# Memory State Feedback Control for Time-Varying Delay Switched Fuzzy Systems

Authors: Le Zhang; Meiyu Jia; Hong Yang; Gang Wu
Journal: Advances in Fuzzy Systems
 (2019)

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## Abstract

Consider the problem of memoryless state feedback controller for time-delay system, which cannot consider both the memoryless and the memory items in the system. Therefore, the memoryless state feedback controller has certain limitations and is more conservative. This paper addresses the memory state feedback control for the time-varying delay switched fuzzy systems based on T-S fuzzy model to overcome the problem discussed above. The state vector and input of the time-varying delay systems contain unknown time-varying delay with known bounds. The designed controller whose parameters are solvable can introduce past state information and reduce the system conservativeness. The more general Lyapunov-Krasovskii functional is selected and the switching law is designed in order to analyze the open-loop system stability, and the memory state feedback controller is designed for the closed-loop system and the criterion for its asymptotic stability. Discuss the solvability of the above two criteria. Finally, a numerical example is given. The simulation results show that the proposed method is more feasible and effective.


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## Body

## 1. Introduction

Hybrid systems are more common in real physical systems. Complex systems in which discrete event subsystems interact with continuous variable subsystems are called hybrid dynamic systems [1–5]. Switched system, as classical hybrid dynamic system, has attracted much attention. And the model of switched system can be interpreted as a group of subsystems and switching laws (logical rules) describing the relationship between the subsystems [6–10]. In practice, many engineering systems, such as network control systems [11, 12], robot systems [13, 14], and electromechanical systems [15], can be described by the switched system models. The T-S fuzzy model provided new method for the nonlinear system problems [16–21]. The fuzzy rules describing the input-output relationship of each subsystem of nonlinear system are important components of the T-S fuzzy model. Each sublinear fuzzy system is connected by membership function. Therefore, the mature linear system theory can be used to solve the problems of nonlinear system based on T-S fuzzy model [22–25].Switched fuzzy systems whose subsystems are composed of T-S fuzzy systems, Yang [26] et al. proposed in 2008, further optimize the switching fuzzy systems. The switching law is used to switch the subfuzzy system controlled by the designed controller. Compared with the switching fuzzy system, this system does not rely on the regional rules and optimizes the secondary fuzzy rules into first-order fuzzy rules. The switched fuzzy system model consisting of “switching” and “fuzzy” ideas does not have the restraint like traditional control system, broadening the performance requirements for control. It is suitable for more complex practical systems. Some researches have made results regarding the proposed switched fuzzy systems [27–34]. The network systems were transformed into switched fuzzy model. The free weighting matrix was used to deal with the delay and packet loss caused by the network in [35]. Authors in [36, 37] focused on stability analysis for the switched fuzzy systems; however, the designed controllers all are memoryless state feedback controllers. This kind of controller cannot have effect in the time-delay systems because it does not introduce the past state information of the systems. That is why memory state feedback controller has attracted much attention [38–44]. For example, in [45], a memory state feedback controller was given to control the affine linear parameter variable systems, which verified that the designed H∞ memory state feedback controller had better immunity and effectively reduced the influence of time delay on the system.Delays have been encountered in almost all real systems, which can be caused in many conditions. Generally, the transmission of information in a system often leads to the occurrence of time delay, such as communication systems and power systems. [46–54]. Scholars are keen to study stability for delay systems because delays in systems often lead to system performance degradation and instability. Reference [55] introduced the nonfragile control of memory state feedback for stochastic systems with time delays and presented the sufficient conditions for the asymptotic stability of closed-loop systems by combining Lyapunov function with linear matrix inequality (LMI). The delay-dependent switched fuzzy system was asymptotically stable by the observer switching method, when the actuator was seriously invalid in [56]. The stability conditions of time-delay-independent system are more conservative than delay-dependent system. It is worth noting that, although there are many studies on memory state feedback control, the control problem of memory state feedback has not been involved in time-varying delay (delay-dependent) switched fuzzy systems.Therefore, this paper uses the Lyapunov function method to study the memory state feedback control problem for switched fuzzy time-varying delay switched fuzzy systems. The stability problem of the open-loop system is discussed, and the criterion for making the open-loop system asymptotically stable is given. The suitable switching law and the memory state feedback controller which has less conservativeness are designed and the more general Lyapunov functionals are selected in order to obtain the sufficient condition for the asymptotic stability of the closed-loop system. This sufficient condition is compared with that under the control of the memoryless state feedback controller for the closed-loop systems to explore the solvability problems. Simulation results testify that the proposed method is better than memoryless state feedback control under the same initial condition.


## 2. Analysis of Time-Varying Delay Switched Fuzzy Systems

Time-varying delay switched fuzzy model based on T-S fuzzy model is proposed. That is, each time-varying delay switched subsystem is composed of T-S fuzzy system. The model is different from the switching fuzzy model, and its switching rule is a first-level switching rule. Consider time-varying delay switched fuzzy systems consisting ofNσ(t) rules as follows:(1)Rσtj:ifzσt1tisMσt1j⋯andzσtptisMσtpj,thenx˙t=Aσtjxt+Aqσtjxt-hσtt+Bσtjuσtt,xt=φt,t∈-d,0,yt=Cσtjxt,j=1,2,⋯,Nσtwhere Rσ(t)j indicates the jth fuzzy rule of the switched fuzzy model; σ(t):M-=1,2,⋯,m is the switching signal to be designed; z(t)=zσt1,zσt2,…,zσtp is the premise variable; Nσ(t) is the number of fuzzy rules; x(t)∈Rn is the state variable vector of the systems; y(t)∈Rs is the output variable vector of the systems; φ(t) is a continuously differentiable vector-valued initial function which satisfies t∈-d,0 and d>0; Mσ(t)pj is fuzzy set; uσ(t)j(t)∈Rq is the input variable of the jth switched subsystem; Aσ(t)j, Aqσ(t)j, Bσ(t)j, Cσ(t)h are constant matrices of known appropriate dimensions; and hσ(t)(t) is time-varying delay function of the jth switched subsystem which meets the following inequalities called slow time-varying delay.Assumption 1.
​(2)0≤hσtt≤d,h˙σtt≤h<1where h is a known constant.


