# Dissipativity-Based Controller Design for Time-Delayed T-S Fuzzy Switched Distributed Parameter Systems

Authors: Xiaona Song; Mi Wang; Shuai Song; Jingtao Man
Journal: Complexity
 (2018)

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## Abstract

This paper studies fuzzy controller design problem for a class of nonlinear switched distributed parameter systems (DPSs) subject to time-varying delay. Initially, the original nonlinear DPSs are accurately described by Takagi-Sugeno fuzzy model in a local region. On the basis of parallel distributed compensation technique, mode-dependent fuzzy proportional and fuzzy proportional-spatial-derivative controllers are constructed, respectively. Subsequently, using single Lyapunov-Krasovskii functional and some matrix inequality methods, sufficient conditions that guarantee the stability and dissipativity of the closed-loop systems are presented in the form of linear matrix inequalities, which allow the control gain matrices to be easily obtained. Finally, numerical examples are provided to demonstrate the validity of the designed controllers.


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## Body

## 1. Introduction

Different from lumped parameter systems, the dynamic behaviors of many industrial processes depend on not only time but also the space, for which the mathematical models are described by various partial differential equations (PDEs) such as parabolic, hyperbolic, elliptic, etc. [1]. In the past decades, considerable research has been conducted for distributed parameter systems (DPSs), especially hyperbolic and parabolic PDE systems [2–10]. To mention a few, the authors in [2] investigated the state estimation problem of linear parabolic PDE systems via using backstepping technique, [3] considered the robust controller design for hyperbolic DPSs, adaptive control approach of parabolic PDE systems was proposed in [4–6], and [10] investigated boundary control problem of parabolic PDE systems with parameter variations. However, most of the mentioned works are reported for linear PDE systems, and the control of nonlinear DPSs is still an open issue.Since Takagi-Sugeno (T-S) fuzzy system was proposed, a huge body studies for nonlinear ordinary differential equation (ODE) systems have been carried out [11–20]. Inspired by the processing method of nonlinear ODE systems, based on T-S fuzzy model, a few results have been obtained for nonlinear PDE systems recently. For instances, Wu et al. [21–23] realized exponential stabilization for a class of nonlinear parabolic PDE systems via designing a fuzzy controller and illustrated the application on nonisothermal plug-flow reactor, Zhu et al. [24] developed fuzzy control method subject to a class of nonlinear coupled ODE-PDE systems with input constraint.It should be pointed out that switched PDE systems have not been paid enough attention. Switched system (SS) is an important class of hybrid systems and plays a key role in industrial process. SSs consist of numbers of subsystems, in which switching rule is utilized to indicate the active subsystem at each instant of time. For switched systems, switching signals are classified into arbitrary switching signal, time-dependent signal, and state-dependent switching signal. As the fundamental problem, the stabilization of switched PDEs has been investigated in [25, 26] and semiconductor power chip is used to illustrate the practical application, whereas, to the best of our knowledge, the control synthesis problem for switched PDE systems has not been fully investigated, which motivates us to do this work.On another research front, proportional-spatial-derivative (P-sD) controller has been considered in many DPSs [27–29] because of its advantages compared with proportional (P) controller. Thus, in this paper, we investigate the problem of control synthesis for switched nonlinear parabolic PDE systems with time delay, including both fuzzy P and P-sD control approaches. Based on single Lyapunov-Krasovskii functional and Wirtinger’s inequality, sufficient conditions for stabilization and dissipative performance are obtained by using linear matrix inequalities. Finally, simulation studies are presented to show the effectiveness of the proposed strategies.We organize the framework as follows: The model description is given in Section2, controllers designs are presented in Section 3, and Section 4 provides simulations to illustrate the validity of the proposed control design approaches. Finally, conclusions are offered in Section 5.Notations. The notation that have been used in this paper are fairly standard. R and Rn denote the set of all real numbers and n dimensional Euclidean space. ∗ stands for symmetric completion, for example, A+∗=A+AT. ωx2=∫l1l2ωT(x)ω(x)dx. Hnl(l1,l2)=Wl,2((l1,l2);Rn) is a real Sobolve space of absolutely continuous vector functions ω(x):(l1,l2)→Rn with square integrable derivatives dlω(x)/dxl of the order l≥1 and(1)ωxHnll1,l2=∫l1l2∑i=0ldiωxdxiTdiωxdxidx.


