# Fuzzy-Model-Based Control for Markov Switching Singularly Perturbed Systems with the Stochastic Communication Protocol

Authors: Zhiguo An; Qijuan Chen; Junxiang Liu; Le Luan; Yong Wang; Kai Zhou; Wenxiong Mo; Huihong Huang
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

This work is concerned with theH∞ control for Markov switching singularly perturbed systems with the stochastic communication protocol. To coordinate the data transmission and save the bandwidth usage, the stochastic communication protocol with a compensator is applied to schedule the information exchange. The goal of this work is to design a joint-Markov-process-based controller such that the resulting system is stochastically stable with prescribed performance. Based on the Lyapunov functional technique, a sufficient condition is derived to ensure the existence of the achieved controller. Finally, the effectiveness and correctness of the developed results are verified by the simulation example.


---

## Body

## 1. Introduction

As a significant component of hybrid systems, Markov switching systems (MSSs) have gained extensive interest due to their capability in modeling subsystems [1–4]. Note that MSSs consist of a finite number of subsystems, and some abrupt variations can be depicted by a Markov process, which is recognized as a key feature of MSSs. Nowadays, owing to their potential practical application, much effort has been devoted, and wonderful fruitful achievements have been gained for both continuous-time MSSs and discrete-time cases [5–8]. Nevertheless, as pointed out in [5, 6], the most existing results are concerned with Markov switching linear systems. Due to the widespread of nonlinear characteristics, it is natural to investigate the Markov switching nonlinear systems. Compared with the standard MSSs, the Markov switching nonlinear systems are more general as they contain high nonlinearity. Lately, the T-S fuzzy model has been tendered to deal with the system’s nonlinearities [9, 10]. Benefit from the T-S fuzzy model, many Markov switching nonlinear systems can be approximated as T-S fuzzy MSSs (FMSSs). Following this excellent result, quantities of valuable results have been forwarded on T-S FMSSs [11–13]. For instance, in [11], a dropout compensation approach has been studied for T-S FMSSs. With respect to the network-induced phenomena, the cyber attack has been considered in FMSSs [13].In many dynamic systems, the system behaviors are involved in multiple-time-scale property. The parasitic parameters, for instance, small-time constants and inductances, may result in the numerical ill-conditioned issues of physical systems. In this regard, the singular perturbation strategy has been employed to tackle the above obstacles. Thanks to singularly perturbed systems (SPSs), the multiple-time-scale-based systems can be transformed into a framework model. Note that the examples of SPSs can be widely found in power systems, airplane systems, etc. Recently, many scholars have drawn their attention to both continuous-time SPSs and discrete-time cases [14–16]. When investigating the SPSs, an extra phenomenon can be encountered, for example, the sudden changes of parameters. To tackle this occurrence, Markov switching SPSs have been studied in [17, 18]. However, the aforementioned results are concerned with linear systems, little attention has been devoted to T-S fuzzy Markov switching SPSs (FMSSPS) except for [19, 20], and this issue remains open and a challenge, which deserves further research.In the networked control systems (NCSs), massive signals are communicated through a shared wireless network. As an unavoidable phenomenon, the NCSs always experience data collisions, fading channels, and input saturation [21]. To prevent the above shortage and mitigate the side effects, many communication protocols have been addressed to govern which sensors can obtain access to send signals such as the popular communication schedule called round-robin protocol [22, 23], try-once-discard protocol [24], and stochastic communication protocol (SCP) [25, 26]. Among them, the SCP is known as an effective method to schedule the signal exchange via a shared channel, in which only one sensor is activated to transmit data. Nevertheless, to our knowledge, no one carries out the exploration of FMSSPSs with the SCP mechanism, which motivates us to this work.Inspired by the aforementioned discussions, our attention focuses on the control issue for FMSSPSs with the communication protocol. The main contributions are outlined as follows: in light of discrete-time FMSSPSs, to coordinate the data transmission and save the bandwidth usage, the SCP is applied to schedule the information exchange. Benefit from the novel Markov process, a mode-dependent Lyapunov functional is formulated such that the resulting system is stochastically stable, and the controller is designed.