### 2.1. Switching Signal Design

Switching signalσ(x(t))=1,xt∈Ω-i;0,x(t)∉Ω-i divides the entire space into m subregions, Ω¯1,Ω¯2,…,Ω¯m, if and only if x(t)∈Ω-i, σ(x(t))=1.Switching law(3)σ=σxt=i:0,+∞→M-=1,2,⋯,mThejth subsystem of time-varying delay switched fuzzy systems is as follows:(4)Rij:ifzi1tisMi1j⋯andziptisMipj,thenx˙t=Aijxt+Aqijxt-hit+Bijuit,xt=φt,t∈-d,0,yt=Cijxt,j=1,2,⋯,Ni,i=1,2,⋯,mThe membership function of the proposed time-varying delay switched fuzzy systems is(5)ηijzt≜∏ρ=1pMiρjzρt∑j=1Ni∏ρ=1pMiρjzρt,M i ρ j z ρ ( t ) denotes the membership function in which zρ(t) belongs to the fuzzy set Miρj. And the membership function ηijz(t) satisfies the following conditions. The membership function ηijz(t) is expressed as ηij in order to simplify the expression:(6)0≤ηijzt≤1,∑h=1Niηijzt=1Therefore, the global model for thejth time-varying delay switched fuzzy systems based on T-S model is obtained:(7)x˙t=∑j=1NiηijAijxt+Aqijxt-hit+Bijuit,xt=φt,t∈-d,0,yt=∑j=1NiηijCijxt,i=1,2,⋯,mThe open-loop system of the proposed systems based on T-S model is given:(8)x˙t=A¯txt+A¯qtxt-hitxt=φt,t∈-d,0,yt=C¯txt,j=1,2,⋯,Ni,i=1,2,⋯,malong with(9)A¯t≜∑i=1m∑j=1NiηijAij,A¯qt≜∑i=1m∑j=1NiηijAqijC¯t≜∑i=1m∑j=1NiηijCij,


### 2.2. The Controller Form

Memoryless state feedback controller exists in most literature, which does not introduce past state information. Therefore, consider the memory state feedback controller form of thejth subsystem of the proposed time-varying delay switched fuzzy system as follows:(10)Rij:ifzi1tisMi1j⋯andziptisMipj,thenui=Kijxt+K1ijxt-hitwhere Kij and K1ij are the two gain matrices of the memory state feedback controller to be sought. The global model of the memory state feedback controller can be described as follows:(11)ui=∑j=1NiηijKijxt+K1ijxt-hitAccording to the given switching law (3) and the memory state feedback controller (11), closed-loop system (12) is obtained:(12)x˙t=∑i=1m∑j=1Niηij∑r=1NiηirAijxt+Aqijxt-hit+BijKirxt+BijK1irxt-hitRemark 2.
In order to better study the proposed time-varying delay switched fuzzy systems, the stability of the open-loop system (8) is considered at first, that is, the characteristics of the system itself.


## 2.1. Switching Signal Design

Switching signalσ(x(t))=1,xt∈Ω-i;0,x(t)∉Ω-i divides the entire space into m subregions, Ω¯1,Ω¯2,…,Ω¯m, if and only if x(t)∈Ω-i, σ(x(t))=1.Switching law(3)σ=σxt=i:0,+∞→M-=1,2,⋯,mThejth subsystem of time-varying delay switched fuzzy systems is as follows:(4)Rij:ifzi1tisMi1j⋯andziptisMipj,thenx˙t=Aijxt+Aqijxt-hit+Bijuit,xt=φt,t∈-d,0,yt=Cijxt,j=1,2,⋯,Ni,i=1,2,⋯,mThe membership function of the proposed time-varying delay switched fuzzy systems is(5)ηijzt≜∏ρ=1pMiρjzρt∑j=1Ni∏ρ=1pMiρjzρt,M i ρ j z ρ ( t ) denotes the membership function in which zρ(t) belongs to the fuzzy set Miρj. And the membership function ηijz(t) satisfies the following conditions. The membership function ηijz(t) is expressed as ηij in order to simplify the expression:(6)0≤ηijzt≤1,∑h=1Niηijzt=1Therefore, the global model for thejth time-varying delay switched fuzzy systems based on T-S model is obtained:(7)x˙t=∑j=1NiηijAijxt+Aqijxt-hit+Bijuit,xt=φt,t∈-d,0,yt=∑j=1NiηijCijxt,i=1,2,⋯,mThe open-loop system of the proposed systems based on T-S model is given:(8)x˙t=A¯txt+A¯qtxt-hitxt=φt,t∈-d,0,yt=C¯txt,j=1,2,⋯,Ni,i=1,2,⋯,malong with(9)A¯t≜∑i=1m∑j=1NiηijAij,A¯qt≜∑i=1m∑j=1NiηijAqijC¯t≜∑i=1m∑j=1NiηijCij,


## 2.2. The Controller Form

Memoryless state feedback controller exists in most literature, which does not introduce past state information. Therefore, consider the memory state feedback controller form of thejth subsystem of the proposed time-varying delay switched fuzzy system as follows:(10)Rij:ifzi1tisMi1j⋯andziptisMipj,thenui=Kijxt+K1ijxt-hitwhere Kij and K1ij are the two gain matrices of the memory state feedback controller to be sought. The global model of the memory state feedback controller can be described as follows:(11)ui=∑j=1NiηijKijxt+K1ijxt-hitAccording to the given switching law (3) and the memory state feedback controller (11), closed-loop system (12) is obtained:(12)x˙t=∑i=1m∑j=1Niηij∑r=1NiηirAijxt+Aqijxt-hit+BijKirxt+BijK1irxt-hitRemark 2.
In order to better study the proposed time-varying delay switched fuzzy systems, the stability of the open-loop system (8) is considered at first, that is, the characteristics of the system itself.


