabilize the closed-loop system. The simulation example is given in Section 7 based on nonlinear simulation illustrating the effectiveness of the proposed schemes. Finally, Section 8 presents some concluding remarks.


## 2. Problem Formulation

Consider an uncertain nonlinear system governed by the following equations:(1)ẋt=φ1xt,ut,ξx,t,fx,u,t,yt=φ2xt,where xt∈Rn is the state vector, ut∈Rm represents the control inputs, yt∈Rp denotes the measurement output vector, and ξx,t∈Rl stands for the uncertainty vector. In the present paper, fx,u,t∈Rq represents the component and/or actuator gain fault which is described as
(2)fx,u,t=∑j=1qθjthjx,u,t,where θjt is a vector of unknown function reflecting the magnitude of the time-varying or constant multiplicative faults. hjx,u,t represents the functional structure of the jth multiplicative faults and usually mixes with system states and/or inputs. Before starting the main results of this paper, we will make the following assumptions.Assumption 1.
The fault vectorθjt is assumed to be unknown but bounded as θjt≤θmax=ρ,∀j=1,…,q, where θmax∈Rq is a known constant vector and ρ is a known positive constant.Assumption 2.
It is assumed that the nonlinear system states and inputs are all bounded before and after the occurrence of a fault, and fault nonlinear function structurehjx,u,t satisfies a Lipschitz condition locally on a set M⊂Rn in which
(3)hjx1,u,t−hjx2,u,t≤γjx1−x2,∀x1,x2∈M,where γj>0 is called a Lipschitz constant and j=1,…,q.


### 2.1. Design Objective

This paper features a robust estimation of real effect factorθjt,∀j=1,…,q, for the uncertain nonlinear system (1) subject both to multiplicative fault fx,u,t given by (2) and to uncertainties ξx,t. It was the main purpose of the paper to solve two problems by (i) estimating multiplicative fault magnitude using a robust adaptive observer and (ii) stabilizing the closed-loop nonlinear system, after the occurrence of multiplicative fault fx,u,t, using a robust adaptive sliding mode controller. To treat this powerful issue, this article introduces two different approaches: separated and integrated FE-based adaptive sliding mode FTC design.We will make the following definition, notation, and lemma in obtaining the main results.Definition.
For an arbitrary matrixX∈Rn×m, if X+∈Rm×n verifies X+X=Im, then X+=XTX−1XT is said to be the left_inverse of X.Notation.
The notation (∗) corresponds to the symmetry matrix block, HeX signifies X+XT, and . stands for the standard norm symbol.Lemma 1.
For matricesX and Y with appropriate dimensions, the following condition holds:
(4)XTY+YTX≤ε−1XTX+εYTY,where ε is the positive scalar.


## 2.1. Design Objective

This paper features a robust estimation of real effect factorθjt,∀j=1,…,q, for the uncertain nonlinear system (1) subject both to multiplicative fault fx,u,t given by (2) and to uncertainties ξx,t. It was the main purpose of the paper to solve two problems by (i) estimating multiplicative fault magnitude using a robust adaptive observer and (ii) stabilizing the closed-loop nonlinear system, after the occurrence of multiplicative fault fx,u,t, using a robust adaptive sliding mode controller. To treat this powerful issue, this article introduces two different approaches: separated and integrated FE-based adaptive sliding mode FTC design.We will make the following definition, notation, and lemma in obtaining the main results.Definition.
For an arbitrary matrixX∈Rn×m, if X+∈Rm×n verifies X+X=Im, then X+=XTX−1XT is said to be the left_inverse of X.Notation.
The notation (∗) corresponds to the symmetry matrix block, HeX signifies X+XT, and . stands for the standard norm symbol.Lemma 1.
For matricesX and Y with appropriate dimensions, the following condition holds:
(4)XTY+YTX≤ε−1XTX+εYTY,where ε is the positive scalar.


## 3. T-S Adaptive Observer Design

Referring to an interpolation mechanism with the convex sum properties [7], the system (1) can be approximated by T-S fuzzy representation with multiplicative faults as follows:
(5)ẋt=∑i=1kμiςtAixt+Biut+Diξx,t+Mi∑j=1qθjthjx,u,t,yt=∑i=1kμiςtCixt,where Ai, Bi, Di, Mi, and Ci are real known matrices with appropriate dimensions. We assume that the pair Ai,Bi is controllable and the pair Ai,Ci is observable. Let µiζt be the normalized fuzzy membership functions which satisfy the properties of the sum convex.
(6)∑i=1kµiζt=1,0≤µiζt≤1,∀i∈1,…,k.Through the approximation of the nonlinear system (1) by augmented T-S fuzzy representation (5), we construct a multiplicative fault estimation adaptive observer as
(7)x^̇t=∑i=1kμiςtAix^t+Biut+Gl,ieyt+Mi∑j=1qθ^jthjx^,u,t,(8)y^t=∑i=1kμiςtCix^t,(9)θ^jt=σjhjTx^,u,tΓeyt,j=1,…,q,where σj are positive scalars, x^t is the observer state, y^t represents the observer output, θ^jt denotes the estimated fault magnitude, and eyt=yt−y^t is the output estimation error. Gl,i are appropriate gain matrices, which can be obtained using LMIs as discussed later. Γ is a design matrix representing the learning rate.From now on, we assume that the state and fault estimation errors are defined, respectively, aset=xt−x^t and eθjt=θjt−θ^jt. It remains to deduce that
(10)ėt=∑i=1kμiςtAi−Gl,iCiet+Diξx,t+Mi∑j=1qθjthjx,u,t−θ^jthjx^,u,t,(11)ėθjt=∑i=1kμiςt−σjhjTx^,u,tΓieyt.The objective is to derive the gains of the robust adaptive observer ((7), (8), and (9)) in order to estimate multiplicative fault magnitudes.


## 4. Sliding Mode Controller Design

### 4.1. Adaptive Sliding Mode Controller Structure

The proposed sliding mode controller with adaptive law is assigned to provide a corrective action in order to compensate multiplicative fault effects and stabilize the nonlinear system described by T-S fuzzy representation. Before starting FTC design, we assume the following:Assumption 3.
rankCiBi=rankBi,∀i∈1,…,k. As the first step, one can define the sliding surface S, when the sliding motion will take place on it, as
(12)S=yt∈ℝp:Sct=0.
Sct∈ℝm is a linear switching function, based on the output feedback information, described as
(13)Sct=∑i=1kµiζtNc,iyt,where Nc,i=CiBi+−hIp−CiBiCiBi+ with an arbitrary matrix h∈ℝm×p. As mentioned above that Ai,Bi is controllable, a nonlinear control input is given by
(14)ut=ult+unt,where ult designs the linear part which is defined as
(15)ult=∑i=1k∑j=1kμiμjςt−Kjx^t−Fa,if^x,u,t,where −Fa,if^x,u,t is designed to compensate multiplicative fault influence. It is assumed that Kj∈ℝm×n and Fa,i=Bi+Mi.As may be seen below, the nonlinear partunt, capable of inducing the sliding motion on the sliding surface S, is proposed with adaptive law as
(16)unt=−ηctSctSct,ifSct=0,0,otherwise,where ηct=ρ^c+ϱc, where ϱc>0 is a small constant and ρ^c is used to determinate ηct such that we will make the following adaptive term as
(17)ρ^̇c=ϵcSct,ρ^c0≥0,where ϵc is the positive gain.