## 2. Preliminary System and T-S Fuzzy Model

In this note, a class of switched nonlinear distributed parameter systems with time delay is investigated, which can be described by the following parabolic partial differential equations (PDEs):(2)ytx,t=Θyxxx,t+fσ1yx,t+fσdyx,t-τt+gσuyx,tux,t+gσωyx,tωx,tzx,t=fσ2yx,tsubject to the Dirichlet boundary condition(3)yl1,t=yl2,t=0and the initial condition(4)yx,t=y0xwhere y(x,t)∈Rn denotes the state vector; x∈[l1,l2]⊂R and t∈[0,∞)⊂R are the spatial position and time, respectively. z(x,t)∈Rn represents the output, u(x,t)∈Rn is the control input, ω(x,t)∈Rn is the disturbance satisfying ∫0∞ωx,t22dt<∞, and τ(t) is the time delay, which satisfies 0≤τ(t)≤h, 0≤τ˙(t)≤d≤1, and Θ is a known matrix. For convenience, define(5)Ayx,t=Θyxxx,tBased on sector nonlinearity method, the switched nonlinear parabolic PDE systems (2) can be represented by Takagi-Sugeno fuzzy switched dynamic model as follows.Plant Rule  Rσi:IF:  θ1(x,t) is Fσ1i, θ2(x,t) is Fσ2i,..., and θp(x,t) is Fσpi,THEN(6)ytx,t=Ayx,t+Aσiyx,t+Adσiyx,t-τt+Bσiux,t+Cσiωx,tzx,t=Lσiyx,twhere θj(x,t) are premise variables and Fσji are fuzzy sets (i∈1,2,…,s, j∈1,2,…,p). σ∈F=1,2,…,l is a switching signal, which is a piecewise constant function.For fuzzy controller design, we represent the fuzzy switched systems as follows:(7)ytx,t=Ayx,t+∑i=1shσiθx,t×Aσiyx,t+Adσiyx,t-τt+Bσiux,t+Cσiωx,tzx,t=∑i=1shσiθx,tLσiyx,twhere hσi(θ(x,t))=∏j=1pFσji(θj(x,t))/∑i=1s∏j=1pFσji(θj(x,t)), 0<hσi(θ(x,t))<1, and ∑i=1shσi(θ(x,t))=1 denote the membership functions with θj(x,t) belonging to the fuzzy set Fσji.In this study, it is assumed that the switching signalσ is arbitrary subjects to νr(t)=1,0(r∈F), which means that the r-th subsystem is active or not, respectively. Thus the fuzzy switched PDE system (7) can be inferred as follows: (8)ytx,t=Ayx,t+∑r=1l∑i=1sνrthriθx,t×Ariyx,t+Adriyx,t-τt+Briux,t+Criωx,tzx,t=∑r=1l∑i=1sνrthriθx,tLriyx,tThe following definition and lemma will be used to obtain the main results.Definition 1.
Given matricesR1=R1T, R2=R2T, system (8) is said to be dissipative if the following inequality holds for a nonnegative real function V(·) with V(0)=0, called the storage function,(9)∫0τ∫l1l1zTx,tR1zx,t+2zTx,tSωx,t+ωTx,tR2ωx,tdxdt+Vy0x≥0,∀τ≥0where y0(x)=y(x,0).Lemma 2 ([30], vector-valued Wirtinger’s inequalities).
Lety∈Hnl(l1,l2) be a vector function with y(l1)=y(l2)=0, then for any n×n real matrix E≥0, we have(10)∫l1l2ysEysds≤l2-l12π-2∫l1l2dysdsTEdysdsdsMoreover, if y(l1)=0 or y(l2)=0, then(11)∫l1l2ysEysds≤4l2-l12π-2∫l1l2dysdsTEdysdsds


## 3. Fuzzy Controllers Design

In this section, two different fuzzy controllers will be proposed for the fuzzy switched distributed parameter system (8).Firstly, we consider designing a proportional control law in Section3.1.


