# RobustH∞ Control for Markovian-Jump-Parameters Takagi–Sugeno Fuzzy Systems

Authors: Wenzhao Qin; Yukai Shen; Lifang Wang
Journal: Complexity
 (2022)

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## Abstract

TheH∞ performance of a class of T-S fuzzy systems with Markovian-jump parameters is analyzed. A state feedback controller is designed for T-S fuzzy systems with Markovian-jump parameters. First, a new modal-dependent Lyapunov function composed of closed-loop functions is constructed, which can fully use system status information. Based on this function, the stability conditions with less conservatism are given by linear matrix inequalities (LMIS). At the same time, a design algorithm for a state feedback controller is proposed for the Markovian-jump-parameters T-S fuzzy systems, which ensures the system’s stability under the condition of H∞−γ performance. Simulation results demonstrate that the mentioned method is accurate and practical.


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## Body

## 1. Introduction

Takagi–Sugeno (T-S) fuzzy systems can provide an effective description of complex nonlinear system problems [1]. The essence of T-S fuzzy systems is nonlinear systems, but each corresponding rule is linear, which is convenient for people to analyze the systems’ stability and study the control problem. In recent decades, T-S fuzzy systems have become a research hotspot and an important method for nonlinear system control. The stability of a static output controller for a discrete-time T-S fuzzy system was studied by introducing relaxation variables and considering multiple Lyapunov matrices in [2]. The equivalence of the weight product index set for fuzzy rules was defined, and a new stability criterion with less conservatism was proposed to guarantee the system’s stability using the equivalence relation in [3]. Two efficient design methods for fuzzy static output feedback controllers were proposed by combining Markovian Lyapunov functions and matrix inequality convexification techniques, and the robust stabilization of a nonlinear hyperbolic partial differential equation (PDE) system represented by a T-S fuzzy system is successfully solved in [4]. At the identical time, in order to make the system stable and ensure its good performance, many scholars have studied comprehensive control methods such as H2 control and H∞ control. In particular, H∞ control can effectively suppress interference, so it is widely used in electric ground vehicle systems [5, 6] and electric train asynchronous motor systems [7]. A new augmented Lyapunov function was proposed to ensure the tracking performance of the T-S fuzzy networked control systems [8]. The nonlinear systems were transformed into continuous-time T-S fuzzy systems with uncertain parameters using the Wertinger integral inequality and the extended inverse convex matrix inequality, and the asymptotic stability condition with less conservativeness was proposed in [9]. A new bounded real lemma was constructed for a class of fuzzy time-delay systems, and a new Lyapunov–Krasovskii functional (LKF) was proposed in [10]. The robust output feedback control problem for a class of interval type 2 T-S fuzzy systems with multiple delays and disturbances was studied, and a delay-dependent dynamic output feedback controller was designed using the generalized redundancy method to make the closed-loop system asymptotically stable and obtain the performance in [11]. On the other side of research, the structure or parameters of increasingly complex automatic control systems inevitably jump randomly after a long period of operation, so stochastic hybrid systems with Markovian-jump parameters have received extensive attention. Markovian-jump systems are very effective in describing environmental changes, system failures, and repairs and therefore are widely used in circuit systems, aircraft systems, manufacturing systems [12–16], and networked control systems under cyber-attacks [17]. In recent years, many scholars are interested in the Markovian-jump fuzzy systems, and these studies have yielded a large number of meaningful results [18–20].Moreover, as digital technology expands rapidly, research on sampled-data control has received high attention. The sampled-data control methods only need to use the instantaneous sampling information of the system, which greatly reduces the information transmission, so it can guarantee a better system performance and reduce the system operating costs. Sampled-data control methods include input delay method [21], discrete-time system method [22], and closed-loop function method [23]. It is worth mentioning that the looped function mentioned in the closed-loop function method is not necessarily a positive function that fully utilizes the information on the intervals xtk to xt. At the jumps, this Lyapunov function does not increase, and the whole sampling interval is sufficiently considered to yield less conservative conditions. These functionals are suitable for impulsive systems analysis as they allow one to express the discrete-time stability conditions in an affine manner, thus enabling the consideration of uncertain and time-varying systems. The research results of sampled-data control are numerous and significant [24–28]. However, the application of the looped function method to Markovian-jump T-S fuzzy systems with sampled data has been rarely investigated. In [29], a new Lyapunov function whose time scheme consists of an exponential closed-loop function was proposed, and linear interpolation and binning techniques were proposed to obtain criteria for mean-square exponential stability, but the conclusions were less conservative. A new input delay-dependent vector method proposed for fuzzy Markovian systems with sampled data was first mentioned in [30].Based on the above discussion, our work is motivated by the issue of exploring the control of Markovian-jumpT-S fuzzy systems based on sampled data. Some of the topics to be pondered are, for example, how to construct the Lyapunov functions to efficiently utilize the system information and how to modify the integral inequality to make the results less conservative. The nonfragile controllers [31–37], which are highly resistant to uncertainty in their parameters, have been widely used in many practical systems, but how we can apply them to Markovian-jump T-S fuzzy nonperiodic sampling control systems.The contributions of this work consist of the following aspects: (1) A new mode-dependent Lyapunov function consisting of bilateral closed-loop functions is used in theT-S fuzzy systems with Markovian-jump parameters, this function fully covers the information on both intervals xtk,xt and xt,xtk+1, whereas the Lyapunov function constructed in the literature [38] only contained ∫tktxTtds and does not contained ∫ttk+1xTtds, however, our constructed function Vt includes both, thus the results derived in this study are less conservative. And we use a modified integral inequality with a free matrix to help deal with an integral part of the derivative of the closed-loop function, enabling the systems to be less conservative. (2) A state feedback controller design algorithm is proposed to ensure the system’s stability under the H∞ performance condition and a truck-trailer model is adopted to verify the effectiveness of the proposed method.The rest of the article is organized as follows: some necessary lemmas and systems depiction are presented in Section2; the stochastic stability condition for Markovian-jump T-S fuzzy nonperiodic sampling control systems is given under the H∞−γ condition and a sampled-data controller with state feedback is designed in Section 3; and the effectiveness of our proposed method is verified with a truck-trailer model in Section 4; finally, we conclude this work with a summary in Section 5.


