# Robust Adaptive Control for a Class of T-S Fuzzy Nonlinear Systems with Discontinuous Multiple Uncertainties and Abruptly Changing Actuator Faults

Authors: Xin Ning; Yao Zhang; Zheng Wang
Journal: Complexity
 (2020)

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## Abstract

In the complex environment, the suddenly changing structural parameters and abrupt actuator failures are often encountered, and the negligence or unproper handling method may induce undesired or unacceptable results. In this paper, taking the suddenly changing structural parameters and abrupt actuator failures into consideration, we focus on the robust adaptive control design for a class of heterogeneous Takagi–Sugeno (T-S) fuzzy nonlinear systems subjected to discontinuous multiple uncertainties. The key point is that the switch modes not only vary with the system time but also vary with the system states, and the intrinsic heterogeneous characteristics make it difficult to design stable controllers. Firstly, the concepts of differential inclusion are introduced to describe the heterogeneous fuzzy systems. Meanwhile, a fundamental lemma is provided to demonstrate the criteria of the boundness for a Filippov solution. Then, by using the set-valued Lie derivative of the Lyapunov function and introducing a vector of specific continuous functions, the closed-loop T-S fuzzy differential inclusion systems are proved to be ultimately bounded. The sufficient conditions for system stability are derived in term of linear matrix inequalities (LMIs), which can be solved directly. Finally, a numerical example is provided to illustrate the effectiveness of the proposed control algorithm.


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## Body

## 1. Introduction

As it is well-known, the T-S fuzzy model is a powerful tool for the analysis and control design of nonlinear systems [1–4]. Therefore, a great wealthy of results has been achieved for T-S fuzzy systems in the past decades. In [5], the parameterized linear matrix inequality technique has been investigated for T-S fuzzy control systems. In [6], by using the LMIs and sum-of-squares-based approach, an output regulator has been constructed for the polynomial fuzzy control systems. In [7, 8], two fuzzy sliding mode control methods have been developed for the T-S fuzzy systems suffering from both matched and unmatched uncertainties. The fault-tolerant control approaches for T-S fuzzy nonlinear systems have been investigated in [9, 10]. As a further development, a finite-time fault-tolerant control structure of T-S fuzzy nonlinear systems has been synthesized in [11]. The event-triggered control structures for T-S fuzzy nonlinear systems can be found in [12, 13]. In [14, 15], the robust filters have been constructed for continuous and discrete-time T-S fuzzy systems, respectively. For the delayed T-S fuzzy systems with and without stochastic perturbations, the stability analysis and stabilization methods have been provided in [16, 17]. Chang et al. [18] focused on the robust adaptive control design for a class of heterogeneous T-S fuzzy nonlinear systems subjected to discontinuous multiple uncertainties.It is also well-known that the uncertainties and disturbances are often encountered in many practical systems. In [19, 20], the active disturbance rejection control methods have been reported for the nonlinear systems with uncertainties. In [21–24], several antidisturbance controllers have been synthesized by constructing the disturbance observers. The composite antidisturbance controllers can be found in [25–29]. Aiming at the uncertainties existing in the quantized control systems, several robust and adaptive controllers have been proposed in [30–32]. For the unknown nonlinearities existing in the control systems, fruitful results have been reported. In [33], a fuzzy adaptive output feedback controller has been proposed for the multi-input and multioutput nonlinear systems with completely unknown nonlinear functions. For a class of switched stochastic nonlinear systems in a pure-feedback form, a fuzzy observer is constructed to approximate unmeasurable states in [34, 35]. In [36], an adaptive fuzzy tracking control problem has been investigated for a class of nonstrict-feedback systems with unmeasured states and unknown nonlinearities. For the state constrained control systems with unknown nonlinear functions, two adaptive control results have been reported in [37, 38]. In [39], the problem of adaptive neural finite-time tracking control for uncertain nonstrict-feedback nonlinear systems with input saturation has been studied.In spite of the progress, in the aforementioned control results, a vital problem on heterogeneous uncertainties, in the sense that the parameters and the structures of the uncertainties keep switching with both the system time and the system states, was omitted. This kind of uncertainties and system nonlinearities are often encountered in many physical systems [40,41], and it is of significant importance to develop an effective controller for the heterogeneous uncertain nonlinear systems. When system-state-based switching uncertainties and system structures are taken into consideration, the traditional analysis and control methods become invalid. Furthermore, the control approaches of conventional switched systems cannot be applied neither because most of the switched systems are required to be purely time-based [42–44]. Moreover, when the fuzzy modeling methods and fuzzy control strategies are introduced in the heterogeneous uncertain system, the control design problem becomes more challenging and interesting. As far as the authors know, no results have been reported for the T-S fuzzy systems subjected to heterogeneous uncertainties and system structures. For the purpose of improving the practicability of the proposed algorithm, the abruptly changing actuator faults are also considered. Motivated by the above considerations, this paper is committed to develop an effective control structure for the T-S fuzzy heterogeneous systems with discontinuous multiple uncertainties and abruptly changing actuator faults. Compared to the existing literature, the main contributions of this paper are as follows:(i)
To the best of the authors’ knowledge, it is the first control solution for the T-S fuzzy systems subjected to heterogeneous uncertainties and system structures, which keep switching with both the system time and the system states.(ii)
A fundamental lemma is provided to demonstrate the criteria of the boundness for a Filippov solution, establishing the mathematical fundamentals for the adaptive control of differential inclusion systems.(iii)
By proposing a specific vector of continuous functions, the closed-loop multivariable T-S fuzzy differential inclusion systems are proved to be ultimately bounded for the first time.