### 3.1. Fuzzy Proportional Controller Design

Using parallel distributed compensation (PDC) strategy, the following fuzzy controller is considered:(12)ux,t=∑r=1l∑i=1sνrthriθx,tKriyx,tThen, from (8) and (12), the following system is got: (13)ytx,t=Ayx,t+∑r=1l∑i=1s∑j=1sνrthriθx,thrjθx,t×Ariyx,t+Adriyx,t-τt+BrjKriyx,t+Criωx,tzx,t=∑r=1l∑i=1sνrthriθx,tLriyx,tTheorem 3.
For the fuzzy switched system (13), if there exist symmetric positive matrices P, Q and scalars εri(i,j=1,2,…,s,r=1,2,…,l) such that (14)PΘ+∗>0and(15)ΞrijPAdriPCri-LriTS∗-1-dQ0∗∗-R2<0where(16)Ξrij=PAri+εriPBrjBriTP-π2l2-l1-2PΘ+∗-LriTR1Lri+QKri=εriBriTPthen for any switching signals, the controller (12) can guarantee that the fuzzy switched system (13) is asymptotically stable with dissipativity performance subject to given matrices S, R1<0, and R2=R2T.Proof.
According to Green’s formula, Lemma2, and Dirichlet boundary condition, we have(17)2∫l1l2yTx,tPAyx,tdx=-∫l1l2yxTx,tPΘ+∗yxx,tdx≤-π2l2-l1-2∫l1l2yTx,tPΘ+∗yx,tdx
For convenience, define(18)Ahri=∑r=1l∑i=1sνrthriθx,tAriChri=∑r=1l∑i=1sνrthriθx,tCriAdhri=∑r=1l∑i=1sνrthriθx,tAdhriand (19)BhrjKhri=∑r=1l∑i=1s∑j=1sνrthriθx,thrjθx,tBrjKrihrij=∑r=1l∑i=1s∑j=1sνrthriθx,thrjθx,tChoose a Lyapunov functional with the following form:(20)Vt=∫l1l2yTx,tPyx,tdx+∫l1l2∫t-τttyTx,αQyx,αdαdxDifferentiating V(t) along the solution of the system (13), we get(21)V˙t≤2∫l1l2yTx,tPytx,tdx+∫l1l2yTx,tQyx,tdx-1-d∫l1l2yTx,t-τtQyx,t-τtdx≤∫l1l2yTx,t-π2l2-l1-2PΘ+PAhri+PBhrjKhri+∗yx,tdx+∫l1l2yTx,tPAdhri+∗yx,t-τtdx+∫l1l2yTx,tPChri+∗ωx,tdx+∫l1l2yTx,tQyx,tdx-1-d∫l1l2yTx,t-τtQyx,t-τtdx
SubstitutingKri=εriBriTP into (21), one has (22)V˙t≤∫l1l2yTx,t-π2l2-l1-2PΘ+PAhri+εriPBhrjBhriTP+∗yx,tdx+∫l1l2yTx,tPAdhri+∗yx,t-τtdx+∫l1l2yTx,tPChri+∗ωx,tdx+∫l1l2yTx,tQyx,tdx-1-d∫l1l2yTx,t-τtQyx,t-τtdx
Inequality (22) can be also rewritten as follows:(23)V˙t≤∫l1l2hrijyx,tyx,t-τtωx,tT×ΨrijPAdriPCri∗-1-dQ0∗∗0yx,tyx,t-τtωx,tdxwhere(24)Ψrij=-π2l2-l1-2PΘ+PAri+εriPBrjBriTP+∗+Q
To obtain the stability analysis results, we setω(x,t)≡0, then the following inequality is got:(25)V˙t≤2∫l1l2yTx,tPAytx,tdx+∫l1l2yTx,tQyx,tdx-1-d∫l1l2yTx,t-τtQyx,t-τtdx≤-π2l2-l1-2∫l1l2yTx,tPΘ+∗yx,tdx+∫l1l2yTx,tPAhri+∗yx,tdx+∫l1l2yTx,tεriPBhrjBhriTP+∗yx,tdx+∫l1l2yTx,tPAdhri+∗yx,t-τtdx+∫l1l2yTx,tQyx,tdx-1-d∫l1l2yTx,t-τtQyx,t-τtdxSimilar to the treatment in (22), inequality (25) implies that(26)V˙t≤∫l1l2hrijyx,tyx,t-τtT×ΨrijPAdri∗-1-dyx,tyx,t-τtdx
Thus, according to Lyapunov stability theory, inequalities (14) and (15) guarantee that the fuzzy switched system (13) is stable for any switching signal.
On the other hand, invoking (15) and (23), the following inequality is obtained:(27)V˙t-∫l1l2yTx,tLriTR1Lriyx,t+2yTx,tLriTSωx,t+ωTx,tR2ωx,tdx<0which can be rewritten as(28)V˙t-∫l1l2zTx,tR1zx,t+2zTx,tSωx,t+ωTx,tR2ωx,tdx<0
Under zero initial condition, integrating (28) on period [0,τ], then it can be derived to(29)0≤Vτ≤∫0τ∫l1l2zTx,tR1zx,t+2zTx,tSωx,t+ωTx,tR2ωx,tdxdt,∀τ≥0
It is obvious that system (13) is also dissipative. This completes the proof.Note that the matrix inequalities in Theorem3 is not linear; we give the following approach to solve it. Using Schur complements, we have(30)ΓrijPAdriPCri-LriTSLriT∗-1-dQ00∗∗-R20∗∗∗R1-1<0where(31)Γrij=PAri+εriPBrjBriTP-π2l2-l1-2PΘ+∗+QPre- and postmultiply both sides of inequality (30) by the matrix diagX,X,I,I, where X=P-1. Define Q~=XQX; then we can obtain(32)ΠrijAdriXCri-XLriTSXLriT∗-1-dQ~00∗∗-R20∗∗∗R1-1<0where(33)Πrij=AriX+εriBrjBriT-π2l2-l1-2ΘX+∗+Q~The matrixX and εri can be obtained by solving inequality (32); substituting X and εri into(34)Kri=εriBriTX-1we can easily obtain the controller gains.Remark 4.
It should be emphasized that the controller designed in Theorem3 requires distributed sensors and actuators. With the development of advanced technique, it is possible to solve this problem, because there are more and more microsensors and microactuators manufactured in control circuitry. However, we are also looking for new method to avoid the drawback of the controller.Remark 5.
It is worth pointed out that the state feedback control method (12) cannot meet the engineering needs in many practical applications, especially for nonlinear spatially distributed processes. Therefore, we will consider the following fuzzy proportional-spatial-derivative (P-sD) controller.