## 2. Preliminaries

### 2.1. Two Useful Lemmas

Lemma 1 (see [23]).
Letx be a differentiable function: α,β⟶ℝn. If there is a vector η∈ℝm, matrix U=UT>0,U∈ℝn×n, and M1,M2∈ℝn×n, then the following inequality holds:(1)−∫αβx˙sTUx˙sds≤ληTM1TU−1M1+λ23M2U−1M1η+2ληTM2T+2ηTM1Tυ−2M2T∫αβxsds,where λ=β−α,υ=xβ−xα,andϱ=xβ+xα.Remark 1.
This inequality is based on the free weight matrix inequality and is improved based on the inequality in [39]. The inequality contains ∫αβxsds instead of 1/β−α∫αβxsds, and this will facilitate the deduction of the main results and degrade the conservatism of the main results.Lemma 2 ((Schur complement lemma) [40]).
If there are symmetric matrixesA, B, and C, then the following three conditions are equivalent:(2)ABBTC<0,A<0,B−CTA−1C<0,B<0,A−CB−1CT<0.


### 2.2. Depiction ofT-S Fuzzy Systems with Markovian-Jump Parameters

Concerning the continuous-timeT-S fuzzy systems with Markovian-jump parameters as follows, whose IF-THEN rules are described:(1)
Rulei: IF φ1t is ζi1, φ2t is ζi2,…, φpt is ζip, THEN(3)x˙t=Airtxt+B1,irtut+D1,irtwt,zt=Cirtxt+B2,irtut+D2,irtwt,wherei∈ℛ=1,2,…,r, r is the number of IF-THEN rules, φ1t, φ2t, ⋯, φpt are premise variables, ζi1,ζi2,…,ζip are fuzzy sets, xt is the vector describing status of system, ut is the vector indicates the control input, wt is the perturbation input taking value in range L20,+∞, zt is the vector of control output, rt is a Markovian process in continuous time taking value in a finite set N=1,2,…,s, and rt with switching transition probability matrix Π=πικs×s is described as follows:(4)Prrt+Δ=κ|rt=ι=πικΔ+oΔ,ι≠κ,1+πικΔ+oΔ,ι=κ,whereΔ>0, limΔ⟶0oΔ/Δ=0. When ι≠κ, πικ>0 indicate the rate of mode transition from mode ι at time t to mode κ at time t+Δ, and πιι=−∑κ=1,κ≠ιsπικ.According to fuzzy logic inference method, system (3) can be reconstructed as follows:(5)x˙t=∑i=1rhiφtAirtxt+B1,irtut+D1,irtwt,zt=∑i=1rhiφtCirtxt+B2,irtut+D2,irtwt,whereφt=φ1tφ2t…φpt, hiφt=ϑiφt/∑j=1rϑjφt, and ϑiφt=∏j=1pζijφjt, ζijφjt denote the membership degree of φjt in ζijj=1,2,…,p, we can easily find out that for all t, ϑiφt≥0, ∑i=1rϑiφt≥0, hiφt≥0, and ∑i=1rhiφt=1. For all the possible rt=ι, we can simplify fuzzy system (5) as follows:(6)x˙t=Aιtxt+B1ιtut+D1ιtwt,zt=Cιtxt+B2ιtut+D2ιtwt,where(7)Aιt=∑i=1rhiφtAιi,B1ιt=∑i=1rhiφtB1ιi,B2ιt=∑i=1rhiφtB2ιi,Cιt=∑i=1rhiφtCιi,D1ιt=∑i=1rhiφtD1ιi,D2ιt=∑i=1rhiφtD2ιi.Consider the zero-order hold function, which gets a set of hold times0=t0<t1<⋯<tk<⋯limk⟶∞tk=+∞, and for any k>0, tk+1−tk=hk∈hm,hM, hm indicate the lower boundary of the sampling periods, and hM indicate the upper bound of the sampling periods, then, the state feedback controller for system (3) is designed as follows.(2)
Rulej：IF φ1t is ζj1, φ2t is ζj2,…, φpt is ζjp, THEN(8)ut=Kjxtk,i=1,2,…,r.Based on the fuzzy logic inference method, the state feedback controllerut is reconstructed as follows:(9)ut=Ktxtk,whereKt=∑j=1rhjφtKj.Substituting theut into system (6), the closed-loop system is as follows:(10)x˙t=Aιtxt+B1ιtKtxtk+D1ιtwt,zt=Cιtxt+B2ιtKtxtk+D2ιtwt,tk≤t<tk+1.