## 2. Problem Formulation and Preliminaries

### 2.1. Problem Statement

Consider the following T-S fuzzy nonlinear systems.Plant Rule: IF θi,1t is μi,1, θi,2t is μi,2, … and θi,pt is μi,p, THEN(1)x˙t=fi1t,xt+Ai1x+Biu,ift,xt∈G1,fi2t,xt+Ai2x+ΔAitx+Biu+dt,ift,xt∈G2,fi3t,xt+Ai3x+ΔAitx+BiΛtu+εut+dt,ift,xt∈G3,where ut∈ℝm is the input signal of the system and xt∈ℝn are the system state vector. θt=θ1t,θ2t,…,θpt are the premise variables. μi,1,μi,2,…,μi,p,i∈Υ=1,2,…,r are the fuzzy sets. r is the number of IF-THEN rules. Gk,k=1,2,3 are the open connected sets satisfying that ∪k=13Gk=ℝ+×ℝn and Gk1∩Gk2=∅,∀k1,k2∈1,2,3,k1≠k2. S=∪k=13∂Gk is of Lebesgue measure zero, where ∂Ω represents the boundary set of a set Ω. For ∀i∈Υ,k=1,2,3,Aik∈ℝn×n,Bi∈ℝn×m,andD∈ℝn×p3 are all known matrices. fikt,xt is a vector of unknown nonlinear functions. ΔAit∈ℝn×n is an unknown matrix varying with the time. Λt is a time-varying diagonal matrix of remanent actuator effectiveness and εut is a vector of actuator deviations when the actuator failure occurs. dt represents the external disturbance. The symbols used in the paper can be found in Table 1.Table 1
The symbols used in the paper.
VariableImplicationθtPremise variablesGkOpen connected sets∂GkLebesgue measure zero∂ΩBoundary set of a setΩfikt,xtUnknown nonlinear functionsAik,Bi,DKnown matricesΔAitUnknown matrix varying with timeΛtMatrix of remanent actuator effectivenessεutVector of actuator deviationsdtExternal disturbanceRemark 1.
It should be highlighted that system model (1) can reflect the actual situation of many practical engineering systems and possesses important research significance. If t,xt stays in G1, the system is under a normal condition. When t,xt enters G2 from G1, the system uncertainties ΔAit suddenly appear and external disturbances dt grow rapidly. Finally, if the system states travel into G3, the actuator faults abruptly come out. Accompanied by t,xt entering into different regions, the system matrix and the nonlinear function vector will change.
It is supposed thatθit never depends on ut, εt, and dt. Define(2)hiθt=∏j=1pμi,jθjt∑i=1r∏j=1pμi,jθjt.
∀i∈Υ,j=1,2,…,p, μi,jθjt denote the degree of membership θjt in μi,j. hiθt is the fuzzy basis function. Clearly, for any i∈Υ, hiθt≥0 and ∑i=1rhiθt=1. Therefore, the dynamics of system (1) can be rewritten as follows:(3)x˙t=gt,xt,gt,xt=∑i=1rhiθtgi1t,xtift,xt∈G1,∑i=1rhiθtgi2t,xtift,xt∈G2,∑i=1rhiθtgi3t,xtift,xt∈G3,where(4)gi1t,xt=fi1t,xt+Ai1x+Biu,gi2t,xt=fi2t,xt+Ai2x,+ΔAitx+Biu+dt,gi3t,xt=fi3t,xt+Ai3x+ΔAitx+BiΛtu+εut+dt.
Our control objective is to design an adaptive controller such that system statesxt can converge into a desired compact set in the presence of the discontinuous multiple uncertainties and abruptly changing actuator faults.
To achieve the control objective, the following assumptions are necessary.Assumption 1.
For anyi∈Υ, k=1,2,3, fikt,xt satisfy fik0,x0=0 and(5)fikt,x1t−fikt,x2t≤Uikx1t−x2t.Assumption 2.
For anyi∈Υ,(6)ΔAit=MiΞitNi,where Ξit are unknown time-varying matrices satisfying ΞiTtΞit≤I and Mi and Ni are known matrices.Assumption 3.
The discontinuous disturbancesdit are Lebesgue measurable and locally bounded, i.e., dit≤Di, where Di is an unknown positive constant.Assumption 4.
It is supposed thatΛt and εut are both Lebesgue measurable, and there exist lower and upper bounds for Λt and εut, respectively. In other words, λminΛt>εΛ>0, εut≤Du. εΛ and Du are positive constants.


### 2.2. Preliminaries

Consider a nonlinear system:(7)x˙=gxt,where x∈ℝ, g:ℝ×0,∞⟶ℝ is Lebesgue measurable, essentially locally bounded and uniformly in t. Moreover, there exist discontinuities in gxt.Definition 1.
A vectorxt is called a Filippov solution of differential equation (7) over 0,∞ if xt is absolutely continuous, and for almost everywhere t∈0,∞,(8)x˙∈Fgxt,where ℱgxt is a upper semicontinuous set-valued map defined by(9)FgxtΔ=⋂ρ>0∩μN=0co−gBxt,ρN,where ∩μN=0 denotes the intersection over sets of Lebesgue measure zero, co− represents the convex closure, and Bxt,ρ=z∈ℝn|xt−z<ρDefinition 2.
(see [45]). Given a locally Lipschitz function Vxt, the generalized gradient of Vxt is defined by(10)∂Vxt=co−lim∇Vx∗t|x∗t⟶xt,x∗t∉ΩV,where Ωv is the set of measure zero and ∇V is not defined. Moreover, the set-valued Lie derivative of a Vxt is defined as(11)DVxtΔ=∩ξ∈∂VxtξTFgxt.Definition 3.
In this paper, the generalized sections for variables, vectors, and matrices are defined. Fora,b∈ℝ, a<b, define b,a=a,b=mina,b,maxa,b. For a,b∈ℝn, define a,b=b,a=ai,bin, where ai is the ith component of a. For A,B∈ℝm×n, define A,B=B,A=Aij,Bijm×n.Lemma 1.
Consider nonlinear system (7). Suppose g⋅ is Lebesgue measurable and x⟶gxt is bounded. Let V:ℝ×0,∞⟶ℝ be locally Lipschitz and regular such that(12)α1xt≤Vxt≤α2xt,∀t≥0,DVxt≤−γ1Vxt+γ2,where α1⋯ and α2⋯ are K∞ functions, γ1>0 and γ2>0 are constants, xt is a Filippov solution of (7) with initial value xt0∈Ω2=x∈ℝn|x<r,α2x≤ρ, and r>0 and ρ>0 are constants satisfying that γ2/γ1<ρ<minxt=rα1xt. Then, xt is bounded and converges to a compact set:(13)Ωf=x∈ℝn|Vx≤γ2γ1.Proof.
DefineΩV=x∈ℝn|x<r,Vx≤ρ. Since α2x≤ρ⟹Vx≤ρ, it is obvious that Ω2⊂ΩV. On the contrary, with the aid of Vx≤ρ⟹α1x≤ρ, we can easily get that ΩV⊂Ω1. Moreover, considering ρ<minxt=rα1xt, we know that Ω1=x∈ℝn|α1x≤ρ is an interior set of Br=x∈ℝn|x<r. Hence,(14)Ω2⊂ΩV⊂Ω1⊂Br.
IfVxt≤γ2/γ1, it is clear that xt stays in Ωf, which is xt∈Ωf⊂ΩV⊂Br. On the contrary, for any xt∉Ωf, we can obtain that Vxt>γ2/γ1 and V˙<a.e.0. Hence, Vxt is nonincreasing. Furthermore, for any xt0∈Ω2, the solution of (7) stays in ΩV, which is xt∈ΩV⊂Br. Therefore, it can be concluded that the solution of system (7) is bounded for all t≥t0. Furthermore, it follows from (12) that(15)∫t0t−γ1Vxτ+γ2dτ≤∫t0tV˙xτdτ=Vxt0−Vxt≤Vxt0.
Therefore,∫t0t−γ1Vxτ+γ2dτ is bounded for all t≥t0. It should be noted that the existence of ∫t0t−γ1Vxτ+γ2dτ can be guaranteed because −γ1Vxt+γ2 is locally Lipschitz and regular by definition. According to Barbalat’s Lemma [46], it can be obtained that −γ1Vxt+γ2⟶0 as t⟶∞. Therefore, we know that xt will converge into Ωf finally. The proof is complete.Lemma 2.
Given any constantε>0 and any vector ξ∈ℝn, the following inequality holds:(16)ξ<ξTξξTξ+ε2+ε.Proof.
Sinceε>0 and ξTξ+ε2>0, it can be easily get that(17)ξTξ+εξTξ+ε22−ξξTξ+ε22=2εξTξξTξ+ε2+ε4>0.
Hence, we know that(18)ξξTξ+ε2<ξTξ+εξTξ+ε2.
By dividingξTξ+ε2 in both sides of (18), inequality (16) can be obtained. The proof is completed.