### 3.2. Fuzzy P-sD Controller Design

In contrast to the proportional state feedback controller proposed in Section3.1, the information of state spatial-derivative is added to fuzzy P-sD controller design, which will bring better dynamic performances.Firstly, using PDC theory, the following fuzzy P-sD controller is constructed(35)ux,t=∑r=1l∑i=1sνrthriθx,tKriyx,t+KDriyxx,tCombining (8) and (35), we have(36)ytx,t=Ayx,t+∑r=1l∑i=1s∑j=1sνrthriθx,thrjθx,t×Ariyx,t+Adriyx,t-τt+BrjKriyx,t+BrjKDriyxx,t+Criωx,tzx,t=∑r=1l∑i=1sνrthriθx,tLriyx,tTheorem 6.
Given matricesS, R1<0, and R2=R2T, then for arbitrary switching signals, the fuzzy switched system (36) is asymptotically stable and dissipative via using controller (35), if there exist matrices P>0, Q>0, Kri, and KDri such that the following inequality holds: (37)ΛrijPBrjKDriPAdriPCri-LriTSLriT∗-PΘ+∗000∗∗-1-dQ00∗∗∗-R20∗∗∗∗R1-1<0where(38)Λrij=PAri+PBrjKri+∗+QProof.
Construct the following Lyapunov functional candidate:(39)Vt=∫l1l2yTx,tPyx,tdx+∫l1l2∫t-τttyTx,αQyx,αdαdxBy differentiating V(t) along the solution of system (36), we can get(40)V˙t≤2∫l1l2yTx,tPytx,tdx+∫l1l2yTx,tQyx,tdx-1-d∫l1l2yTx,t-τtQyx,t-τtdx
Substituting (36) into inequality (40), one has(41)V˙t≤2∫l1l2yTx,tPAyx,tdx+∫l1l2yTx,tPAhri+PBhrjKhri+∗yx,tdx+∫l1l2yTx,tPBhrjKDhri+∗yxx,tdx+∫l1l2yTx,tPAdhri+∗yx,t-τtdx+∫l1l2yTx,tPChri+∗ωx,tdx+∫l1l2yTx,tQyx,tdx-1-d∫l1l2yTx,t-τtQyx,t-τtdx
According to Green’s formula and Dirichlet boundary condition, it follows that(42)2∫l1l2yTx,tPAyx,tdx=-∫l1l2yxTx,tPΘ+∗yxx,tdx
In light of (41) and (42), one can obtain (43)V˙t≤-∫l1l2yxTx,tPΘ+∗yxx,tdx+∫l1l2yTx,tPAhri+PBhrjKhri+∗yx,tdx+∫l1l2yTx,tPBhrjKDhri+∗yxx,tdx+∫l1l2yTx,tPAdhri+∗yx,t-τtdx+∫l1l2yTx,tPChri+∗ωx,tdx+∫l1l2yTx,tQyx,tdx-1-d∫l1l2yTx,t-τtQyx,t-τtdx
Defineηx,t=yT(x,t)yxT(x,t)yT(x,t-τ(t))ωT(x,t); inequality (43) can be also written as follows:(44)V˙t≤∫l1l2hrijηx,t×ΛrijPBrjKDriPAdriPCri∗-PΘ+∗00∗∗-1-dQ0∗∗∗0ηTx,t
Setω(x,t)≡0 to obtain the stability analysis results as follows:(45)V˙t≤∫l1l2hrijyx,tyxx,tyx,t-τtT×ΛrijPBrjKDriPAdri∗-PΘ+∗0∗∗-1-dQyx,tyxx,tyx,t-τt
Based on inequality (37), the fuzzy switched system (36) is stable for any switching signal. The following proof process is similar to Theorem 3; thus it is omitted.SettingMDri=KDriP-1 and Nri=KriP-1, following the similar line in Theorem 3, one has(46)ΩrijPBrjMDriPPAdriPCri-LriTSLriT∗-PΘ+∗000∗∗-1-dQ00∗∗∗-R20∗∗∗∗R1-1<0where(47)Ωrij=PAri+PBrjNriP+∗+QPre- and postmultiplying both sides of the inequality (46) by matrix (48)diagX,X,X,I,Iwhere X=P-1 and setting Q~=XQX, one can obtain(49)HrijBrjMDriAdriXCri-XLriTSLriT∗-ΘX+∗000∗∗-1-dQ~00∗∗∗-R20∗∗∗∗R1-1<0where(50)Hrij=AriX+BrjNri+∗+Q~The matrices MDri, Nri, X, and Q~ can be obtained by solving inequalities (49), then substituting X, MDri, and Nri into(51)KDri=MDriX-1,Kri=NriX-1we can easily get the controller gains.