## 2. Problem Formulations

Consider theith discrete-time Markov switching system modeled by the T-S fuzzy model.Plant Rulep: IF ξ1k is Mp1, and ξ2k is Mp2, and ⋯, and ξgk is Mpg, THEN(1)x1ι+1=Ap,φι11x1ι+εAp,φι12x2ι+Bp,φι1uι+Cp,φι1ωι,x2ι+1=Ap,φι21x1ι+εAp,φι22x2ι+Bp,φι2uι+Cp,φι2ωι,zι=Dp,φι1x1ι+εDp,φι2x2ι+Mp,φιuι+Gp,φιωι,where x1ι∈ℝns and x2ι∈ℝnf are the fast state and the slow state, respectively. zk∈ℝnz and uι∈ℝnu are the output vector and control input, respectively. ωι∈l20,∞ means the disturbance signal. The sequence φι,ι≥0 renders a discrete-time Markov chain (DTMC) subject to a finite set Ns=1,2,…,Ns. Here, φι describes a homogeneous DTMC with the transition probability matrix of FMSSPS (1) inferred as(2)πij=Prφι+1=j|φι=i,where πij≥0, ∑j∈Nsπij=1, ∀i,j∈Ns, and TPM Π=πijNs×Ns.For technique analysis,∀i∈Ns, Ap,φι11, Ap,φι12, Ap,φι21, Ap,φι22, Bp,φι1, Bp,φι2, Cp,φι1, Cp,φι2, Dp,φι1, Dp,φι2, Mp,φι, and Gp,φι are denoted by Api11, Api12, Api21, Api22, Bpi1, Bpi2, Cpi1, Cpi2, Dpi1, Dpi2, Mpi, and Gpi, respectively.Recall the fuzzy weighting functionℏpξι=∏s=1tMpsξsι/∑p=1r∏s=1tMpsξsι, where Mpsξsι refers to the grade of the membership degree of ξsι in Mps. In general, assume ∑p=1rℏpξι=1 and ℏpξι≥0.Letxι=x1⊤ιx2⊤ι⊤; by virtue of T-S fuzzy technique, FMSSPS (1) is derived as(3)xι+1=∑p=1rℏpξιApiEεxι+Bpiuι+Cpiωι,zι=∑p=1rℏpξιDpiEεxι+Mpiuι+Gpiωι,where Eϵ=diagIns,ϵInf, Api=Api11Api12Api21Api22, Bpi=Bpi1Bpi2, Cpi=Cpi1Cpi2, and Dpi=Dpi1Dpi2.In the NCSs, some redundant signals are communicated in the conventional data transmission manner, which may result in unfavorite phenomena, for instance, data collisions. The control signalvk and the actuators uk share the same communication network (CN). To prevent such unfavorite factors, the SCP scheduling is used to regulate the node order in transmitting data. Note that only one sensor is borrowed to release the signal each time, and the sensors are chosen in a stochastic way. In general, letting ψι∈1,2,…,Nc signifies the chosen actuator which gains the permission to access the CN at the time interval ι. Notably, ψι can be recognized as a stochastic process regulated by another DTMC obeying a set Nc=1,2,…,Nc, and TPM Ψ=τmnNc×Nc is determined by(4)τmn=Prψι+1=n|ψι=m,where ∀m,n∈Nc, τmn∈0,1, and ∑n∈Ncτmn=1.Letvι=v1⊤ιv2⊤ι⋯vnu⊤ι and uι=u1⊤ιu2⊤ι⋯unu⊤ι, where vmι denotes the mth control input vector and un signifies the nth actuator. Firstly, assume that a set of zero-order hold is employed in the signal transmission. Accordingly, the mth actuator umι is updated by the following principle:(5)umι=vmι,ifψι=m,umι−1,otherwise.Aiming at describing the data transmission strategy of actuators mathematically, a Kronecker sign function is inferred as(6)δx−y=1,ifx=y,0,otherwise.As indicated from the updating principle (5), the mth actuator umι is updated when ψι=m. Consequently, for ∀ι, the updated actuator uι can be devised as(7)uι=Ψψιvι+I−Ψψιuι−1,where Ψm≜diagδm1,δm2,…,δmnum=1,2,…,nu and δxy=δx−y.The control lawvk in this work is constructed as follows:(8)vι=∑q=1rℏqξιKq,φιEεxι,where Kq,φι are matrices to be designed.Substituting (8) into (3), the closed-loop FMSPS (9) is formulated as(9)xι+1=∑p=1rℏpξι∑q=1rℏqξιApqimEεxι+BpiI−Ψmuι−1+Cpiωι,zι=∑p=1rℏpξι∑q=1rℏqξιDpqimEεxι+MpiI−Ψmuι−1+Gpiωι,where(10)Apqim=Api+BpiΨmKqi,Dpqim=Dpi+MpiΨmKqi.Before proceeding further, some lemmas and definitions are provided.Definition 1 (see [27]).
The FMSSPS (9) with ωk=0 is named stochastic stable (SS) if for any δ0,ϑ0, one has(11)Ε∑k=0∞δk2|δ0,ϑ0<∞.Definition 2 (see [27]).
The FMSSPS (9) is named SS with a prescribed ℋ∞ performance level γ if the FMSSPS (18) is SS and under zero initial condition such that(12)∑k=0∞Εδk2<γ2∑k=0∞Εϱk2.Lemma 1 (see [18]).
For given a scalarϵ¯>0, W1, W2, and W3 are matrices with suitable dimensions. For any ϵ∈0,ϵ¯, W1+ϵW1+ϵ2W1>0 such that(13)W1>0,W1+ϵ¯W1>0,W1+ϵ¯W2+ϵ¯2W3>0.Lemma 2 (see [18]).
For any symmetric matricesℛtt=1,2,3,4 and matrix ℛ5 which meets(14)ℛ1>0,ℛ1+ϵ¯ℛ3ϵ¯ℛ5⊤∗ϵ¯ℛ2>0,ℛ1+ϵ¯ℛ3ϵ¯ℛ5⊤∗ϵ¯ℛ2+ϵ¯2ℛ4>0,one has Eϵℛϵ=ℛϵ⊤Eϵ>0 for any ϵ∈0,ϵ¯, where(15)ℛε=ℛ1+ϵℛ3ϵℛ5⊤∗ϵℛ2+ϵ2ℛ4.


