# H∞ Fault Estimation and Fault-Tolerant Control for T-S Fuzzy Systems with Actuator and Sensor Faults Using Sliding Mode Observer

Authors: Slim Dhahri; Essia Ben Alaïa; Faycal Ben Hmida; Anis Sellami
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This paper investigates the problems of the robust fault estimation (FE) and fault-tolerant control (FTC) for the Takagi-Sugeno (T-S) fuzzy systems with unmeasurable premise variables (PVs) subject to external disturbances, actuator, and sensor faults. An adaptive fuzzy sliding mode observer (SMO) with estimated PVs is designed to reconstruct the state, actuator, and sensor faults simultaneously. Compared with the existing results, the proposed observer is with a wider application range since it does not require the knowledge of the upper bound of faults that some FE methods demand. Based on the FE information, a dynamic output-feedback fault-tolerant controller (DOFFTC) is designed to compensate the effect of faults by stabilizing the closed-loop systems. By using theH∞ filtering method, sufficient conditions for the existence of the proposed SMO and DOFFTC are derived in terms of linear matrix inequalities (LMIs) optimization. Finally, a nonlinear inverted pendulum system is given to validate the proposed methods.


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## Body

## 1. Introduction

In industrial applications, the increasing demand of higher performance, safety, reliability, maintainability, and survivability represent a major concern. So, it is important to support research on fault estimation (FE) and fault-tolerant control (FTC) for a class of nonlinear systems specifically Takagi-Sugeno (T-S) fuzzy models [1–5]. Many approaches have been developed in recent decades, such as sliding mode observer (SMO) [6–12], unknown input observer [13–15], adaptive observer [16, 17], and descriptor observer [18, 19].The sliding mode (SM) scheme is a powerful tool to overcome uncertainties and external disturbances in dynamic systems, due to its good robustness, simple structure, and strong applicability. Therefore, it has a good application prospect in the field of FE and FTC and has attracted more and more attention from both academia and industry. Fruitful works can be found regarding this issue. For instance, in [8], a sliding mode observer (SMO) is designed for the estimation of actuator faults in T-S fuzzy models with digital communication channel, but the sensor faults were not considered. While in [9], the estimation of simultaneously actuator and sensor faults is realized for T + S fuzzy systems using nonquadratic Lyapunov function. However, the FTC problem was not treated. While, in [20], a FTC design for T-S fuzzy systems was established using unknown input observer approach. Moreover, a FTC scheme based on SMO for T-S fuzzy systems with local nonlinear models is proposed in [6]. Sufficient conditions are derived to calculate observer and controller gains which are solved using linear matrix inequalities (LMIs). In [21], an adaptive sliding mode FTC design is developed for a class of uncertain T-S fuzzy systems affected by multiplicative faults. Unfortunately, these previous results needed that the premise variables of the T-S fuzzy systems are measurable and the upper bounds of faults are known. Therefore, how to design a suitable SMO to overcome the above drawbacks?Motivated by the above discussion, an adaptive fuzzy SMO is developed for the T-S fuzzy systems with unmeasurable premise variables in order to estimate the state, actuator, and sensor faults. Then, based on FE a dynamic output-feedback fault-tolerant controller (DOFFTC) is designed to compensate the fault effects by stabilizing the closed-loop system. Finally, the simulation result of a nonlinear inverted pendulum system is given to illustrate the effectiveness of the proposed method.Compared with the existing results, the advantages of our work can be summarized as follows. First of all, we investigate the problem of fault estimation and fault-tolerant control based on the dynamic output-feedback controller for T-S fuzzy systems with external disturbances and actuator and sensor faults. Whereas, many researchers have considered only actuator faults [22, 23] or senor faults [24 + 26]. Then, the sliding mode observer is designed using adaptive law to avoid the hypothesis concerning knowledge of the upper bound of faults, which give less conservative results and offer more freedom in comparison with [27 + 29]. Moreover, authors in [6, 24] assume that the premise variables of the T-S fuzzy systems are measurable. Whereas, in this work, we consider the unmeasurable premise variables. The gains of the observer and controller are computed separately to avoid the coupling and reduce the computation complexity. Whereas, in [25], a single step solving algorithm is needed.The rest of this paper is organized as follows. In Section2, we describe the system and the problem studied in this paper. In Section 3, we present the design of the adaptive fuzzy sliding mode observer and the analysis of the stability of the error system. The fault estimation is studied in Section 4. In Section 5, the dynamic output-feedback fault-tolerant controller is developed. Finally, simulation example in Section 6 validates the efficiency of the proposed methods.


### 1.1. Notations

Throughout the paper, the following notations are used.In denotes an identity matrix with dimension of n×n. Rn and Rn×m denote the n-dimensional Euclidean space and n×m real matrices. The pseudoinverse of a matrix A is denoted by A+. For a real matrix A, A>0 indicates that A is symmetric positive definite, and A>0 indicates that A is symmetric negative definite. . denotes the Euclidean norm or its induced spectral norm. The symmetric terms in a symmetric matrix are denoted by ∗. Finally, the space of square integrable functions is denoted by L2, that is, for any ξ.∈L20∞, ξ.2=∫0∞ξT.ξ.dt.


## 1.1. Notations

Throughout the paper, the following notations are used.In denotes an identity matrix with dimension of n×n. Rn and Rn×m denote the n-dimensional Euclidean space and n×m real matrices. The pseudoinverse of a matrix A is denoted by A+. For a real matrix A, A>0 indicates that A is symmetric positive definite, and A>0 indicates that A is symmetric negative definite. . denotes the Euclidean norm or its induced spectral norm. The symmetric terms in a symmetric matrix are denoted by ∗. Finally, the space of square integrable functions is denoted by L2, that is, for any ξ.∈L20∞, ξ.2=∫0∞ξT.ξ.dt.


