# Observer-Based Controller Design for a Class of Nonlinear Networked Control Systems with Random Time-Delays Modeled by Markov Chains

Authors: Yanfeng Wang; Peiliang Wang; Zuxin Li; Huiying Chen
Journal: Journal of Control Science and Engineering
 (2017)

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## Abstract

This paper investigates the observer-based controller design problem for a class of nonlinear networked control systems with random time-delays. The nonlinearity is assumed to satisfy a global Lipschitz condition and two dependent Markov chains are employed to describe the time-delay from sensor to controller (S-C delay) and the time-delay from controller to actuator (C-A delay), respectively. The transition probabilities of S-C delay and C-A delay are both assumed to be partly inaccessible. Sufficient conditions on the stochastic stability for the closed-loop systems are obtained by constructing proper Lyapunov functional. The methods of calculating the controller and the observer gain matrix are also given. Two numerical examples are used to illustrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

Networked control systems (NCSs) are spatially distributed systems where the communication between sensor, controller, and actuator is carried out by a shared band limited digital network [1, 2]. NCSs are used in a wide range of areas such as robots, industrial manufacturing plants, and remote surgery due to their advantages in practical applications, for example, flexible architectures, the reduced weight, simple installation, and maintenance as well as high flexibility and reliability [3, 4]. However, the communication networks also present some constraints such as time-delays and packet dropouts result from the limited bandwidth. It is generally known that the time-delay maybe degrades the performance or even causes instability [5, 6].The Markov chain which is a discrete-time stochastic process with the Markov property can be effectively used to model the time-delay in NCSs. The random time-delays in NCSs modeled as Markov chains have been researched in the past several years, and many results have been reported [7–17]. In [7], the time-delay of NCSs was modeled as a Markov chain, and further a LQG optimal controller design method was proposed. In [8], the NCSs were molded as Markov jump linear systems (MJLSs) where the S-C delay was molded as a finite state Markov chain, and a V-K iteration method was proposed to get a stabilizing controller. In [9], a buffer was added ahead of the actuator, and the time-delays from sensor to actuator were lumped together which was molded as a Markov chain, and then the mean-square stability of the closed-loop system was derived. In [10–12], for the NCSs with S-C delay, the problem of H∞ control was investigated using the Bounded Real Lemma and the Markov jump theory. In [13, 14], the S-C delay and C-A delay were modeled as two independent Markov chains. The resulting closed-loop systems were transformed to control systems which contain two Markov chains. The sufficient and necessary conditions for the stochastic stability of the resulting closed-loop systems were established, and the mode-dependent state feedback and output feedback controller were designed, respectively.The transition probabilities of time-delays in [7–14] were assumed to be completely accessible. However, in practical applications, this assumption is too ideal and hence will limit the application of the derived results due to the difficulty in obtaining all the transition probabilities of time-delay precisely. Some results have been obtained when the transition probabilities of the time-delays (data packet dropout) are partly inaccessible. In [15], the H∞ control problem was investigated for the NCSs with random data packet dropouts. The closed-loop systems were modeled as MJLSs with four modes and partly inaccessible transition probabilities. In [16, 17], the closed-loop systems were modeled as MJLSs with partly inaccessible transition probabilities of the S-C delay, and the stabilization controller was designed though the linear matrix inequality (LMI) method. In [18], the transition probabilities of the time-delay were assumed to be partly inaccessible, and the fault-tolerant controller for the discrete-time NCSs was designed. Unfortunately, in [16–18], only the S-C delay was considered and an improved controller should take both S-C delay and C-A delay into consideration.It is well known that nonlinearities usually exist in practical systems. Hence, research about nonlinear NCSs is important in both application and theory. To the best of the authors’ knowledge, up to now, involving both S-C delay and C-A delay to design the controller for nonlinear NCSs when transition probabilities of S-C delay and C-A delay are both partly inaccessible has not been investigated, which motivates our investigation.In this paper, we propose two controller design methods for a kind of nonlinear NCSs with both S-C delay and C-A delay based on observer. Compared to the previous relevant works, the main contribution of this paper is that the proposed methods can deal with the situations of both complete accessible transition probabilities and partly inaccessible transition probabilities. The rest of this paper is organized as follows. The closed-loop system model with Markov delays is obtained in Section2. The main results and proofs are given in Section 3. Section 4 presents the simulation results, and the conclusions are provided in Section 5.


