# Fuzzy Model-Based Asynchronous Control for Markov Switching Systems with Stochastic Fading Channels

Authors: Fayuan Wu; Jinhui Tang; Zhuang Liu; Qi Xiao; Xiaodong Zheng; Shuangsi Xue
Journal: Complexity
 (2020)

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## Abstract

This work investigates the asynchronous control for fuzzy Markov switching systems (MSSs) with randomly occurring fading channel. By resorting to a T-S fuzzy model, the nonlinear MSSs can be handled. Meanwhile, in the unreliable network, the Rice fading model is proposed to capture the randomly occurring channel fading, which covers packet dropouts and network-induced delays as special cases. In light of the hidden Markov model, the asynchronous phenomenon of controller is taken into consideration, and asynchronous fuzzy controller is obtained. In the end, a numerical example and a single-link robotic arm model are applied to demonstrate the validity of the derived results.


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## Body

## 1. Introduction

In physical applications, such as power station monitoring systems, fire-fighting operating systems, and electric networks, the signals are transmitted through a shared wireless/wired communication channel. The stability of the systems is weakened by uncontrollable accidents on some occasions and external complex environment, and it is of importance to make the systems have certain robustness. However, because of the limited resources, the communication quality and computing load are affected, and the unexpected phenomenon occurs, such as networked-induced delay, quantization, and fading channel (FC). Normally, when signals are transmitted by wireless communication link, the phenomena of reflection/diffraction can be encountered, which results in multichannel-based fading. Recently, many scholars have devoted their attention to hybrid systems with FC [1–3]. It is remarkable that, compared with packet dropouts, FC is more general, which contains the finite coefficients. Lately, FC is widely applied in time-dependent probabilistic process, and many valuable results are reported in robust control [4], sliding mode control [5], and H∞ filtering [6, 7].As a special type of switching systems, Markov switching systems (MSSs) consist of finite subsystems, and different subsystems are activated by a random Markov process [8–10]. Note that MSSs have been successfully applied in practical systems with probabilistic parameter changes, for instance, component failures. Added by its powerful application, many fruitful results have been gained in the issues of admissibility and stabilization, robust control, and filtering [11–14]. As one can be seen from [11–14] that, the control designed scheme has major concerns on synchronous situation. Owing to the networked mechanism and time delay, it is unreasonable to consider the synchronous scenario. By resorting to the hidden Markov model (HMM) [15–17], the asynchronous controller/filter can be solved in an application perspective.On the contrary, it can be observed that aforementioned literatures are constrained to linear systems, which limited the practical application [18–23]. In general, MSSs tend to nonlinearities for external nonlinear environments and other factors, and it is urgent to take nonlinear MSSs into consideration [24–27]. Following this trend, many efforts have been devoted to nonlinear systems. Among them, Takagi–Sugeno (T-S) fuzzy model has been proved to be an efficient tool in tacking with nonlinear systems, in which by resorting to fuzzy inference strategy, the nonlinear systems are separated into finite local linear parts [28–30]. Recently, many meaningful results of T-S fuzzy MSSs (FMSSs) have been expressed in [20–22]. However, the FC issue has not been extended to FMSSs, not to mention asynchronous scenario.Inspired by the above observation, we focus on the asynchronous control for FMSSs with randomly occurring FC. The major contributions are summarized as follows: (1) the FC model with disturbance is absorbed, which covers packet dropouts and network-induced delays as special cases; (2) added by parallel distributed compensation strategy and HMM scheme, the asynchronous control law is developed, which relaxes the limitation in existing results; and (3) by means of Lyapunov functional and stochastic analysis, sufficient criteria are gained and SLRAM is applied to illustrate the effectiveness of the proposed asynchronous control law.The rest of this study is organized as below. The system descriptions are given in Section2, and main results including stochastic stability of closed-loop system and controller gains are provided in Section 3. Computational experiments are expressed in Section 4. In the end, conclusions are shown in Section 5.Notations: the notations utilized in this work are same as that in [10]. Furthermore, symL=L+L⊤ and diag⋅ means the block-diagonal matrix.