## 2.1. Two Useful Lemmas

Lemma 1 (see [23]).
Letx be a differentiable function: α,β⟶ℝn. If there is a vector η∈ℝm, matrix U=UT>0,U∈ℝn×n, and M1,M2∈ℝn×n, then the following inequality holds:(1)−∫αβx˙sTUx˙sds≤ληTM1TU−1M1+λ23M2U−1M1η+2ληTM2T+2ηTM1Tυ−2M2T∫αβxsds,where λ=β−α,υ=xβ−xα,andϱ=xβ+xα.Remark 1.
This inequality is based on the free weight matrix inequality and is improved based on the inequality in [39]. The inequality contains ∫αβxsds instead of 1/β−α∫αβxsds, and this will facilitate the deduction of the main results and degrade the conservatism of the main results.Lemma 2 ((Schur complement lemma) [40]).
If there are symmetric matrixesA, B, and C, then the following three conditions are equivalent:(2)ABBTC<0,A<0,B−CTA−1C<0,B<0,A−CB−1CT<0.


## 2.2. Depiction ofT-S Fuzzy Systems with Markovian-Jump Parameters

Concerning the continuous-timeT-S fuzzy systems with Markovian-jump parameters as follows, whose IF-THEN rules are described:(1)
Rulei: IF φ1t is ζi1, φ2t is ζi2,…, φpt is ζip, THEN(3)x˙t=Airtxt+B1,irtut+D1,irtwt,zt=Cirtxt+B2,irtut+D2,irtwt,wherei∈ℛ=1,2,…,r, r is the number of IF-THEN rules, φ1t, φ2t, ⋯, φpt are premise variables, ζi1,ζi2,…,ζip are fuzzy sets, xt is the vector describing status of system, ut is the vector indicates the control input, wt is the perturbation input taking value in range L20,+∞, zt is the vector of control output, rt is a Markovian process in continuous time taking value in a finite set N=1,2,…,s, and rt with switching transition probability matrix Π=πικs×s is described as follows:(4)Prrt+Δ=κ|rt=ι=πικΔ+oΔ,ι≠κ,1+πικΔ+oΔ,ι=κ,whereΔ>0, limΔ⟶0oΔ/Δ=0. When ι≠κ, πικ>0 indicate the rate of mode transition from mode ι at time t to mode κ at time t+Δ, and πιι=−∑κ=1,κ≠ιsπικ.According to fuzzy logic inference method, system (3) can be reconstructed as follows:(5)x˙t=∑i=1rhiφtAirtxt+B1,irtut+D1,irtwt,zt=∑i=1rhiφtCirtxt+B2,irtut+D2,irtwt,whereφt=φ1tφ2t…φpt, hiφt=ϑiφt/∑j=1rϑjφt, and ϑiφt=∏j=1pζijφjt, ζijφjt denote the membership degree of φjt in ζijj=1,2,…,p, we can easily find out that for all t, ϑiφt≥0, ∑i=1rϑiφt≥0, hiφt≥0, and ∑i=1rhiφt=1. For all the possible rt=ι, we can simplify fuzzy system (5) as follows:(6)x˙t=Aιtxt+B1ιtut+D1ιtwt,zt=Cιtxt+B2ιtut+D2ιtwt,where(7)Aιt=∑i=1rhiφtAιi,B1ιt=∑i=1rhiφtB1ιi,B2ιt=∑i=1rhiφtB2ιi,Cιt=∑i=1rhiφtCιi,D1ιt=∑i=1rhiφtD1ιi,D2ιt=∑i=1rhiφtD2ιi.Consider the zero-order hold function, which gets a set of hold times0=t0<t1<⋯<tk<⋯limk⟶∞tk=+∞, and for any k>0, tk+1−tk=hk∈hm,hM, hm indicate the lower boundary of the sampling periods, and hM indicate the upper bound of the sampling periods, then, the state feedback controller for system (3) is designed as follows.(2)
Rulej：IF φ1t is ζj1, φ2t is ζj2,…, φpt is ζjp, THEN(8)ut=Kjxtk,i=1,2,…,r.Based on the fuzzy logic inference method, the state feedback controllerut is reconstructed as follows:(9)ut=Ktxtk,whereKt=∑j=1rhjφtKj.Substituting theut into system (6), the closed-loop system is as follows:(10)x˙t=Aιtxt+B1ιtKtxtk+D1ιtwt,zt=Cιtxt+B2ιtKtxtk+D2ιtwt,tk≤t<tk+1.