## 2. System Description and Problem Statement

Considering the following T-S fuzzy model: RuleI: IF z1 is M1i, …, and zg is Mgi, THEN(1)x˙p=Apixp+Bpiu+fa+Epid.,yp=Cpxp+Npfs+Dpd,where xp∈ℝn represents the state vector; u∈ℝm is the input; yp∈ℝp is the measured output; Api, Bpi, Epi, Cp, Np, and Dp are known matrices with compatible dimensions; fa:ℝ+⟶ℝq and fs:ℝ+⟶ℝh represent the additive faults generated by actuator and sensor, respectively; d∈ℝl is the external disturbance which belong to L20∞. It is supposed that matrices Bpi are of full column rank, i.e., rankBpi=m, matrix Np is of full column rank, Cp is of full row rank, the pairs Api,Cp are observable, and the pairs Api,Bpi are controllable. Besides, z=z1…zg denotes the unmeasurable premise variable, Mji are the fuzzy sets, k is the number of IF-THEN rules, and g is the number of the premise variable.Using the technique [26], the overall model of system (1) is given by(2)x˙p=∑i=1khizApixp+Bpiu+fa+Epid,yp=Cpxp+Npfs+Dpd,where hiz=wiz/∑i=1kwiz and wiz=∏j=1gMjizj, here Mji is the grade of the membership function of zj. We assume that wiz≥0, i=1,…,k. Then, it easy to see that ∑i=1kwiz>0, for any z. Hence, hiz satisfies hiz≥0 and ∑i=1khiz=1.Lemma 1 (see [27]).
For matricesA and B with appropriate dimensions, we have(3)ATB+BTA≤εATA+ε−1BTB,for any ε>0.Lemma 2 (see [28]).
If the following inequalities hold,(4)Φii<0,1≤i≤k.2k−1Φii+Φij+Φji<0,1≤i≠j≤k,we have(5)∑i=1k∑j=1khihjΦij<0.Introduce the statexf such that(6)x˙f=−Afxf+Afyp,where −Af∈ℝh×h is an arbitrary stable matrix. Let x=xpxf; then, the following augmented T-S fuzzy system is constructed:(7)x˙=∑i=1khizAix+Biu+fa+Nfs+Eid,y=Cx,where(8)Ai=Api0AfCp−Af,Bi=Bpi0,N=0AfNp,Ei=EpiAfDp.C=0Ih,y=ypxf.Combining the actuator faultfa and the sensor fault fs in the same unknown vector f=fafsT and assuming it is bounded and satisfies f⩽w, where w>0 is an unknown real constant, then system (8) can be re-expressed as(9)x˙=∑i=1khizAix+Biu+Mif+Eid,y=Cx,where Mi=Bpi00AfNp.For designing observers, it is often assumed, in the literature, that the premise variablez is available for measurement. In this paper, it is interesting to develop a sliding mode observer for the unmeasurable premise variable T-S fuzzy system. A sliding mode observer for system (9) is in the form(10)x^˙=∑i=1khiz^Aix^+Biu+Gliey+Gniϑ.,y^=Cx^,where z^ is the estimation of unmeasured premise variable, x^ is the state estimation of x, y^ is the observer output, ey:=y−y^ is the output estimation error, Gli and Gni are the observer gain matrices, and ϑ is the discontinuous vector to be designed.Define the state estimation error,e=x−x^. Based on (9) and (10), we get the following error dynamics system:(11)e˙=∑i=1khiz^Aoie+ϕ+Mif−Gniϑ+Eid,where Aoi=Ai−GliC and(12)ϕ=∑i=1khiz−hiz^Aix+Biu+Mif+Eid.Note thatϕ⟶0 when e⟶0, that is, ϕ is treated as an unstructured vanishing perturbation which is supposed to be growth-bounded for x,x^∈ℝn such that(13)ϕ≤γx−x^,where γ is a small Lipschitz scalar. For the simplicity, h^i denotes hiz^.Authors in [29] have proven that the necessary and sufficient conditions for the existence of sliding mode observer when the system includes faults are(A1)