## 2.1. Problem Statement

Consider the following T-S fuzzy nonlinear systems.Plant Rule: IF θi,1t is μi,1, θi,2t is μi,2, … and θi,pt is μi,p, THEN(1)x˙t=fi1t,xt+Ai1x+Biu,ift,xt∈G1,fi2t,xt+Ai2x+ΔAitx+Biu+dt,ift,xt∈G2,fi3t,xt+Ai3x+ΔAitx+BiΛtu+εut+dt,ift,xt∈G3,where ut∈ℝm is the input signal of the system and xt∈ℝn are the system state vector. θt=θ1t,θ2t,…,θpt are the premise variables. μi,1,μi,2,…,μi,p,i∈Υ=1,2,…,r are the fuzzy sets. r is the number of IF-THEN rules. Gk,k=1,2,3 are the open connected sets satisfying that ∪k=13Gk=ℝ+×ℝn and Gk1∩Gk2=∅,∀k1,k2∈1,2,3,k1≠k2. S=∪k=13∂Gk is of Lebesgue measure zero, where ∂Ω represents the boundary set of a set Ω. For ∀i∈Υ,k=1,2,3,Aik∈ℝn×n,Bi∈ℝn×m,andD∈ℝn×p3 are all known matrices. fikt,xt is a vector of unknown nonlinear functions. ΔAit∈ℝn×n is an unknown matrix varying with the time. Λt is a time-varying diagonal matrix of remanent actuator effectiveness and εut is a vector of actuator deviations when the actuator failure occurs. dt represents the external disturbance. The symbols used in the paper can be found in Table 1.Table 1
The symbols used in the paper.
VariableImplicationθtPremise variablesGkOpen connected sets∂GkLebesgue measure zero∂ΩBoundary set of a setΩfikt,xtUnknown nonlinear functionsAik,Bi,DKnown matricesΔAitUnknown matrix varying with timeΛtMatrix of remanent actuator effectivenessεutVector of actuator deviationsdtExternal disturbanceRemark 1.
It should be highlighted that system model (1) can reflect the actual situation of many practical engineering systems and possesses important research significance. If t,xt stays in G1, the system is under a normal condition. When t,xt enters G2 from G1, the system uncertainties ΔAit suddenly appear and external disturbances dt grow rapidly. Finally, if the system states travel into G3, the actuator faults abruptly come out. Accompanied by t,xt entering into different regions, the system matrix and the nonlinear function vector will change.
It is supposed thatθit never depends on ut, εt, and dt. Define(2)hiθt=∏j=1pμi,jθjt∑i=1r∏j=1pμi,jθjt.
∀i∈Υ,j=1,2,…,p, μi,jθjt denote the degree of membership θjt in μi,j. hiθt is the fuzzy basis function. Clearly, for any i∈Υ, hiθt≥0 and ∑i=1rhiθt=1. Therefore, the dynamics of system (1) can be rewritten as follows:(3)x˙t=gt,xt,gt,xt=∑i=1rhiθtgi1t,xtift,xt∈G1,∑i=1rhiθtgi2t,xtift,xt∈G2,∑i=1rhiθtgi3t,xtift,xt∈G3,where(4)gi1t,xt=fi1t,xt+Ai1x+Biu,gi2t,xt=fi2t,xt+Ai2x,+ΔAitx+Biu+dt,gi3t,xt=fi3t,xt+Ai3x+ΔAitx+BiΛtu+εut+dt.
Our control objective is to design an adaptive controller such that system statesxt can converge into a desired compact set in the presence of the discontinuous multiple uncertainties and abruptly changing actuator faults.
To achieve the control objective, the following assumptions are necessary.Assumption 1.
For anyi∈Υ, k=1,2,3, fikt,xt satisfy fik0,x0=0 and(5)fikt,x1t−fikt,x2t≤Uikx1t−x2t.Assumption 2.
For anyi∈Υ,(6)ΔAit=MiΞitNi,where Ξit are unknown time-varying matrices satisfying ΞiTtΞit≤I and Mi and Ni are known matrices.Assumption 3.
The discontinuous disturbancesdit are Lebesgue measurable and locally bounded, i.e., dit≤Di, where Di is an unknown positive constant.Assumption 4.
It is supposed thatΛt and εut are both Lebesgue measurable, and there exist lower and upper bounds for Λt and εut, respectively. In other words, λminΛt>εΛ>0, εut≤Du. εΛ and Du are positive constants.


## 2.2. Preliminaries

Consider a nonlinear system:(7)x˙=gxt,where x∈ℝ, g:ℝ×0,∞⟶ℝ is Lebesgue measurable, essentially locally bounded and uniformly in t. Moreover, there exist discontinuities in gxt.Definition 1.
A vectorxt is called a Filippov solution of differential equation (7) over 0,∞ if xt is absolutely continuous, and for almost everywhere t∈0,∞,(8)x˙∈Fgxt,where ℱgxt is a upper semicontinuous set-valued map defined by(9)FgxtΔ=⋂ρ>0∩μN=0co−gBxt,ρN,where ∩μN=0 denotes the intersection over sets of Lebesgue measure zero, co− represents the convex closure, and Bxt,ρ=z∈ℝn|xt−z<ρDefinition 2.
(see [45]). Given a locally Lipschitz function Vxt, the generalized gradient of Vxt is defined by(10)∂Vxt=co−lim∇Vx∗t|x∗t⟶xt,x∗t∉ΩV,where Ωv is the set of measure zero and ∇V is not defined. Moreover, the set-valued Lie derivative of a Vxt is defined as(11)DVxtΔ=∩ξ∈∂VxtξTFgxt.Definition 3.
In this paper, the generalized sections for variables, vectors, and matrices are defined. Fora,b∈ℝ, a<b, define b,a=a,b=mina,b,maxa,b. For a,b∈ℝn, define a,b=b,a=ai,bin, where ai is the ith component of a. For A,B∈ℝm×n, define A,B=B,A=Aij,Bijm×n.Lemma 1.
Consider nonlinear system (7). Suppose g⋅ is Lebesgue measurable and x⟶gxt is bounded. Let V:ℝ×0,∞⟶ℝ be locally Lipschitz and regular such that(12)α1xt≤Vxt≤α2xt,∀t≥0,DVxt≤−γ1Vxt+γ2,where α1⋯ and α2⋯ are K∞ functions, γ1>0 and γ2>0 are constants, xt is a Filippov solution of (7) with initial value xt0∈Ω2=x∈ℝn|x<r,α2x≤ρ, and r>0 and ρ>0 are constants satisfying that γ2/γ1<ρ<minxt=rα1xt. Then, xt is bounded and converges to a compact set:(13)Ωf=x∈ℝn|Vx≤γ2γ1.Proof.
DefineΩV=x∈ℝn|x<r,Vx≤ρ. Since α2x≤ρ⟹Vx≤ρ, it is obvious that Ω2⊂ΩV. On the contrary, with the aid of Vx≤ρ⟹α1x≤ρ, we can easily get that ΩV⊂Ω1. Moreover, considering ρ<minxt=rα1xt, we know that Ω1=x∈ℝn|α1x≤ρ is an interior set of Br=x∈ℝn|x<r. Hence,(14)Ω2⊂ΩV⊂Ω1⊂Br.
IfVxt≤γ2/γ1, it is clear that xt stays in Ωf, which is xt∈Ωf⊂ΩV⊂Br. On the contrary, for any xt∉Ωf, we can obtain that Vxt>γ2/γ1 and V˙<a.e.0. Hence, Vxt is nonincreasing. Furthermore, for any xt0∈Ω2, the solution of (7) stays in ΩV, which is xt∈ΩV⊂Br. Therefore, it can be concluded that the solution of system (7) is bounded for all t≥t0. Furthermore, it follows from (12) that(15)∫t0t−γ1Vxτ+γ2dτ≤∫t0tV˙xτdτ=Vxt0−Vxt≤Vxt0.
Therefore,∫t0t−γ1Vxτ+γ2dτ is bounded for all t≥t0. It should be noted that the existence of ∫t0t−γ1Vxτ+γ2dτ can be guaranteed because −γ1Vxt+γ2 is locally Lipschitz and regular by definition. According to Barbalat’s Lemma [46], it can be obtained that −γ1Vxt+γ2⟶0 as t⟶∞. Therefore, we know that xt will converge into Ωf finally. The proof is complete.Lemma 2.
Given any constantε>0 and any vector ξ∈ℝn, the following inequality holds:(16)ξ<ξTξξTξ+ε2+ε.Proof.
Sinceε>0 and ξTξ+ε2>0, it can be easily get that(17)ξTξ+εξTξ+ε22−ξξTξ+ε22=2εξTξξTξ+ε2+ε4>0.
Hence, we know that(18)ξξTξ+ε2<ξTξ+εξTξ+ε2.
By dividingξTξ+ε2 in both sides of (18), inequality (16) can be obtained. The proof is completed.