## 3. Main Results

We propose a sufficient condition to solve the control problem formulated in the previous section.First, two definitions are necessary.Definition 1 (see [41]).
Whenut=0, the system (10) is stochastic stable, if the continuous function Φt∈ℝn defined on the interval −τ,0, and the initial mode r0∈N, then the following inequality holds:(11)limt⟶∞ε∫0txTs,Φ,r0xs,Φ,r0ds<∞,where ε⋅ denotes the mathematical expectation and xs,Φ,r0 indicates the solution of system (10) at the instant t under the initial condition Φt and r0.Definition 2.
For a control law that anywt≠0, wt∈L20,∞, system (10) is stochastic stable under the H∞−γ condition. That is to say, for some scalar γ>0, system (10) is stochastic stable if the inequality(12)ε∫0∞zTtztdt<γ2∫0∞wTtwtdt,holds under zero initial condition.
Next, we focus on the following system:(13)x˙t=Aιtxt+B1ιtKtxtk,tk≤t<tk+1.
If the controller is a constant, we will give the stochastic stability conditions as follows:Theorem 1.
Given a scalarρ>0, T-S fuzzy system (13) is stochastic stable if there are matrices Pi>0, U1>0, U2>0, H1=H1T, H2, H3=H3T, and H4, and arbitrary matrices Y1, Y2, Y3, M1t,tk, M2t,tk, N1t,tk, and N2t,tk with appropriate dimensions such that the following inequation is true for each i∈ℝ, ι∈N, hk∈hm,hM:(14)Ξq<0,q=1,2,where Ξ1=Δ1t,tkhkM1Tt,tkhkhMM2Tt,tk∗−U20∗∗−3U2, Ξ2=Δ2t,tkhkN1Tt,tkhkhMN2Tt,tk∗−U10∗∗−3U1, Δ1t,tk=Θ1t,tk+Θ2t,tk+hkΘ3t,tk, Δ2t,tk=Θ1t,tk+Θ2t,tk+hkΘ4t,tk, Θ1t,tk=Syme1TPie6−Π1TH2Π2+Π4TH4Π2+M1Tt,tkΠ6−2M2Tt,tke4−N1Tt,tkΠ7−2N2Tt,tke5+e1T∑j=1sπijPje1−Π1TH1Π2+Π4TH3Π4, Θ2t,tk=SymΠ10TΦ1t,tk, Θ3t,tk=SymΠ1TH1Π3+Π3TH2Π2+N2Tt,tkΠ9+e6TU1e6, Θ4t,tk=SymΠ4TH3Π5+Π5TH2Π2+M2Tt,tkΠ8+e6TU2e6, ej=0n×j−1nIn0n×6−jn,j=1,2,…,6, Π1=e1T−e2Te4TT,Π2=e2Te3TT,Π3=e6Te1TT,Π4=e1T−e3Te5TT,Π5=e6T−e1TTΠ6=e1T−e2TT,Π7=e1T−e3TT,Π8=e1+e2,Π9=e1+e3,Π10=e1TY1+e2TY2+e6TY3T, and Φ1t,tk=Aιte1+B1ιtKtke2−e6.Proof.
Let us take the Lyapunov function for system (13) as follows:(15)Wrt,t=Vrt,t+V0t,t∈tk,tk+1,k∈N,where Vrt,t is a quadratic Lyapunov function defined as Vrt,t=xTtPrtxt, and V0t=∑p=12Vpt with V1t=tk+1−tη1TH1η1t+2H2η3t+t−tkη2TH3η2t+2H4η3t, V2t=tk+1−t∫tktx˙TsU1x˙sds−t−tk∫ttk+1x˙TsU2x˙sds, where ζ1t=∫tktxsds,ζ2t=∫ttk+1xsds,ζ3t=xt−xtk,ζ4t=xt−xtk+1, η1t=colζ3t,ζ1t,η2t=colζ4t,ζ2t,η3t=colxtk,xtk+1, and ηt=colxt,η3t,ζ1t,ζ2t,x˙t.