### 4.2. Reaching Condition

As mentioned earlier, it must be proven, using the nonlinear part structure ofunt, that the system will be forced to reach and slide onto the corresponding sliding mode surface S in a finite time. In this way, we design a Lyapunov function as
(18)Vct=12ScTtSct+12ϵcρ~c2,where ρ~c=ρc−ρ^c is the estimated error of ρc.The derivative of (18) with respect to time gives
(19)V̇ct=∑i=1k∑j=1kμiμjζtNc,iCiAi−Kjxt+ρc−ηctSct−ρ~cSct=∑i=1k∑j=1kμiμjζtNc,iCiAi−Kjxt−ϱc−εcSct.Define the subset system as(20)Ωc=xxt≤κc.The reachability condition, which guarantees to force the system to attain the sliding surfaceS, is satisfied if the scalar ϱc is chosen to satisfy ϱc>Nc,iCiAi−Kjκc such that
(21)ScTtSct≤−εcSct.Furthermore, the proposed sliding mode controller with adaptive law ensures the existence of an ideal sliding motion in finite time; that is,Sct=Ṡct=0,∀t≥tc.


## 4.1. Adaptive Sliding Mode Controller Structure

The proposed sliding mode controller with adaptive law is assigned to provide a corrective action in order to compensate multiplicative fault effects and stabilize the nonlinear system described by T-S fuzzy representation. Before starting FTC design, we assume the following:Assumption 3.
rankCiBi=rankBi,∀i∈1,…,k. As the first step, one can define the sliding surface S, when the sliding motion will take place on it, as
(12)S=yt∈ℝp:Sct=0.
Sct∈ℝm is a linear switching function, based on the output feedback information, described as
(13)Sct=∑i=1kµiζtNc,iyt,where Nc,i=CiBi+−hIp−CiBiCiBi+ with an arbitrary matrix h∈ℝm×p. As mentioned above that Ai,Bi is controllable, a nonlinear control input is given by
(14)ut=ult+unt,where ult designs the linear part which is defined as
(15)ult=∑i=1k∑j=1kμiμjςt−Kjx^t−Fa,if^x,u,t,where −Fa,if^x,u,t is designed to compensate multiplicative fault influence. It is assumed that Kj∈ℝm×n and Fa,i=Bi+Mi.As may be seen below, the nonlinear partunt, capable of inducing the sliding motion on the sliding surface S, is proposed with adaptive law as
(16)unt=−ηctSctSct,ifSct=0,0,otherwise,where ηct=ρ^c+ϱc, where ϱc>0 is a small constant and ρ^c is used to determinate ηct such that we will make the following adaptive term as
(17)ρ^̇c=ϵcSct,ρ^c0≥0,where ϵc is the positive gain.


## 4.2. Reaching Condition

As mentioned earlier, it must be proven, using the nonlinear part structure ofunt, that the system will be forced to reach and slide onto the corresponding sliding mode surface S in a finite time. In this way, we design a Lyapunov function as
(18)Vct=12ScTtSct+12ϵcρ~c2,where ρ~c=ρc−ρ^c is the estimated error of ρc.The derivative of (18) with respect to time gives
(19)V̇ct=∑i=1k∑j=1kμiμjζtNc,iCiAi−Kjxt+ρc−ηctSct−ρ~cSct=∑i=1k∑j=1kμiμjζtNc,iCiAi−Kjxt−ϱc−εcSct.Define the subset system as(20)Ωc=xxt≤κc.The reachability condition, which guarantees to force the system to attain the sliding surfaceS, is satisfied if the scalar ϱc is chosen to satisfy ϱc>Nc,iCiAi−Kjκc such that
(21)ScTtSct≤−εcSct.Furthermore, the proposed sliding mode controller with adaptive law ensures the existence of an ideal sliding motion in finite time; that is,Sct=Ṡct=0,∀t≥tc.


## 5. FE-Based Fault-Tolerant Control Design: A Separated Approach

For several years, great effort has been devoted to study the FE-based FTC problem in a precise and effective way. The focus of previous studies has been on the division of this issue into separate steps:Step 1.
Conceive an observer to estimate faults and state variables.Step 2.
Conceive a controller to stabilize the closed-loop systems.Figure1 illustrates the separated FE-based FTC design for uncertain nonlinear systems subject to process faults and multiplicative actuator faults.Figure 1
FE-based fault-tolerant control: a separated approach.Now, we propose to design the separated FE-based FTC problem in order to compute adaptive observer gainsGl,i and controller gains Kj such that a robust stability of the closed-loop nonlinear system, described by the T-S form, is achieved despite the presence of multiplicative faults and uncertainties.