## 3. Main Results

### 3.1. Control Design

In the following text, the robust adaptive control problem for the concerned T-S fuzzy discontinuous nonlinear systems will be addressed.In view of (3), according to Definition 1, we can get the following differential inclusion:(19)x˙∈∑i=1rhiθtFfit,xt+FAixt+FΔA¯itxt+BiFΛ¯tut+BiFd¯t,where(20)fi=fikt,xtift,xt∈Gk,k=1,2,3,Ai=Aikift,xt∈Gk,k=1,2,3,ΔA¯it=0ift,xt∈G1,ΔAitift,xt∈G2∪G3,Λ¯t=Iift,xt∈G1∪G2,Λtift,xt∈G3,d¯t=0ift,xt∈G1,dtift,xt∈G2,εut+dtift,xt∈G3.DefineD=supt≥0εut+dt,ϑ=1/inft≥0λminΛt. Considering system (19), the control law is designed as follows.Controller RuleRp,i: IF θ1t is μi,1, θ2t is μi,2 and … and θpt is μi,p, THEN(21)ut=−ϑ^φϑxt,vt,vt=Kixt−D^φDxt,where Ki is the control gain matrix to be designed and ϑ^ and D^ are the estimations of ϑ and D, respectively. The continuous functions φϑxt,vt and φDxt are defined as(22)φϑxt,vt=φϑ,1t,φϑ,2t,…,φϑ,mtT,φϑ,lxt,vt=vltzltϑ^vltzltϑ^vlt2+εv2,l=1,2,…,m,φDxt=BTPxtxTtPBBTPxt+εD2,where(23)B=∑i=1rhiθtBi,P∈ℝn×n is a symmetric positive matrix. εv,εD>0 are design constants. zlt is the lth component of zt=xTtPℬT. The adaptive parameters are updated by(24)D^˙=ηDxTtPBφDxt−ηDσDD^,ϑ^˙=−ηϑxTtPBvt−ηϑσϑϑ^,where ηD,ηϑ>0 are the gains of adaptive laws and σD,σϑ>0 are design constants.Remark 2.
Note that it is improper to separately design the control laws for eachGk,k=1,2,3, and a universal controller has to be developed for all the three modes. In practical, the condition of multiple uncertainties suddenly changes or actuator failure abruptly occurs which cannot be determined easily. Moreover, the condition is concerned not only with the system time but also with the system states, which makes this problem more complex. Since the boundaries of Gk are unknown in practical, the separately design methods cannot be applied and a universal control law which is applicable for all the three modes is necessary. In this paper, Gk,k=1,2,3, are only used for analysis, but are not used in control design.Remark 3.
The considered system cannot be controlled by using the controllers of conventional switched systems possessing switching signals those only vary with time. The three modes of the concerned system are distinguished by using the conditions concerned with both the system time and the system states. In this paper, the switchings among the three modes are more intrinsic and are difficult to be dealt with. In fact, the proposed controller can degrade into an asynchronous control law ifGk,k=1,2,3 is only concerned with time. For the considered Gk, the proposed controller can be thought of as a deep asynchronous controller.