## 2. Problem Formulations

Consider the T-S FMSS (1) as follows:Plant rulei: IF ϕ1k is Mi1,…,ϕfk is Mif, THEN(1)δk+1=Ai,rkδk+Bi,rkuk+Ci,rkωk,zk=Di,rkδk+Fi,rkωk,where xk∈ℝnx is the state vector, uk∈ℝnu is the control input, zk∈ℝnz is the output vector, and ωk∈ℝnω is the external disturbance. Stochastic variable (SV) rk is a discrete Markov chain (DMC) and rk∈ℛ=1,2,…,R. For any rk∈ℛ, Ark, Brk, Crk, Drk, and Frk indicate the predetermined matrices. Mij are the fuzzy sets, and ϕlk∈ϕ1k,ϕ2k,…,ϕfk means the premise variable. Here, i∈1,2,…,κ and κ implies the IF-THEN rules number.In (1), for any μ,ν∈ℛ, the transition probability matrix (TPM) of original state is inferred by Γ=ϑμν:(2)Prrk+1=νrk=μ=ϑμν,where μ,ν∈ℛ, ϑμν∈0,1, and ∀μ∈ℛ, ∑ν∈ℛϑμν=1. For rk=μ, one has ℛi,rk=ℛiμ, where ℛi,rk=Ai,rk,Bi,rk,Ci,rk,Di,rk,Fi,rk.By means of the T-S fuzzy model, FMSS (1) can be deduced as follows:(3)δk+1=∑i=1κℏiηkAiμδk+Biμuk+Ciμωk,zk=∑i=1κℏiηkDiμδk+Fiμωk,where Mijηjk symbolizes the grade of membership of ηjk in Mij, ℑiηk=∏j=1κMijηjk, and ℏiηk=ℑiηk/∑i=1κℑiηk; here, ℏiηk≥0, and ∑i=1κℏiηk=1. For simplification, denoting(4)Aℏμ=∑i=1κℏiηkAiμ,Bℏμ=∑i=1κℏiηkBiμ,Cℏμ=∑i=1κℏiηkCiμ,Dℏμ=∑i=1κℏiηkDiμ,Fℏμ=∑i=1κℏiηkFiμ,apparently, the FMSS (3) can be rewritten as(5)δk+1=Aℏμδk+Bℏμuk+Cℏμωk,zk=Dℏμδk+Fℏμωk.It is remarkable that the signalsδk are sending out via unreliable network; some unpredictable factors occur, such as channel fading (CF) and packet dropout. To model the randomly occurring CF, Lth-order CF model is proposed:(6)δ¯k=∑l=0Lαlkδk−l+Hζk,where δ¯k is the signal arrived at controller and αlkl=0,1,…,L are mutually independent SVs with εαlk=αl subject to variances εαlk−αl2=α¯l2. ζk represents the external disturbance and H is the known matrix.Remark 1.
In the unreliable network, aiming to depict the unexpected networked phenomenon including scattering and other factors, fading channel is proposed, which is characterized by the Rice fading model. The stochastic variablesαlkl=1,2,…,L can describe the variety of fading channel, which are applied and make the signal transmission more general.
In this work, by considering the unreliable of network medium, one establishes the following asynchronous control law (ACL):
Controller rulei: IF φnk is Min,n∈l=1,2,…,f, THEN(7)uk=Kiτkδ¯k,where Kiτk indicates the unsolved controller gains. The SV τk is another DMC. DMC τk∈T=1,2,…,T represents HMM and obeys the conditional probability matrix CPM Λ=φμt:(8)Prτk=trk=μ=φμt,where φμt∈0,1 and ∑t∈Tφμt=1. Accordingly, the ACL (7) is inferred as(9)uk=Kℏtδ¯k,where Kℏt=∑i=1κℏiηkKit.Remark 2.
Note that asynchronous phenomenon comes from the network-induced delay, quantization, and other factors, which results in unreasonable of traditional synchronous control law [17, 31]. To tackle with such shortage, HMM is employed in designing controller, which is elaborated by rk,τk,Γ,Λ.
Substituting (6) and (9) into (5), the closed-loop FMSS can be acquired:(10)δk+1=Aℏμ+α0BℏμKℏtδk+Cℏμωk+∑l=1LαlBℏμKℏtδk−l+BℏμKℏtHζk+α˜0kBℏμKℏtδk+∑l=1Lα˜lkBℏμKℏtδk−l,zk=Dℏμδk+Fℏμωk,where α˜lk=αl−αlk.Definition 1.
(see [12]). Under the initial conditions δ0, r0, the FMSS (10) is called stochastically stable (SS), such that(11)Ε∑k=0∞δk2δ0,r0<∞.Definition 2.
(see [12]). The FMSS (10) is SS with ℋ∞ performance γ such that(12)∑k=0∞zk2<γ2∑k=0∞ϖ⊤kϖk.