## 2. Problem Formulation

The configuration of the NCSs considering time-delays is depicted in Figure1 where τk and dk denote the S-C delay and C-A delay, respectively.Figure 1
Diagram of the NCSs.In this paper,τk and dk are modeled as two homogeneous Markov chains which take value in the set Ω={τm,…,τM} and Γ={dm,…,dM}, and their transition probability matrices are Ξ=[λij] and Π=[πrs], respectively. That is, τk and dk jump from mode i to j and from r to s, respectively, with probabilities λij and πrs, which are defined by λij=Prτk+1=j∣τk=i, πrs=Prdk+1=s∣dk=r, where λij≥0, πrs≥0, and ∑j=τmτMλij=1, ∑s=dmdMπrs=1, for all i,j∈Ω and r,s∈Γ.In this paper, the transition probabilities ofτk and dk are both considered to be partly accessible; that is, some elements in matrix Ξ and Π are unknown. For notational clarity, ∀i∈Ω, we denote Ω=Ωki+Ωuki with Ωki={j:λijisknown}, Ωuki={j:λijisunknown}. Moreover, if Ωki≠⌀, it is further described as Ωki={Ωk1i,Ωk2i,…,Ωkμi} (1≤μ≤e), and Ωuki is described as Ωuki={Ωk-1i,Ωk-2i,…,Ωk-e-μi}, where e is the number of elements in the set Ω.Similarly,∀r∈Γ, we denote Γ=Γkr+Γukr with Γkr={s:πrsisknown}, Γukr={s:πrsisunknown}. If Γkr≠⌀, it is further described as Γkr={Γk1r,Γk2r,…,Γkθr} (1≤θ≤ϕ) and Γukr is described as Γukr={Γk-1r,Γk-2r,…,Γk-ϕ-θr}, where ϕ is the number of elements in the set Γ.Considering the following controlled plant after sampling,(1)xk+1=Axk+fk,xk+Bu~k,yk=Cxk,where xk∈Rn is the state vector, u~k∈Rm is the control input, and yk∈Rp is the measured output. A, B, and C are known real constant matrices with appropriate dimensions. f(k,xk) is a nonlinear vector function which satisfies the following global Lipschitz conditions:(2)fk,xk≤gxk,fk,xk-fk,zk≤gxk-zk,where g is a known real scalar.The dynamic observer-based control scheme is given by(3)Observer:x^k+1=Ax^k+fk,x^k+Buk+Lyk-τk-y^k-τk,y^k=Cx^k,(4)Controller:uk=Kx^k,u~k=uk-dk,where x^k∈Rn, uk∈Rm, and y^k∈Rp are the state vector, control input, and output vector of the observer, respectively. K∈Rm×n is the controller gain and L∈Rn×p is the observer gain.Remark 1.
It should be pointed out that the control inputu~k of the controlled plant (1) is different from the control input uk of the observer (3) due to the existence of the C-A delay dk, while, in the most of the observer-based controller design problem, u~k and uk were assumed to be identical.
Define the state estimate errors as(5)ek=xk-x^k.Substituting (3) and (4) into (1) and (5), the closed-loop system can be obtained as follows:(6)xk+1=Axk+fk,xk+BKxk-τk-ek-τk,ek+1=A+BKek+Fk-BKxk+BKxk-dk-ek-dk-LCek-τk,where Fk=f(k,xk)-f(k,x^k).
By definingηkT=xkTekTT, the closed-loop system (6) can be written in a compact form as follows:(7)ηk+1=A--B1KI1ηk+B2KI1ηk-dk-I2LC-ηk-τk+F-k,where (8)A-=A00A,B1=0B,B2=BB,I1=I-I,I2=0I,C-=0C,F-=fk,xkFk.Definition 2 (see [13]).
System (7) is stochastically stable if, for every finite η0 and τ0∈Ω, d0∈Γ, there exists a finite matrix R>0 such that(9)E∑k=0∞ηk2∣η0,τ0,d0<η0TRη0.In this paper, our objective is to design the dynamic observer-based control scheme (3) and (4), such that the closed-loop system (7) is stochastically stable on condition that the transition probabilities of τk and dk are both partly inaccessible.


## 3. Main Results and Proofs

In this section, we will present the main results. To proceed, we will need the following three lemmas.Lemma 3 (see [19]).
For any positive-definite matrixW, scalars δ,δ0 satisfying δ≥δ0≥1 and vector function υl, one has (∑l=δ0δυl)TW∑l=δ0δυl≤(δ-δ0+1)∑l=δ0δυlTWυl.Lemma 4 (see [20], S-procedure).