## 3. Main Results

In this section, sufficient conditions are elicited to ensure the SS and a prescribedℋ∞ performance level of the FMSSPS (9).Theorem 1.
The closed FMSSPS (9) is called SS with a prescribed ℋ∞ performance level γ if there exist symmetric matrices P¯i>0, Q¯i>0, ℛ1, ℛ2, ℛ3, and ℛ4 and matrices ℛ5, K¯ϵqi, U1, and U2 such that(16)ℛ1>0,(17)ℛ1+ϵ¯ℛ3ϵ¯ℛ5⊤ϵ¯ℛ5ϵ¯ℛ2>0,(18)ℛ1+ϵ¯ℛ3ϵ¯ℛ5⊤ϵ¯ℛ5ϵ¯ℛ2+ϵ¯2ℛ4>0,(19)Γppimt<0,1≤p≤r,t=ℓ,ı,J,(20)Γpqimt+Γqpimt<0,1≤p<q≤r,t=ℓ,ı,J.Meanwhile, theϵ-dependent controller gains are achieved as(21)Kqi=K¯ϵqiU1⊤+ϵU2⊤−1,q=1,2,…,r,i∈Ns,where(22)Γpqimt=Γpqi1tΓpqim2t⊤TfΓpqim3t⊤TfΓpqim4t⊤∗Γpqi500∗∗Γpqi60∗∗∗−I,t=ℓ,ı,J,Γpqi1ℓ=diagP¯i−V1+V1⊤,Q¯i−Yi+Yi⊤,−γ2I,Γpqi1ı=diagP¯i−ℛϵ¯+ℛϵ¯¯⊤,Qm¯i−Yi+Yi⊤,−γ2I,Γpqi1J=diagP¯i−ℛϵ¯+ℛϵ¯⊤,Q¯i−Yi+Yi⊤,−γ2I,Γpqim2ℓ=ApiU1+BpiΨmK¯ϵqiE0BpiI−ΨmYiCpi,Γpqim2ı=ApiU2+BpiΨmK¯ϵqiEϵ¯BpiI−ΨmYiCpi,Γpqim2J=ApiU3+BpiΨmK¯ϵqiEϵ¯BpiI−ΨmYiCpi,Γpqim3ℓ=ΨmK¯ϵqiE0I−ΨmYi0,Γpqim3ı=Γpqi3J=ΨmK¯ϵqiEϵ¯I−ΨmYi0,Γpqim4ℓ=DpiU1+MpiΨmK¯ϵqiE0MpiI−ΨmYiGpi,Γpqim4ı=DpiU2+MpiΨmK¯ϵqiEε¯MpiI−ΨmYiGpi,Γpqim4J=DpiU3+MpiΨmK¯ϵqiEϵ¯MpiI−Ψmℛϵ¯Gpi,Γpqi5=diag−P¯1,−P¯2,…,−P¯Nr,Γpqi6=diag−Q¯1,−Q¯2,…,−Q¯Nr,Ti=θf1Iθf2I⋯θfNrI,ℛϵ¯=V1+ϵ¯V2,U1=W1,U2=W1+ϵ¯W2,U3=W1+ϵ¯W2+ϵ¯2W3,V1=ℛ10ℛ5ℛ2,V2=ℛ3ℛ5⊤0ℛ4,W1=ℛ1000,W2=ℛ3ℛ5⊤ℛ5ℛ2,W3=000ℛ4,E0=Ins000,Eϵ¯=Ins00ϵ¯Ini.Proof.
Combining with the linear matrix inequalities (LMIs) (17)–(19) and Lemma 2, for any ϵ∈0,ϵ¯, it yields that(23)Γ˜pqi1Γ˜pqim2⊤TiΓ˜pqim3⊤TiΓ˜pqim4⊤∗Γpqi500∗∗Γpqi60∗∗∗−I<0,where(24)Γ˜pqi1=diagP¯i−ℛϵ+ℛϵ⊤,Q¯i−ℛϵ+ℛϵ⊤,−γ2I,Γ˜pqim2=ApiEϵℛϵ+BpiΨmKqiℛϵ⊤EϵBpiI−ΨmℛϵCpi,Γ˜pqim3=ΨmKqiℛϵ⊤EϵI−Ψmℛϵ0,Γ˜pqim4=DpiEϵℛϵ+MpiΨmKqiℛϵ⊤EϵMpiI−ΨmℛϵGpi.