## 3. Main Results

In this section, the stability of the open-loop system and the control of the closed-loop system with feedback are mainly studied.Lemma 3.
Given the constant matricesX and Y of the appropriate dimension, for any constant ε>0, the following inequality holds:(13)XTY+YTX≤εXTX+ε-1YTY


### 3.1. Stability Analysis for Open-Loop System

Theorem 4.
Consider switched fuzzy systems (8); suppose there exist constants βiλ (i,λ∈M-) (either all nonnegative or all nonpositive); hσ(t)(t) satisfies Equation (2); there exist positive definite symmetric matrices PLi, WLi, QLi, and positive scalars εi, ε1i, d, such that the matrix inequality (14) is satisfied; then system (8) is symptomatically stable: (14)AijTPLi+PLiAij+εi-1PLiPLi+AijTWLi+ε1iIAij+QLi+∑λ=1mβiλPLi-PLλ00εiAqijTAqij+ε1i-1AqijTWLiWLiAqij+AqijTWLiAqij-1-hitQLi<0i=1,2,⋯,m,j,r=1,2,⋯,Ni.Proof.
Without loss of generality, supposeβiλ≥0. For any i∈M-, if xT(t)(PLi-PLλ)x(t)≥0, ∀λ∈M-. From (14), the inequality (15) is given:(15)AijTPLi+PLiAij+εi-1PLiPLi+AijTWLi+ε1iIAij]+QLi00εiAqijTAqij+ε1i-1AqijTWLiWLiAqij+AqijTWLiAqij-1-hitQLi<0
Obviously, for∀xt∈Rn∖0, there certainly is an i∈M- such that xT(t)(PLi-PLλ)x(t)≥0, ∀λ∈M-, For arbitrary i∈M-, let Ωi=xt∈Rn∖0∣xT(t)(PLi-PLλ)x(t)≥0,∀λ∈M-. Then ⋃i=1mΩi=Rn∖0. Further, construct the sets Ω-1=Ω1,⋯, Ω-i=Ωi-⋃λ=1i-1Ω-λ,⋯,Ω-m=Ωm-⋃λ=1m-1Ω-λ. Obviously, we have ⋃i=1mΩi=Rn∖0, and Ω-i∩Ω-λ=Φ, i≠λ.
Consider the following Lyapunov functional:(16)VLixt=VL1xt+VL2xt+VL3xtwhere(17)VL1xt=xTtPLixtVL2xt=d∫-d0∫t+θtx˙TsWLix˙sdsdθVL3xt=∫t-hittxTsQLixsdsPLi,WLi,QLi are all positive definite symmetric matrices, which satisfy matrix inequality (14). When x(t)∈Ω-i, i∈M¯, then(18)V˙Lixt=V˙L1xt+V˙L2xt+V˙L3xtAssuming that inequality (2) is established, and according to Lemma 3, the following formulas are obtained:(19)V˙L1xt=x˙TtPLixt+xTtPLix˙t=∑j=1NiηijAijxt+Aqijxt-hitTPLixt+xTtPLiAijxt+Aqijxt-hit≤∑j=1NiηijxTtAijTPLi+PLiAij+εi-1PLiPLixt+xt-hitTεiAqijTAqijxt-hit(20)V˙L2xt=d2xT.tWLix.t-d∫t-dtxT.sWLix.sds≤d2x˙TtWLix˙t-d∫t-hittxT.sWLix.sds=∑j=1Niηijd2Aijxt+Aqijxt-hitTWLiAijxt+Aqijxt-hit-d∫t-hittxT.sWLix.sds≤d2∑j=1NiηijxTtAijTWLiAijxt+ε1ixTtAijTAijxt+ε1i-1xt-hitTAqijTWLiWLiAqijxt-hit+xt-hitTAqijTWLiAqijxt-hit-d∫t-hittxT.sWLix.sds=d2∑j=1NiηijxTtAijTWLi+ε1iIAij×xt+xt-hitTε1i-1AqijTWLiWLiAqij+AqijTWLiAqijxt-hit-d∫t-hittxT.sWLix.sds
According to Assumption1, (1-hi.(t))e-αhi(t)≥(1-hi)e-αd>0; then(21)V˙L3xt=xTtQLixt-1-h.itxTt-hitQLixt-hit≤∑j=1NiηijxTtQLixt-1-hitxTt-hitQLixt-hit
According to formula (19), formula (20), and formula (21), V˙i(x(t)) can be written in the following form(22)V˙Lixt=V˙L1xt+V˙L2xt+V˙L3xt≤∑j=1NiηijxTtAijTPLi+PLiAij+εi-1PLiPLi+AijTWLi+ε1iIAij]+QLixt+xt-hitTεiAqijTAqij+ε1i-1AqijTWLiWLiAqij+AqijTWLiAqij-1-hitQLi×xt-hit-d∫t-τtxT.sWLix.sds=∑j=1NiηijxTtxt-hitT×AijTPLi+PLiAij+εi-1PLiPLi+AijTWLi+ε1iIAij]+QLi00εiAqijTAqij+ε1i-1AqijTWLiWLiAqij+AqijTWLiAqij-1-hitQLi×xtxt-hit<0
We know inequality (22) is established from inequality (15). Therefore, VLi(x(t)) is the Lyapunov functional and the open-loop system (8) is asymptotically stable.Remark 5.
The Lyapunov functions selected in this paper are composed of basic independent matrices, and the constant coefficientd is added to the VL2(x(t)) functions. So, the study of nonlinear systems (8) is more general.