## 3. Main Results

Theorem 1.
If there exists a scalarγ>0, matrices Pμ>0μ∈ℛ, Uμt>0μ∈ℛ,t∈T, and Ql>0l=1,2,…,L, the FMSS (10) is SS with ℋ∞ performance γ, for any μ,ν∈ℛ, t∈T, such that(13)∑t∈TφμtUμt<Pμ,(14)Ωiiμt<0,(15)Ωijμt+Ωjiμt<0,where(16)Ωijμt=Ωijμt1Ωijμt2Ωμ3,Pμ=∑ν∈ℛϑμνPν,Ωijμt1=diagΩijμt11,Ωijμt12,Ωijμt13,Ωijμt14,Ωijμt2=Ωijμt21Ωijμt22Ωijμt23Ωijμt24,Ωμ3=diag−Pμ,−Pμ,−IL⊗Pμ,−I,Ωijμt11=−Uμt+∑q=1LQq,Ωijμt13=−γ2I,Ωijμt12=diag−Q1,−Q2,…,−QL,Ωijμt14=−γ2I,Ωijμt21=Aiμ+α0BiμKjt⊤PμBjμKjtZ⊤PμCiμ⊤PμBiμKjtH⊤Pμ,Ωijμt22=α¯0BiμKjt⊤Pμ000,Z=α1Iα2I⋯αLI,Ωijμt23=0Θ¯⊗BiμKjt⊤Pμ00,Ωijμt24=Diμ0Fiμ0⊤,Θ¯=diagα¯1,α¯2,…,α¯L.Proof.
It follows from (14) and (15) that(17)Ωℏμt=∑i=1κ∑j=1κℏiηkℏjηkΩijμt=∑i=1κℏi2ηkΩiiμt+∑i=1κ−1∑j=i+1κℏiηkℏjηkΩijμt+Ωjiμt<0,where(18)Ωℏμt=Ωℏμt1Ωℏμt2Ωμ3,Ωℏμt1=diagΩℏμt11,Ωℏμt12,Ωℏμt13,Ωℏμt14,Ωℏμt2=Ωℏμt21Ωℏμt22Ωℏμt23Ωℏμt24,Ωℏμ3=diag−pμ,−pμ,−IL⊗pμ,−I,Ωℏμt11=−Pμ+∑q=1LQq,Ωℏμt13=−γ2I,Ωℏμt12=diag−Q1,−Q2,…,−QL,Ωℏμt21=Aℏμ+α0BℏμKℏtTpμBℏμKℏtZTpμCℏμTpμBℏμKℏtHTpμ,Ωℏμt22=α¯0BℏμKℏtTpμ000,Ωℏμt14=−γ2I,Ωℏμt23=0Θ¯⊗BℏμKℏtTpμ00,Ωℏμt24=Dℏμ0Fℏμ0T.
Construct the Lyapunov functional for FMSS (10):(19)Vk,δk=V1k,δk+V2k,δk,where(20)V1k,δk=δ⊤kPμδk,V2k,δk=∑q=1L∑p=k−qk−1δ⊤pQpδp.
Calculating the difference ofVk,δk and taking expectation, one has(21)ΕΔVk=ΕVk+1,δk+1δk,rk−Vk,δk.
Along system (10), it yields that(22)εΔV1k=εδTk+1pμδk+1−δTkpμδk=εAℏμ+α0BℏμKℏtδk+Cℏμωk+∑l=1LαlBℏμKℏtδk−l+BℏμKℏtHςkTpuAℏμ+α0BℏμKℏtδk+Cℏμωk+∑l=1LαlBℏμKℏtδk−1+BℏμKℏtHςk+α˜02kδTkKℏtTBℏμTpμBℏμKℏtδk+∑l=1Lα˜lkBℏμKℏtδk−lTpμ×∑l=1Lα˜lkBℏμKℏtδk−l−δTkPμδk.
Added by the variances in fading channelsα¯ll=0,1,…,L, (22) can be reformulated as(23)εΔV1k≤εAℏμ+α0BℏμKℏtδk+Cℏμωk+BℏμKℏtZδLk+BℏμKℏtHζk⊤×PμAℏμ+α0BℏμKℏtδk+Cℏμωk+BℏμKℏtZδLk+BℏμKℏtHζk+α¯02δ⊤kKℏt⊤Bℏμ⊤PμBℏμKℏtδk+δL⊤kΘ¯2⊗Kℏt⊤Bℏμ⊤PμBℏμKℏtδLk−δ⊤kPμδk,where(24)δLk=δ⊤k−1δ⊤k−2⋯δ⊤k−L⊤.
On the contrary, for termV2k,δk, one has(25)εΔV2k=ε∑q=1L∑p=k−q+1kδ⊤pQpδp−∑q=1L∑p=k−qk−1δ⊤pQpδp=δ⊤k∑q=1LQqδk−∑q=1Lδ⊤k−qQqδk−q.
For simplification, denotingϖk=ω⊤kζ⊤k⊤. Firstly, we will prove the SS of system (10) with ϖk=0. Combining with (23) and (25), one obtains(26)εΔVk≤εAℏμ+α0BℏμKℏtδk+BℏμKℏtZδLk⊤×PμAℏμ+α0BℏμKℏtδk+BℏμKℏtZδLk+α¯02δ⊤kKℏt⊤Bℏμ⊤PμBℏμKℏtδk+δL⊤kΘ¯2⊗Kℏt⊤Bℏμ⊤PμBℏμKℏtδLk−δ⊤kPμδk+δ⊤k∑q=1LQqδk−∑q=1Lδ⊤k−qQqδk−q.