### 5.1. LMI Optimization-Based Observer Stability

Theorem1 establishes the sufficient conditions for the stability of the observer errors ((10) and (11)) with prescribed ℋ∞ performances by using Lyapunov stability and LMI technique.Theorem 1.
The state estimation error is robustly stable with simultaneously maximized admissible Lipschitz constantγj>0 and minimized gain ς>0 for the system uncertainties ξx,t, if there exist constants, 0≤λ≤1, ε>0, and αj>0, and matrices Pe=PeT>0,Wi, such that the following multiobjective optimization problem has a solution:
(22)minλε+∑j=1qαj+1−λς,subject to the following LMI
(23)Ξ¯i+CeTCePeDiPeMiIn∗−ςIl00∗∗−εIq0∗∗∗−∑j=1qαj<0,where Ξ¯i=AiTPe+PeAi−WiCi−CiTWiT. One can prove that the observer gain can be obtained from Gl,i=Pe−1Wi. In addition, the T-S adaptive observer ((7), (8), and (9)) ensures that the estimated x^t and θ^jt converge to the nonlinear system state xt and the multiplicative fault magnitude θjt.Proof.
One can start by investigating the following Lyapunov function:(24)Vet=eTtPeet+∑j=1qσj−1eθjTteθjt>0,where Pe∈ℝn×n is the design Lyapunov matrix. The time derivative of Vet is handled as
(25)V̇et=ėTtPeet+eTtPeėt+∑j=1qσj−1ėθjTteθjt+eθjTtėθjt.
Based on state and fault estimation errors ((10) and (11)), V̇et can be written as
(26)V̇et=∑i=1kμiζteTtAl,iTPe+PeAl,iet+∑j=1q2eTtPeMiθjthjx,u,t−θ^jthjx^,u,t+∑j=1q2σj−1eθjtθ^̇jt+2eTtPeDiξx,t,where Al,i=Ai−Gl,iCi. From this, one can conclude that
(27)V̇et=∑i=1kμiζteTtAl,iTPe+PeAl,iet+∑j=1q2eTtPeMiθjthjx,u,t−θjthjx^,u,t+2eTtPeDiξx,t.
Referring to Lemma1 and according to the Lipschitz condition (3), one can further derive
(28)2eTtPeMiθjthjx,u,t−θjthjx^,u,t≤1εeTtPeMiMiTPeet+εhjx,u,t−hjx^,u,tTθjTtθjthjx,u,t−hjx^,u,t≤1εeTtPeMiMiTPeet+εθjt2hjx,u,t−hjx^,u,t2≤1εeTtPeMiMiTPeet+ερ2γj2et2=1εeTtPeMiMiTPeet+εγ~j2et2.where γ~j=ργj. Now, according to (28), one can derive Vet as
(29)V̇et≤∑i=1kμiζteTtAl,iTPe+PeAl,iet+1εeTtPeMiMiTPeet+∑j=1qeTtεγ~j2et+2eTtPeDiξx,t.
To attain the robustness of the proposed multiplicative fault estimation adaptive observer ((7), (8), and (9)) against system uncertainties ξx,t, we investigate the controlled estimation error rt as
(30)rt=Ceet.
Consider the following worst-case performance measure:(31)H∞=supξ2≠0rt22ξx,t22≤ς.
One can now proceed with the presence of the following variableJet as
(32)Jet=V̇et+rTtrt−ςξTx,tξx,t.
Obviously, one can write the above expression (32) by
(33)Jet≤∑i=1kμiζteTtAl,iTPe+PeAl,i+1εPeMiMiTPe+∑j=1qεγ~j2+CeTCeet+2eTtPeDiξx,t−ςξTx,tξx,t.
Let one define the following new variable as(34)αj=1εγ~j2.
From now on, one can get(35)γ~j=1εαj.
The stability of the T-S fuzzy system (5) is achieved for any fault Lipschitz nonlinear function with Lipschitz constant less than or equal to an unknown maximized constant γ~j,∀j=1,…,q. Maximization of γ~j and minimization of ς can be accomplished by simultaneous minimization of αj, ε, and ς, ∀j=1,…,q. In this way, one can obtain a multiobjective optimization design.
As is clear from (34), one has
(36)Jet≤∑i=1kμiζteTtAl,iTPe+PeAl,i+1εPeMiMiTPe+∑j=1qαj−1+CeTCeet+2eTtPeDiξx,t−ςξTx,tξx,t=ΦTtΔΦt,where
(37)Φt=etξx,t,Δ=∑i=1kμiζtΞij+CeTCePeDiDiTPe−ς,Ξij=Al,iTPe+PeAl,i+1εPeMiMiTPe+∑j=1qαj−1.
One can conclude that if∆<0, it implies that Jet<0.
According to the Schur complement, the previous inequality is satisfied if and only if we checked for the presence of this relation:(38)∑i=1kμiζtΩij<0,where Ωij are the same as those in (23).
In this way, the state estimation error is asymptotically stable with the attenuation levelς as follows:
(39)rt22≤ςξx,t22.
It is obvious thatet=xt−x^t→0. Due to the fault, nonlinear function hjx,u,t satisfies the Lipschitz condition; the T-S multiplicative fault estimation adaptive observer ((7), (8), and (9)) ensures that eθjt→0. From this, one can deduce that, according to (9), the estimation of multiplicative fault magnitude θjt, for the uncertain nonlinear system (1) described by T-S fuzzy structure (5), can be achieved.
This completes the proof.


### 5.2. LMI Optimization-Based Closed-Loop Stability

Once the sliding mode is obtained, we consider to analyze the stability of the closed-loop T-S fuzzy system. Let the equivalent controlueqt, such that Ṡct is equal to zero, be
(40)ueqt=∑i=1kμiζt−Nc,iCiAixt+Diξx,t+ult.The dynamic of the closed-loop system with the equivalent control law (40) takes the form
(41)ẋt=∑i=1k∑j=1kμiμjζtΘiAi−BiKjxt+B¯i,jϕt,(42)yt=∑i=1kμiζtCixt,where B¯i,j=BiKjMiΘiDi, Θi=In−BiNc,iCi, and ϕt=eTtefTx,u,tξTx,tT, where efx,u,t=fx,u,t−f^x,u,t.The objective now is to develop a sufficient condition to achieve the stability of the closed-loop T-S fuzzy system ((41) and (42)) on the sliding surface S despite the occurrence of multiplicative faults and the presence of uncertainties.Theorem 2.
The closed-loop T-S fuzzy system ((41) and (42)) is robustly stable with the H∞ attenuation level ςs>0, if there exist the matrices P¯x=P¯xT>0, Qj, and Y=YT, such that
(43)minςs
satisfying the following LMI constraints:(44)ΔSepaFTC=ϒi,jBiQjMiΘiDiP¯xCiT∗−2Y+ςsIn000∗∗−ςsIq00∗∗∗−ςsIl0∗∗∗∗−Ip<0,where ϒi,j=P¯xAiTΘiT+ΘiAiP¯x−BiQj−QjTBiT, Y=ςsP¯x, and Kj=QjP¯x−1.Proof.
Consider the following Lyapunov function for the closed-loop system as(45)Vxt=xTtPxxt,where Px∈ℝn×n is the symmetric positive definite matrix. The time derivative of Vxt is handled as
(46)V̇xt=∑i−1k∑j=1kμiμjζtxTtΘiAi−BiKjTPx+PxΘiAi−BiKjxt+2xTtPxB¯i,jϕt.
To achieve the robustness withH∞ performance of the closed-loop T-S fuzzy system ((41) and (42)) to ϕt, the following inequality must then hold:
(47)Jxt=V̇xt+yTtyt−ςsϕTtϕt<0.
Insertion of (46) in (47) yields
(48)Jxt=∑i=1k∑j=1kμiμjζtxTtΘiAi−BiKjTPx+PxΘiAi−BiKj+CiTCixt+2xTtPxB¯i,jϕt−ςsϕTtϕt.
Consequently (in the matrix form), it remains to prove thatJxt<0, if
(49)ψi,jPxB¯i,jB¯i,jTPx−ςsI<0,where ψi,j=ΘiAi−BiKjTPx+PxΘiAi−BiKj+CiTCi. Using the Schur complement, the relation (49) can be reformulated as
(50)∏i,j=∑i,jPxBiKjPxMiPxΘiDiCiT∗−ςsIn000∗∗−ςsIq00∗∗∗−ςsIl0∗∗∗∗−Ip<0,where ∑i,j=ΘiTAiTPx+PxΘiAi−KjTBiTPx−PxBiKj.
Inequality (50) contains several nonlinear terms. One can design in the next step to formulate this as an LMI problem. To effect the necessary change of variables, one will define the following matrix X with the special diagonal structure as X=diagPx−1,Px−1,Iq,Il,Ip. Then, X×Πi,j×XT<0 is true, and it is obvious that
(51)ϒi,jBiQjMiΘiDiP¯xCiT∗−ςsP¯xP¯x000∗∗−ςsIq00∗∗∗−ςsIl0∗∗∗∗−Ip<0,where ϒi,j=P¯xAiTΘiT+ΘiAiP¯x−BiQj−QiTBiT, P¯x=Px−1, and Qj=KjPx−1. According to Lemma 1, it is evident to check the presence of the following relation
(52)P¯x+P¯x≤P¯xP¯x+In.
Obviously, (52) is true for ςc>0 as
(53)−ςsP¯xP¯x≤−2ςsP¯x+ςsIn.
After some manipulations, one can get(54)∑i=1k∑j=1kμiμjζtΔSepaFTC<0,where ∆SepaFTC has the same structure with (44). Clearly, a stability proof of the closed-loop T-S fuzzy system ((41) and (42)) is required with respect to the H∞ performance level ςs.
This completes the proof.