### 3.2. Memory State Feedback Controller Design

The sufficient conditions for the stability of the open-loop system (8) have been obtained by the Lyapunov functional method. Suppose that the state of the proposed time-varying time-delay switched fuzzy system is measurable, suitable controller and switching law can be designed to make the closed-loop system (12) asymptotically stable.Theorem 6.
Suppose there exist constantsβiλ (i,λ∈M-) (either all nonnegative or all nonpositive); there exist positive scalars ε2i,ε3i,ε4i,d, and positive definite symmetric matrices Pi,Wi,Qi, hσ(t)(t) satisfy Equation (2), such that the matrix inequalities (23) are satisfied; then there are memory state feedback controller (11) and switching law σ(t): M-=1,2,⋯,m to make system (12) symptomatically stable:(23)∏1100∏22<0i=1,2,⋯,m,j,r=1,2,⋯,Nialong with(24)∏11=AijTPi+PiAij+KirTBijTPi+PiBijKir+ε2i-1PiPi+ε3i-1PiPi+d2AijT+KirTBijTWi+ε4iIAij+BijKir+Qi+∑λ=1mβiλPi-Pλ∏22=ε2iAqijTAqij+ε3iK1irTBijTBijK1ir-1-hitQi+d2AqijT+K1irTBijTWiAqij+BijK1ir+ε4i-1AqijT+K1irTBijTWiWiAqij+BijK1irProof.
Without loss of generality, supposeβiλ≥0. For any i∈M-, if xT(t)(Pi-Pλ)x(t)≥0, ∀λ∈M-. From (23), we have (25)Γ1100Γ22<0where(26)Γ11=AijTPi+PiAij+KirTBijTPi+PiBijKir+ε2i-1PiPi+ε3i-1PiPi+d2AijT+KirTBijTWi+ε4iIAij+BijKir]+QiΓ22=∏22
Obviously, for∀x(t)∈Rn∖0, there certainly is an i∈M- such that xT(t)(Pi-Pλ)x(t)≥0, ∀λ∈M-, For arbitrary i∈M-, let Ωi=x(t)∈Rn∖0xT(t)(Pi-Pλ)x(t)≥0,∀λ∈M-. Then ⋃i=1mΩi=Rn∖0. Further, construct the sets Ω-1=Ω1,⋯, Ω-i=Ωi-⋃λ=1i-1Ω-λ,⋯,Ω-m=Ωm-⋃λ=1m-1Ω-λ. Obviously, we have ⋃i=1mΩi=Rn∖0, and Ω-i∩Ω-λ=Φ, i≠λ.
Consider more general Lyapunov functional:(27)Vixt=V1xt+V2xt+V3xtwhere(28)V1xt=xTtPixtV2xt=d∫-d0∫t+θtx˙TsWix˙sdsdθV3xt=∫t-hittxTsQixsdsPi, Wi, Qi are positive definite symmetric matrices satisfying the matrix inequality (23). When x(t)∈Ω-i, i∈M¯, there is(29)V˙ixt=V˙1xt+V˙2xt+V˙3xtAssuming that inequality (2) is established, and according to Lemma 3, the following formulas are obtained:(30)V˙1xt=x˙TtPixt+xTtPix˙t=∑j=1Niηij∑r=1NiηirAijxt+Aqijxt-hit+BijKirxt+BijK1irxt-hitTPixt+xTtPiAijxt+Aqijxt-hit+BijKirxt+BijK1irxt-hit≤∑j=1Niηij∑r=1NiηirxTtAijTPLi+PLiAij+PiBijKir+ε2i-1PiPi+ε3i-1PiPi+KirTBijTPixt+xt-hitTε3iK1irTBijTBijK1ir+ε2iAqijTAqijxt-hit(31)V˙2xt=d2xT.tWix.t-d∫t-dtxT.sWix.sds≤d2x˙TtWix˙t-d∫t-hittxT.sWix.sds=∑j=1Niηij∑r=1Niηird2Aijxt+Aqijxt-hit+BijKirxt+BijK1irxt-hitTWiAijxt+Aqijxt-hit+BijKirxt+BijK1irxt-hit-d∫t-hittxT.sWix.sds=d2∑j=1Niηij∑r=1NiηirxTtAijT+KirTBijTWi×Aij+BijKirxt+xTtAijT+KirTBijTWi×Aqij+BijK1irxt-hit+xt-hitTAqijT+K1irTBijTWi×Aij+BijKirxt+xt-hitTAqijT+K1irTBijTWi×Aqij+BijK1irxt-hit-d∫t-hittxT.sWix.sds≤d2∑j=1Niηij∑r=1NiηirxTtAijT+KirTBijT×Wi+ε4iI×Aij+BijKirxt+xt-hitTAqijT+K1irTBijTWi×Aqij+BijK1ir+ε4i-1AqijT+K1irTBijT×WiWiAqij+BijK1irxt-hit-d∫t-hittxT.sWix.sds(32)V˙3xt=xTtQixt-(1-h.itxTt-hitQixt-hit≤∑j=1Niηij∑r=1NiηirxTtQixt-1-hitxTt-hitQixt-hitFrom formulas (30)–(32), we get formula (33):(33)V˙ixt=V˙1xt+V˙2xt+V˙3xt≤∑j=1Niηij∑r=1NiηirxTtAijTPi+PiAij+KirTBijTPi+PiBijKir+ε3i-1PiPi+ε2i-1PiP+Qi+d2AijT+KirTBijTWi+ε4iIAij+BijKirxt+xt-hitTε2iAqijTAqij+ε3iK1irTBijTBijK1ir-1-hitQi+d2[AqijT+K1irTBijTWiAqij+BijK1ir+ε4i-1AqijT+K1irTBijTWiWiAqij+BijK1irxt-hit-d∫t-hittxT.sWLix.sdsEquation (33) can be rewritten as(34)V˙it≤∑j=1Niηij∑r=1Niηirxtxt-hitTΘijrxtxt-hitin which(35)Θijr=AijTPi+PiAij+KirTBijTPi+PiBijKir+ε2i-1PiPi+ε3i-1PiPi+Qi+d2AijT+KirTBijT×Wi+ε4iIAij+BijKir00ε2iAqijTAqij+ε3iK1irTBijTBijK1ir-1-hitQi+d2AqijT+K1irTBijT×WiAqij+BijK1ir+ε4i-1AqijT+K1irTBijTWi×WiAqij+BijK1ir
When EquationΘijr<0, V˙(x(t))<0 can be obtained. Therefore, the controller (11) can ensure that system (12) is asymptotically stable under the switching law σ(x(t))=i.Remark 7.
The integral term-d∫t-τtxT.(s)Wix.(s)ds<0 is obtained, because 0<d<1, Wi is a positive definite symmetry matrix and the physical meaning of the integral term is the area.There are Lyapunov functional matricesPi, Wi and closed-loop system product terms in Equation (23),which can be seen from Theorem 6. In order to solve this problem, an additional matrix is introduced to decouple, and feasible solution in which Equation (23) depends on linear matrix inequalities is obtained.Theorem 8.
Suppose there exist constantsβiλ (i,λ∈M-) (either all nonnegative or all nonpositive); there exist positive scalars ε2i,ε3i,ε4i,d and positive definite symmetric matrices Pi,Wi,Qi,Si, continuously differentiable symmetry matrices Yir,Y1ir, hσ(t)(t) satisfy Equation (2), such that the matrix inequalities (36) are satisfied; then there are memory state feedback controller (42) and switching law σ(t): M-=1,2,⋯,m to make system (12) symptomatically stable:(36)γ11γ12γ1300000∗γ22000000∗0γ3300000000γ44γ45γ46γ47γ48000∗γ55000000∗0γ6600000∗00γ770000∗000γ88<0i=1,2,⋯,m,j,r=1,2,⋯,Nialong with (37)γ11=XiAijT+AijXi+YirTBijT+BijYir+ε2i-1+ε3i-1I+Si+∑λ=1mβiλPi-Pλγ12=XiAijT+YirTBijTγ13=XiAijT+YirTBijTγ22=-d2Wi-1γ33=-ε4id2I-1γ44=-1-hitSiγ45=XiAqijTγ46=Y1irTBijTγ47=XiAqijT+Y1irTBijTγ48=XiAqijT+Y1irTBijTγ55=-ε2iI-1γ66=-ε3iI-1γ77=-d2Wi-1γ88=-d2-1ε4iIProof.
Using the Schur method for the matrix inequality (35)(38)ψ11ψ12ψ1300000∗γ22000000∗0γ3300000000ψ44ψ45ψ46ψ47ψ48000∗γ55000000∗0γ6600000∗00γ770000∗000γ88<0in which(39)ψ11=AijTPi+PiAij+KirTBijTPi+PiBijKir+ε2i-1PiPi+ε3i-1PiPi+Qiψ12=AijT+KirTBijTψ13=AijT+KirTBijTψ44=-1-hitQiψ45=AqijTψ46=K1irTBijTψ47=AqijT+K1irTBijTψ48=AqijT+K1irTBijT
Multiply matrixdiagPi-1,I,I,I,Pi-1,I,I,I,Wi-1 by Equation (38) and define it:(40)Xi=Pi-1,Si=XiQiXi,Yir=KirXi,Y1ir=K1irXi
Then, formula (23) is equivalent to formula (36). Proof is to be true.The gain matrices of the memory state feedback controller are obtained by solving LMI (36):(41)Kir=YirXi-1,K1ir=Y1irXi-1The global model (11) of the memory state feedback controller can be written in the following form:(42)ui=∑j=1NiηijYijXi-1xt+Y1ijXi-1xt-hitInference 1. Suppose there exist constants βiλ (i,λ∈M-) (either all nonnegative or all nonpositive); there exist positive scalars ε2i,ε3i,ε4i,d and positive definite symmetric matrices Pi,Wi,Qi,Si; continuously differentiable symmetry matrices Yir,Y1ir, hσ(t)(t) satisfy Equation (2) and then the matrix inequalities(43)Ε11γ12γ130000∗γ2200000∗0γ330000000γ44γ45Ε46Ε47000∗γ5500000∗0Ε660000∗00Ε77<0where(44)Ε11=XiAijT+AijXi+YirTBijT+BijYir+ε2i-1I+Si+∑λ=1mβiλPi-Pλγ12=XiAijT+YirTBijTγ13=XiAijT+YirTBijTγ22=-d2Wi-1γ33=-ε4id2I-1γ44=-1-hitSiγ45=XiAqijTΕ46=XiAqijTΕ47=XiAqijTγ55=-ε2iI-1Ε66=γ77=-d2Wi-1Ε77=γ88=-d2-1ε4iIThe closed-loop system is asymptotically stable, when the memoryless state feedback controller isui=KqijXi-1x(t), Kqir=YirXi-1 and the switching law is σ(t):M-=1,2,⋯,m.Remark 9.
This paper discusses the situation that the designed controller can stabilize the original system regardless of time-varying delay of the systems. Comparing the linear matrix inequalities (36) with (43), it can be seen that when the inequality (36) has a solution, Equation (43) must have a solution (take Y1ir to 0); on the contrary, it is not necessarily true. Formula (36) has better solvability than Formula (43). Therefore, the memory state feedback controller can control the system in which the memoryless state feedback controller is not controllable.Remark 10.
Equation (23) is transformed into a strict linear matrix inequality (36) by Schur theorem and the matrix inequality method. The transformed LMI matrix increases the conservativeness of the result but makes the final result MATLAB solvable.