Applying Schur complement to (26), it is clear that(27)εΔVk≤εψ˜⊤kΩ˜ℏμψ˜k,where(28)ψ˜k=δ⊤kδL⊤k⊤,Ω˜ℏμt=Ω˜ℏμt1Ω˜ℏμt2Ω˜μ3,Ω˜ℏμt1=diagΩℏμt11,Ωℏμt12,Ω˜ℏμt2=Ωℏμt21Ωℏμt22Ωℏμt23,Ω˜μ3=diag−Pμ,−Pμ,−Pμ.
In light of (27), it achieves that(29)εΔVk≤ρεδ⊤kδk,where ρ=λmaxΩ˜ℏμt, ∀i=1,2,…,κ,μ∈ℛ,t∈T. By (17), it is clear that ρ<0, which indicates(30)εδ⊤kδk≤1ρεΔVk.
ForN⟶∞, it yields(31)ε∑k=0∞δ⊤kδk≤−1ρV0,δ0<∞.
By Definition1, it is obviously that FMSS (10) is SS with ϖk=0.
Next, the FMSS (10) is SS with the ℋ∞ performance index will be verified. For ϖk≠0 under zero-initial condition, denoting Jk=∑k=0∞εzk2−γ2εϖk2, one has(32)Jk≤∑k=0∞εzTkzk−γ2ϖTkϖk+ΔVk≤∑k=0∞εAℏμ+α0BℏμKℏtδk+Cℏμωk+BℏμKℏtZδLk+BℏμKℏtHζkT×PμAℏμ+α0BℏμKℏtδk+Cℏμωk+BℏμKℏtZδLk+BℏμKℏtHζk+α¯02δTkKℏtTBℏtTPμBℏμKℏtδk+δLTkΘ¯2⊗KℏμTBℏμTPμBℏμKℏtδLk−δTkPμδk+δTk∑q=1LQqδk−∑q=1LδTk−qQqδk−q+Dℏμδk+FℏμωkTDℏμδk+Fℏμωk−γ2ϖTkϖk.
Deploy Schur complement to (32), which results in(33)Jk≤∑k=0∞ψ⊤kΩℏμψk,where ψk=δ⊤kδL⊤kω⊤kζ⊤k⊤.
In light of (17) and (33), it can be concluded that(34)limN⟶∞ε∑k=0Nz⊤kzk<γ2∑k=0Nω⊤kωk,which indicates ε∑k=0∞z⊤kzk<γ2∑k=0∞ω⊤kωk. By Definition 2, the FMSS (10) is SS with ℋ∞ performance γ.Theorem 2.
If there exists a scalarγ>0, matrices Pμ>0μ∈ℛ, Uμt>0μ∈ℛ,t∈T, Ql>0l=1,2,…,L, and matrices Wtt∈T, Ytt∈T, K¯jtj=1,2,…,κ,t∈T, the FMSS (10) is SS with ℋ∞ performance γ, for any μ,ν∈ℛ, t∈T, such that(35)WtBiμ=BiμYt,(36)Υiiμt<0,(37)Υijμt+Υjiμt<0,where(38)ϒijμt=Ωijμt1ϒijμt2ϒμt3,ϒijμt2=ϒijμt21ϒijμt22ϒijμt23ϒijμt24,ϒijμt21=AiμTWtT+α0K¯jtTBiμTzTK¯jtTBiμTCiμTWtTHTK¯jtTBiμT,ϒijμt22=α¯0K¯jtTBiμT000,ϒijμt230Θ⊗K¯jtTBiμT00,ϒμt3=diagPμ−symWt,Pμ−symWt,IL⊗Pμ−symWt,−I.
The controller parameters can be obtained by(39)Kjt=Yt−1K¯jt,j=1,2,…,κ,t∈T.Proof.
Define(40)ℳμt=diagI,I,…,I,Pμ−1Wt⊤,Pμ−1Wt⊤,IL⊗Pμ−1Wt⊤,I.
Applying a congruence transformation to (14), it yields that(41)Υ¯iiμt=ℳμt⊤Ωiiμtℳμt<0,where(42)Υ¯iiμt=Ωiiμt1Υ¯iiμt2Υ¯μt3,Υ¯iiμt2=Υ¯iiμt21Υ¯iiμt22Υ¯iiμt23Ωiiμt24,Υ¯iiμt21=Aiμ⊤Wt⊤+α0K¯it⊤Biμ⊤Z⊤Kit⊤Biμ⊤Wt⊤Ciμ⊤Wt⊤H⊤Kit⊤Biμ⊤Wt⊤,Υ¯iiμt22=α¯0Kit⊤Biμ⊤Wt⊤000,Υ¯iiμt23=0Θ¯⊗Kit⊤Biμ⊤Wt⊤00,Υ¯μt3=diag−WtPμ−1Wt⊤,−WtPμ−1Wt⊤,−IL⊗WtPμ−1Wt⊤,−I.
FromPμ−WtPμ−1Pμ−Wt⊤>0, it follows that(43)−WtPμ−1Wt⊤<Pμ−symWt.
By resorting to (35), (39), and (43), (35) can be guaranteed if (14) holds. By the same way, (37) and (39) can be guaranteed if (15) and (17) hold.