LetTl∈Rn×n(l=0,1,…,z) be symmetric matrices. The conditions on ξTT0ξ<0, ∀ξ≠0 s.t. ξTTsξ≤0,(l=0,1,…,z) hold if there exist scalars εl≥0(l=0,1,…,z) such that T0-∑l=1zεlTl<0.Lemma 5 (see [21]).
For given scalarsλi≥0(i=1,…,N1), πr≥0(r=1,…,N2) and matrix Pi,r≥0, the following inequality always holds:(10)∑i=1N1∑r=1N2λiπrPi,r≤∑i=1N1∑r=1N2λiπr∑i=1N1∑r=1N2Pi,r.In the following, a sufficient condition such that the closed-loop system (7) is stochastically stable will be derived.Theorem 6.
Taking the controller gain matrixK and observer gain matrix L, if there exist positive-definite matrices Pi,r>0, Pj,s>0, Q1>0, Q2>0, Q3>0, Q4>0, Q5>0, Q6>0, Z1>0, Z2>0, Z3>0, Z4>0 and scalar ε1≥0, ε2≥0 such that the following inequality(11)Φ≜Φ11∗∗∗∗∗∗∗Φ21Φ22∗∗∗∗∗∗Φ31Φ32Φ33∗∗∗∗∗Φ41Φ42Φ43Φ44∗∗∗∗Z2Z100Φ55∗∗∗0Z1000Φ66∗∗Z40Z3000Φ77∗00Z30000Φ88<0,where (12)Φ11=dM-dm+1Q1+τM-τm+1Q2+Q3+Q4+Q5+Q6+dM-dm2A--B1KI1-ITZ1A--B1KI1-I+dm2A--B1KI1-ITZ2A--B1KI1-I+τM-τm2A--B1KI1-ITZ3A--B1KI1-I+τm2A--B1KI1-ITZ4A--B1KI1-I+A--B1KI1TP-j,sA--B1KI1-Pi,r+ε1g2I3TI3+ε2g2I4TI4,Φ21=dM-dm2B2KI1TZ1A--B1KI1-I+dm2B2KI1TZ2A--B1KI1-I+τM-τm2B2KI1TZ3A--B1KI1-I+τm2B2KI1TZ4A--B1KI1-I+B2KI1TP-j,sA--B1KI1,Φ22=dM-dm2B2KI1TZ1B2KI1+dm2B2KI1TZ2B2KI1+τM-τm2B2KI1TZ3B2KI1+τm2B2KI1TZ4B2KI1+B2KI1TP-j,sB2KI1-Q1-2Z1,Φ31=-dM-dm2I2LC-TZ1A--B1KI1-I-dm2I2LC-TZ2A--B1KI1-I-τM-τm2I2LC-TZ3A--B1KI1-I-τm2I2LC-TZ4A--B1KI1-I-I2LC-TP-j,sA--B1KI1,Φ32=-dM-dm2I2LC-TZ1B2KI1-dm2I2LC-TZ2B2KI1-τM-τm2I2LC-TZ3B2KI1-τm2I2LC-TZ4B2KI1-I2LC-TP-j,sB2KI,Φ33=dM-dm2I2LC-TZ1I2LC-+dm2I2LC-TZ2I2LC-+τM-τm2I2LC-TZ3I2LC-+τm2I2LC-TZ4I2LC-+I2LC-TP-j,sI2LC--Q2-2Z3,Φ41=dM-dm2Z1A--B1KI1-I+dm2Z2A--B1KI1-I+τM-τm2Z3A--B1KI1-I+τm2Z4A--B1KI1-I+P-j,sA--B1KI1,Φ42=dM-dm2Z1B2KI1+dm2Z2B2KI1+τM-τm2Z3B2KI1+τm2Z4B2KI1+P-j,sB2KI1,Φ43=-dM-dm2Z1I2LC--dm2Z2I2LC--τM-τm2Z3I2LC--τm2Z4I2LC--P-j,sI2LC-,Φ44=dM-dm2Z1+dm2Z2+τM-τm2Z3+τm2Z4+P-j,s-ε1I3TI3-ε2I4TI4,Φ55=-Q3-Z2-Z1,Φ66=-Q4-Z1,Φ77=-Q5-Z4-Z3,Φ88=-Q6-Z3,I3=I0,I4=0I,P-j,s=∑j=τmτM∑s=dmdMλijπrsPj,s,holds for all i,j∈Ω and r,s∈Γ, the closed-loop system (7) is stochastically stable.Proof.