## 3.1. Stability Analysis for Open-Loop System

Theorem 4.
Consider switched fuzzy systems (8); suppose there exist constants βiλ (i,λ∈M-) (either all nonnegative or all nonpositive); hσ(t)(t) satisfies Equation (2); there exist positive definite symmetric matrices PLi, WLi, QLi, and positive scalars εi, ε1i, d, such that the matrix inequality (14) is satisfied; then system (8) is symptomatically stable: (14)AijTPLi+PLiAij+εi-1PLiPLi+AijTWLi+ε1iIAij+QLi+∑λ=1mβiλPLi-PLλ00εiAqijTAqij+ε1i-1AqijTWLiWLiAqij+AqijTWLiAqij-1-hitQLi<0i=1,2,⋯,m,j,r=1,2,⋯,Ni.Proof.
Without loss of generality, supposeβiλ≥0. For any i∈M-, if xT(t)(PLi-PLλ)x(t)≥0, ∀λ∈M-. From (14), the inequality (15) is given:(15)AijTPLi+PLiAij+εi-1PLiPLi+AijTWLi+ε1iIAij]+QLi00εiAqijTAqij+ε1i-1AqijTWLiWLiAqij+AqijTWLiAqij-1-hitQLi<0
Obviously, for∀xt∈Rn∖0, there certainly is an i∈M- such that xT(t)(PLi-PLλ)x(t)≥0, ∀λ∈M-, For arbitrary i∈M-, let Ωi=xt∈Rn∖0∣xT(t)(PLi-PLλ)x(t)≥0,∀λ∈M-. Then ⋃i=1mΩi=Rn∖0. Further, construct the sets Ω-1=Ω1,⋯, Ω-i=Ωi-⋃λ=1i-1Ω-λ,⋯,Ω-m=Ωm-⋃λ=1m-1Ω-λ. Obviously, we have ⋃i=1mΩi=Rn∖0, and Ω-i∩Ω-λ=Φ, i≠λ.
Consider the following Lyapunov functional:(16)VLixt=VL1xt+VL2xt+VL3xtwhere(17)VL1xt=xTtPLixtVL2xt=d∫-d0∫t+θtx˙TsWLix˙sdsdθVL3xt=∫t-hittxTsQLixsdsPLi,WLi,QLi are all positive definite symmetric matrices, which satisfy matrix inequality (14). When x(t)∈Ω-i, i∈M¯, then(18)V˙Lixt=V˙L1xt+V˙L2xt+V˙L3xtAssuming that inequality (2) is established, and according to Lemma 3, the following formulas are obtained:(19)V˙L1xt=x˙TtPLixt+xTtPLix˙t=∑j=1NiηijAijxt+Aqijxt-hitTPLixt+xTtPLiAijxt+Aqijxt-hit≤∑j=1NiηijxTtAijTPLi+PLiAij+εi-1PLiPLixt+xt-hitTεiAqijTAqijxt-hit(20)V˙L2xt=d2xT.tWLix.t-d∫t-dtxT.sWLix.sds≤d2x˙TtWLix˙t-d∫t-hittxT.sWLix.sds=∑j=1Niηijd2Aijxt+Aqijxt-hitTWLiAijxt+Aqijxt-hit-d∫t-hittxT.sWLix.sds≤d2∑j=1NiηijxTtAijTWLiAijxt+ε1ixTtAijTAijxt+ε1i-1xt-hitTAqijTWLiWLiAqijxt-hit+xt-hitTAqijTWLiAqijxt-hit-d∫t-hittxT.sWLix.sds=d2∑j=1NiηijxTtAijTWLi+ε1iIAij×xt+xt-hitTε1i-1AqijTWLiWLiAqij+AqijTWLiAqijxt-hit-d∫t-hittxT.sWLix.sds
According to Assumption1, (1-hi.(t))e-αhi(t)≥(1-hi)e-αd>0; then(21)V˙L3xt=xTtQLixt-1-h.itxTt-hitQLixt-hit≤∑j=1NiηijxTtQLixt-1-hitxTt-hitQLixt-hit
According to formula (19), formula (20), and formula (21), V˙i(x(t)) can be written in the following form(22)V˙Lixt=V˙L1xt+V˙L2xt+V˙L3xt≤∑j=1NiηijxTtAijTPLi+PLiAij+εi-1PLiPLi+AijTWLi+ε1iIAij]+QLixt+xt-hitTεiAqijTAqij+ε1i-1AqijTWLiWLiAqij+AqijTWLiAqij-1-hitQLi×xt-hit-d∫t-τtxT.sWLix.sds=∑j=1NiηijxTtxt-hitT×AijTPLi+PLiAij+εi-1PLiPLi+AijTWLi+ε1iIAij]+QLi00εiAqijTAqij+ε1i-1AqijTWLiWLiAqij+AqijTWLiAqij-1-hitQLi×xtxt-hit<0
We know inequality (22) is established from inequality (15). Therefore, VLi(x(t)) is the Lyapunov functional and the open-loop system (8) is asymptotically stable.Remark 5.
The Lyapunov functions selected in this paper are composed of basic independent matrices, and the constant coefficientd is added to the VL2(x(t)) functions. So, the study of nonlinear systems (8) is more general.