## 4. Computational Experiments

### 4.1. Numerical Example

Consider the FMSS (1) with 3 modes, its parameters are chosen as follows:(44)A11=−0.7933−0.0250−0.40650.4676,B11=−1.35691.6143,C11=−0.2241−0.1587,D11=1.2622−0.3065,A12=0.5916−1.5784−0.60350.1609,B12=−0.31910.9530,C12=−1.43490.9430,D12=−0.48980.6855,A13=0.62550.05200.2561−0.7916,B13=0.72250.2704,C13=1.4089−0.4147,D13=1.03352.2099,A21=−0.3922−0.0717−0.4506−0.0868,B21=−0.66640.5477,C21=−0.08160.3037,D21=−0.35042.6857,A22=0.80480.56030.6455−0.6569,B22=0.5714−1.1684,C22=1.98910.8228,D22=−2.09490.4153,A23=−0.31070.06270.47320.7070,B23=0.0078−0.8450,C23=0.01121.0132,D23=−0.0921−0.5206,Fiμ=0.01,i=1,2,μ=1,2,3,H=0.050.05T.The TPMΓ and CPM Λ are inferred as(45)Γ=0.40.50.10.050.050.90.20.40.4,Λ=0.80.20.50.50.70.3.One assumes that theLth-order of CF model (6) with L=1 and α0=0.78, α1=0.36, α¯0=0.3, and α¯1=0.3. ωk=1.6sin2k/1+k2, ζk=0.8exp−k2. By solving the inequalities of Theorem 2, a set of controller gains are acquired:(46)K11K12K21K22=−0.1301−0.1029−0.1329−0.10610.18550.14420.17090.1330.By applying the aforementioned controller gains, Figures1 and 2 demonstrate the state mode and controller mode, respectively. The state trajectories with CF are shown in Figure 3, and the control input is depicted in Figure 4. Defining γdk=∑s=0kzs2/∑s=0kϖs2, Figure 5 plots the evolution of γdk.Figure 1
State mode.Figure 2
Controller mode.Figure 3
The state trajectories with CF.Figure 4
Control input.Figure 5
Evolution of γd(k).