## 5.1. LMI Optimization-Based Observer Stability

Theorem1 establishes the sufficient conditions for the stability of the observer errors ((10) and (11)) with prescribed ℋ∞ performances by using Lyapunov stability and LMI technique.Theorem 1.
The state estimation error is robustly stable with simultaneously maximized admissible Lipschitz constantγj>0 and minimized gain ς>0 for the system uncertainties ξx,t, if there exist constants, 0≤λ≤1, ε>0, and αj>0, and matrices Pe=PeT>0,Wi, such that the following multiobjective optimization problem has a solution:
(22)minλε+∑j=1qαj+1−λς,subject to the following LMI
(23)Ξ¯i+CeTCePeDiPeMiIn∗−ςIl00∗∗−εIq0∗∗∗−∑j=1qαj<0,where Ξ¯i=AiTPe+PeAi−WiCi−CiTWiT. One can prove that the observer gain can be obtained from Gl,i=Pe−1Wi. In addition, the T-S adaptive observer ((7), (8), and (9)) ensures that the estimated x^t and θ^jt converge to the nonlinear system state xt and the multiplicative fault magnitude θjt.Proof.
One can start by investigating the following Lyapunov function:(24)Vet=eTtPeet+∑j=1qσj−1eθjTteθjt>0,where Pe∈ℝn×n is the design Lyapunov matrix. The time derivative of Vet is handled as
(25)V̇et=ėTtPeet+eTtPeėt+∑j=1qσj−1ėθjTteθjt+eθjTtėθjt.
Based on state and fault estimation errors ((10) and (11)), V̇et can be written as
(26)V̇et=∑i=1kμiζteTtAl,iTPe+PeAl,iet+∑j=1q2eTtPeMiθjthjx,u,t−θ^jthjx^,u,t+∑j=1q2σj−1eθjtθ^̇jt+2eTtPeDiξx,t,where Al,i=Ai−Gl,iCi. From this, one can conclude that
(27)V̇et=∑i=1kμiζteTtAl,iTPe+PeAl,iet+∑j=1q2eTtPeMiθjthjx,u,t−θjthjx^,u,t+2eTtPeDiξx,t.
Referring to Lemma1 and according to the Lipschitz condition (3), one can further derive
(28)2eTtPeMiθjthjx,u,t−θjthjx^,u,t≤1εeTtPeMiMiTPeet+εhjx,u,t−hjx^,u,tTθjTtθjthjx,u,t−hjx^,u,t≤1εeTtPeMiMiTPeet+εθjt2hjx,u,t−hjx^,u,t2≤1εeTtPeMiMiTPeet+ερ2γj2et2=1εeTtPeMiMiTPeet+εγ~j2et2.where γ~j=ργj. Now, according to (28), one can derive Vet as
(29)V̇et≤∑i=1kμiζteTtAl,iTPe+PeAl,iet+1εeTtPeMiMiTPeet+∑j=1qeTtεγ~j2et+2eTtPeDiξx,t.
To attain the robustness of the proposed multiplicative fault estimation adaptive observer ((7), (8), and (9)) against system uncertainties ξx,t, we investigate the controlled estimation error rt as
(30)rt=Ceet.
Consider the following worst-case performance measure:(31)H∞=supξ2≠0rt22ξx,t22≤ς.
One can now proceed with the presence of the following variableJet as
(32)Jet=V̇et+rTtrt−ςξTx,tξx,t.
Obviously, one can write the above expression (32) by
(33)Jet≤∑i=1kμiζteTtAl,iTPe+PeAl,i+1εPeMiMiTPe+∑j=1qεγ~j2+CeTCeet+2eTtPeDiξx,t−ςξTx,tξx,t.
Let one define the following new variable as(34)αj=1εγ~j2.
From now on, one can get(35)γ~j=1εαj.
The stability of the T-S fuzzy system (5) is achieved for any fault Lipschitz nonlinear function with Lipschitz constant less than or equal to an unknown maximized constant γ~j,∀j=1,…,q. Maximization of γ~j and minimization of ς can be accomplished by simultaneous minimization of αj, ε, and ς, ∀j=1,…,q. In this way, one can obtain a multiobjective optimization design.
As is clear from (34), one has
(36)Jet≤∑i=1kμiζteTtAl,iTPe+PeAl,i+1εPeMiMiTPe+∑j=1qαj−1+CeTCeet+2eTtPeDiξx,t−ςξTx,tξx,t=ΦTtΔΦt,where
(37)Φt=etξx,t,Δ=∑i=1kμiζtΞij+CeTCePeDiDiTPe−ς,Ξij=Al,iTPe+PeAl,i+1εPeMiMiTPe+∑j=1qαj−1.
One can conclude that if∆<0, it implies that Jet<0.
According to the Schur complement, the previous inequality is satisfied if and only if we checked for the presence of this relation:(38)∑i=1kμiζtΩij<0,where Ωij are the same as those in (23).
In this way, the state estimation error is asymptotically stable with the attenuation levelς as follows:
(39)rt22≤ςξx,t22.
It is obvious thatet=xt−x^t→0. Due to the fault, nonlinear function hjx,u,t satisfies the Lipschitz condition; the T-S multiplicative fault estimation adaptive observer ((7), (8), and (9)) ensures that eθjt→0. From this, one can deduce that, according to (9), the estimation of multiplicative fault magnitude θjt, for the uncertain nonlinear system (1) described by T-S fuzzy structure (5), can be achieved.
This completes the proof.