### 3.2. Stability Analysis

Defineϑ˜t=ϑ^t−ϑt,D˜t=D^t−Dt. Combining (19)–(21) yields(25)x˙∈∑i=1rhiθt∑j=1rhjθtFfixt+FAi+BiKjxt+FΔA¯itxt+BiFd¯t−BiD^φDxt+BiFΛ¯tut−Bivt.Theorem 1.
Consider the closed-loop fuzzy differential inclusion (25) under Assumptions 1–4. The fuzzy controller is designed as (21) and the adaptive parameters are updated by (24). Given scalars α,λ>0. For any i,j∈Υ and k∈1,2,3, if there exist matrices P,K such that(26)Πi,jk<0,i=1,2,…,r,Πi,jk+Πi,jk<0,1≤i<j≤r,where(27)Πi,jk=Πi,j,11kPP−1λ2I,Πi,j,11k=PAik+BiKj+Aik+BiKjTP+1λ2UikTUik+PMiMiTP+NiTNi+αP,then, for any initial conditions, the Filippov solution of closed-loop fuzzy differential inclusion (25) is bounded and converge to a compact set:(28)Ωf=x∈ℝn|Vx≤εf/β,where(29)εf=σDD22+σϑϑ2+DεD+mεvϑ,β=minα,ηDσD,ηϑ,σϑ.Proof.
Select a Lyapunov functional candidate as follows:(30)Vxt=12xTtPxt+12ηDD˜2+12ηϑϑ˜2,where P is a positive definite matrix. According to Definition 2, the set-valued Lie derivative of a Vxt can be taken as(31)DVxt=∩ξ∈∂VxtξTψxt,D^,vt,D^˙,ϑ^˙T,where(32)ψxt,D^,vt=∑i=1rhiθt∑j=1rhjθtFfixt+FAi+BiKjxt+FΔA¯itxt+Fd¯t−BiD^φDxt+BiFΛ¯tut−Bivt.
By definingπfi,πAij,πΔAi,πΛ,andπd such that(33)πfi∈fikift,xt∈Gk,fik1fik2ift,xt∈∂Gk1∩∂Gk2,πAij∈Aik+BiKjift,xt∈Gk,Aik1+BiKj,Aik2+BiKjift,xt∈∂Gk1∩∂Gk2,πΔAi∈0ift,xt∈G1,ΔAiift,xt∈G2∪G3,0,ΔAiift,xt∈∂G1∩∂G2∪G3,πΛ∈Iift,xt∈G1∪G2,Λtift,xt∈G3,Λt,Iift,xt∈∂G1∪G2∩∂G3,πd∈0ift,xt∈G1,dift,xt∈G2,εu+dift,xt∈G3,0,dift,xt∈∂G1∩∂G2,d,εu+dift,xt∈∂G2∩∂G3,0,εu+dift,xt∈∂G1∩∂G3,we can rewrite (30) as(34)DVxt=D1Vxt+D2Vxt,where(35)D1Vxt=∑i=1rhiθt∑j=1rhjθtxTtPπfi+xTtPπAijxt+xTtPπΔAixt,(36)D2Vxt=∑i=1rhiθtxTtPBiπd−xTtPBiD^φDxt+xTtPBiπΛut−xTtPBivt+1ηDD˜D^˙+1ηϑϑ˜ϑ^˙.
Firstly, the analysis ofD1Vxt are given. Since P is nonsingular, it can be proved that PπΔAi∈0,PΔAi holds for all t,xt. Hence, based on Assumption 2, it can be known that, for any t,xt, xTtPπΔAixt∈0,xTtPΔAixt and(37)xTtPπΔAixt≤12xTtPMiMiTPxt+12xTtNiTNixt.
Fort,xt∈Gk,k=1,2,3, it follows from Assumption 1 that(38)πfiTπfi≤xTtUikTUikxt,PπAij=12PAik+BiKj+Aik+BiKjTP.
By combining (35), (37), and (38), we know that, for t,xt∈Gk,k=1,2,3, the following inequality holds:(39)D1Vxt≤12∑i=1rhiθt∑j=1rhjθtxtπfiTΦi,jkxtπfi,where(40)Φi,jk=Φi,j,11kPP−1λ2,Φi,j,11k=PAik+BiKj+Aik+BiKjTP+1λ2UikTUik+PMiMiTP+NiTNi.
By using (26), it can be checked that, for t,xt∈Gk,k=1,2,3,(41)D1Vxt≤−α2xTtPxt.
On the contrary, fort,xt∈∂Gk1∩∂Gk2,k1,k2∈1,2,3, it can be proved that(42)xTtPπAijxt∈xTtPAik1+BiKjxt,xTtPAik2+BiKjxt,πfiTπfi∈fik1Tfik1,fik2Tfik2∈xTtUik1TUik1xt,xTtUik2TUik2xt.
Accordingly, the following inequalities can be obtained:(43)xTtPπAijxt≤maxk∈1,2,3xTtPAik+BiKjxt,πfiTπfi≤maxk∈1,2,3xTtUik2TUik2xt.
By combining (35), (37), and (43), we know that, for t,xt∈∂Gk1∩∂Gk2,k1,k2∈1,2,3,(44)D1Vxt≤12∑i=1rhiθt∑j=1rhjθtxtπfiTΨi,jkxtπfi,where(45)Ψi,jk=Ψi,j,11kPP−1λ2,Ψi,j,11k=maxk∈1,2,3PAik+BiKj+Aik+BiKjTP+1λ2UikTUik+PMiMiTP+NiTNi.
Since (26) holds for any k∈1,2,3, we know that, for t,xt∈∂Gk1∩∂Gk2,k1,k2∈1,2,3, D1Vxt≤−α/2xTtPxt also holds. Then, by considering (41), it can be concluded that, for any t,xt,(46)D1Vxt≤−α2xTtPxt.
Next texts provide the analysis ofD2Vxt. From (33), we know that, for t,xt,(47)πd∈0,εu+d,πΛ,i∈Λit,1,where πΛ,i and Λit represent the ith component of πΛ and Λt on the diagonal line, respectively. Hence, it can be obtained that, for any t,xt,(48)∑i=1rhiθtxTtPBiπd∈0,xTtPBεu+d,∑i=1rhiθtxTtPBiπΛut∈xTtPBΛtut,xTtPBut.
From Lemma2, it is easy to know that(49)∑i=1rhiθtxTtPBiπd−xTtPBiD^φDxt≤DxTtPB−xTtPBD^φDxt≤DxTtPBxTtPBTxTtPBxTtPBT+εD2−xTtPBD^φDxt+DεD≤−D˜xTtPBφDxt+DεD.
Meanwhile, simple computation shows that(50)∑i=1rhiθtxTtPBiπΛut−xTtPBivt=−ϑ^∑l=1mzlπΛ,lφϑ,lxt,vt−xTtPBvt.
Since0<1/ϑ≤πΛ,i∈Λit,1, we can get that(51)−ϑ^zlπΛ,lφϑ,lxt,vt=−zlπΛ,lϑ^vltzltϑ^vltzltϑ^vlt2+εv2≤−1ϑzltϑ^vlt2zltϑ^vlt2+εv2.
Hence, it follows from Lemma2 that(52)−ϑ^∑l=1mzlπΛ,lφϑ,lxt,vt−xTtPBvt≤∑l=1m−1ϑzltϑ^vlt2zltϑ^vlt2+εv2−zltvlt≤∑l=1m1ϑ−zltϑ^vlt+εv−zltvlt≤∑l=1mϑ^ϑzltvlt−zltvlt+εvϑ≤ϑ˜ϑxTtPBvt+mεvϑ.
By combining (36) and (49)–(52), it can be obtained that(53)D2Vxt≤−D˜xTtPBφDxt+ϑ˜ϑxTtPBvt+1ηDD˜D^˙+1ηϑϑϑ˜ϑ^˙+DεD+mεvϑ,where m is the dimension of the system input signal ut. Substituting (24) into (53) yields(54)D2Vxt≤−σDD˜D^−σϑϑ˜ϑ^ϑ+DεD+mεvϑ.
By using the following inequalities,(55)2D^D˜≤−D˜2+D2,ϑ˜ϑ^≤−ϑ˜2+ϑ2.
We know that(56)D2Vxt≤−σD2D˜2−σϑϑ˜22ϑ+εf.
By combining (34), (46), and (56), we know that(57)DVxt≤−α2xTtPxt−σD2D˜2−σϑϑ˜22ϑ+εf,which means(58)DVxt≤−βVxt+εf,where βandεf are defined in (29). According to Lemma 1, it can be proved that the Filippov solution of closed-loop fuzzy differential inclusion (25) is bounded and converge to Ωf. The proof is complete.Remark 4.
It should be noted that the final compact setΩf can be an arbitrarily small neighborhood of the origin by adjusting the control gains and adaptive parameters. Moreover, in most of the adaptive control results, the ultimate boundness of the closed-loop control system is finally ensured. However, for the differential inclusion systems, the criteria of ultimately boundness have never been provided. Hence, it can be concluded that Lemma 1 lays the mathematical fundamentals for the adaptive control of differential inclusion systems.
Next, we will provide the computation method of control gainK and matrix P which is necessary in adaptive laws.Theorem 2.
Consider the closed-loop fuzzy differential inclusion (25) under Assumptions 1–4. The fuzzy controller is designed as (20), and the adaptive parameters are updated by (24). Given scalars α,λ>0. For any i,j∈Υ and k∈1,2,3, if there exist matrices Q,Rj such that(59)Γi,jk<0,i=1,2,…,r,Γi,jk+Γi,jk<0,1≤i<j≤r,where(60)Γi,jk=Γi,j,11kQUikTQNiTI∗−λ2I00∗∗−I0∗∗∗−1λ2I,Γi,j,11k=AikQ+QAik+BiRj+RjTBiT+αQ+MiMiT,then, for any initial conditions, the Filippov solution of closed-loop fuzzy differential inclusion (25) is bounded and converges to the compact set Ωf. Moreover, if condition (59) is feasible, it can be obtained that(61)Kj=RjQ−1,P=Q−1.Proof.
DefineQ=P−1andRj=KjP−1. By performing a congruence transformation to with Πi,jk, we can get the following matrix:(62)Π⌢i,jk=Π⌢i,j,11kII−1λ2I,where(63)Π⌢i,j,11k=AikQ+QAik+BiRj+RjTBiT+αQ+1λ2QUikTUikQ+MiMiT+QNiTNiQ.
By performing a Schur complement transformation to (61), Γi,jk can be obtained. Since(64)Γi,jk<0⇔Π⌢i,jk<0⇔Πi,jk<0,we know that (26) is satisfied by using (58). According to Theorem 1, the boundedness and convergence of the Filippov solution can be guaranteed. The proof is complete.Remark 5.
System who involves state-based switching usually has more complex dynamic behaviors which motivates various useful applications, while the construction of rigorous stability for these systems is challenging. Different to classic analysis methods in switching systems such as average dwell time (ADT), differential-inclusion-based approaches provide feasible alternative solutions to the stability analysis of stated-based switching systems. Instead of analyzing the value of the vector field at individual points, differential-inclusion-based methods focus on the behavior of vector field at the neighborhood of each point. This idea is at the core of constructing continuous Filippov solution which is a general solution to the original differential equations, where the discontinuities is covered by set-valued mapping which is a convex combination of vector field around them. As a result, rigorous stability result can be established for discontinuous vector field as long as the Lebesgue measure of the discontinuities is zero. From the illustration above, it is clear that, for state-based switching systems, the superiority of utilizing differential inclusion is significant.