rankCMi=rankMi=q˜(A2)
The invariant zeros ofAi,Mi,C are stableUnder A1, there exists a coordinate transformationx⟶x¯=T1ix such that system (8) is transformed into(14)x¯˙=∑i=1kh^iA¯ix¯+ϕ¯+B¯iu+M¯ifa+E¯id,y¯=C¯x¯,where(15)A¯i=T1iAiT1i−1=A¯1iA¯2iA¯3iA¯4i,ϕ¯=T1iϕ,B¯i=T1iBi=B¯1iB¯2i,M¯i=T1iMi=0M¯2i,M¯2i=0M¯22i,E¯i=T1iEi=E¯1iE¯2i,C¯=CT1i−1=0C¯2,where A¯1i∈ℝn+h−p×n+h−p, B¯2i∈ℝp×m, M¯22i∈ℝq¯×q+h, E¯2i∈ℝp×l, and C¯2∈ℝp×p is nonsingular.Applying the linear change of coordinatesT1i to the error system (11), then we obtain(16)e¯˙=∑i=1kh^iA¯oie¯+ϕ¯+M¯if−G¯niϑ+E¯id,where A¯oi=T1iAoiT1i−1. We suppose the observer gain G¯ni has been the following form to facilitate the analysis:(17)G¯ni=−K¯iC¯2−1C¯2−1,K¯i=K¯1i0n+h−p×q˜,where K¯1i∈ℝn+h−p×p−q˜. It is noted that(18)K¯iM¯i=K¯1i0n+h−p×q˜0p−q˜×q+hM¯22i=0,which motivates us to consider a coordinate transformation e¯⟶e˜:=T2ie¯, where(19)T2i=In+h−pK¯i0p×n+h−pC¯2.In the new set of coordinates, (16) becomes(20)e˜˙=∑i=1kh^iA˜oie˜+ϕ˜+M˜if−G˜niϑ+E˜id,where(21)A˜oi=T2iT−1A¯oiT2i−1=A˜i−G˜liC˜and(22)A˜i=T2iA¯iT2i−1=A˜1iA˜2iA˜3iA˜4i,G˜li=T2iG¯li=G˜l1iG˜l2i,ϕ˜=T2iϕ¯:=ϕ˜1ϕ˜2,M˜i=T2iM¯i=0M˜2i,E˜i=T2iE¯i=E˜1iE˜2i,C˜=C¯T2i−1=0Ip,G˜ni=T2iG¯ni=0Ip,with(23)A˜1i=A¯1i+K¯iA¯3i,A˜2i=A¯2i+K¯iA¯4iC¯2−1−A¯1i+K¯iA¯3iC¯2−1−A¯1iK¯iC¯2−1,A˜3i=C¯2A¯3i,A˜4i=C¯2A¯4i−A¯3iK¯iC¯2−1,M˜2i=C¯2M¯2i,E˜1i=E¯1i+K¯iE¯2i,E˜2i=C¯2E¯2i.Define the matrixG˜li=G˜l1iG˜l2i. If the observer gain matrices G˜l1i and G˜l2i are chosen as G˜l1i=A˜2i and G˜l2i=A˜4i−A˜4s, where A˜4s is an arbitrary negative definite matrix, it can be concluded that(24)A˜oi=A˜i−G˜liC˜=A¯1i+K¯iA¯3i0C¯2A¯3iA˜4s.According to our choice,A˜4s is stable. Therefore, A˜oi is stable from the stability of A¯1i+K¯iA¯3i.Partitioning the error system conformably toe˜≔e1TeyTT with e1∈ℝn+h−p and ey∈ℝp yields(25)e˙1=∑i=1kh^iA˜1ie1+ϕ˜1+E˜1id,e˙y=∑i=1kh^iA˜4sey+A˜3ie1+ϕ˜2+M˜2if−ϑ+E˜2,id.Considering the following sliding mode surface as(26)S=e1,ey|ey=0,we now design the discontinuous error injection ϑ as follows:(27)ϑ=∑i=1kh^iρiP2ieyP2iey,ifey≠0,0,otherwise,where P2i>0∈ℝp×p is the unique solution to the Lyapunov equation for A˜4s with the design matrix Q0>0∈ℝp×p:(28)P2iA˜4s+P2iA˜4sT=−Q0and(29)ρi=C¯2M¯2iw^+ρ0,with the adaptive law,(30)w^˙=−q∑i=1kh^iC¯2M¯2iP2iey.ρ0 is a small positive constant and q is designed constant.Remark 1.
The presence of bothf and d in (18) poses the main challenge to the SMO design problem in this paper. To tackle this difficulty, in addition to the design of ϑ for the estimation of f in the SMO, we use the H∞ approach for the stability analysis of the error system.
Define the controlled output of the error systemm as(31)m=He1ey,where H=diagH1,H2 is a full rank design matrix.
We will utilize theH∞ approach to analyse the stability of the error system (18) so that(i)
limt⟶∞e˜=0 if d=0, ∀t≥0(ii)
∫0∞mTmdt≤μ2∫0∞dTddt, otherwise
μ>0 is the attenuation level to be minimized.