Then, for eachrt=ι,ι∈S, we say that L is a weak infinitesimal generator of Wrt,t, it follows that LVit=2ηTte1TPie6ηt+ηTte1T∑j=1sπijPje1ηt, LV1t=ηTt−Π1TH1Π2+2H2Π2−2tk+1−tΠ1TH1Π3+Π3TH2Π2+Π4TH3Π4+2H4Π2+2t−tkΠ4TH3Π5+Π5TH4Π2ηt, LV2t=tk+1−tηtTe6TU1e6ηt−∫tktx˙TsU1x˙sds+t−tkηtTe6TU2e6ηt−∫ttk+1x˙TsU2x˙sds,
By Lemma1, the following two inequalities hold:(16)−∫tktx˙TsU1x˙sds≤ηTtt−tkM1Tt,tkU1−1M1t,tk+h23M2Tt,tkU1−1M2t,tk+SymM1Tt,tkΠ6−2M2Tt,tke3+t−tkM2Tt,tkΠ8ηt,(17)−∫ttk+1x˙TsU2x˙sds≤ηTttk+1−tN1Tt,tkU2−1N1t,tk+h23N2Tt,tkU2−1N2t,tk+SymN1Tt,tkΠ7−2N2Tt,tke5+tk+1−tN2Tt,tkΠ9ηt,where M1t,tk=∑i=1r∑j=1rhiφthjφtkM11,ij,M12,ij,…,M1k,ij, M2t,tk=∑i=1r∑j=1rhiφthjφtkM21,ij,M22,ij,…,M2k,ij, N1t,tk=∑i=1r∑j=1rhiφthjφtkN11,ij,N12,ij,…,N1k,ij, andN2t,tk=∑i=1r∑j=1rhiφthjφtkN21,ij,N22,ij,…,N2k,ij.
Moreover, based on the system (13), for arbitrary matrices Y1, Y2, and Y3 with appropriate dimensions, the following equation is true:(18)2xtTY1+xtkTY2+x˙tTY3−x˙t+Aιtxt+B1ιtKtxtk=0.
Observing the two inequalities (16) and (17), let’s add LWιt to the right side of the equation (18), thus, for any t∈tk,tk+1, there is the following conclusion:(19)LWιt≤ηtTtk+1−thkΞ1+t−tkhkΞ2ηt.where Ξ1=Δ1t,tk+hkN1Tt,tkU2−1N1t,tk+hM2/3N2Tt,tkU2−1N2t,tk and Ξ2=Δ2t,tk+hkM1Tt,tkU1−1M1t,tk+hM2/3M2Tt,tkU1−1M2t,tk.
We can conclude from Lemma2 and (14) that(20)Ξ1<−ρI2,Ξ2<−ρI2.This means(21)LWιt≤−ρxtk2.
By Dynkin formula, for anyt≥0, we get the following equation:(22)εWrt,t−εWr0,t0≤−ρε∫0txs2ds.
Hence,(23)ε∫0txs2ds≤ρ−1εWr0,t0.
After the analysis above, we finally reach the conclusion.(24)limt⟶∞ε∫0txs2ds≤dsup−∞<s≤0Φs.
Then, we can conclude from Definition1 that system (13) is stochastically stable.Remark 2.
Inspired by literature [38], we propose a new Markovian modal-dependent Lyapunov function, this function fully utilizes the system sampled-date information from both intervals xtk,xt and xt,xtk+1, which satisfies the closed-loop function condition (V0tk=V0tk+1=0). However, the Lyapunov function constructed in the literature [38] only contained ∫tktxTtds and does not contain ∫ttk+1xTtds whereas our constructed function Vt includes both, thus the results derived in this study are less conservative.
Next, based on Theorem1, we give the stochastic stability condition of system (10) under the H∞−γ condition.Theorem 2.