## 3.1. Fuzzy Proportional Controller Design

Using parallel distributed compensation (PDC) strategy, the following fuzzy controller is considered:(12)ux,t=∑r=1l∑i=1sνrthriθx,tKriyx,tThen, from (8) and (12), the following system is got: (13)ytx,t=Ayx,t+∑r=1l∑i=1s∑j=1sνrthriθx,thrjθx,t×Ariyx,t+Adriyx,t-τt+BrjKriyx,t+Criωx,tzx,t=∑r=1l∑i=1sνrthriθx,tLriyx,tTheorem 3.
For the fuzzy switched system (13), if there exist symmetric positive matrices P, Q and scalars εri(i,j=1,2,…,s,r=1,2,…,l) such that (14)PΘ+∗>0and(15)ΞrijPAdriPCri-LriTS∗-1-dQ0∗∗-R2<0where(16)Ξrij=PAri+εriPBrjBriTP-π2l2-l1-2PΘ+∗-LriTR1Lri+QKri=εriBriTPthen for any switching signals, the controller (12) can guarantee that the fuzzy switched system (13) is asymptotically stable with dissipativity performance subject to given matrices S, R1<0, and R2=R2T.Proof.
According to Green’s formula, Lemma2, and Dirichlet boundary condition, we have(17)2∫l1l2yTx,tPAyx,tdx=-∫l1l2yxTx,tPΘ+∗yxx,tdx≤-π2l2-l1-2∫l1l2yTx,tPΘ+∗yx,tdx
For convenience, define(18)Ahri=∑r=1l∑i=1sνrthriθx,tAriChri=∑r=1l∑i=1sνrthriθx,tCriAdhri=∑r=1l∑i=1sνrthriθx,tAdhriand (19)BhrjKhri=∑r=1l∑i=1s∑j=1sνrthriθx,thrjθx,tBrjKrihrij=∑r=1l∑i=1s∑j=1sνrthriθx,thrjθx,tChoose a Lyapunov functional with the following form:(20)Vt=∫l1l2yTx,tPyx,tdx+∫l1l2∫t-τttyTx,αQyx,αdαdxDifferentiating V(t) along the solution of the system (13), we get(21)V˙t≤2∫l1l2yTx,tPytx,tdx+∫l1l2yTx,tQyx,tdx-1-d∫l1l2yTx,t-τtQyx,t-τtdx≤∫l1l2yTx,t-π2l2-l1-2PΘ+PAhri+PBhrjKhri+∗yx,tdx+∫l1l2yTx,tPAdhri+∗yx,t-τtdx+∫l1l2yTx,tPChri+∗ωx,tdx+∫l1l2yTx,tQyx,tdx-1-d∫l1l2yTx,t-τtQyx,t-τtdx
SubstitutingKri=εriBriTP into (21), one has (22)V˙t≤∫l1l2yTx,t-π2l2-l1-2PΘ+PAhri+εriPBhrjBhriTP+∗yx,tdx+∫l1l2yTx,tPAdhri+∗yx,t-τtdx+∫l1l2yTx,tPChri+∗ωx,tdx+∫l1l2yTx,tQyx,tdx-1-d∫l1l2yTx,t-τtQyx,t-τtdx
Inequality (22) can be also rewritten as follows:(23)V˙t≤∫l1l2hrijyx,tyx,t-τtωx,tT×ΨrijPAdriPCri∗-1-dQ0∗∗0yx,tyx,t-τtωx,tdxwhere(24)Ψrij=-π2l2-l1-2PΘ+PAri+εriPBrjBriTP+∗+Q