## 3. Sliding Mode Observer Design

In this section, we will discuss the design method of SMO. The main results are presented as follows.


### 3.1. Stability Analysis

Theorem 1.
Consider system (9) with conditions A1 and A2. The error system (18) is asymptotically stable with a minimal μ if there exist matrices P1i>0, P2i>0, K¯i, and small positive scalar ε such that the following convex optimization problem is solved:
minμ¯ subject to(32)Λ¯1iΛ2iP1iE¯1i+WiE¯2iP1i0∗Λ¯4iP2iC¯2E¯2i0P2i∗∗−μ¯I00∗∗∗−εI0∗∗∗0−εI<0,where(33)Λ¯1i=A¯1iTP1i+P1iA¯1i+WiA¯3i+A¯3iTWiT+εγ2I+H1TH1,Λ2i=A¯3iTC¯2TP2i,Λ¯4i=A˜4sTP2i+P2iA˜4s+εγ2Ip+H2TH2.
K¯i and μ are obtained as K¯i=P1i−1Wi and μ=μ¯.Proof.
Consider the Lyapunov functional candidate:(34)Ve˜=e˜TPie˜+1qw˜2,where Pi=diagP1i,P2i, P1i∈ℝn+h−p×n+h−p, and w˜=w−w^. By deviating Ve˜, we obtain(35)V˙e˜=∑i=1kh^ie˜TA˜oiTPi+PiA˜oie˜+2e˜TPiϕ˜+2e˜TPiE˜id+2e˜TPiM˜if−2e˜TPiG˜niϑ+2q−w˜w^˙.
According to Lemma1, we obtain(36)2e˜TPiϕ˜≤1εe˜TPi2e˜+εϕ˜2≤1εe˜TPi2e˜+εγ2e˜2.
Hence, from (33), we have(37)V˙e˜≤∑i=1kh^ie˜TA˜oiTPi+PiA˜oi+1εPi2e˜Te˜+εγ2Ie˜+2e˜TPiϕ˜+2e˜TPiE˜id+2e˜TPiM˜if−2e˜TPiG˜niϑ+2q−w˜w^˙︸termi.
Substituting the adaptive laww^˙ and the discontinuous vector ϑ into the termi, we have(38)∑i=1kh^ie˜TPiM˜if−e˜TPiG˜niϑ+1qw^˙w˜=∑i=1kh^ieyTP2iC¯2M¯2if−eyTP2iϑ+1qw^˙w˜=∑i=1kh^ieyTP2iC¯2M¯2if−eyTP2iC¯2M¯2iw^+ρ0×P2ieyP2iey+1qw^˙w˜=∑i=1kh^ieyTP2iC¯2M¯2if−C¯2M¯2iw^+ρ0P2iey+1qw−w^−q∑i=1kh^iC¯2M¯2iP2iey⩽∑i=1kh^iP2ieyC¯2M¯2iw−ρ0+C¯2M¯2iw=−ρ0∑i=1kh^iP2iey<0.
When substituting obtained expressions (35) into (34), we obtain(39)V˙e˜≤∑i=1kh^ie˜TA˜oiTPi+PiA˜oi+1εPi2+εγ2Ie˜+2e˜TPiE˜id.
For the cased=0, then(40)V˙e˜≤∑i=1kh^ie1eyTΛie1ey,where(41)Λi=Λ1iΛ2i∗Λ4i,with(42)Λ1i=A¯1iTP1i+P1iA¯1i+WiA¯3i+A¯3iTWiT+1εP1i2+εγ2I,Λ2i=A¯3iTC¯2TP2i,Λ4i=A˜4sTP2i+P2iA˜4s+1εP2i2+εγ2I.
Denote(43)Wi=P1iK¯i,μ¯=μ2.
Using the Schur complement, if (32) holds, we have(44)Λi=Λ1iΛ2i∗Λ4i<0,which implies that V˙e˜<0, i.e., limt⟶∞e˜=0.
On the contrary, for the case thatd≠0, let(45)J1=V˙+mTm−μ2dTd.
Substituting (31) and (39) into (45) yields(46)J1=V˙e˜+e˜THTHe˜−μ2dTd≤∑i=1kh^ie˜TΛi+HTHe˜+2e˜TPiE˜id−μ2dTd≤∑i=1kh^ie1eydTΛ¯1iΛ2iP1iE¯1i+WiE¯2i∗Λ¯4iP2iC¯2E¯2i∗∗−μ¯Ie1eyd,where Λ¯1i=Λ1i+H1TH1 and Λ¯4i=Λ4i+H2TH2.
Using the Schur complement, we have that if (30) holds(47)υi=Λ¯1iΛ2iP1iE¯1i+WiE¯2i∗Λ¯4iP2iC¯2E¯2i∗∗−μ¯I<0,which means that J1<0; i.e., the error system (23) is asymptotically stable with the H∞ performance μ.
From (43), K¯i and μ are computed as K¯i=P1i−1Wi and μ=μ¯.Remark 2.
According to Theorem1, the error system is asymptotically stable with H∞ performance. Thus, for some small ϖ>0, we have e˜≤ϖ. In addition, one has e1≤e˜.


#### 3.1.1. Sliding Motion Analysis

In this section, the gain parameterρ0 in (29) will be determined to demonstrate a sliding motion occurs on S in finite time.Theorem 2.
Ifρ0 in (27) is chosen to satisfy(48)ρ0−ρ¯≥ςmax+γϖ+νmaxd,where ρ¯ is positive scalar, then a sliding motion occurs on the surface S, for all t≥ts, where ts is the finite time at which sliding is established.Proof.
Define(49)Vs=12eyTP2iey+12qw˜2,where the matrix P2i is proposed in Theorem 1; then, the derivative of Vs satisfies(50)V˙s=∑i=1kh^ieyTP2iA˜4sey+A˜3ie1+ϕ˜+E˜2id+M˜2if−ϑ+1qw^˙w˜.
According to (35), we have(51)V˙s≤∑i=1kh^ieyTP2iA˜4sey+A˜3ie1+ϕ˜+E˜2id−ρ0P2iey.
SinceA˜4s is a stable design matrix and P2i is solution to Lyapunov (28), we have(52)eyTP2iA˜22sey=12eyTP2iA˜4s+P2iA˜4sT︸−Q0iey=−12eyTQ0iey⩽0.
Using relation (46), we have(53)V˙s≤∑i=1kh^ieyTP2iA˜3ie1+ϕ˜+E˜2id−ρ0P2iey≤∑i=1kh^iP2ieyA˜3i+γe˜+E˜2id−ρ0≤P2ieyςmax+γϖ+νmaxd−ρ0,where(54)ςmax=maxλmaxA˜31,λmaxA˜32,…,λmaxA˜3k,νmax=maxλmaxE˜21,λmaxE˜22,…,λmaxE˜2k,where λmaxA is the maximum eigenvalue of A.
If condition (42) holds, then the sliding mode reaching condition(55)V˙s⩽−ρ0P2ieyt⩽−ρ0λminP2iVs1/2is guaranteed. Applying the chain rule d/dtVs=1/2VsV˙s, the reaching condition (47) can be integrated and rearranged to obtain an estimate for ts:(56)12VsV˙s≤−ρ0λminP2i⇒∫0tsddtVsdt≤−ρ0λminP2i∫0tsdt⇒Vsts−Vs0≤ρ0λminP2its⇒ts≤1ρ0Vs0λminP2i.
This proves that the sliding surfaceS is thus reached in finite time ts.