Choose the Lyapunov function candidate asV(ηk,τk,dk)=∑ρ=15Vρ(ηk,τk,dk), where(13)V1ηk,τk,dk=ηkTPτk,dkηk,V2ηk,dk,τk=∑l=k-dkk-1ηlTQ1ηl+∑l=k-τkk-1ηlTQ2ηl,V3ηk,dk,τk=∑l=k-dmk-1ηlTQ3ηl+∑l=k-dMk-1ηlTQ4ηl+∑l=k-τmk-1ηlTQ5ηl+∑l=k-τMk-1ηlTQ6ηl,V4ηk,dk,τk=∑n=-dM+1-dm∑l=k+nk-1ηlTQ1ηl+∑n=-τM+1-τm∑l=k+nk-1ηlTQ2ηl,V5ηk,dk,τk=∑n=-dM+1-dm∑l=k+nk-1dM-dmζlTZ1ζl+∑n=-dm-1∑l=k+nk-1dmζlTZ2ζl+∑n=-τM+1-τm∑l=k+nk-1τM-τmζlTZ3ζl+∑n=-τm-1∑l=k+nk-1τmζlTZ4ζl,ζl=ηl+1-ηl.In the following, we denote Pdk,τk as Pr,i when dk=r and τk=i. Then along the solution of system (7), we have (14)EΔV1=Eηk+1TPτk+1,dk+1ηk+1∣τk=i,dk=r-ηkTPi,rηk=A--B1KI1ηk+B2KI1ηk-dk-I2LC-ηk-τk+F-kT∑j=τmτM∑s=dmdMλijπrsPj,sA--B1KI1ηk+B2KI1ηk-dk-I2LC-ηk-τk+F-k-ηkTPi,rηk,EΔV2=ηkTQ1ηk-ηk-rTQ1ηk-r+∑l=k+1-dk+1k-1ηlTQ1ηl-∑l=k+1-dkk-1ηlTQ1ηl+ηkTQ2ηk-ηk-iTQ2ηk-i+∑l=k+1-τk+1k-1ηlTQ2ηl-∑l=k+1-τkk-1ηlTQ2ηl.It is noticed that(15)∑l=k+1-dk+1k-1ηlTQ1ηl=∑l=k+1-dmk-1ηlTQ1ηl+∑l=k+1-dk+1k-dmηlTQ1ηl≤∑l=k+1-dkk-1ηlTQ1ηl+∑l=k+1-dMk-dmηlTQ1ηl,∑l=k+1-τk+1k-1ηlTQ2ηl=∑l=k+1-τmk-1ηlTQ2ηl+∑l=k+1-τk+1k-τmηlTQ2ηl≤∑l=k+1-τkk-1ηlTQ2ηl+∑l=k+1-τMk-τmηlTQ2ηl.Therefore, we obtain (16)EΔV2≤ηkTQ1ηk-ηk-rTQ1ηk-r+∑l=k+1-dMk-dmηlTQ1ηl+ηkTQ2ηk-ηk-iTQ2ηk-i+∑l=k+1-τMk-τmηlTQ2ηl,EΔV3=ηkTQ3ηk-ηk-dmTQ3ηk-dm+ηkTQ4ηk-ηk-dMTQ4ηk-dM+ηkTQ5ηk-ηk-τmTQ5ηk-τm+ηkTQ6ηk-ηk-τMTQ6ηk-τM,EΔV4=dM-dmηkTQ1ηk-∑l=k+1-dMk-dmηlTQ1ηl+τM-τmηkTQ2ηk-∑l=k+1-τMk-τmηlTQ2ηl,EΔV5=dM-dm2ζkTZ1ζk-∑l=k-dMk-dm-1dM-dmζlTZ1ζl+dm2ζkTZ2ζk-dm∑l=k-dmk-1ζlTZ2ζl+τM-τm2ζkTZ3ζk-∑l=k-τMk-τm-1τM-τmζlTZ3ζl+τm2ζkTZ4ζk-τm∑l=k-τmk-1ζlTZ4ζl.Note that (17)-dm∑l=k-dmk-1ζlTZ2ζl-∑l=k-dMk-dm-1dM-dmζlTZ1ζl=-dm∑l=k-dmk-1ζlTZ2ζl-∑l=k-rk-dm-1dM-dmζlTZ1ζl-∑l=k-dMk-r-1dM-dmζlTZ1ζl≤-dm∑l=k-dmk-1ζlTZ2ζl-∑l=k-rk-dm-1r-dmζlTZ1ζl-∑l=k-dMk-r-1dM-rζlTZ1ζl.