## 5.2. LMI Optimization-Based Closed-Loop Stability

Once the sliding mode is obtained, we consider to analyze the stability of the closed-loop T-S fuzzy system. Let the equivalent controlueqt, such that Ṡct is equal to zero, be
(40)ueqt=∑i=1kμiζt−Nc,iCiAixt+Diξx,t+ult.The dynamic of the closed-loop system with the equivalent control law (40) takes the form
(41)ẋt=∑i=1k∑j=1kμiμjζtΘiAi−BiKjxt+B¯i,jϕt,(42)yt=∑i=1kμiζtCixt,where B¯i,j=BiKjMiΘiDi, Θi=In−BiNc,iCi, and ϕt=eTtefTx,u,tξTx,tT, where efx,u,t=fx,u,t−f^x,u,t.The objective now is to develop a sufficient condition to achieve the stability of the closed-loop T-S fuzzy system ((41) and (42)) on the sliding surface S despite the occurrence of multiplicative faults and the presence of uncertainties.Theorem 2.
The closed-loop T-S fuzzy system ((41) and (42)) is robustly stable with the H∞ attenuation level ςs>0, if there exist the matrices P¯x=P¯xT>0, Qj, and Y=YT, such that
(43)minςs
satisfying the following LMI constraints:(44)ΔSepaFTC=ϒi,jBiQjMiΘiDiP¯xCiT∗−2Y+ςsIn000∗∗−ςsIq00∗∗∗−ςsIl0∗∗∗∗−Ip<0,where ϒi,j=P¯xAiTΘiT+ΘiAiP¯x−BiQj−QjTBiT, Y=ςsP¯x, and Kj=QjP¯x−1.Proof.
Consider the following Lyapunov function for the closed-loop system as(45)Vxt=xTtPxxt,where Px∈ℝn×n is the symmetric positive definite matrix. The time derivative of Vxt is handled as
(46)V̇xt=∑i−1k∑j=1kμiμjζtxTtΘiAi−BiKjTPx+PxΘiAi−BiKjxt+2xTtPxB¯i,jϕt.
To achieve the robustness withH∞ performance of the closed-loop T-S fuzzy system ((41) and (42)) to ϕt, the following inequality must then hold:
(47)Jxt=V̇xt+yTtyt−ςsϕTtϕt<0.
Insertion of (46) in (47) yields
(48)Jxt=∑i=1k∑j=1kμiμjζtxTtΘiAi−BiKjTPx+PxΘiAi−BiKj+CiTCixt+2xTtPxB¯i,jϕt−ςsϕTtϕt.
Consequently (in the matrix form), it remains to prove thatJxt<0, if
(49)ψi,jPxB¯i,jB¯i,jTPx−ςsI<0,where ψi,j=ΘiAi−BiKjTPx+PxΘiAi−BiKj+CiTCi. Using the Schur complement, the relation (49) can be reformulated as
(50)∏i,j=∑i,jPxBiKjPxMiPxΘiDiCiT∗−ςsIn000∗∗−ςsIq00∗∗∗−ςsIl0∗∗∗∗−Ip<0,where ∑i,j=ΘiTAiTPx+PxΘiAi−KjTBiTPx−PxBiKj.
Inequality (50) contains several nonlinear terms. One can design in the next step to formulate this as an LMI problem. To effect the necessary change of variables, one will define the following matrix X with the special diagonal structure as X=diagPx−1,Px−1,Iq,Il,Ip. Then, X×Πi,j×XT<0 is true, and it is obvious that
(51)ϒi,jBiQjMiΘiDiP¯xCiT∗−ςsP¯xP¯x000∗∗−ςsIq00∗∗∗−ςsIl0∗∗∗∗−Ip<0,where ϒi,j=P¯xAiTΘiT+ΘiAiP¯x−BiQj−QiTBiT, P¯x=Px−1, and Qj=KjPx−1. According to Lemma 1, it is evident to check the presence of the following relation
(52)P¯x+P¯x≤P¯xP¯x+In.
Obviously, (52) is true for ςc>0 as
(53)−ςsP¯xP¯x≤−2ςsP¯x+ςsIn.
After some manipulations, one can get(54)∑i=1k∑j=1kμiμjζtΔSepaFTC<0,where ∆SepaFTC has the same structure with (44). Clearly, a stability proof of the closed-loop T-S fuzzy system ((41) and (42)) is required with respect to the H∞ performance level ςs.
This completes the proof.