Recalling Lemma2 and LMIs (17)–(19), for any ϵ∈0,ϵ¯, it is clear that Eϵℛϵ=ℛϵ⊤Eϵ>0. On the contrary, with respect to the fact that inequality ℛϵ⊤−P¯iPiℛϵ−P¯i≥0, ℛϵ⊤−Q¯iQiℛϵ−Q¯i≥0, P¯i=Pi−1, and Q¯i=Qi−1, which yields(25)Γ^pqi1Γ^pqim2⊤TiΓ^pqim3⊤TiΓ^pqim4⊤∗Γpqi500∗∗Γpqi60∗∗∗−I<0,where(26)Γ^pqi1=diag−ℛϵ⊤Piℛϵ,−ℛϵ⊤Qiℛϵ,−γ2I,Γ^pqim2=ApqiEϵℛϵBpiI−ΨmℛϵCpi,Γ^pqim3=ΨmKqiEϵℛϵI−Ψmℛϵ0,Γ^pqim4=DpqiEϵℛϵMpiI−ΨmℛϵGpi,Γ^pqi5=diag−P1−1,−P2−1,…,−PNs−1,Γ^pqi6=diag−Q1−1,−Q2−1,…,−QNs−1.
Premultiplying and postmultiplying (25) with diagℛϵ−⊤,ℛϵ−⊤,I,…,I and its transpose, where ℛϵ=U1+ϵU2, yield(27)Γ¯pqi1Γ¯pqim2⊤TiΓ¯pqim3⊤TiΓ¯pqim4⊤∗Γpqi500∗∗Γpqi60∗∗∗−I<0,where(28)Γ¯pqi1=diag−Pi,−Qi,−γ2I,Γ¯pqim2=ApqiEϵBpiI−ΨmCpi,Γ¯pqim3=ΨmKqiEϵI−Ψm0,Γ¯pqim4=DpqiEϵMpiI−ΨmGpi.
In the following, a Lyapunov functional for FMSSPS (9) is established:(29)Vι,xι,uι,φι=x⊤ιPφιxι+u⊤k−1Qφιuk−1.
By calculating the difference ofVι,xι,uι,φι, one has(30)EΔVι=EVι+1,xι+1,uι+1,φι+1=g|ι,xι,uι,f−Vι,xι,uι,φι.
Recalling FMSSPS (9), EΔVι can be derived as(31)EΔVι=x⊤ι+1Pixι+1−x⊤ιPixι+u⊤ιQiuι−u⊤k−1Qiuk−1,where(32)Pi=∑j∈NsπijPj,Qi=∑j∈NsθijQj.
In (31), the first term can be further devised as(33)x⊤ι+1Pixι+1=∑p=1rℏpξι∑q=1rℏqξιx⊤ιPi+EϵApqim⊤PiApqimEϵxι+x⊤ιEϵApqim⊤PiBpiI−Ψmuι−1+u⊤ι−1I−ΨmBpi⊤PiApqimEϵxι+x⊤ιEϵApqim⊤PiCpiωk+ω⊤kCpi⊤PiApqimEϵxι+u⊤ι−1I−Ψm⊤Bpi⊤PiBpiI−Ψmuι−1+u⊤ι−1I−ΨmBpi⊤PiCpiωk+ω⊤kCpi⊤PiBpiI−Ψmuι−1+ω⊤kCpi⊤PiCpiωk.