To obtain the stability analysis results, we setω(x,t)≡0, then the following inequality is got:(25)V˙t≤2∫l1l2yTx,tPAytx,tdx+∫l1l2yTx,tQyx,tdx-1-d∫l1l2yTx,t-τtQyx,t-τtdx≤-π2l2-l1-2∫l1l2yTx,tPΘ+∗yx,tdx+∫l1l2yTx,tPAhri+∗yx,tdx+∫l1l2yTx,tεriPBhrjBhriTP+∗yx,tdx+∫l1l2yTx,tPAdhri+∗yx,t-τtdx+∫l1l2yTx,tQyx,tdx-1-d∫l1l2yTx,t-τtQyx,t-τtdxSimilar to the treatment in (22), inequality (25) implies that(26)V˙t≤∫l1l2hrijyx,tyx,t-τtT×ΨrijPAdri∗-1-dyx,tyx,t-τtdx
Thus, according to Lyapunov stability theory, inequalities (14) and (15) guarantee that the fuzzy switched system (13) is stable for any switching signal.
On the other hand, invoking (15) and (23), the following inequality is obtained:(27)V˙t-∫l1l2yTx,tLriTR1Lriyx,t+2yTx,tLriTSωx,t+ωTx,tR2ωx,tdx<0which can be rewritten as(28)V˙t-∫l1l2zTx,tR1zx,t+2zTx,tSωx,t+ωTx,tR2ωx,tdx<0
Under zero initial condition, integrating (28) on period [0,τ], then it can be derived to(29)0≤Vτ≤∫0τ∫l1l2zTx,tR1zx,t+2zTx,tSωx,t+ωTx,tR2ωx,tdxdt,∀τ≥0
It is obvious that system (13) is also dissipative. This completes the proof.Note that the matrix inequalities in Theorem3 is not linear; we give the following approach to solve it. Using Schur complements, we have(30)ΓrijPAdriPCri-LriTSLriT∗-1-dQ00∗∗-R20∗∗∗R1-1<0where(31)Γrij=PAri+εriPBrjBriTP-π2l2-l1-2PΘ+∗+QPre- and postmultiply both sides of inequality (30) by the matrix diagX,X,I,I, where X=P-1. Define Q~=XQX; then we can obtain(32)ΠrijAdriXCri-XLriTSXLriT∗-1-dQ~00∗∗-R20∗∗∗R1-1<0where(33)Πrij=AriX+εriBrjBriT-π2l2-l1-2ΘX+∗+Q~The matrixX and εri can be obtained by solving inequality (32); substituting X and εri into(34)Kri=εriBriTX-1we can easily obtain the controller gains.Remark 4.
It should be emphasized that the controller designed in Theorem3 requires distributed sensors and actuators. With the development of advanced technique, it is possible to solve this problem, because there are more and more microsensors and microactuators manufactured in control circuitry. However, we are also looking for new method to avoid the drawback of the controller.Remark 5.
It is worth pointed out that the state feedback control method (12) cannot meet the engineering needs in many practical applications, especially for nonlinear spatially distributed processes. Therefore, we will consider the following fuzzy proportional-spatial-derivative (P-sD) controller.