We can obtain the following inequality by Lemma3:(18)-dm∑l=k-dmk-1ζlTZ2ζl-∑l=k-dMk-dm-1dM-dmζlTZ1ζl≤-ηk-ηk-dmTZ2ηk-ηk-dm-ηk-dm-ηk-rTZ1ηk-dm-ηk-r-ηk-r-ηk-dMTZ1ηk-r-ηk-dM.Similarly, we have (19)-τm∑l=k-τmk-1ζlTZ4ζl-∑l=k-τMk-τm-1τM-τmζlTZ3ζl≤-ηk-ηk-τmTZ4ηk-ηk-τm-ηk-τm-ηk-iTZ3ηk-τm-ηk-i-ηk-i-ηk-τMTZ3ηk-i-ηk-τM.From (14)–(19), we have(20)EΔVηk,dk,τk∣τk=i,dk=r≤ξkTΦ-ξk,where(21)Φ-=Φ-11∗∗∗∗∗∗∗Φ21Φ22∗∗∗∗∗∗Φ31Φ32Φ33∗∗∗∗∗Φ41Φ42Φ43Φ-44∗∗∗∗Z2Z100Φ55∗∗∗0Z1000Φ66∗∗Z40Z3000Φ77∗00Z30000Φ88,Φ-11=dM-dm+1Q1+τM-τm+1Q2+Q3+Q4+Q5+Q6+dM-dm2A--B1KI1-ITZ1A--B1KI1-I+dm2A--B1KI1-ITZ2A--B1KI1-I+τM-τm2A--B1KI1-ITZ3A--B1KI1-I+τm2A--B1KI1-ITZ4A--B1KI1-I+A--B1KI1TP-j,sA--B1KI1-Pi,r,Φ-44=dM-dm2Z1+dm2Z2+τM-τm2Z3+τm2Z4+P-j,s,ξkT=ηkTηk-rTηk-iTF-kηk-dmTηk-dMTηk-τmTηk-τMT.It follows from (2) that (22)fk,xkTfk,xk≤g2xkTxk,FkTFk≤g2ekTek,which imply that(23)fk,xkTfk,xk-g2xkTxk=F-kTI3TI3F-k-g2ηkTI3TI3ηk=ξkT-g2I3TI3∗∗∗∗∗∗∗00∗∗∗∗∗∗000∗∗∗∗∗000I3TI3∗∗∗∗00000∗∗∗000000∗∗0000000∗00000000ξk≜ξkTΛ1ξk≤0,FkTFk-g2ekTek=F-kTI4TI4F-k-g2ηkTI4TI4ηk=ξkT-g2I4TI4∗∗∗∗∗∗∗00∗∗∗∗∗∗000∗∗∗∗∗000I4TI4∗∗∗∗00000∗∗∗000000∗∗0000000∗00000000ξk≜ξkTΛ2ξk≤0.By the well-known S-procedure, that is, Lemma 4, we can get E{ΔV(ηk,dk,τk)}≤ξkTΦ-ξk<0 with constraints (23) holding if there exist nonnegative real scalars ε1≥0, ε2≥0 such that(24)Φ--ε1Λ1-ε2Λ2=Φ<0.From (24), we have(25)EΔVηk,dk,τk∣τk=i,dk=r≤-λmin-ΦξkTξk=-λmin-ΦηkTηk+ηk-rTηk-r+ηk-iTηk-i+F-kTF-k+ηk-dmTηk-dm+ηk-dMTηk-dM+ηk-τmTηk-τm+ηk-τMTηk-τM≤-αηk2,where α=inf⁡-λmin-Φ>0.
From (25), we can see that, for any T≥1,(26)E∑k=0∞ηk2≤1αEVη0,d0,τ0-1αEVηT+1,dT+1,τT+1≤1αEVη0,d0,τ0.According to Definition 2, the closed-loop system (7) is stochastically stable, which completes the proof.Theorem 7.