## 3.1. Stability Analysis

Theorem 1.
Consider system (9) with conditions A1 and A2. The error system (18) is asymptotically stable with a minimal μ if there exist matrices P1i>0, P2i>0, K¯i, and small positive scalar ε such that the following convex optimization problem is solved:
minμ¯ subject to(32)Λ¯1iΛ2iP1iE¯1i+WiE¯2iP1i0∗Λ¯4iP2iC¯2E¯2i0P2i∗∗−μ¯I00∗∗∗−εI0∗∗∗0−εI<0,where(33)Λ¯1i=A¯1iTP1i+P1iA¯1i+WiA¯3i+A¯3iTWiT+εγ2I+H1TH1,Λ2i=A¯3iTC¯2TP2i,Λ¯4i=A˜4sTP2i+P2iA˜4s+εγ2Ip+H2TH2.
K¯i and μ are obtained as K¯i=P1i−1Wi and μ=μ¯.Proof.
Consider the Lyapunov functional candidate:(34)Ve˜=e˜TPie˜+1qw˜2,where Pi=diagP1i,P2i, P1i∈ℝn+h−p×n+h−p, and w˜=w−w^. By deviating Ve˜, we obtain(35)V˙e˜=∑i=1kh^ie˜TA˜oiTPi+PiA˜oie˜+2e˜TPiϕ˜+2e˜TPiE˜id+2e˜TPiM˜if−2e˜TPiG˜niϑ+2q−w˜w^˙.
According to Lemma1, we obtain(36)2e˜TPiϕ˜≤1εe˜TPi2e˜+εϕ˜2≤1εe˜TPi2e˜+εγ2e˜2.
Hence, from (33), we have(37)V˙e˜≤∑i=1kh^ie˜TA˜oiTPi+PiA˜oi+1εPi2e˜Te˜+εγ2Ie˜+2e˜TPiϕ˜+2e˜TPiE˜id+2e˜TPiM˜if−2e˜TPiG˜niϑ+2q−w˜w^˙︸termi.
Substituting the adaptive laww^˙ and the discontinuous vector ϑ into the termi, we have(38)∑i=1kh^ie˜TPiM˜if−e˜TPiG˜niϑ+1qw^˙w˜=∑i=1kh^ieyTP2iC¯2M¯2if−eyTP2iϑ+1qw^˙w˜=∑i=1kh^ieyTP2iC¯2M¯2if−eyTP2iC¯2M¯2iw^+ρ0×P2ieyP2iey+1qw^˙w˜=∑i=1kh^ieyTP2iC¯2M¯2if−C¯2M¯2iw^+ρ0P2iey+1qw−w^−q∑i=1kh^iC¯2M¯2iP2iey⩽∑i=1kh^iP2ieyC¯2M¯2iw−ρ0+C¯2M¯2iw=−ρ0∑i=1kh^iP2iey<0.
When substituting obtained expressions (35) into (34), we obtain(39)V˙e˜≤∑i=1kh^ie˜TA˜oiTPi+PiA˜oi+1εPi2+εγ2Ie˜+2e˜TPiE˜id.
For the cased=0, then(40)V˙e˜≤∑i=1kh^ie1eyTΛie1ey,where(41)Λi=Λ1iΛ2i∗Λ4i,with(42)Λ1i=A¯1iTP1i+P1iA¯1i+WiA¯3i+A¯3iTWiT+1εP1i2+εγ2I,Λ2i=A¯3iTC¯2TP2i,Λ4i=A˜4sTP2i+P2iA˜4s+1εP2i2+εγ2I.
Denote(43)Wi=P1iK¯i,μ¯=μ2.
Using the Schur complement, if (32) holds, we have(44)Λi=Λ1iΛ2i∗Λ4i<0,which implies that V˙e˜<0, i.e., limt⟶∞e˜=0.
On the contrary, for the case thatd≠0, let(45)J1=V˙+mTm−μ2dTd.
Substituting (31) and (39) into (45) yields(46)J1=V˙e˜+e˜THTHe˜−μ2dTd≤∑i=1kh^ie˜TΛi+HTHe˜+2e˜TPiE˜id−μ2dTd≤∑i=1kh^ie1eydTΛ¯1iΛ2iP1iE¯1i+WiE¯2i∗Λ¯4iP2iC¯2E¯2i∗∗−μ¯Ie1eyd,where Λ¯1i=Λ1i+H1TH1 and Λ¯4i=Λ4i+H2TH2.
Using the Schur complement, we have that if (30) holds(47)υi=Λ¯1iΛ2iP1iE¯1i+WiE¯2i∗Λ¯4iP2iC¯2E¯2i∗∗−μ¯I<0,which means that J1<0; i.e., the error system (23) is asymptotically stable with the H∞ performance μ.
From (43), K¯i and μ are computed as K¯i=P1i−1Wi and μ=μ¯.Remark 2.
According to Theorem1, the error system is asymptotically stable with H∞ performance. Thus, for some small ϖ>0, we have e˜≤ϖ. In addition, one has e1≤e˜.