## 3.2. Fuzzy P-sD Controller Design

In contrast to the proportional state feedback controller proposed in Section3.1, the information of state spatial-derivative is added to fuzzy P-sD controller design, which will bring better dynamic performances.Firstly, using PDC theory, the following fuzzy P-sD controller is constructed(35)ux,t=∑r=1l∑i=1sνrthriθx,tKriyx,t+KDriyxx,tCombining (8) and (35), we have(36)ytx,t=Ayx,t+∑r=1l∑i=1s∑j=1sνrthriθx,thrjθx,t×Ariyx,t+Adriyx,t-τt+BrjKriyx,t+BrjKDriyxx,t+Criωx,tzx,t=∑r=1l∑i=1sνrthriθx,tLriyx,tTheorem 6.
Given matricesS, R1<0, and R2=R2T, then for arbitrary switching signals, the fuzzy switched system (36) is asymptotically stable and dissipative via using controller (35), if there exist matrices P>0, Q>0, Kri, and KDri such that the following inequality holds: (37)ΛrijPBrjKDriPAdriPCri-LriTSLriT∗-PΘ+∗000∗∗-1-dQ00∗∗∗-R20∗∗∗∗R1-1<0where(38)Λrij=PAri+PBrjKri+∗+QProof.
Construct the following Lyapunov functional candidate:(39)Vt=∫l1l2yTx,tPyx,tdx+∫l1l2∫t-τttyTx,αQyx,αdαdxBy differentiating V(t) along the solution of system (36), we can get(40)V˙t≤2∫l1l2yTx,tPytx,tdx+∫l1l2yTx,tQyx,tdx-1-d∫l1l2yTx,t-τtQyx,t-τtdx
Substituting (36) into inequality (40), one has(41)V˙t≤2∫l1l2yTx,tPAyx,tdx+∫l1l2yTx,tPAhri+PBhrjKhri+∗yx,tdx+∫l1l2yTx,tPBhrjKDhri+∗yxx,tdx+∫l1l2yTx,tPAdhri+∗yx,t-τtdx+∫l1l2yTx,tPChri+∗ωx,tdx+∫l1l2yTx,tQyx,tdx-1-d∫l1l2yTx,t-τtQyx,t-τtdx
According to Green’s formula and Dirichlet boundary condition, it follows that(42)2∫l1l2yTx,tPAyx,tdx=-∫l1l2yxTx,tPΘ+∗yxx,tdx
In light of (41) and (42), one can obtain (43)V˙t≤-∫l1l2yxTx,tPΘ+∗yxx,tdx+∫l1l2yTx,tPAhri+PBhrjKhri+∗yx,tdx+∫l1l2yTx,tPBhrjKDhri+∗yxx,tdx+∫l1l2yTx,tPAdhri+∗yx,t-τtdx+∫l1l2yTx,tPChri+∗ωx,tdx+∫l1l2yTx,tQyx,tdx-1-d∫l1l2yTx,t-τtQyx,t-τtdx
Defineηx,t=yT(x,t)yxT(x,t)yT(x,t-τ(t))ωT(x,t); inequality (43) can be also written as follows:(44)V˙t≤∫l1l2hrijηx,t×ΛrijPBrjKDriPAdriPCri∗-PΘ+∗00∗∗-1-dQ0∗∗∗0ηTx,t
Setω(x,t)≡0 to obtain the stability analysis results as follows:(45)V˙t≤∫l1l2hrijyx,tyxx,tyx,t-τtT×ΛrijPBrjKDriPAdri∗-PΘ+∗0∗∗-1-dQyx,tyxx,tyx,t-τt
Based on inequality (37), the fuzzy switched system (36) is stable for any switching signal. The following proof process is similar to Theorem 3; thus it is omitted.SettingMDri=KDriP-1 and Nri=KriP-1, following the similar line in Theorem 3, one has(46)ΩrijPBrjMDriPPAdriPCri-LriTSLriT∗-PΘ+∗000∗∗-1-dQ00∗∗∗-R20∗∗∗∗R1-1<0where(47)Ωrij=PAri+PBrjNriP+∗+QPre- and postmultiplying both sides of the inequality (46) by matrix (48)diagX,X,X,I,Iwhere X=P-1 and setting Q~=XQX, one can obtain(49)HrijBrjMDriAdriXCri-XLriTSLriT∗-ΘX+∗000∗∗-1-dQ~00∗∗∗-R20∗∗∗∗R1-1<0where(50)Hrij=AriX+BrjNri+∗+Q~The matrices MDri, Nri, X, and Q~ can be obtained by solving inequalities (49), then substituting X, MDri, and Nri into(51)KDri=MDriX-1,Kri=NriX-1we can easily get the controller gains.