## 3.1. Control Design

In the following text, the robust adaptive control problem for the concerned T-S fuzzy discontinuous nonlinear systems will be addressed.In view of (3), according to Definition 1, we can get the following differential inclusion:(19)x˙∈∑i=1rhiθtFfit,xt+FAixt+FΔA¯itxt+BiFΛ¯tut+BiFd¯t,where(20)fi=fikt,xtift,xt∈Gk,k=1,2,3,Ai=Aikift,xt∈Gk,k=1,2,3,ΔA¯it=0ift,xt∈G1,ΔAitift,xt∈G2∪G3,Λ¯t=Iift,xt∈G1∪G2,Λtift,xt∈G3,d¯t=0ift,xt∈G1,dtift,xt∈G2,εut+dtift,xt∈G3.DefineD=supt≥0εut+dt,ϑ=1/inft≥0λminΛt. Considering system (19), the control law is designed as follows.Controller RuleRp,i: IF θ1t is μi,1, θ2t is μi,2 and … and θpt is μi,p, THEN(21)ut=−ϑ^φϑxt,vt,vt=Kixt−D^φDxt,where Ki is the control gain matrix to be designed and ϑ^ and D^ are the estimations of ϑ and D, respectively. The continuous functions φϑxt,vt and φDxt are defined as(22)φϑxt,vt=φϑ,1t,φϑ,2t,…,φϑ,mtT,φϑ,lxt,vt=vltzltϑ^vltzltϑ^vlt2+εv2,l=1,2,…,m,φDxt=BTPxtxTtPBBTPxt+εD2,where(23)B=∑i=1rhiθtBi,P∈ℝn×n is a symmetric positive matrix. εv,εD>0 are design constants. zlt is the lth component of zt=xTtPℬT. The adaptive parameters are updated by(24)D^˙=ηDxTtPBφDxt−ηDσDD^,ϑ^˙=−ηϑxTtPBvt−ηϑσϑϑ^,where ηD,ηϑ>0 are the gains of adaptive laws and σD,σϑ>0 are design constants.Remark 2.
Note that it is improper to separately design the control laws for eachGk,k=1,2,3, and a universal controller has to be developed for all the three modes. In practical, the condition of multiple uncertainties suddenly changes or actuator failure abruptly occurs which cannot be determined easily. Moreover, the condition is concerned not only with the system time but also with the system states, which makes this problem more complex. Since the boundaries of Gk are unknown in practical, the separately design methods cannot be applied and a universal control law which is applicable for all the three modes is necessary. In this paper, Gk,k=1,2,3, are only used for analysis, but are not used in control design.Remark 3.
The considered system cannot be controlled by using the controllers of conventional switched systems possessing switching signals those only vary with time. The three modes of the concerned system are distinguished by using the conditions concerned with both the system time and the system states. In this paper, the switchings among the three modes are more intrinsic and are difficult to be dealt with. In fact, the proposed controller can degrade into an asynchronous control law ifGk,k=1,2,3 is only concerned with time. For the considered Gk, the proposed controller can be thought of as a deep asynchronous controller.