## 3.2. Memory State Feedback Controller Design

The sufficient conditions for the stability of the open-loop system (8) have been obtained by the Lyapunov functional method. Suppose that the state of the proposed time-varying time-delay switched fuzzy system is measurable, suitable controller and switching law can be designed to make the closed-loop system (12) asymptotically stable.Theorem 6.
Suppose there exist constantsβiλ (i,λ∈M-) (either all nonnegative or all nonpositive); there exist positive scalars ε2i,ε3i,ε4i,d, and positive definite symmetric matrices Pi,Wi,Qi, hσ(t)(t) satisfy Equation (2), such that the matrix inequalities (23) are satisfied; then there are memory state feedback controller (11) and switching law σ(t): M-=1,2,⋯,m to make system (12) symptomatically stable:(23)∏1100∏22<0i=1,2,⋯,m,j,r=1,2,⋯,Nialong with(24)∏11=AijTPi+PiAij+KirTBijTPi+PiBijKir+ε2i-1PiPi+ε3i-1PiPi+d2AijT+KirTBijTWi+ε4iIAij+BijKir+Qi+∑λ=1mβiλPi-Pλ∏22=ε2iAqijTAqij+ε3iK1irTBijTBijK1ir-1-hitQi+d2AqijT+K1irTBijTWiAqij+BijK1ir+ε4i-1AqijT+K1irTBijTWiWiAqij+BijK1irProof.
Without loss of generality, supposeβiλ≥0. For any i∈M-, if xT(t)(Pi-Pλ)x(t)≥0, ∀λ∈M-. From (23), we have (25)Γ1100Γ22<0where(26)Γ11=AijTPi+PiAij+KirTBijTPi+PiBijKir+ε2i-1PiPi+ε3i-1PiPi+d2AijT+KirTBijTWi+ε4iIAij+BijKir]+QiΓ22=∏22
Obviously, for∀x(t)∈Rn∖0, there certainly is an i∈M- such that xT(t)(Pi-Pλ)x(t)≥0, ∀λ∈M-, For arbitrary i∈M-, let Ωi=x(t)∈Rn∖0xT(t)(Pi-Pλ)x(t)≥0,∀λ∈M-. Then ⋃i=1mΩi=Rn∖0. Further, construct the sets Ω-1=Ω1,⋯, Ω-i=Ωi-⋃λ=1i-1Ω-λ,⋯,Ω-m=Ωm-⋃λ=1m-1Ω-λ. Obviously, we have ⋃i=1mΩi=Rn∖0, and Ω-i∩Ω-λ=Φ, i≠λ.
Consider more general Lyapunov functional:(27)Vixt=V1xt+V2xt+V3xtwhere(28)V1xt=xTtPixtV2xt=d∫-d0∫t+θtx˙TsWix˙sdsdθV3xt=∫t-hittxTsQixsdsPi, Wi, Qi are positive definite symmetric matrices satisfying the matrix inequality (23). When x(t)∈Ω-i, i∈M¯, there is(29)V˙ixt=V˙1xt+V˙2xt+V˙3xtAssuming that inequality (2) is established, and according to Lemma 3, the following formulas are obtained:(30)V˙1xt=x˙TtPixt+xTtPix˙t=∑j=1Niηij∑r=1NiηirAijxt+Aqijxt-hit+BijKirxt+BijK1irxt-hitTPixt+xTtPiAijxt+Aqijxt-hit+BijKirxt+BijK1irxt-hit≤∑j=1Niηij∑r=1NiηirxTtAijTPLi+PLiAij+PiBijKir+ε2i-1PiPi+ε3i-1PiPi+KirTBijTPixt+xt-hitTε3iK1irTBijTBijK1ir+ε2iAqijTAqijxt-hit(31)V˙2xt=d2xT.tWix.t-d∫t-dtxT.sWix.sds≤d2x˙TtWix˙t-d∫t-hittxT.sWix.sds=∑j=1Niηij∑r=1Niηird2Aijxt+Aqijxt-hit+BijKirxt+BijK1irxt-hitTWiAijxt+Aqijxt-hit+BijKirxt+BijK1irxt-hit-d∫t-hittxT.sWix.sds=d2∑j=1Niηij∑r=1NiηirxTtAijT+KirTBijTWi×Aij+BijKirxt+xTtAijT+KirTBijTWi×Aqij+BijK1irxt-hit+xt-hitTAqijT+K1irTBijTWi×Aij+BijKirxt+xt-hitTAqijT+K1irTBijTWi×Aqij+BijK1irxt-hit-d∫t-hittxT.sWix.sds≤d2∑j=1Niηij∑r=1NiηirxTtAijT+KirTBijT×Wi+ε4iI×Aij+BijKirxt+xt-hitTAqijT+K1irTBijTWi×Aqij+BijK1ir+ε4i-1AqijT+K1irTBijT×WiWiAqij+BijK1irxt-hit-d∫t-hittxT.sWix.sds(32)V˙3xt=xTtQixt-(1-h.itxTt-hitQixt-hit≤∑j=1Niηij∑r=1NiηirxTtQixt-1-hitxTt-hitQixt-hitFrom formulas (30)–(32), we get formula (33):(33)V˙ixt=V˙1xt+V˙2xt+V˙3xt≤∑j=1Niηij∑r=1NiηirxTtAijTPi+PiAij+KirTBijTPi+PiBijKir+ε3i-1PiPi+ε2i-1PiP+Qi+d2AijT+KirTBijTWi+ε4iIAij+BijKirxt+xt-hitTε2iAqijTAqij+ε3iK1irTBijTBijK1ir-1-hitQi+d2[AqijT+K1irTBijTWiAqij+BijK1ir+ε4i-1AqijT+K1irTBijTWiWiAqij+BijK1irxt-hit-d∫t-hittxT.sWLix.sdsEquation (33) can be rewritten as(34)V˙it≤∑j=1Niηij∑r=1Niηirxtxt-hitTΘijrxtxt-hitin which(35)Θijr=AijTPi+PiAij+KirTBijTPi+PiBijKir+ε2i-1PiPi+ε3i-1PiPi+Qi+d2AijT+KirTBijT×Wi+ε4iIAij+BijKir00ε2iAqijTAqij+ε3iK1irTBijTBijK1ir-1-hitQi+d2AqijT+K1irTBijT×WiAqij+BijK1ir+ε4i-1AqijT+K1irTBijTWi×WiAqij+BijK1ir
When EquationΘijr<0, V˙(x(t))<0 can be obtained. Therefore, the controller (11) can ensure that system (12) is asymptotically stable under the switching law σ(x(t))=i.Remark 7.