## 6. FE-Based Fault-Tolerant Control Design: An Integrated Approach

Several publications have appeared in recent years documenting FE-based FTC design with a single step in order to achieve an optimal robustness interaction between observer and controller models. Figure2 illustrates the integrated FE-based FTC design for uncertain nonlinear systems subject to process fault and multiplicative actuator fault.Figure 2
FE-based fault-tolerant control: an integrated approach.In this section, we explore the possibility of the integrated FE-based FTC design to compute, in a single step, adaptive observer gainsGl,i and controller gains Kj in the sense that it ameliorates the robustness of the closed-loop nonlinear system despite the presence of multiplicative faults and uncertainties. Combining (10), (11), and (41) gives the following augmented closed-loop system, including fault estimation with fault compensation control, expressed as
(55)ẋt=∑i=1k∑j=1kμiμjζtΘiAi−BiKjxt+B¯i,jϕt,(56)ėt=∑i=1kμiζtAi−Gl,iCiet+D¯i1ϕt+Mi∑j=1qθjthjx,u,t−θ^thjx^,u,t,(57)ėθjt=∑i=1kμiζt−σjhjTx^,u,tΓieyt,(58)yLt=∑i=1kμiζtCL,ixt+Ce,iet,where B¯i,j=BiKjMiΘiDi, D¯i1=00Di, Θi=In−BiNc,iCi, and ϕt−etTefTx,u,tξTx,tT..Theorem 3.
Under the sliding mode input structure (14), the closed-loop T-S fuzzy system ((55), (56), (57), and (58)) is robustly stable with both maximized admissible Lipschitz constant γe,j>0 and minimized gain HyLϕ∞<ςc, if there exist constants, 0≤λc≤1, εe>0, and αe,j>0, and matrices Pe=PeT>0, P¯x=P¯xT>0, Wi, Qj, and Y=YT, such that the multiobjective LMI optimization problem admits a solution as
(59)minλcεe+∑j=1qαe,j+1−λcςc,subjecttoΞ¯ij=Ξ¯11,ijΞ¯12,ij∗Ξ¯22,ij<0,where
(60)Ξ¯11,ij=Ξ11,ij0∗Ξ22,i,Ξ11,ij=HeΘiAiP¯x−HeBiQj,Ξ22,i=HePeAi−HeWiCi+Ce,iTCe,iWi=PeGl,iΞ¯12,ij=BiQjMiΘiDiP¯xCL,iT0000PeDi0PeMiIn,Ξ¯22,ij=−diag2Y−ςcIn,ςcIq,ςcIl,Ip,εeIq,∑j=1qαe,j−1,Y=ςcP¯x,P¯x=Px−1,Qj=KjPx−1.
The gain matrices of the adaptive sliding mode controllerKj and observer Gl,i are given by
(61)Kj=QjP¯x−1,Gl,i=Pe−1Wi.Proof.
Stability analysis: In order to assure the stability of the augmented closed-loop system ((55), (56), (57), and (58)), one can start by investigating the following Lyapunov function as
(62)Vt=Vet+Vxt,where Vet=eTtPeet+∑j=1qσj−1eθjTteθjt>0 and Vxt=xTtPxxt>0, and Px∈ℝn×n is the symmetric positive definite matrix.
As first, one can proceed analogously to Theorem1. Hence, the time derivative of Vet is bounded as
(63)V̇et≤∑i=1kμiζteTtAl,iTPe+PeAl,i+1εePeMiMiTPe+∑j=1qαe,j−1et+2eTtPeD¯i1ϕt.
On the other hand, similar to Theorem2 and by taking into account the closed-loop T-S fuzzy system (55), the time derivative of Vxt is expressed as
(64)V̇xt=∑i=1k∑j=1kμiμjζtxTtΘiAi−BiKjTPx+PxΘiAi−BiKjxt+2xTtPxB¯ijϕt.
Robust performance index: Let(65)Jt=V̇t+yLTtyLt−ςcϕTtϕt<0.
The inequality (65), after substituting (63) and (64), becomes
(66)Jt≤∑i=1k∑j=1kμiμjζtxTtBij1TPx+PxBij1+CL,iTCL,ixt+eTtAl,iTPe+PeAl,i+1εePeMiMiTPe+∑j=1qαe,j−1+Ce,iTCe,iet+2xTtPxB¯ijϕt+2eTtPeD¯i1ϕt−ςcϕTtϕt.where Bij1=ΘiAi−BiKj. Equivalently, in the matrix form, one can obtain the following expression as
(67)Jt≤∑i=1k∑j=1kμiμjζtχTtΓijχt,where χt=xTteTtϕTtT. The variable Γij is defined as
(68)Γij=Γ11,ij0PxBiKjPxMiPxΘiDi∗Γ22,i00PeDi∗∗−ςcIn00∗∗∗−ςcIq0∗∗∗∗−ςcIl,such that
(69)Γ11,ij=HeΘiAi−BiKjTPx+CL,iTCL,i,Γ22,i=HeAl,iTPe+1εePeMiMiTPe+∑j=1qαe,j−1+Ce,iTCe,i.
To effect the necessary change of variables, one will make the following matrixX with the special diagonal structure as X=diagPx−1,In,Px−1,Iq,Il. After pre- and postmultiplying by X and its transpose in Γij, then it is obvious that
(70)Γ¯ij=Γ¯11,ij0BiQjMiΘiDi∗Γ¯22,i00PeDi∗∗−ςcP¯xP¯x00∗∗∗−ςcIq0∗∗∗∗−ςcIl,where
(71)Γ¯11,ij=HeΘiAiP¯x−HeBiQj+P¯xCL,iTCL,iP¯x,Γ¯22,i=HePeAi+HeWiCi+1εePeMiMiTPe+∑j=1qαe,j−1+Ce,iTCe,i,P¯x=Px−1,Qj=KjP¯x−1,Wi=PeGl,i.
After simple manipulation by using Lemma1, it is evident to obtain the relation (59). From this, one can conclude that the augmented closed-loop T-S fuzzy system ((55), (56), (57), and (58)) is robustly stable against et, efx,u,t, and ξx,t with respect to the H∞ performance level ςc.
This completes the proof.


## 7. Illustrative Example

In the present section, the design of the separated and integrated sliding mode fault-tolerant control based on adaptive observer information requirement is performed by considering the nonlinear model of a single-link flexible-joint robot arm taken from [13]. Firstly, let us consider the nonlinear model without faults defined by
(72)θ̇m=ωm,ω̇m=kJmθl−θm−BνJmωm+KτJmut,θ̇l=ωl,ω̇l=kJlθl−θm−mghJlsinθl,where θm and ωm are the position and angular velocity of the DC motor, respectively, and θl and ωl represent the position and angular velocity of the link. The values of the parameters are given in Table 1.Table 1
Nonlinear system parameters.
System parameters
Values
Units
Motor inertiaJm
3.7 × 10−3
kg m2
Link inertiaJl
9.3 × 10−3
kg m2
Pointer massm
0.21
kg
Link lengthh
0.15
m
Torsional springk
0.18
Nm/rad
Viscous frictionBv
4.6 × 10−3
M
Amplifier gainKτ
8 × 10−2
Nm/VWe choose thatx1=θm, x2=ωm, x3=θl, and x4=ωl.The flexible-joint robot arm system is described in the nonlinear form as follows:(73)ẋt=Axt+But+Γx,t+Mfx,u,t+Dξx,t,yt=Cxt,with
(74)A=0100−48.6−1.2548.60000119.50−19.50,B=M=021.6200,D=00.2500,C=100001000010,Γx,u,t=021.62ut03.33sinx3t,whereΓx,u,t encapsulates the nonlinearities present in the DC motor. The schematic diagram of a single-link flexible-joint robot arm is shown in Figure 3.Figure 3
Schematic diagram of a single-link flexible-joint robot arm.To evaluate the performances of the proposed FE-based adaptive sliding mode fault-tolerant control, we consider the presence of two types of multiplicative faults affecting the considered nonlinear system, which are described in the following.(1) Actuator gain fault (loss of effectiveness): multiplicative fault occurs in the actuator which is defined as a partial loss of effectiveness. We suppose that fault magnitudeθ1t has the following structure:
(75)θ1t=0,t<5sec,0.8,5sec≤t<20sec,0,t>15sec.The fault functionh1x,u,t is expressed as
(76)h1x,u,t=0ut00T.In this case, the first multiplicative fault is modeled as(77)f1x,u,t=03θ1tut00T.(2) Abnormal friction subject to process fault: an abnormal friction appears in the DC motor where it leads to parameter changes in the nonlinear system state matrix. This multiplicative process fault has the following special structure:(78)f2x,u,t=0−5θ2tωmt00T,which corresponds to the structure function handled as
(79)h2x,u,t=0−5ωmt00T.We suppose that the viscous friction constantBν increases by 80% at t=2.5sec, that is, θ2t=0 at t<2.5sec and θ2t=0.8 at t≥2.5sec.The flexible-joint robot arm system can be formulated in the T-S representation (5), where k=2, with the system matrices:
(80)A1=0100−48.6−1.2548.60000119.50−22.830,B1=M1=021.6200,A2=0100−48.6−1.2448.60000119.50−18.770,B2=M2=021.6200D1=D2=00.2500,C1=C2=100001000010.The parametersμixt are given by
(81)μ1xt=ϑt+0.211.21,μ2xt=1−ϑt1.21,where ϑt=sinx3t/x3t.Comparative simulations are given using the separated and integrated multiplicative FE-based FTC design with the same system parameters and initial conditions.