## 3.2. Stability Analysis

Defineϑ˜t=ϑ^t−ϑt,D˜t=D^t−Dt. Combining (19)–(21) yields(25)x˙∈∑i=1rhiθt∑j=1rhjθtFfixt+FAi+BiKjxt+FΔA¯itxt+BiFd¯t−BiD^φDxt+BiFΛ¯tut−Bivt.Theorem 1.
Consider the closed-loop fuzzy differential inclusion (25) under Assumptions 1–4. The fuzzy controller is designed as (21) and the adaptive parameters are updated by (24). Given scalars α,λ>0. For any i,j∈Υ and k∈1,2,3, if there exist matrices P,K such that(26)Πi,jk<0,i=1,2,…,r,Πi,jk+Πi,jk<0,1≤i<j≤r,where(27)Πi,jk=Πi,j,11kPP−1λ2I,Πi,j,11k=PAik+BiKj+Aik+BiKjTP+1λ2UikTUik+PMiMiTP+NiTNi+αP,then, for any initial conditions, the Filippov solution of closed-loop fuzzy differential inclusion (25) is bounded and converge to a compact set:(28)Ωf=x∈ℝn|Vx≤εf/β,where(29)εf=σDD22+σϑϑ2+DεD+mεvϑ,β=minα,ηDσD,ηϑ,σϑ.Proof.
Select a Lyapunov functional candidate as follows:(30)Vxt=12xTtPxt+12ηDD˜2+12ηϑϑ˜2,where P is a positive definite matrix. According to Definition 2, the set-valued Lie derivative of a Vxt can be taken as(31)DVxt=∩ξ∈∂VxtξTψxt,D^,vt,D^˙,ϑ^˙T,where(32)ψxt,D^,vt=∑i=1rhiθt∑j=1rhjθtFfixt+FAi+BiKjxt+FΔA¯itxt+Fd¯t−BiD^φDxt+BiFΛ¯tut−Bivt.
By definingπfi,πAij,πΔAi,πΛ,andπd such that(33)πfi∈fikift,xt∈Gk,fik1fik2ift,xt∈∂Gk1∩∂Gk2,πAij∈Aik+BiKjift,xt∈Gk,Aik1+BiKj,Aik2+BiKjift,xt∈∂Gk1∩∂Gk2,πΔAi∈0ift,xt∈G1,ΔAiift,xt∈G2∪G3,0,ΔAiift,xt∈∂G1∩∂G2∪G3,πΛ∈Iift,xt∈G1∪G2,Λtift,xt∈G3,Λt,Iift,xt∈∂G1∪G2∩∂G3,πd∈0ift,xt∈G1,dift,xt∈G2,εu+dift,xt∈G3,0,dift,xt∈∂G1∩∂G2,d,εu+dift,xt∈∂G2∩∂G3,0,εu+dift,xt∈∂G1∩∂G3,we can rewrite (30) as(34)DVxt=D1Vxt+D2Vxt,where(35)D1Vxt=∑i=1rhiθt∑j=1rhjθtxTtPπfi+xTtPπAijxt+xTtPπΔAixt,(36)D2Vxt=∑i=1rhiθtxTtPBiπd−xTtPBiD^φDxt+xTtPBiπΛut−xTtPBivt+1ηDD˜D^˙+1ηϑϑ˜ϑ^˙.
Firstly, the analysis ofD1Vxt are given. Since P is nonsingular, it can be proved that PπΔAi∈0,PΔAi holds for all t,xt. Hence, based on Assumption 2, it can be known that, for any t,xt, xTtPπΔAixt∈0,xTtPΔAixt and(37)xTtPπΔAixt≤12xTtPMiMiTPxt+12xTtNiTNixt.
Fort,xt∈Gk,k=1,2,3, it follows from Assumption 1 that(38)πfiTπfi≤xTtUikTUikxt,PπAij=12PAik+BiKj+Aik+BiKjTP.
By combining (35), (37), and (38), we know that, for t,xt∈Gk,k=1,2,3, the following inequality holds:(39)D1Vxt≤12∑i=1rhiθt∑j=1rhjθtxtπfiTΦi,jkxtπfi,where(40)Φi,jk=Φi,j,11kPP−1λ2,Φi,j,11k=PAik+BiKj+Aik+BiKjTP+1λ2UikTUik+PMiMiTP+NiTNi.
By using (26), it can be checked that, for t,xt∈Gk,k=1,2,3,(41)D1Vxt≤−α2xTtPxt.
On the contrary, fort,xt∈∂Gk1∩∂Gk2,k1,k2∈1,2,3, it can be proved that(42)xTtPπAijxt∈xTtPAik1+BiKjxt,xTtPAik2+BiKjxt,πfiTπfi∈fik1Tfik1,fik2Tfik2∈xTtUik1TUik1xt,xTtUik2TUik2xt.
Accordingly, the following inequalities can be obtained:(43)xTtPπAijxt≤maxk∈1,2,3xTtPAik+BiKjxt,πfiTπfi≤maxk∈1,2,3xTtUik2TUik2xt.
By combining (35), (37), and (43), we know that, for t,xt∈∂Gk1∩∂Gk2,k1,k2∈1,2,3,(44)D1Vxt≤12∑i=1rhiθt∑j=1rhjθtxtπfiTΨi,jkxtπfi,where(45)Ψi,jk=Ψi,j,11kPP−1λ2,Ψi,j,11k=maxk∈1,2,3PAik+BiKj+Aik+BiKjTP+1λ2UikTUik+PMiMiTP+NiTNi.
Since (26) holds for any k∈1,2,3, we know that, for t,xt∈∂Gk1∩∂Gk2,k1,k2∈1,2,3, D1Vxt≤−α/2xTtPxt also holds. Then, by considering (41), it can be concluded that, for any t,xt,(46)D1Vxt≤−α2xTtPxt.
Next texts provide the analysis ofD2Vxt. From (33), we know that, for t,xt,(47)πd∈0,εu+d,πΛ,i∈Λit,1,where πΛ,i and Λit represent the ith component of πΛ and Λt on the diagonal line, respectively. Hence, it can be obtained that, for any t,xt,(48)∑i=1rhiθtxTtPBiπd∈0,xTtPBεu+d,∑i=1rhiθtxTtPBiπΛut∈xTtPBΛtut,xTtPBut.
From Lemma2, it is easy to know that(49)∑i=1rhiθtxTtPBiπd−xTtPBiD^φDxt≤DxTtPB−xTtPBD^φDxt≤DxTtPBxTtPBTxTtPBxTtPBT+εD2−xTtPBD^φDxt+DεD≤−D˜xTtPBφDxt+DεD.
Meanwhile, simple computation shows that(50)∑i=1rhiθtxTtPBiπΛut−xTtPBivt=−ϑ^∑l=1mzlπΛ,lφϑ,lxt,vt−xTtPBvt.
Since0<1/ϑ≤πΛ,i∈Λit,1, we can get that(51)−ϑ^zlπΛ,lφϑ,lxt,vt=−zlπΛ,lϑ^vltzltϑ^vltzltϑ^vlt2+εv2≤−1ϑzltϑ^vlt2zltϑ^vlt2+εv2.
Hence, it follows from Lemma2 that(52)−ϑ^∑l=1mzlπΛ,lφϑ,lxt,vt−xTtPBvt≤∑l=1m−1ϑzltϑ^vlt2zltϑ^vlt2+εv2−zltvlt≤∑l=1m1ϑ−zltϑ^vlt+εv−zltvlt≤∑l=1mϑ^ϑzltvlt−zltvlt+εvϑ≤ϑ˜ϑxTtPBvt+mεvϑ.
By combining (36) and (49)–(52), it can be obtained that(53)D2Vxt≤−D˜xTtPBφDxt+ϑ˜ϑxTtPBvt+1ηDD˜D^˙+1ηϑϑϑ˜ϑ^˙+DεD+mεvϑ,where m is the dimension of the system input signal ut. Substituting (24) into (53) yields(54)D2Vxt≤−σDD˜D^−σϑϑ˜ϑ^ϑ+DεD+mεvϑ.
By using the following inequalities,(55)2D^D˜≤−D˜2+D2,ϑ˜ϑ^≤−ϑ˜2+ϑ2.
We know that(56)D2Vxt≤−σD2D˜2−σϑϑ˜22ϑ+εf.
By combining (34), (46), and (56), we know that(57)DVxt≤−α2xTtPxt−σD2D˜2−σϑϑ˜22ϑ+εf,which means(58)DVxt≤−βVxt+εf,where βandεf are defined in (29). According to Lemma 1, it can be proved that the Filippov solution of closed-loop fuzzy differential inclusion (25) is bounded and converge to Ωf. The proof is complete.Remark 4.
It should be noted that the final compact setΩf can be an arbitrarily small neighborhood of the origin by adjusting the control gains and adaptive parameters. Moreover, in most of the adaptive control results, the ultimate boundness of the closed-loop control system is finally ensured. However, for the differential inclusion systems, the criteria of ultimately boundness have never been provided. Hence, it can be concluded that Lemma 1 lays the mathematical fundamentals for the adaptive control of differential inclusion systems.
Next, we will provide the computation method of control gainK and matrix P which is necessary in adaptive laws.Theorem 2.
Consider the closed-loop fuzzy differential inclusion (25) under Assumptions 1–4. The fuzzy controller is designed as (20), and the adaptive parameters are updated by (24). Given scalars α,λ>0. For any i,j∈Υ and k∈1,2,3, if there exist matrices Q,Rj such that(59)Γi,jk<0,i=1,2,…,r,Γi,jk+Γi,jk<0,1≤i<j≤r,where(60)Γi,jk=Γi,j,11kQUikTQNiTI∗−λ2I00∗∗−I0∗∗∗−1λ2I,Γi,j,11k=AikQ+QAik+BiRj+RjTBiT+αQ+MiMiT,then, for any initial conditions, the Filippov solution of closed-loop fuzzy differential inclusion (25) is bounded and converges to the compact set Ωf. Moreover, if condition (59) is feasible, it can be obtained that(61)Kj=RjQ−1,P=Q−1.Proof.
DefineQ=P−1andRj=KjP−1. By performing a congruence transformation to with Πi,jk, we can get the following matrix:(62)Π⌢i,jk=Π⌢i,j,11kII−1λ2I,where(63)Π⌢i,j,11k=AikQ+QAik+BiRj+RjTBiT+αQ+1λ2QUikTUikQ+MiMiT+QNiTNiQ.
By performing a Schur complement transformation to (61), Γi,jk can be obtained. Since(64)Γi,jk<0⇔Π⌢i,jk<0⇔Πi,jk<0,we know that (26) is satisfied by using (58). According to Theorem 1, the boundedness and convergence of the Filippov solution can be guaranteed. The proof is complete.Remark 5.
System who involves state-based switching usually has more complex dynamic behaviors which motivates various useful applications, while the construction of rigorous stability for these systems is challenging. Different to classic analysis methods in switching systems such as average dwell time (ADT), differential-inclusion-based approaches provide feasible alternative solutions to the stability analysis of stated-based switching systems. Instead of analyzing the value of the vector field at individual points, differential-inclusion-based methods focus on the behavior of vector field at the neighborhood of each point. This idea is at the core of constructing continuous Filippov solution which is a general solution to the original differential equations, where the discontinuities is covered by set-valued mapping which is a convex combination of vector field around them. As a result, rigorous stability result can be established for discontinuous vector field as long as the Lebesgue measure of the discontinuities is zero. From the illustration above, it is clear that, for state-based switching systems, the superiority of utilizing differential inclusion is significant.