### 3.1.1. Sliding Motion Analysis

In this section, the gain parameterρ0 in (29) will be determined to demonstrate a sliding motion occurs on S in finite time.Theorem 2.
Ifρ0 in (27) is chosen to satisfy(48)ρ0−ρ¯≥ςmax+γϖ+νmaxd,where ρ¯ is positive scalar, then a sliding motion occurs on the surface S, for all t≥ts, where ts is the finite time at which sliding is established.Proof.
Define(49)Vs=12eyTP2iey+12qw˜2,where the matrix P2i is proposed in Theorem 1; then, the derivative of Vs satisfies(50)V˙s=∑i=1kh^ieyTP2iA˜4sey+A˜3ie1+ϕ˜+E˜2id+M˜2if−ϑ+1qw^˙w˜.
According to (35), we have(51)V˙s≤∑i=1kh^ieyTP2iA˜4sey+A˜3ie1+ϕ˜+E˜2id−ρ0P2iey.
SinceA˜4s is a stable design matrix and P2i is solution to Lyapunov (28), we have(52)eyTP2iA˜22sey=12eyTP2iA˜4s+P2iA˜4sT︸−Q0iey=−12eyTQ0iey⩽0.
Using relation (46), we have(53)V˙s≤∑i=1kh^ieyTP2iA˜3ie1+ϕ˜+E˜2id−ρ0P2iey≤∑i=1kh^iP2ieyA˜3i+γe˜+E˜2id−ρ0≤P2ieyςmax+γϖ+νmaxd−ρ0,where(54)ςmax=maxλmaxA˜31,λmaxA˜32,…,λmaxA˜3k,νmax=maxλmaxE˜21,λmaxE˜22,…,λmaxE˜2k,where λmaxA is the maximum eigenvalue of A.
If condition (42) holds, then the sliding mode reaching condition(55)V˙s⩽−ρ0P2ieyt⩽−ρ0λminP2iVs1/2is guaranteed. Applying the chain rule d/dtVs=1/2VsV˙s, the reaching condition (47) can be integrated and rearranged to obtain an estimate for ts:(56)12VsV˙s≤−ρ0λminP2i⇒∫0tsddtVsdt≤−ρ0λminP2i∫0tsdt⇒Vsts−Vs0≤ρ0λminP2its⇒ts≤1ρ0Vs0λminP2i.
This proves that the sliding surfaceS is thus reached in finite time ts.


## 3.1.1. Sliding Motion Analysis

In this section, the gain parameterρ0 in (29) will be determined to demonstrate a sliding motion occurs on S in finite time.Theorem 2.
Ifρ0 in (27) is chosen to satisfy(48)ρ0−ρ¯≥ςmax+γϖ+νmaxd,where ρ¯ is positive scalar, then a sliding motion occurs on the surface S, for all t≥ts, where ts is the finite time at which sliding is established.Proof.
Define(49)Vs=12eyTP2iey+12qw˜2,where the matrix P2i is proposed in Theorem 1; then, the derivative of Vs satisfies(50)V˙s=∑i=1kh^ieyTP2iA˜4sey+A˜3ie1+ϕ˜+E˜2id+M˜2if−ϑ+1qw^˙w˜.
According to (35), we have(51)V˙s≤∑i=1kh^ieyTP2iA˜4sey+A˜3ie1+ϕ˜+E˜2id−ρ0P2iey.
SinceA˜4s is a stable design matrix and P2i is solution to Lyapunov (28), we have(52)eyTP2iA˜22sey=12eyTP2iA˜4s+P2iA˜4sT︸−Q0iey=−12eyTQ0iey⩽0.
Using relation (46), we have(53)V˙s≤∑i=1kh^ieyTP2iA˜3ie1+ϕ˜+E˜2id−ρ0P2iey≤∑i=1kh^iP2ieyA˜3i+γe˜+E˜2id−ρ0≤P2ieyςmax+γϖ+νmaxd−ρ0,where(54)ςmax=maxλmaxA˜31,λmaxA˜32,…,λmaxA˜3k,νmax=maxλmaxE˜21,λmaxE˜22,…,λmaxE˜2k,where λmaxA is the maximum eigenvalue of A.
If condition (42) holds, then the sliding mode reaching condition(55)V˙s⩽−ρ0P2ieyt⩽−ρ0λminP2iVs1/2is guaranteed. Applying the chain rule d/dtVs=1/2VsV˙s, the reaching condition (47) can be integrated and rearranged to obtain an estimate for ts:(56)12VsV˙s≤−ρ0λminP2i⇒∫0tsddtVsdt≤−ρ0λminP2i∫0tsdt⇒Vsts−Vs0≤ρ0λminP2its⇒ts≤1ρ0Vs0λminP2i.
This proves that the sliding surfaceS is thus reached in finite time ts.


## 4. Robust Reconstruction of Actuator and Sensor Faults

In Theorems1 and 2, one has proven that the error system is asymptotically stable and can thus be driven onto sliding surface S at some time instant ts; the error system (23) is reduced to(57)0=∑i=1kh^ieyTP2iA˜3ie1+ϕ˜2+M˜2if+E˜2id−ϑeq,∀t≥ts,where(58)ϑeq=∑i=1kh^iρiP2ieyP2iey+δ,with δ>0 is small scalar. Define(59)ge1,x,x^,u=A˜3ie1+ϕ˜2+E˜2id.Computing the norm of (59) yields(60)ge1,x,x^,u≤A˜21i+γ2e1+E˜2id≤ςmax+γe˜+νmaxd≤μ¯ςmax+γσmaxH−1+νmaxd,where σmaxA for a matrix A denotes the maximum singular value of the matrix. The result (52) follows by keeping in mind that e˜≤σmaxH−1μ¯d. Therefore, it follows that(61)supd≠0=ge1,x,x^,ud=β1μ¯+β2,where β1=ςmax+γσmaxH−1 and β2=νmax. Thus, for a small β1μ¯+β2, both actuator and sensor faults are estimated by(62)f^=f^af^s=∑i=1kh^iM˜2i+ρiP2ieyP2iey+δ.