### 7.1. Separated Multiplicative FE-Based FTC Design

#### 7.1.1. First Step: Adaptive Observer Design

The design parameters were chosen asλ=0.9 and H=I4×4. By solving Theorem 1 with the MATLAB LMI Toolbox, the adaptive observer ((7), (8), and (9)) design is achieved as
(82)Gl,1=104×0.0001−0.0021−0.00884.04220.11573.82220.0117−0.00270.00010.0077−0.0013−0.0021,Gl,2=104×0.00060.0023−0.0050−2.56151.1566−0.66570.00660.00050.00150.02280.0003−0.0170,where we find that
(83)Pe=4.167300000.002200003.7929−2.066900−2.06694.1673.


#### 7.1.2. Second Step: Sliding Mode Controller Design

From Theorem2, the sliding mode controller gains (14) are described as
(84)K1=K2=101.573611.6305−34.496811.4298,such that
(85)Px=0.2650−0.94670.2119−0.8697−0.946712.35890.03130.06340.21190.03130.4505−0.5907−0.86970.0634−0.59076.2204.


### 7.2. Integrated Multiplicative FE-Based FTC Design

By solving the LMI conditions given in Theorem3, using the “mincx” function of the MATLAB LMI toolbox, the matrix gains of the adaptive observer ((7), (8), and (9)) and the sliding mode controller (14) are computed in a single step as
(86)Gl,1=0.50010.4705−4.4445386.9460475.486224.29934.44460.02950.955921.22200.0114−21:4797,Gl,2=0.50010.4705−4.4445386.9496475.486224.30044.44460.02950.955921.22200.0114−17.4097,K1=K2=102.656111.4223−35.648212.0718,where we find that
(87)Pe=104×0.903500000.001100001.0631−0.411900−0.41191.0631,Px=0.2689−0.94360.2134−0.8727−0.943626.47510.03440.05970.21340.03440.4366−0.5899−0.87270.0597−0.58995.9295.Additionally, the LMI optimization gains of the integrated and separated FE-based FTC designs are listed in Table2. As can be seen, the uncertainty attenuation level, which refers to the integrated approach, is much less than that of the separated approach. This latter loses a certain robustness degree against uncertainties illustrating the better multiplicative fault estimation and compensation using the integrated FE-based FTC approach.Table 2
LMI optimization gains.
Integrated FTC
Separated FTC
Observer
Controller
Uncertainty attenuation level
ςc=0.1588
ς=0.0612
ςs=0.7337
Admissible Lipschitz constant
γ1=γ2=0.7916
γe,1=γe,2=0.3911We learned that the fault nonlinear function satisfies the Lipschitz condition. In this way, the admissible Lipschitz constant, which refers to the integrated approach, is greater than the one given by the separated approach, thus illustrating the superiority of the integrated FE-based FTC approach to treat a large range of fault nonlinear function.It is worth pointing that simulation results are given with online multiplicative fault estimation and compensation for the closed-loop nonlinear system in the presence of uncertainties asξx,t=0.1×3sin0.3t such that
(i)
the sliding mode controller (14) is considered where Fa,1=Fa,2=1, N1=N2=−0.75000.0463−0.7500, and ηct=ρ^c+0.1 such that the adaptive term is given by ρ^̇Sc=2×Sct,
(ii)
the initial conditions arex10=Π/15, x20=0.2, x30=Π/12, x40=0, θ^1t=0=θ^2t=0=0, and ρ^ct=0=0,
(iii)
σ1=σ2=10 and Γ1=Γ2=0.75×111.From Figure4, it is quite clear to see that the state estimation errors remain zero in finite time; that is, the T-S adaptive observer proposed in this paper can robustly estimate nonlinear system states with acceptable performances. Figure 5 illustrates the first multiplicative fault estimation error (partial loss of actuator effectiveness).Figure 4
Nonlinear system state estimation errors.Figure 5
Actuator gain fault estimation error.The results display that the proposed adaptive observer ((7), (8), and (9)), using the integrated FE-based FTC design, is capable of estimating multiplicative actuator fault with better performances compared to the separated FE-based FTC design. Note that abrupt changes of fault can generate small peak in Figure 5 at time 5 sec. Meanwhile, an abnormal friction in the DC motor has been introduced, at t=2.5sec, to show the capability of the proposed FE strategy to handle process faults.Consider Figure6, which plots process fault estimation error generated using the separated FE-based FTC scheme against this variable using the integrated FE-based FTC scheme. It should be highlighted that the proposed adaptive observer design, using a single-step LMI formulation, can still track the considered process fault with better performances compared to the separated FE-based FTC design in terms of precision and robustness against the uncertainties.Figure 6
Process fault estimation error.Figures7–9 outline a comparison between the nonlinear single-link flexible-joint robot output responses referring to two different cases: output responses with the separated FE-based FTC and output responses with the integrated FE-based FTC.Figure 7
First closed-loop system output response: output response with the separated FE-based FTC (red line) and output response with the integrated FE-based FTC (blue line).Figure 8
Second closed-loop system output response: output response with the separated FE-based FTC (red line) and output response with the integrated FE-based FTC (blue line).Figure 9
Third closed-loop system output response: output response with the separated FE-based FTC (red line) and output response with the integrated FE-based FTC (blue line).Based on the figures shown above, the conceived adaptive sliding mode controller (14) can stabilize with satisfactory performances the closed-loop nonlinear single-link flexible-joint robot based on the integrated and separated FE-based sliding mode FTC. More precisely, as can be seen in zoomed versions from Figures 7–9, the proposed integrated FE-based FTC is capable of compensating real multiplicative faults with better performances compared to the separated FE-based FTC in terms of precision and robustness against uncertainties.


## 7.1. Separated Multiplicative FE-Based FTC Design

### 7.1.1. First Step: Adaptive Observer Design

The design parameters were chosen asλ=0.9 and H=I4×4. By solving Theorem 1 with the MATLAB LMI Toolbox, the adaptive observer ((7), (8), and (9)) design is achieved as
(82)Gl,1=104×0.0001−0.0021−0.00884.04220.11573.82220.0117−0.00270.00010.0077−0.0013−0.0021,Gl,2=104×0.00060.0023−0.0050−2.56151.1566−0.66570.00660.00050.00150.02280.0003−0.0170,where we find that
(83)Pe=4.167300000.002200003.7929−2.066900−2.06694.1673.