The integral term-d∫t-τtxT.(s)Wix.(s)ds<0 is obtained, because 0<d<1, Wi is a positive definite symmetry matrix and the physical meaning of the integral term is the area.There are Lyapunov functional matricesPi, Wi and closed-loop system product terms in Equation (23),which can be seen from Theorem 6. In order to solve this problem, an additional matrix is introduced to decouple, and feasible solution in which Equation (23) depends on linear matrix inequalities is obtained.Theorem 8.
Suppose there exist constantsβiλ (i,λ∈M-) (either all nonnegative or all nonpositive); there exist positive scalars ε2i,ε3i,ε4i,d and positive definite symmetric matrices Pi,Wi,Qi,Si, continuously differentiable symmetry matrices Yir,Y1ir, hσ(t)(t) satisfy Equation (2), such that the matrix inequalities (36) are satisfied; then there are memory state feedback controller (42) and switching law σ(t): M-=1,2,⋯,m to make system (12) symptomatically stable:(36)γ11γ12γ1300000∗γ22000000∗0γ3300000000γ44γ45γ46γ47γ48000∗γ55000000∗0γ6600000∗00γ770000∗000γ88<0i=1,2,⋯,m,j,r=1,2,⋯,Nialong with (37)γ11=XiAijT+AijXi+YirTBijT+BijYir+ε2i-1+ε3i-1I+Si+∑λ=1mβiλPi-Pλγ12=XiAijT+YirTBijTγ13=XiAijT+YirTBijTγ22=-d2Wi-1γ33=-ε4id2I-1γ44=-1-hitSiγ45=XiAqijTγ46=Y1irTBijTγ47=XiAqijT+Y1irTBijTγ48=XiAqijT+Y1irTBijTγ55=-ε2iI-1γ66=-ε3iI-1γ77=-d2Wi-1γ88=-d2-1ε4iIProof.
Using the Schur method for the matrix inequality (35)(38)ψ11ψ12ψ1300000∗γ22000000∗0γ3300000000ψ44ψ45ψ46ψ47ψ48000∗γ55000000∗0γ6600000∗00γ770000∗000γ88<0in which(39)ψ11=AijTPi+PiAij+KirTBijTPi+PiBijKir+ε2i-1PiPi+ε3i-1PiPi+Qiψ12=AijT+KirTBijTψ13=AijT+KirTBijTψ44=-1-hitQiψ45=AqijTψ46=K1irTBijTψ47=AqijT+K1irTBijTψ48=AqijT+K1irTBijT
Multiply matrixdiagPi-1,I,I,I,Pi-1,I,I,I,Wi-1 by Equation (38) and define it:(40)Xi=Pi-1,Si=XiQiXi,Yir=KirXi,Y1ir=K1irXi
Then, formula (23) is equivalent to formula (36). Proof is to be true.The gain matrices of the memory state feedback controller are obtained by solving LMI (36):(41)Kir=YirXi-1,K1ir=Y1irXi-1The global model (11) of the memory state feedback controller can be written in the following form:(42)ui=∑j=1NiηijYijXi-1xt+Y1ijXi-1xt-hitInference 1. Suppose there exist constants βiλ (i,λ∈M-) (either all nonnegative or all nonpositive); there exist positive scalars ε2i,ε3i,ε4i,d and positive definite symmetric matrices Pi,Wi,Qi,Si; continuously differentiable symmetry matrices Yir,Y1ir, hσ(t)(t) satisfy Equation (2) and then the matrix inequalities(43)Ε11γ12γ130000∗γ2200000∗0γ330000000γ44γ45Ε46Ε47000∗γ5500000∗0Ε660000∗00Ε77<0where(44)Ε11=XiAijT+AijXi+YirTBijT+BijYir+ε2i-1I+Si+∑λ=1mβiλPi-Pλγ12=XiAijT+YirTBijTγ13=XiAijT+YirTBijTγ22=-d2Wi-1γ33=-ε4id2I-1γ44=-1-hitSiγ45=XiAqijTΕ46=XiAqijTΕ47=XiAqijTγ55=-ε2iI-1Ε66=γ77=-d2Wi-1Ε77=γ88=-d2-1ε4iIThe closed-loop system is asymptotically stable, when the memoryless state feedback controller isui=KqijXi-1x(t), Kqir=YirXi-1 and the switching law is σ(t):M-=1,2,⋯,m.Remark 9.
This paper discusses the situation that the designed controller can stabilize the original system regardless of time-varying delay of the systems. Comparing the linear matrix inequalities (36) with (43), it can be seen that when the inequality (36) has a solution, Equation (43) must have a solution (take Y1ir to 0); on the contrary, it is not necessarily true. Formula (36) has better solvability than Formula (43). Therefore, the memory state feedback controller can control the system in which the memoryless state feedback controller is not controllable.Remark 10.
Equation (23) is transformed into a strict linear matrix inequality (36) by Schur theorem and the matrix inequality method. The transformed LMI matrix increases the conservativeness of the result but makes the final result MATLAB solvable.