## 4. Simulation Examples

Here, we consider the following fuzzy switched parabolic PDE systems with time delay:(52)ytx,t=Ayx,t+∑r=12∑i=12νrthriθx,t×Ariyx,t+Adriyx,t-τt+Briux,t+Criωx,tzx,t=∑r=12∑i=12νrthriθx,tLriyx,tsubject to the Dirichlet boundary conditions(53)y1l1,t=y2l1,t=0,y1l2,t=y2l2,t=0,and the initial conditions(54)y1x,0=0.5sin⁡2x,y2x,0=0.3sin⁡2xwhere(55)A11=3,0;0,3,A12=3,0;0,3,B11=-1,2;0,-3,B12=-1,1;1,-3,C11=C12=L11=L12=Ad11=Ad12=I,and(56)A21=3,0;0,3,A22=-3,0;0,-3,B21=1,2;0,-3,B22=1,2;1,-3,C21=C22=L21=L22=Ad21=Ad22=I.The membership functions are chosen as(57)h11=h21=y2x,t+1.21.7,h12=h22=1-y2x,t+1.21.7Finally, assume(58)R1=-1,0;0,-1,R2=1,0;0,1S=1,0;0,1,Θ=1,0;0,1,τt=0.2.Now, we give the open-loop trajectories ofy1(x,t)2, y2(x,t)2, y1(x,t), y2(x,t), and the switching signal of system (52) in Figures 1–4. It is obvious that the system is unstable, which motives us to design controllers in Examples 7 and 8.Figure 1
The open-loop trajectories ofy1(x,t)2 and y2(x,t)2.Figure 2
The open-loop trajectory ofy1(x,t).Figure 3
The open-loop trajectory ofy2(x,t).Figure 4
The trajectory of switching signal.Example 7.
In this part, we choose the fuzzy controller as(59)ux,t=∑r=12∑i=12νrthriy2x,tKriyx,tBy solving the inequality in Theorem 3, we can obtain the controller gains as(60)K11=3.66012.8822-7.807713.1527,K12=4.9581-2.8964-3.673413.0819and(61)K21=-5.01991.9283-6.50978.7999,K22=-3.0219-1.0872-3.91854.9103
Figures5–7 present the closed-loop trajectories of y1(x,t)2, y2(x,t)2, y1(x,t), and y2(x,t) of system (52) with the controller (59); the switching signal is shown in Figure 4. It is obvious that the system is stable.Figure 5
The closed-loop trajectories ofy1(x,t)2 and y2(x,t)2.Figure 6
The closed-loop trajectory ofy1(x,t).Figure 7
The closed-loop trajectory ofy2(x,t).Example 8.
In this part, the following fuzzy controller is selected:(62)ux,t=∑r=12∑i=12νrthriθx,tKriyx,t+KDriyxx,tSolving inequality in Theorem 6, one can obtain the controller gains:(63)K11=10.94673.68392.52543.3732,K12=10.94673.68392.52543.3732K21=-7.1588-3.0161-0.06012.1910,K22=-1.1820-0.63190.01860.1704and(64)KD11=-1.5514-0.4746-0.3547-0.3268,KD12=-1.5514-0.4746-0.3547-0.3268KD21=1.06490.0780-0.01600.2146,KD22=0.97870.0478-0.02880.2191The switching signal is given in Figure4; Figures 8–10 present the closed-loop trajectories of y1(x,t)2, y2(x,t)2, y1x,t, and y2(x,t), respectively. From Figures 5 and 8, it is easy to observe that system (52) has better performances via using the fuzzy P-sD controller (62).Figure 8
The closed-loop trajectories ofy1(x,t)2 and y2(x,t)2.Figure 9
The closed-loop trajectory ofy1(x,t).Figure 10
The closed-loop trajectory ofy2(x,t).


## 5. Conclusions

In this paper, we have developed two fuzzy control laws for a class of T-S fuzzy switched distributed parameter systems (DPSs) with time delay. Using single Lyapunov-Krasovskii functional, sufficient conditions that ensure the DPSs stable and dissipative were established based on linear matrix inequalities. Furthermore, simulation studies have been shown to support the proposed techniques. It should be stressed that this is only a limited work for fuzzy switched DPSs. In future work, we will further investigate fuzzy control problem for Markovian jump DPSs and multi-Lyapunov functionals will be introduced to reduce the conservatism.



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*Source: 6215945-2018-10-17.xml*