## 5. Dynamic Output-Feedback Fault-Tolerant Control Design

In this section, a fuzzy dynamic output-feedback FTC (DOFFTC) will be constructed to guarantee the stability of closed-loop system (2). The following corrected output will be given which is obtained by subtracting the reconstructed sensor faults from the (faulty) outputs:(63)yc=Cpxp+Npfs−f^s+Dpd,where f^s is estimation of fs obtained from the FE scheme. System (2) becomes(64)x˙p=∑i=1khiApixp+Bpiu+fa+Epid,yc=Cpxp+Npefs+Dpd,where efs=fs−f^s is the sensor fault estimation error. The fuzzy DOFFTC for system (56) is constructed as follows:(65)x˙c=∑i=1k∑j=1khihjAcijxc+Bciyc,u=∑i=1khiCcixc+Dciyc−f^a,where xc∈ℝn×n is the controller state, f^a is the estimation of fa, and Acij∈ℝn×n, Bci∈ℝn×p, Cci∈ℝm×n, and Dci∈ℝm×p denote controller matrices to be obtained later, respectively.Substituting (63) into (65), we obtain(66)x˙c=∑i=1k∑j=1khihjAcijxc+BciCpxp+BciNpefs+BciDpd,u=∑i=1khiCcixc+DciCpxp+DciNpefs+DciDpd−f^a.Then, substituting (64) into (66), we further obtain(67)x˙p=∑i=1khiApixp+BpiCcixc+BpiDciCpxp+BpiDciNpefs+BpiDciDpd+Bpiefa+Epid,where efa=fa−f^a is the actuator fault estimation error.The dynamic equation of the closed-loop system is obtained as follows:(68)x˜˙=∑i=1k∑j=1khihjAijx˜+Eijϖ,yc=Cx˜+Dϖ,where x˜=xpxc, ϖ=efaTefsTdTT, and(69)Aij=Api+BpiDciCpBpiCciBciCpAcij,Eij=BpiBpiDciNpBpiDciDp+Epi0BciNpBciDp,C=Cp0,D=0NpDp.So far, the control purpose in this paper for the closed-loop system (60) is to design the controller gain matrices of (57) such that the corrected output satisfies the H∞ performance as follows:(70)∫0∞ycTycdt≤μc∫0∞ϖTϖdt,for ϖ∈L20∞ and attenuation level μc>0.Theorem 3.
The closed-loop system (60) is asymptotically stable with a minimal μc in (61) if there exist matrices X>0, Y>0, and A^cij, B^ci, C^ci, and D^ci, i,j=1,…,k, such that the following convex optimization problem is solved:
minμ¯c subject to(71)Φii<0,1≤i≤k,2k−1Φii+Φij+Φji<0,1≤i<j≤k,where(72)Φij=Υ1ijΥ2ijBpiΥ3ijΥ4ijEpiXCpT∗Υ5ijYBpiΥ6ijΥ7ijYEpiCpT∗∗−μ¯cI0000∗∗∗−μ¯cI000∗∗∗∗−μ¯cI0NpT∗∗∗∗∗−μ¯cIDpT∗∗∗∗∗∗−μ¯cI,with(73)Υ1ij=ApiX+XApiT+BpiC^ci+C^ciTBpiT,Υ2ij=A^cijT+Api+BpiD^ciCp,Υ3ij=BpiD^ciNp,Υ4ij=BpiD^ciDp,Υ5ij=YApi+ApiTYT+B^ciCp+CpTB^ciT,Υ6ij=B^ciNp,Υ7ij=B^ciDp.
The gain matrices of the DOFFTC are as follows:(74)Dci=D^ci,Acij=S−1A^cij−YApi+BpiD^ciCpXQ−T−S−1YBpiCcj−BciCpXQ−T,Cci=C^ci−DciCpXQ−T,Bci=S−1B^ci−YBpiDci.
μc is obtained as μc=μ¯c.Proof.
Consider the Lyapunov functionVx=x˜TPxx˜, where Px>0; then, the derivative of Vx can be obtained as(75)V˙x=∑i=1k∑j=1khihjx˜TAijTPx+PxAijx˜T+2x˜TPxEijϖ.
Let(76)J2=V˙x+ycTyc−μc2ϖTϖ.
Substituting (76) into (75), we obtain(77)J2=∑i=1k∑j=1khihjx˜TAijTPx+PxAijx˜T+2x˜TPxEijϖ+ycTyc−μc2ϖTϖ=∑i=1k∑j=1khihjx˜TAijTPx+PxAijx˜T+2x˜TPxEijϖ+x˜TCTCx˜+2x˜TCTDϖ+ϖTDTDϖ−μc2ϖTϖ=∑i=1k∑j=1khihjx˜TAijTPx+PxAij+CTCx˜T+2x˜TPxEijϖ+2x˜TCTDϖ+ϖTDTDϖ−μc2ϖTϖ=x˜ϖTΘx˜ϖ,where(78)Θ=∑i=1k∑j=1khihjAijTPx+PxAij+CTCPxEij+CTD∗DTD−μc2I.
It is easy to find thatJ2<0 if Θ<0. Using the Schur complement, Θ<0 is equivalent to(79)∑i=1k∑j=1khihjAijTPx+PxAijPxEijCT∗−μc2IDT∗∗−I<0.
Let us define the matrixPx and its inverse Px−1:(80)Px=YSSTW,Px−1=XQQTZ.
Due toPxPx−1=I2n, we have(81)PxXQT=In0,PxXInQT0=InY0ST.
We will also define the matrices:(82)Π1=XIQT0,Π2=PxΠ1=IY0ST.
Pre- and post-multiplying (79) by diagΠ1T,I,I and its transpose and by using the variable change:(83)A^ij=YApi+BpiDciCpX+SBciCpX+YBpiCcjQT+SAcijQT,B^ci=YBpiDci+SBci,C^ci=DciCX+CciQT,D^ci=Dci.
Inequality (72) can thus be easily obtained.