### 7.1.2. Second Step: Sliding Mode Controller Design

From Theorem2, the sliding mode controller gains (14) are described as
(84)K1=K2=101.573611.6305−34.496811.4298,such that
(85)Px=0.2650−0.94670.2119−0.8697−0.946712.35890.03130.06340.21190.03130.4505−0.5907−0.86970.0634−0.59076.2204.


## 7.1.1. First Step: Adaptive Observer Design

The design parameters were chosen asλ=0.9 and H=I4×4. By solving Theorem 1 with the MATLAB LMI Toolbox, the adaptive observer ((7), (8), and (9)) design is achieved as
(82)Gl,1=104×0.0001−0.0021−0.00884.04220.11573.82220.0117−0.00270.00010.0077−0.0013−0.0021,Gl,2=104×0.00060.0023−0.0050−2.56151.1566−0.66570.00660.00050.00150.02280.0003−0.0170,where we find that
(83)Pe=4.167300000.002200003.7929−2.066900−2.06694.1673.


## 7.1.2. Second Step: Sliding Mode Controller Design

From Theorem2, the sliding mode controller gains (14) are described as
(84)K1=K2=101.573611.6305−34.496811.4298,such that
(85)Px=0.2650−0.94670.2119−0.8697−0.946712.35890.03130.06340.21190.03130.4505−0.5907−0.86970.0634−0.59076.2204.


## 7.2. Integrated Multiplicative FE-Based FTC Design

By solving the LMI conditions given in Theorem3, using the “mincx” function of the MATLAB LMI toolbox, the matrix gains of the adaptive observer ((7), (8), and (9)) and the sliding mode controller (14) are computed in a single step as
(86)Gl,1=0.50010.4705−4.4445386.9460475.486224.29934.44460.02950.955921.22200.0114−21:4797,Gl,2=0.50010.4705−4.4445386.9496475.486224.30044.44460.02950.955921.22200.0114−17.4097,K1=K2=102.656111.4223−35.648212.0718,where we find that
(87)Pe=104×0.903500000.001100001.0631−0.411900−0.41191.0631,Px=0.2689−0.94360.2134−0.8727−0.943626.47510.03440.05970.21340.03440.4366−0.5899−0.87270.0597−0.58995.9295.Additionally, the LMI optimization gains of the integrated and separated FE-based FTC designs are listed in Table2. As can be seen, the uncertainty attenuation level, which refers to the integrated approach, is much less than that of the separated approach. This latter loses a certain robustness degree against uncertainties illustrating the better multiplicative fault estimation and compensation using the integrated FE-based FTC approach.Table 2
LMI optimization gains.
Integrated FTC
Separated FTC
Observer
Controller
Uncertainty attenuation level
ςc=0.1588
ς=0.0612
ςs=0.7337
Admissible Lipschitz constant
γ1=γ2=0.7916
γe,1=γe,2=0.3911We learned that the fault nonlinear function satisfies the Lipschitz condition. In this way, the admissible Lipschitz constant, which refers to the integrated approach, is greater than the one given by the separated approach, thus illustrating the superiority of the integrated FE-based FTC approach to treat a large range of fault nonlinear function.It is worth pointing that simulation results are given with online multiplicative fault estimation and compensation for the closed-loop nonlinear system in the presence of uncertainties asξx,t=0.1×3sin0.3t such that
(i)
the sliding mode controller (14) is considered where Fa,1=Fa,2=1, N1=N2=−0.75000.0463−0.7500, and ηct=ρ^c+0.1 such that the adaptive term is given by ρ^̇Sc=2×Sct,
(ii)
the initial conditions arex10=Π/15, x20=0.2, x30=Π/12, x40=0, θ^1t=0=θ^2t=0=0, and ρ^ct=0=0,
(iii)
σ1=σ2=10 and Γ1=Γ2=0.75×111.From Figure4, it is quite clear to see that the state estimation errors remain zero in finite time; that is, the T-S adaptive observer proposed in this paper can robustly estimate nonlinear system states with acceptable performances. Figure 5 illustrates the first multiplicative fault estimation error (partial loss of actuator effectiveness).Figure 4
Nonlinear system state estimation errors.Figure 5
Actuator gain fault estimation error.The results display that the proposed adaptive observer ((7), (8), and (9)), using the integrated FE-based FTC design, is capable of estimating multiplicative actuator fault with better performances compared to the separated FE-based FTC design. Note that abrupt changes of fault can generate small peak in Figure 5 at time 5 sec. Meanwhile, an abnormal friction in the DC motor has been introduced, at t=2.5sec, to show the capability of the proposed FE strategy to handle process faults.Consider Figure6, which plots process fault estimation error generated using the separated FE-based FTC scheme against this variable using the integrated FE-based FTC scheme. It should be highlighted that the proposed adaptive observer design, using a single-step LMI formulation, can still track the considered process fault with better performances compared to the separated FE-based FTC design in terms of precision and robustness against the uncertainties.Figure 6
Process fault estimation error.Figures7–9 outline a comparison between the nonlinear single-link flexible-joint robot output responses referring to two different cases: output responses with the separated FE-based FTC and output responses with the integrated FE-based FTC.Figure 7
First closed-loop system output response: output response with the separated FE-based FTC (red line) and output response with the integrated FE-based FTC (blue line).Figure 8
Second closed-loop system output response: output response with the separated FE-based FTC (red line) and output response with the integrated FE-based FTC (blue line).Figure 9
Third closed-loop system output response: output response with the separated FE-based FTC (red line) and output response with the integrated FE-based FTC (blue line).Based on the figures shown above, the conceived adaptive sliding mode controller (14) can stabilize with satisfactory performances the closed-loop nonlinear single-link flexible-joint robot based on the integrated and separated FE-based sliding mode FTC. More precisely, as can be seen in zoomed versions from Figures 7–9, the proposed integrated FE-based FTC is capable of compensating real multiplicative faults with better performances compared to the separated FE-based FTC in terms of precision and robustness against uncertainties.


## 8. Conclusion

This paper has proposed two adaptive sliding mode FTC schemes for an uncertain nonlinear system subject to multiplicative and process faults. In the first scheme, the separated FE-based FTC is constructed to compensate real fault effects based on output feedback information and to ensure robust stability of the closed-loop system. In the second scheme, the integrated FE-based sliding mode FTC is conceived in order to achieve an optimal robustness interaction between observer and controller models. The novelty of the proposed approach is that the observer and controller gains are obtained by solving a single-step multiobjective LMI optimization problem in order to offer a solution to stabilize the closed-loop nonlinear system despite the occurrence of real fault effects. As shown in the comparative simulations of the flexible-joint robot arm system described by the T-S fuzzy model, the proposed integrated FE-based FTC improves the best robustness interactions between multiplicative fault estimation and sliding mode control. Summing up the results, we can conclude that the research into the integrated FE-based sliding mode FTC for an uncertain nonlinear system has been very successful.



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*Source: 1462594-2018-06-26.xml*