Besides, the third term in (31) can be rewritten as(34)u⊤ιQiuι=∑q=1rℏqξιΨmKqiEϵxι+I−Ψmuι−1⊤Qi×ΨmKqiEϵxι+I−Ψmuι−1=∑q=1rℏqξιx⊤ιEϵKqi⊤Ψm⊤QiΨmKqiEϵxι+x⊤ιEϵKqi⊤Ψm⊤QiI−Ψmuι−1+u⊤ι−1I−ΨmQiΨmKqiEϵxι+u⊤ι−1I−ΨmQiI−Ψmuι−1.
Combining (29)–(34) yields(35)EΔVι=ϑ⊤k∑p=1rℏpξι∑q=1rℏqξιΓ⟶pqimϑk,where(36)ϑk=x⊤ιu⊤ι−1ω⊤k,Γ⟶pqi1=diag−Pi,−Qi,0,Γ⟶pqim=Γ⟶pqi1+Γ¯pqim2⊤PiΓ¯pqim2+Γ¯pqim3⊤QiΓ¯pqim3.
Whenωk=0, it follows from inequality (35) that(37)EΔVι≤ϑ←⊤ι∑p=1rℏpξι∑q=1rℏqξιΓ←pqimϑ←ι≤−χExι2,where ϑ←k=x⊤ιu⊤ι−1, Γ←pqim=Γ←pqi1+Γ←pqim2⊤PiΓ←pqim2+Γ←pqim3⊤QiΓ←pqim3, Γ←pqi1=diag−Pi,−Qi, Γ←pqim2=ApqimEϵBpiI−Ψm, Γ←pqim3=DpqimEϵMpiI−Ψm, and χ=minf∈Ns,p,q∈1,2,…,rλminΓ←pqim. Clearly, recalling (27), one gets χ>0. Consequently, one concludes that(38)E∑ι=0∞xι2<−1χE∑ι=0∞ΔVι≤1χEV0,x0,u0,r0<∞.
Recalling Definition1, when ωk=0, FMSSPS (16) is SS.
Next, in the case ofωk≠0, we will provide the analysis of H∞ performance for FMSSPS (16). Define the H∞ performance index:(39)JT=E∑k=0Tz⊤ιzι−γ2ω⊤ιωι.
Substituting (35) into (39), JT can be formulated as(40)JT≤E∑k=0Tz⊤ιzι−γ2ω⊤ιωι+ΔVι≤ϑ⊤ι∑p=1rℏpξι∑q=1rℏqξιΓpqim′ϑι,where(41)Γpqim′=Γ⟶pqim+Γpqim4⊤Γpqim4,Γpqim4=DpqimEϵMpiI−ΨmGpi.
Additionally, by applying the Schur complement to (27) and (40), one gets(42)JT<0.
LettingT⟶∞, it is directly attained from (42) that(43)∑ι=0∞Ezι2≤γ2∑ι=0∞Eωι2.
Accordingly, by means of Definition2, one concludes that FMSSPS (16) is SS with ℋ∞ performance index γ. This completes the proof.