Taking the controller gain matrixK and observer gain matrix L, the closed-loop system (7) is stochastically stable if there exist positive-definite matrices Pi,r>0, Pj,s>0, Fj,s>0, Q1>0, Q2>0, Q3>0, Q4>0, Q5>0, Q6>0, Z1>0, Z2>0, Z3>0, Z4>0, Y1>0, Y2>0, Y3>0, Y4>0 and scalar ε1≥0, ε2≥0 such that(27)Λ11∗∗∗Λ21Λ22∗∗Λ310Λ33∗Λ4100Λ44<0,(28)Pj,sFj,s=I,YlZl=I,l∈1,2,3,4,where(29)Λ11=Λ-11∗∗∗0-Q1-2Z1∗∗00-Q2-2Z3∗000-τ1I3TI3-τ2I4TI4,Λ21=Z2Z1000Z100Z40Z3000Z30,Λ22=-Q3-Z2-Z1∗∗∗0-Q4-Z1∗∗00-Q5-Z4-Z3∗000-Q6-Z3,Λ31=dM-dmΥdM-dmB2KI1-dM-dmI2LC-dM-dmIdmΥdmB2KI1-dmI2LC--dmIτM-τmΥτM-τmB2KI1-τM-τmI2LC-τM-τmIτmΥτmB2KI1-τmI2LC--τmI,Λ33=-Y1∗∗∗0-Y2∗∗00-Y3∗000-Y4,Λ41=λiΩk1iπrΓk1rA--B1KI1T⋯1-∑l∈Ωkiλil1-∑l∈ΓkrπrlA--B1KI1TλiΩk1iπrΓk1rB2KI1T⋯1-∑l∈Ωkiλil1-∑l∈ΓkrπrlB2KI1T-λiΩk1iπrΓk1rI2LC-T⋯-1-∑l∈Ωkiλil1-∑l∈ΓkrπrlI2LC-TλiΩk1iπrΓk1rI⋯1-∑l∈Ωkiλil1-∑l∈ΓkrπrlIT,Λ44=-FΩk1i,Γk1r∗∗0⋱∗00-FΩk-e-μi,Γk-ϕ-θr,Λ-11=dM-dm+1Q1+τM-τm+1Q2+Q3+Q4+Q5+Q6-Z2-Z4+ε1g2I3TI3+ε2g2I4TI4-Pi,r,Υ=A--B1KI1-Ihold for all i,j∈Ω and r,s∈Γ.Proof.
Making use of Lemma5, we have(30)∑j∈Ωuki∑s∈ΓkrλijπrsPj,s≤1-∑l∈Ωkiλil∑s∈Γkrπrs∑j∈Ωuki∑s∈ΓkrPj,s,(31)∑j∈Ωki∑s∈ΓukrλijπrsPj,s≤1-∑l∈Γkrπrl∑j∈Ωkiλij∑j∈Ωki∑s∈ΓukrPj,s,(32)∑j∈Ωuki∑s∈ΓukrλijπrsPj,s≤1-∑l∈Γkrπrl1-∑l∈Ωkiλil∑j∈Ωuki∑s∈ΓukrPj,s.Letting Pj,s-1=Fj,s, j∈Ω, s∈Γ, Zl-1=Yl, l∈{1,2,3,4} and by Schur complement Lemma, (27) can be obtained from (11) and (30)–(32), which complete the proof.In Theorem7, in order to get a desired controller (3)-(4) for the closed-loop NCS (7) with partly inaccessible transition probabilities, Lemma 5 is used to separate λijπrs from λijπrsPj,s. However, this could lead to conservativeness to some extents as shown in the Numerical Example. In the following, a less conservative result will be given in the following theorem.Theorem 8.