## 6. Inverted Pendulum Example

Consider the nonlinear inverted pendulum system from [30]:(84)x˙1=x2,x˙2=gsinx1−mlax22sin2x1/2−bacosx1x4−acosx1F−fc4l/3−mlacosx12,x˙3=x4,x˙4=−mgasin2x1/2+4mla/3x22sinx1−bax4+4a/3F−fc4/3−macosx12,where m=2kg, M=8kg, l=0.5m, b=0.06Ns/rad, ρ=0.05, g=9.81ms−2, L=2m, a=1/m+M, and fc=ρsignx4. In [30], the above nonlinear system is expressed by two-rule T + S fuzzy model (2) with(85)Ap1=0100g4l/3−mla00ba4l/3−mla0001−mga4/3−ma00−ba4/3−ma,Bp1=0−a4l/3−mla04a/34/3−ma,Ep1=1101,Ap2=0100g22/π4l/3−mla/200ba2/24l/3−mla/20001−mga2/π4/3−ma/200−ba4/3−ma/2,Bp2=0−a2/24l/3−mla/204a/34/3−ma/2,Ep2=1101,Cp=100000100001,Np=010,Dp=00.50,With membership functions areh1x1=1−1/1+exp−14x1−π/8/1+exp−14x1+π/8 and h2x1=1−h1x1. The membership functions are chosen based on the method of sector nonlinearity [30].ChoosingAf=1, A˜4s=diag−3,−5,−7, H1=I2×2, and H2=I3×3 and solving LMI optimization problem given in Theorem 1, we can calculate the H∞ performance level μ=0.6128 and the following observer gains:(86)K¯11=1.52833.8106,K¯12=0.47470.7659,Gl1=−1.500−8.81060−4.3941−36.4793014.322602.9471−3.614900−1.08076,Gl2=−1.0607−5.76590−3.1132−18.1749011.342702.9514−0.649300−0.33576,Gn1=0−10−1.5000−3.8106001.080701−1.08070001,Gn2=0−10−1.0607−0.7659000.335701−0.33570001.According to Theorem3, we obtain the H∞ performance level, μc=1.0746, and the following controller gain matrices:(87)X=6.5946−22.8284−1.4260−0.3703−22.828489.05756.3415−10.5593−1.42606.34157.6086−7.8653−0.3703−10.5593−7.865321.3122,Y=106.4288−22.7293−12.3762−22.2923−22.72938.67294.14007.7688−12.37624.140083.78546.0534−22.29237.76886.053420.5865,Ac11=−4.8054−4.6433−29.6951186.83130.1944−1.9736−6.009913.6829−1.0255−3.2484−24.8435194.3215−0.20020.36652.2852−20.0596,Ac12=−4.8921−4.8143−31.5827397.35240.1805−1.9966−6.262344.6510−1.1243−3.4524−27.0566463.3002−0.18810.39102.5087−56.4758,Ac21=−3.7054−1.8250−9.847115.88560.1188−1.8469−6.206428.5907−0.7277−1.1952−15.1229198.4365−0.3068−0.0084−0.1040−1.6435,Ac22=−3.6976−1.8083−9.6846−4.72040.1008−1.8815−6.593779.3446−0.8225−1.3877−17.2258451.0355−0.3028−0.0007−0.0702−11.9604,Bc1=−152.14000.0680−193.0902−14.66857.4849−16.7177−180.1739−8.0082−296.1359−46.74320.705430.6986,Bc2=13.01366.172818.4474−30.5119−3.3066−14.3595−153.5522−52.0403−169.8636−61.53541.80053.2691,Cc1=0.06560.22331.5563−14.3353, Cc2=0.05470.08631.0121−32.8360, Dc1=12.98810.387916.8240, and Dc2=11.48623.78567.6871. In the corresponding simulation, the parameters associated with the equivalent output error injection ϑeq have been chosen to be ρ1=10, ρ2=15, ρ0=5, ρ¯=0.1, and δ=0.01 and initial conditions x10=π/20, x20=0, x30=2, and x40=0. The considered actuator and sensor faults have, respectively,(88)fa=0,t<2,0.5sin0.25t,2≤t<10,0.5sin0.75t,t≥10,fs=0,t<5,0.75,5≤t<15,1,t≥15.And the external disturbancesd are supposed to be random noises from −0.1 and 0.1. The simulation results are provided with online simultaneous actuator and sensor faults’ injection. Figures 1 and 2 indicate that the adaptive SMO can estimate existing faults simultaneously with satisfactory precision by rejecting the effects of disturbances.Figure 1
The actuator faultfa and its estimation.Figure 2
The sensor faultfs and its estimation.Simulation result for the output responsey2 (considered faulty) is provided in Figure 3. It is observed that the output without FTC does not converge to the output of the fault-free model (i.e., without any fault). However, the output trajectory of y2 with FTC reaches the output of the nominal model. Therefore, the proposed fuzzy DOFFTC design achieves the performance under faults and disturbances, and the stability of the closed-loop system is guaranteed while satisfying the prescribed ℋ∞ performance.Figure 3
Second closed-loop system output response: output response with free faults (black line), output response without FTC (red line), output response with FTC without adaptive law (orange line), and output response with the proposed FTC (blue line).


## 7. Conclusion

This paper focuses on the problems of FE and FTC for T-S fuzzy systems with unmeasurable PVs and having external disturbances, actuator, and sensor faults. An adaptive fuzzy SMO is designed for estimating the state, actuator, and sensor faults, simultaneously. Using the FE scheme, a DOFFTC is designed to compensate the faults and to stabilize the closed-loop system. Finally, simulation results show the effectiveness of the proposed methods.



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