## 4. Simulation Results

Consider the switched fuzzy model with time-varying delay as follows in order to prove the effectiveness of the proposed control method:(45)R11:ifξisM111,thenx˙t=A11xt+Aq11xt-h1t+B11u1tyt=C11xtR12:ifξisM112,thenx˙t=A12xt+Aq12xt-h1t+B12u1tyt=C12xtR21:ifδisM211,thenx˙t=A21xt+Aq21xt-h2t+B21u2tyt=C21xtR22:ifδisM212,thenx˙t=A22xt+Aq22xt-h2t+B22u2tyt=C22xtwhere(46)A11=-11112-18,A12=-216190,A21=-11.3314-17A22=-116180,B11=-173228,B12=-18-2-2238B21=-15.2343.0127,B22=-19-1-2137,Aq11=-35413113,Aq12=-370.30.114,Aq21=-27402915Aq22=-360.20.316Select the time-varying delay functions depending on [48] as follows:(47a)h1t=0.003-0.003sin⁡t(47b)h2t=0.01-0.01sin⁡tThen,h1≤0.003, h2≤0.01, and d≤0.02M 11 1 , M 11 2 , M 21 1 , M 21 2 are represented by the following membership functions:(48)M111x2t=η11zt=1-11+e-2zt,M112zt=η12zt=11+e-2zt,M211δt=η21δt=1-11+e-2σt-0.3,M212δt=η22δt=11+e-2δt-0.3The parameters in the stability criterion are as follows:ε 2 i = ε 3 i = ε 4 i = 0.5, βiλ=1, r,j,i=1,2. Using MATLAB to solve the linear matrix inequality (36), we get(49)P1=0.10140.04110.04110.0531,P2=0.09870.02800.02800.0411,K11=9.93514.8792-1.0104-2.7214,K12=9.93514.8792-1.0104-2.7214K21=9.41193.3242-0.6760-2.4021,K22=9.41193.3242-0.6760-2.4021K111=0.02530.01520.00390.0041,K112=0.02530.01520.00390.0041K121=0.01930.00900.00050.0020,K122=0.01930.00900.00050.0020And the gain of the memoryless feedback controller:(50)Kq11=0.89580.2771-0.0165-0.0911,Kq12=0.89580.2771-0.0165-0.0911Kq21=0.73280.11810.0246-0.0647,Kq22=0.73280.11810.0246-0.0647Figures1 and 2 show the state response curves of the closed-loop system (12) controlled by the memory state feedback controller and the memoryless state feedback controller, with the same initial conditions and the switching laws, under the simulink simulation of MATLAB. Figures 3 and 4 show the control curves of the two controllers, where x(0)=[5,-5].Figure 1
State response curve of closed-loop system controlled by memory state feedback controller.Figure 2
State response curve of closed-loop system under control of memoryless feedback controller.Figure 3
Control curve of memory state feedback controller.Figure 4
Control curve of memoryless state feedback controller.The state response curve in Figure1 quickly converges to 0, and the closed-loop system is asymptotically stable; the state response curve in Figure 2 is divergent and the closed-loop system is unstable, which are shown from Figures 1 and 2 under the same initial conditions. The control curve of the memory state feedback controller converges rapidly and the control curve of the memoryless state feedback controller diverges in Figures 3 and 4 under the same initial conditions. Therefore, the control effect with the memory state feedback controller is better than the memoryless state feedback controller for the time-varying delay switched fuzzy system under the given initial conditions.The initial value is changed to 100 times; that is,x(0)=[500,-500]; we get the state response curve and control curve of the time-varying delay switched fuzzy system in Figures 5 and 6 under the memory state feedback controller and the switching law in order to further verify the large-scale stability of the system. The results show that the changing initial value has little effect on the convergence speed of the system.Figure 5
The state response curve of the closed-loop system under the control of the memory state feedback controller when the initial value is expanded.Figure 6
The state control curve of the closed-loop system under the control of the memory state feedback controller when the initial value is expanded.


## 5. Conclusion

In this paper, memory state feedback controller and switching law are designed for a class of time-varying delay switched fuzzy systems. The stability problem of closed-loop systems based on memory state feedback control is studied. According to the definition of asymptotic stability of the system, the Lyapunov functional is constructed, and the LMI solvable criterion for asymptotic stability of the closed-loop system is obtained. Finally, numerical examples and comparative analysis experiments are carried out to verify the effectiveness of the proposed method for the proposed control of time-varying switched fuzzy systems.



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*Source: 4787925-2019-05-02.xml*