There exists dynamic observer-based control scheme (3)-(4) such that the resulting closed-loop system (7) is stochastically stable if, for all i,j∈Ω and r,s∈Γ, there exist matrices Pi,r>0, Pj,s>0, Fj,s>0, Q1>0, Q2>0, Q3>0, Q4>0, Q5>0, Q6>0, Z1>0, Z2>0, Z3>0, Z4>0, Y1>0, Y2>0, Y3>0, Y4>0 and scalar ε1≥0, ε2≥0 such that(33)∑j∈Ωki∑s∈ΓkrλijπrsΛ∗ΘΩki,Γkr-ΔΩki,Γkr<0,(34)∑s∈ΓkrλijπrsΛ∗ΘΩuki,Γkr-ΔΩuki,Γkr<0,∀j∈Ωuki,(35)∑j∈ΩkiλijΛ∗ΘΩki,Γukr-ΔΩki,Γukr<0,∀s∈Γukr,(36)Λ∗ΘΩuki,Γukr-ΔΩuki,Γukr<0,∀j∈Ωuki,∀s∈Γukr,(37)Fj,sPj,s=I,YlZl=I,l∈1,2,3,4,where(38)Λ=Λ11∗∗Λ21Λ22∗Λ310Λ33,Λ11=Λ-11∗∗∗0-Q1-2Z1∗∗00-Q2-2Z3∗000-τ1I3TI3-τ2I4TI4,Λ21=Z2Z1000Z100Z40Z3000Z30,Λ22=-Q3-Z2-Z1∗∗∗0-Q4-Z1∗∗00-Q5-Z4-Z3∗000-Q6-Z3,Λ31=dM-dmΥdM-dmB2KI1-dM-dmI2LC-dM-dmIdmΥdmB2KI1-dmI2LC--dmIτM-τmΥτM-τmB2KI1-τM-τmI2LC-τM-τmIτmΥτmB2KI1-τmI2LC--τmI,Λ33=-Y1∗∗∗0-Y2∗∗00-Y3∗000-Y4,Λ-11=dM-dm+1Q1+τM-τm+1Q2+Q3+Q4+Q5+Q6-Z2-Z4+ε1g2I3TI3+ε2g2I4TI4-Pi,r,ΘΩki,ΓkrT=λiΩk1iπrΓk1rΨ~T⋯λiΩkμiπrΓkθrΨ~T,ΔΩki,Γkr=diag-FΩk1iΓk1r,…,-FΩkμiΓkθr,ΘΩuki,ΓkrT=πrΓk1rΨ~T⋯πrΓkθrΨ~T,ΔΩuki,Γkr=diag-Fj,Γk1r,…,-Fj,Γkθr,ΘΩki,ΓukrT=λiΩk1iΨ~T⋯λiΩkμiΨ~T,ΔΩki,Γukr=diag-FΩk1i,s,…,-FΩkμi,s,ΘΩuki,ΓukrT=Ψ~T⋯Ψ~T,ΔΩuki,Γukr=diag-Fj,s,…,-Fj,s,Ψ~=Ψ00,Ψ=A--B1KI1B2KI1-I2LC-I,Υ=A--B1KI1-I.Proof.
Equation (11) is equivalent to (39)Λ+Ψ~T∑j=τmτM∑s=dmdMλijπrsPj,sΨ~<0,that is (40)∑j∈Ωki∑s∈Γkrλijπrs+∑j∈Ωuki∑s∈Γkrλijπrs+∑j∈Ωki∑s∈Γukrλijπrs+∑j∈Ωuki∑s∈ΓukrλijπrsΛ+Ψ~T∑j∈Ωki∑s∈ΓkrλijπrsPj,s+∑j∈Ωuki∑s∈ΓkrλijπrsPj,s+∑j∈Ωki∑s∈ΓukrλijπrsPj,s+∑j∈Ωuki∑s∈ΓukrλijπrsPj,sΨ<0.The above inequality can be written as (41)∑j∈Ωki∑s∈ΓkrλijπrsΛ+Ψ~T∑j∈Ωki∑s∈ΓkrλijπrsPj,sΨ+∑j∈Ωukiλij∑s∈ΓkrπrsΛ+Ψ~T∑s∈ΓkrπrsPj,sΨ+∑s∈Γukrπrs∑j∈ΩkiλijΛ+Ψ~T∑j∈ΩkiπrsPj,sΨ+∑j∈Ωuki∑s∈ΓukrλijπrsΛ+Ψ~TPj,sΨ<0.By Schur complement Lemma,(42)∑j∈Ωki∑s∈ΓkrλijπrsΛ+∑j∈Ωki∑s∈ΓkrλijπrsΨ~TPj,sΨ<0,is equivalent to the following inequality(43)∑j∈Ωki∑s∈ΓkrλijπrsΛ∗ΘΩki,Γkr-Δ~Ωki,Γkr<0,where Δ~ΩkiΓkr=diag{-P-1Ωk1iΓk1r,…,-P-1ΩkμiΓkθr}. Let Pj,s-1=Fj,s, j∈Ω, s∈Γ, and Zl-1=Yl, l∈{1,2,3,4}; we can obtain (33) and (37). Hence, (42) holds if (33) holds. Because λij≥0, πrs≥0, it can be concluded that if (33)–(36) holds, (39) holds, which completes the proof.The conditions in Theorem7 are in fact a set of linear matrix inequalities (LMIs) with some inversion constraints. Though they are nonconvex which brings difficulties in solving them using the existing convex optimization tool, we can use the cone complementarity linearization (CCL) algorithm [22] to transform this problem into the nonlinear minimization problem with LMIs constraints as follows:(44)mintr∑l=14ZlYl+∑j=τmτM∑s=dmdMPj,sFj,ss.t.27,45,and46(45)ZlIIYl>0,l∈1,2,3,4,(46)Pj,sIIFj,s>0,j∈Ω,s∈Γ.Furthermore, the iterative algorithm which can be used to calculate the controller gain matrix K and observer gain matrix L is given as follows.Algorithm 9.