For a given scalarγ>0, T-S fuzzy system (10) is stochastic stable under the H∞−γ condition, if there are matrices Pi>0, U1>0, U2>0, H1=H1T, H2, H3=H3T, and H4, and arbitrary matrices Y1, Y2, Y3, M1,ij, M2,ij, N1,ij, N2,ij such that inequation (25) is true for each i∈ℝ, ι∈N, and hk∈hm,hM:(25)Ξq,ij<0,q=1,2,where Ξ1,ij=Δ1,ijhkM1,ijThkhMM2,ijTΦ2,jT∗−U200∗∗−3U20∗∗∗−I, Ξ2,ij=Δ2,ijhkN1,ijThkhMN2,ijTΦ2,jT∗−U100∗∗−3U10∗∗∗−I, Δ1,ij=Θ1,ij+Θ2,ij+hkΘ3,ij, Δ2,ij=Θ1,ij+Θ2,ij+hkΘ4,ij, Θ1,ij=Syme^1TPie^6−Π^1TH2Π^2+Π^4TH4Π^2+M1,ijTΠ^6−2M2,ijTe^4−N1,ijTΠ^7−2N2,ijTe^5+e^1T∑j=1sπijPje^1−Π^1TH1Π^2+Π^4TH3Π^4, Θ2,ij=SymΠ^10TΦ1,ij−γ2e^7Te^7, Θ3,ij=SymΠ^1TH1Π^3+Π^3TH2Π^2+N2,ijTΠ^9+e^6TU1e^6, Θ4,ij=SymΠ^4TH3Π^5+Π^5TH2Π^2+M2,ijTΠ^8+e^6TU2e^6, e^j=0n×j−1nIn0n×6−jn0n×q,j=1,2,…,6, e^7=0q×6nIq, Π^1=e^1T−e^2Te^4TT,Π^2=e^2Te^3TT,Π^3=e^6Te^1TT,Π^4=e^1T−e^3Te^5TT,Π^5=e^6T−e^1TT, Π^6=e^1T−e^2TT,Π^7=e^1T−e^3TT,Π^8=e^1+e^,Π^9=e^1+e^3, Π^10=e^1TY1+e^2TY2+e^6TY3T, andΦ1,j=Aιe^1+B1ιKje^2−e^6+D1ιe^7, Φ2,j=Cιe^1+B2ιKje^2+D2ιe^7, and the definitions of M1,ij, M2,ij, N1,ij, N2,ij are similar to that in Theorem 1.
Next, we give the proof of Theorem2.Proof.
Observe the statement in Theorem2, assume that there is a scalar ρ>0 satisfying(26)Ξ¯q,ij<0,q=1,2,where Ξ¯1,ij=Δ1,ijhkM1,ijThkhMM2,ijT∗−U20∗∗−3U2, Ξ¯2,ij=Δ2,ijhkN1,ijThkhMN2,ijT∗−U10∗∗−3U1, Δ1,ij=Θ1,ij+Θ2,ij+hkΘ3,ij, Δ2,ij=Θ1,ij+Θ2,ij+hkΘ4,ij, Θ1,ij=Syme1TPie6−Π1TH2Π2+Π4THΠ2+M1,ijTΠ6−2M2,ijTe4−N1,ijTΠ7−2N2,ijTe5+e1T∑j=1sπijPje1−Π1TH1Π2+Π4TH3Π4, Θ2,ij=SymΠ10TΦ1,j, Θ3,ij=SymΠ1TH1Π3+Π3TH2Π2+N2,ijTΠ9+e6TU1e6, and Θ4,ij=SymΠ4TH3Π5+Π5TH2Π2+M2,ijTΠ8+e6TU2e6, Φ1,j=Aιe1+B1ιKje2−e6.
Even more, the termsΞ1 and Ξ2 in inequation (14) can be described as follows:(27)Ξq=∑i=1rhiφt∑j=1rhjφtΞ¯q,ij,q=1,2.
Hence, we can conclude from inequation (26) that inequation (14) is true, that is to say, Ξq<0,q=1,2, then we know by Theorem 1 that system (10) is stochastically stable when wt=0.
Next, we will prove that the system (10) is stochastic stable under the H∞−γ condition.
Let(28)Jzw=LWιt+zTtzt−γ2wTtwt.
Using the same proof as in Theorem1, for any t∈tk,tk+1, the following inequation holds:(29)Jzw≤ηtwtTtk+1−thkΞ¯1+t−tkhkΞ¯2ηtwt,where Ξ1t,tk=Δ1t,tkhkM1Tt,tkhkhMM2Tt,tkΦ2Tt,tk∗−U200∗∗−3U20∗∗∗−I, Ξ2t,tk=Δ2t,tkhkN1Tt,tkhkhMN2Tt,tkΦ2Tt,tk∗−U100∗∗−3U10∗∗∗−I, Δ1t,tk=Θ1t,tk+Θ2t,tk+hkΘ3t,tk, Δ2t,tk=Θ1t,tk+Θ2t,tk+hkΘ4t,tk, Θ1t,tk=Syme^1TPie^6−Π^1TH2Π^2+Π^4TH4Π^2+M1Tt,tkΠ^6−2M2Tt,tke^4−N1Tt,tkΠ^7−2N2Tt,tke^5+e^1T∑j=1sπijPje^1−Π^1TH1Π^2+Π^4TH3Π^4, Θ2t,tk=SymΠ^10TΦ1t,tk−γ2e^7Te^7, Θ3t,tk=SymΠ^1TH1Π^3+Π^3TH2Π^2+N2Tt,tkΠ^9+e^6TU1e^6, Θ4t,tk=SymΠ^4TH3Π^5+Π^5TH2Π^2+M2Tt,tkΠ^8+e^6TU2e^6, Φ1t,tk=Aιte^1+B1ιtKte^2−e^6+D1ιte^7, andΦ2t,tk=Cιte^1+B1ιtKte^2+D2ιte^7.