### 4.2. Practical Example

Robot arms are essential parts in fire-fighting operating systems. In this case, we consider a single-link robotic arm model (SLRAM) stemmed from [12], which is inferred by(47)ϕ¨t=−gℒℳkJksinϕt−ℛJkϕ˙t+1Jkut+ωt,where the meaning of g, ϕt, ℒ, Jk, and ℛ are explained in [12] and g=9.81m/s2, ℒ=0.5m, ℛ=1kg⋅m2/s, ℳ1=1kg, ℳ2=5kg, J1=1kg⋅m2, and J2=5kg⋅m2.According to [12], for sampling period T=0.01s, the SLRAM is reestablished as the T-S FMSSs.Plant rule 1: IFδ1k is about 0 rad, THEN(48)δk+1=A1,μδk+B1,μuk+C1,μωk,zk=D1,μδk+F1,μωk.Plant rule 2: IFδ2k is about ±π rad, THEN(49)δk+1=A2,μδk+B2,μuk+C2,μωk,zk=D2,μδk+F2,μωk,where(50)A1μ=1TTgℒℳμJμ1−TℛJμ,B1μ=0TJμ,C1μ=0T,D1μ=10,F1μ=1,A2μ=1TεTgℒℳμJμ1−TℛJμ,B2μ=0TJμ,C2μ=0T,D2μ=10,F2μ=1,H=11⊤.The TPMΓ and CPM Λ are expressed as follows:(51)Γ=0.40.60.750.25,Λ=0.70.30.450.55.Assuming thatL=1 and α0=0.81, α1=0.2236, α¯0=0.13, and α¯1=0.2236. Meanwhile, ωk and ζk are chosen as Example 4.1. Solving the inequalities of Theorem 2, the controller gains are achieved:(52)K11K12K21K22=2.4702−10.30372.5196−9.9710−0.4982−8.7620−0.4263−7.9807.For initial conditionδ0=0.2−0.3⊤, by resorting to the obtained filter gains, Figures 6 and 7 demonstrate the state mode and controller mode, respectively. The state trajectories with CF are shown in Figure 8, and control input is depicted in Figure 9. In the end, Figure 10 plots the evolution of γdk. It can be observed from Figures 8–10 that the proposed controller is effective.Figure 6
State mode.Figure 7
Controller mode.Figure 8
The state trajectories with CF.Figure 9
Control input.Figure 10
Evolution of γd(k).