## 4. Simulation Study

In this section, we will present a numerical example with two fuzzy subsystems to demonstrate the effectiveness of the proposed control method. The switching regionsGi of system (1) are defined as(65)G1=x∈ℝ2|x≥1,G2=x∈ℝ2|0.5≤x<1,G3=x∈ℝ2|x<0.5.The system-related matrices are given as follows:(66)A11=−2.71.50.51.2,A21=−2.10.60.21.8,A12=−1.92.31.52.8,A22=−3.62.70.11.5,A13=−1.20.72.10.5,A23=−2.81.73.51.9,B1=−1.93.6,B2=−1.52.3.The unknown nonlinear functions are set as(67)f11=0.016cos2x1x20.022sinx1+x2,f21=0.018sinx120.02cosx22,f12=0.06sinx1cosx20.025sinx1,f22=0.028cosx1cosx20.025cosx2,f13=0.024sinx1cosx20.025cosx1+x2,f23=0.01sinx1x20.05cosx1x2.To compute the control gains, we take(68)M1=0.012000.08,M2=0.015000.06,N1=0.069000.087,N2=0.055000.072,U11=U12=U13=0.25000.25,U21=U22=U23=0.3000.3.By lettingα=2andλ=1, we can solve (59) and obtain that(69)P=4.05906.91226.912214.0675,K1=−5.6326−10.8441,K2=−9.0548−17.6541.In the simulation, the initial values of the system states and adaptive parameters are set asx10=1.2,x20=−0.5,D^0=0,andϑ^0=1. The corresponding adaptive gains and constants are selected as ηD=ηϑ=5,σD=σϑ=2,andεD=εv=0.001. The disturbance is taken as(70)dt=20sin2πt−π6+0.5.The actuator-related settings are(71)Λt=0.7+0.1sint2+π3;εu=0.001.Moreover, to reveal the advantages of the proposed method, the advanced neural network based adaptive control method (NN-AC) and the disturbance observer based control method (DOBC) have been employed in the simulation experiments. The parameters of the NN-AC method are set asηΘ=5andσΘ=2. The disturbance observation gain of the DOBC method is selected as L=1.2,0;0,1.2. The control gains of the NN-AC and DOBC methods are set as the same as the proposed method.The simulation results are provided in Figures1–3. It is obvious that the proposed method can force both x1t and x2t to converge towards zero under switching between Gi and finally stay inside a small region near the equilibrium point. Differently, the NN-AC and the DOBC method will cause violent shock, demonstrating that these two methods may possess worse adaptability for the state-dependent switching regions compared with the proposed method. Moreover, it can be found that, by using the proposed method, the adaptive parameters D^t and ϑ^t also converge to a stable value after the initial transient. It can be concluded that, using the proposed method, the closed-loop stability can be guaranteed even under the worst situations (where disturbances and actuator faults both occur). The advantages of the proposed method can be revealed therefore.Figure 1
The trajectories of the system statex1t under NN-AC, DOBC, and proposed method.Figure 2
The trajectories of the system statex2t under NN-AC, DOBC, and proposed method.Figure 3
The trajectories of the adaptive parameters under the proposed method.Furthermore, to show the robustness of the proposed method, three cases are considered. The parameters of the three cases are given by Table2.Table 2
The parameters of the three simulation cases.
∖Switching regionsFault parametersDisturbance parametersCase 1G1=xt∈ℝ2|x≥1G2=xt∈ℝ2|0.5≤x≤1G3=xt∈ℝ2|x≤0.5Λt=0.7+0.1sint/2+π/3dt=20sin2πt−π/6+0.5Case 2G1=xt∈ℝ2|x≥1.2G2=xt∈ℝ2|0.5≤x≤1.2G3=xt∈ℝ2|x≤0.5Λt=0.5+0.1sint/2+π/3dt=30sin2πt−π/6+0.5Case 3G1=xt∈ℝ2|x≥1.2G2=xt∈ℝ2|0.8≤x≤1.2G3=xt∈ℝ2|x≤0.8Λt=0.3+0.1sint/2+π/3dt=50sin2πt−π/6+0.5Under the three cases, the simulation results using the proposed method are given in Figures4–7. It can be found that although the switching regions, the fault parameters, and the disturbance parameters have changed, the proposed method can still achieve desired control performance.Figure 4
The trajectories of the system statex1t of the proposed method under three cases.Figure 5
The trajectories of the system statex2t of the proposed method under three cases.Figure 6
The trajectories of the system stateD^t of the proposed method under three cases.Figure 7
The trajectories of the system stateϑ^t of the proposed method under three cases.


## 5. Conclusions

A novel robust adaptive controller is given in this paper for solving one of the motivating problems in nonlinear fuzzy systems, that is, to appropriately describe the behavior of the system and to guarantee the stability of the system under discontinuous multiple uncertainties and state-based switching. The proposed differential-inclusion-based method provides a constructive procedure for the controller design and analysis of a class of heterogeneous T-S fuzzy nonlinear systems with suddenly changing structural parameters and abrupt actuator failures where switching of system dynamics is related with both time and system states. The stability of the resulting closed-loop differential inclusion system is rigorously discussed by virtue of introducing a new fundamental stability lemma for adaptive discontinuous systems, and our results are validated by carefully designed simulations. It should be noted that our control scheme can be easily extended to other T-S fuzzy nonlinear systems with discontinuities and state-based switching, which may provide useful insights for further future research.



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