We take(30)Ξ¯q=∑i=1Rhiφt∑j=1RhjφtΞq,ij,q=1,2,where Ξq,ij<0q=1,2; then, we know that(31)Ξ¯q<0,q=1,2.
Observing inequation (29), we conclude that(32)Jzw≤0.
Therefore, for anyτ≥0, there must be a non-negative integer α such that tα≤τ<tα; then let us observe formula (28) and inequation (32), and we obtain that(33)∫0τzTtztdt−γ2∫0τwTtwtdt≤−Wιτ≤0.
By the Dynkin formula, for anyt≥0, the following inequation holds:(34)ε∫0∞zTtztdt<−γ2∫0∞wTtwtdt.
As stated in Definition2, we know that system (10) is stochastic stable under the H∞−γ condition.
Next, let’s give the sufficient condition for the existence of state feedback controllers for sampled-data follows according to Theorem2.Theorem 3.
3.For a given scalarγ>0, the T-S fuzzy system (10) is stochastic stable under the H∞−γ condition, if there are matrices P˜i>0, U˜1>0, U˜2>0, H˜1=H˜1T, H˜2, H˜3=H˜3T, H˜4, Y˜, and Lι, and arbitrary matrices M˜1,ij, M˜2,ij, N˜1,ij, and N˜2,ij such that inequations (30) are true for each i,j∈ℝ, ι∈N, hk∈hm,hM, and(35)Ξ˜q,ij<0,q=1,2,where Ξ˜1,ij=Δ˜1,ijhkM˜1,ijThkhMM˜2,ijTΦ˜2,jT∗−U˜200∗∗−3U˜20∗∗∗−I, Ξ˜2,ij=Δ˜2,ijhkN˜1,ijThkhMN˜2,ijTΦ˜2,jT∗−U˜100∗∗−3U˜10∗∗∗−I, Δ˜1,ij=Θ˜1,ij+Θ˜2,ij+hkΘ˜3,ij, Δ˜2,ij=Θ˜1,ij+Θ˜2,ij+hkΘ˜4,ij, Θ˜1,ij=Syme^1TPi˜e^6−Π^1TH˜2Π^2+Π^4TH˜4Π^2+Π^6TX˜e^6−2M˜2,ijTe^4−N˜1,ijTΠ^7−2N˜2,ijTe^5+e^1T∑j=1sπijP˜je^1−Π^1TH˜1Π^2+Π^4TH˜3Π^4, Θ˜2,ij=SymΠ^10TΦ˜1,j−γ2e^7Te^7, Θ˜3,ij=SymΠ^1TH˜1Π^3+Π^3TH˜2Π^2+N˜2,ijTΠ^9+e^6TU˜1e^6, Θ˜4,ij=SymΠ^4TH˜3Π^5+Π^5TH˜2Π^2+M˜2,ijTΠ^8+e^6TU˜2e^6, Π˜10=e^1T+ϵ1e^2T+ϵ2e^6TT, Φ˜1,j=AιY˜Te^1+B1ιLje^2−Y˜Te^6+D1ιe^7,Φ˜2,j=CιY˜Te^1+B1ιLje^2+D2ιte^7, and Π^θθ=1,2,…,9 are the same definitions as in Theorem 2; in this case, we give the expected controller gains as follows:(36)Ki=LiY−T,i=1,2,…,r.Proof.
.LetY˜=Y1−1, L=KY˜T, Y2=ϵ1Y1, Y3=ϵ2Y1, Pi˜=Y˜PiY˜T, U˜1=Y˜U1Y˜T, U˜2=Y˜U2Y˜T, H˜1=Y˜H1Y˜T, H˜2=Y˜H2Y˜T, H˜3=Y˜H3Y˜T, H˜4=Y˜H4Y˜T, M˜1,ij=Y˜M1,ijY˜T, M˜2,ij=Y˜M2,ijY˜T, N˜1,ij=Y˜N1,ijY˜T, N˜2,ij=Y˜N2,ijY˜T, and(37)Γ=diagY˜,Y˜,Y˜,Y˜,Y˜,Y˜,I,Y˜,Y˜,I.
By the statements in Theorem2, we know that Ξ1,ij<0,Ξ2,ij<0, let’s pre-multiply Ξ1,ij by Γ, and postmultiply Ξ2,ij by ΓT, we can conclude that inequations (35) are accurate. This concludes the proof process.