## 4.1. Numerical Example

Consider the FMSS (1) with 3 modes, its parameters are chosen as follows:(44)A11=−0.7933−0.0250−0.40650.4676,B11=−1.35691.6143,C11=−0.2241−0.1587,D11=1.2622−0.3065,A12=0.5916−1.5784−0.60350.1609,B12=−0.31910.9530,C12=−1.43490.9430,D12=−0.48980.6855,A13=0.62550.05200.2561−0.7916,B13=0.72250.2704,C13=1.4089−0.4147,D13=1.03352.2099,A21=−0.3922−0.0717−0.4506−0.0868,B21=−0.66640.5477,C21=−0.08160.3037,D21=−0.35042.6857,A22=0.80480.56030.6455−0.6569,B22=0.5714−1.1684,C22=1.98910.8228,D22=−2.09490.4153,A23=−0.31070.06270.47320.7070,B23=0.0078−0.8450,C23=0.01121.0132,D23=−0.0921−0.5206,Fiμ=0.01,i=1,2,μ=1,2,3,H=0.050.05T.The TPMΓ and CPM Λ are inferred as(45)Γ=0.40.50.10.050.050.90.20.40.4,Λ=0.80.20.50.50.70.3.One assumes that theLth-order of CF model (6) with L=1 and α0=0.78, α1=0.36, α¯0=0.3, and α¯1=0.3. ωk=1.6sin2k/1+k2, ζk=0.8exp−k2. By solving the inequalities of Theorem 2, a set of controller gains are acquired:(46)K11K12K21K22=−0.1301−0.1029−0.1329−0.10610.18550.14420.17090.1330.By applying the aforementioned controller gains, Figures1 and 2 demonstrate the state mode and controller mode, respectively. The state trajectories with CF are shown in Figure 3, and the control input is depicted in Figure 4. Defining γdk=∑s=0kzs2/∑s=0kϖs2, Figure 5 plots the evolution of γdk.Figure 1
State mode.Figure 2
Controller mode.Figure 3
The state trajectories with CF.Figure 4
Control input.Figure 5
Evolution of γd(k).


## 4.2. Practical Example

Robot arms are essential parts in fire-fighting operating systems. In this case, we consider a single-link robotic arm model (SLRAM) stemmed from [12], which is inferred by(47)ϕ¨t=−gℒℳkJksinϕt−ℛJkϕ˙t+1Jkut+ωt,where the meaning of g, ϕt, ℒ, Jk, and ℛ are explained in [12] and g=9.81m/s2, ℒ=0.5m, ℛ=1kg⋅m2/s, ℳ1=1kg, ℳ2=5kg, J1=1kg⋅m2, and J2=5kg⋅m2.According to [12], for sampling period T=0.01s, the SLRAM is reestablished as the T-S FMSSs.Plant rule 1: IFδ1k is about 0 rad, THEN(48)δk+1=A1,μδk+B1,μuk+C1,μωk,zk=D1,μδk+F1,μωk.Plant rule 2: IFδ2k is about ±π rad, THEN(49)δk+1=A2,μδk+B2,μuk+C2,μωk,zk=D2,μδk+F2,μωk,where(50)A1μ=1TTgℒℳμJμ1−TℛJμ,B1μ=0TJμ,C1μ=0T,D1μ=10,F1μ=1,A2μ=1TεTgℒℳμJμ1−TℛJμ,B2μ=0TJμ,C2μ=0T,D2μ=10,F2μ=1,H=11⊤.The TPMΓ and CPM Λ are expressed as follows:(51)Γ=0.40.60.750.25,Λ=0.70.30.450.55.Assuming thatL=1 and α0=0.81, α1=0.2236, α¯0=0.13, and α¯1=0.2236. Meanwhile, ωk and ζk are chosen as Example 4.1. Solving the inequalities of Theorem 2, the controller gains are achieved:(52)K11K12K21K22=2.4702−10.30372.5196−9.9710−0.4982−8.7620−0.4263−7.9807.For initial conditionδ0=0.2−0.3⊤, by resorting to the obtained filter gains, Figures 6 and 7 demonstrate the state mode and controller mode, respectively. The state trajectories with CF are shown in Figure 8, and control input is depicted in Figure 9. In the end, Figure 10 plots the evolution of γdk. It can be observed from Figures 8–10 that the proposed controller is effective.Figure 6
State mode.Figure 7
Controller mode.Figure 8
The state trajectories with CF.Figure 9
Control input.Figure 10
Evolution of γd(k).


## 5. Conclusions

In this work, the asynchronous control for FMSSs with randomly occurring FC is designed. By resorting to the T-S fuzzy model, the nonlinear MSSs can be handled. Furthermore, the Rice fading model is proposed to capture the randomly occurring channel fading, which covers packet dropouts and network-induced delays as special cases. In light of HMM, the asynchronous phenomenon of controller is taken into consideration, and asynchronous fuzzy controller is obtained. In the end, two examples are applied to demonstrate the validity of the derived results. It can be observed that, compared with the existing approach, the obtained methodology is more general, and less conservatism results are expressed. In the networked scenario, sliding mode control for FMSSs with FC is significant in future works [32].



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*Source: 8840784-2020-09-18.xml*