## 4. Simulation Examples

Example 1.
Consider FMSSPS (16) with the following parameters:(44)A11=0.31.70.30.80.10.31.40.20.9,B11=0.50.41.20.20.20.3,C11=0.60.20.7⊤,D11=0.30.40.9,M11=0.50.8,G11=0.7,A12=0.10.41.11.50.20.10.81.10.6,B12=1.40.60.70.10.50.8,C12=0.30.51⊤,D12=0.60.20.3,M12=1.50.5,G12=0.4,A21=0.21.10.10.21.11.20.60.91.1,B21=0.30.20.40.50.20.3,C21=1.80.70.2⊤,D21=0.60.90.1,M21=0.30.5,G21=0.5,A22=1.11.52.41.50.70.60.11.10.8,B22=0.60.20.50.30.70.5,C22=0.60.30.4⊤,D22=0.70.60.7,M22=0.50.6,G22=0.3.
The TPM of the corresponding FMSSPS (1) is selected as Π=0.250.750.650.35. For another MP Φ in SCP (7), the TPM is chosen as Φ=0.40.60.50.5.
Letγ=5, ϵ=0.002, ℏ1ξι=15−x2ι/30, and ℏ2ξι=1−ℏ1ξι. In view of the LMIs of Theorem 1, the controller gains Kqiq=1,2;f=1,2,3,4 can be derived as follows:(45)K1,1=−0.0008−0.0016−0.0001−0.0010−0.00110.0005,K1,2=−0.0011−0.0063−0.0088−0.0010−0.0061−0.0085,K2,1=−0.00070.00320.0022−0.00060.00170.0015,K2,2=−0.00090.01460.0249−0.00100.01480.0244.
To carry on the simulation study, the external disturbance and the initial condition are selected asωk=0.9exp−0.4ιsin50ι and x0=−0.10.20.3⊤, respectively. The possible mode switching of Markov process φι and the evolution of states x1ι, x2ι, and x3ι over 100 realizations are depicted in Figure 1, respectively. The control input over 100 realizations is plotted in Figure 2. Furthermore, the evolution of the output zι over 100 realizations is shown in Figure 3.Figure 1
The dynamics of FMSSPS (18) in Example 1. (a) The possible mode switching of MP ′φι. (b) The evolution of state x1ι. (c) The evolution of state x2ι. (d) The evolution of state x3ι.
(a)(b)(c)(d)Figure 2
The control inputuι over 100 realizations in Example 1.Figure 3
The outputzι over 100 realizations in Example 1.


## 5. Conclusions

In this work, theH∞ control problem has been discussed for FMSSPSs with the SCP. In order to coordinate the data transmission and save the bandwidth usage, the SCP with a compensator is applied to schedule the information exchange. Furthermore, some sufficient criteria have been forwarded such that the resulting system is SS. Finally, one example is exhibited to show the effectiveness and correctness of the developed results. In addition, some advanced techniques including the sliding mode-based filter will be researched in our following work [28, 29].



---

*Source: 6690729-2021-02-18.xml*