## 4. Numerical Simulation Example

We provide an example in this work to verify that the proposed approach is effective.Example: The Truck-Trailer model is mentioned in the literature [42, 43]. Here, we take it as an example to verify the results. The Truck-Trailer model is shown as follows:(38)x˙1t=−w¯pt¯Lt¯0x1t+w¯pt¯l0t¯0ut+0.1w1t+0.1w2t,x˙2t=w¯pt¯Lt¯0x1t+0.1w1t+0.1w2t,x˙3t=w¯pt¯t¯0sinx2t+w¯pt¯2Lx1t,where x1t is the angle difference between the trailer and the truck, x3t is the vertical position of the rear of the trailer, x2t is the angle of the trailer, w1t and w2t are disturbance input, and the model parameters can be set as t¯=2, w¯1=−1, w¯2=−1.05, w¯3=−0.95, t¯0=0.5, l=2.8, L=5.5. Assume that the switching process between the three modes follows a Markov process, and the transfer probability matrix is Π: Π=−0.30.10.20.4−0.60.20.80.1−0.9.LetΘt=x2t+w¯t¯/2Lx1t, the membership function is defined as follows:h1Θt=sinΘt−g¯Θt/Θt1−g¯,ifΘt≠0,1,ifΘt=0 where g¯=10−2/π, and let h2Θt=1−h1Θt.TheT-S fuzzy systems of the model (38) can be described as follows:(39)x˙t=∑i=13hιx1tAi,ιxt+B1i,ιut+D1i,ιwt.The parameters of which we refer to the literature [42, 43] are defined as follows: A1,ι=−w¯pt¯/Lt¯000w¯pt¯/Lt¯000w¯p2t¯2/2Lt¯0w¯pt¯/t¯00,A2,ι=−w¯pt¯/Lt¯000w¯pt¯/Lt¯000g¯w¯p2t¯2/2Lt¯0g¯w¯pt¯/t¯00, B11,ι=B12,ι=w¯pt¯/lt¯000,D11,ι=D12,ι=0.10.10.10.100.Then, the output vectorzt is reconstructed as follows:(40)zt=∑i=13hιx1tCi,ιxt+B2i,ιut+D2i,ιwt,where C1,ι=C2,ι=0.10.20.20.30.20.1,B21,ι=B22,ι=1.21.1,andD21,ι=D22,ι=0.15000.2.First of all, the uncertain sampling period is considered. Based on Theorem3, let ε1=ε2=0.5, and hm=0, γmin with different largest sampling periods hM are given in Table 1. It i’s easy to find that a larger hM corresponds to a larger γmin.Table 1
Different performanceγmin for different hM, where hm=0.
hM0.050.150.250.35γmin1.73171.76781.82511.9298Next, the situation of a specific sampling period is analyzed. In other words,hm=hM=h, the H∞ performances γmin with different h are given in Table 2. It is easy to find that a larger h corresponds to a larger γmin.Table 2
Different performanceγmin for different h, where hm=hM=h.
h0.100.150.200.25γmin1.74271.75781.77601.7986From the two tables above, we can see that when the sampling period increases, the corresponding performanceγmin also increases. This phenomenon inspires us to improve the H∞ performance of the system by increasing the sampling frequency.In addition, we select sampling periodhm=0.1, hM=0.3 to verify Theorem 3, and we can obtain the following controller: K1=12.1392−22.37144.0967 and K2=11.7478−20.28204.1563.Based on the above controller, the initial value of the system is selected asx0=−0.5π−0.75π−4. Based on the nonperiodic sampled-data controller obtained from Theorem 3, the state response curve and Markovian-jump signals of the closed-loop system in Figure 1 show that the system state response is approaching zero. Figure 2 shows the sampled-data control input and the system aperiodic sampling intervals, as we see that the synchronization error of the Truck-Trailer model is approaching zero. Furthermore, each stem represents the sampling time tk, and the value of each stem indicates the range of the sampling interval hk. We can thus conclude that the closed-loop system is stable under the action of our proposed nonperiodic sampled-data controller, which shows the effectiveness of our proposed method.Figure 1
System state response and Markovian-jump signals in example.Figure 2
Sampled-data control input and aperiodic sampling intervals in example.


## 5. Conclusion

In this paper, a design algorithm of the state feedback controller is proposed, which ensures the stability of theT-S fuzzy systems with Markovian-jump parameters under H∞ performance condition, and the stability condition with less conservative is given in linear matrix inequality form. At the same time, a Truck-Trailer model is used to simulate the T-S fuzzy systems with Markovian-jump parameters, and the simulation results also prove that our presented method is real and effective.



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