Step  1. Find a feasible solution satisfying (27), (45), and (46) and set as (Z10,Z20,Z30,Z40,Y10,Y20,Y30,Y40,Pj,s0,Fj,s0,K0,L0); Let k=0.
Step  2. Solve the following LMI optimization problem for variables (Z1,Z2,Z3,Z4,Y1,Y2,Y3,Y4,Pj,s,Fj,s,K,L). mintr(∑l=14(ZlkYl+RlYlk) + ∑j=τmτM∑s=dmdM(Pj,skFj,s+Pj,sFj,sk)), subject to (27), (45), and (46); set (Z1k=Z1, Y1k=Y1, Z2k=Z2, Y2k=Y2, Z3k=Z3, Y3k=Y3, Z4k=Z4, Y4k=Y4, Pj,sk=Pj,s, Fj,sk=Fj,s, Kk=K, Lk=L).
Step  3. If (27) and (28) are satisfied, then exit the iteration. Otherwise, let k=k+1, and return to Step  2.The conditions in Theorem8 can be solved by the similar procedures.Remark 10.
Making use of Matlab LMI tool box, a feasible solution satisfying (27), (45), and (46) can be obtained by the functions feasp() and dec2mat(). Once the feasible solution (Z10,Z20,Z30,Z40,Y10,Y20,Y30,Y40,Pj,s0,Fj,s0,K0,L0) is obtained, then the LMI optimization problem mintr(∑l=14(ZlkYl+ZlYlk)+∑j=τmτM∑s=dmdM(Pj,skFj,s+Pj,sFj,sk)), subject to (27), (45), and (46), can be solved by the functions defcx() and mincx().


## 4. Numerical Example

In this section, we present two examples to illustrate the effectiveness of the proposed method.Example 1.
Consider the nonlinear controlled plant with the following parameter:(47)A=0.52-0.6900.19,B=0.30.2,C=1.50.70.20.3,g=0.1,fk,xk=0.1sin⁡xk.
We can see that the linear part of the control plant is stable itself. The random S-C delayτk∈{0,1} and C-A delay dk∈{0,1}. Their transition probability matrices are as follows, respectively:(48)Ξ=0.70.3??,Π=??0.20.8.By Theorem 7, we can obtain the controller gain matrix K and observer gain matrix L as follows:(49)K=-0.04240.1725,L=0.5519-2.63830.1065-0.6730.The initial value x0=-11T, x^0=-0.9-1.1T, τ0=d0=0. The trajectories of the closed-loop system’s states and the corresponding estimated value are shown in Figures 2 and 3 which indicate that the closed-loop system is stochastically stable.
By Theorem8, the controller gain matrix K and observer gain matrix L can also be obtained as follows:(50)K=-0.08020.2425,L=0.2968-1.70360.1569-1.0017.The trajectories of the closed-loop system’s states and the corresponding estimated value are omitted.Figure 2
Statex1 and its estimated value x^1.Figure 3
Statex2 and its estimated value x^2.Example 2.
Consider the classical angular positioning system shown in Figure4, which consists of a rotating antenna at the origin of the plane driven by a motor [23].
Assume that the angular position of the antennaφ, the angular position of the moving object φr, and the angular velocity of the antenna φ˙ are measurable. The state variables are chosen as φφ˙T; then state equation of the angular positioning system can be obtained with the following parameter, where f(k,xk) can be treated as the external disturbance:(51)A=10.099500.99,B=0.00390.0783,C=1.40.8-0.20.4,g=0.2,fk,xk=0.2sin⁡xkk.
Obviously, it can be seen that the linear part of the control plant is unstable. The random S-C delayτk∈{0,1} and C-A delay dk∈{0,1}. Their transition probability matrices are as follows, respectively:(52)Ξ=??0.60.4,Π=0.20.8??.For this unstable plant, we cannot get the controller gain matrix K and observer gain matrix L by Theorem 7 due to its conservativeness. However, by Theorem 8, we can obtain K and L as follows:(53)K=-1.1957-1.1143,L=0.1943-0.27150.11300.6251.The initial value x0=-1.61.7T, x^0=-1.71.6T, τ0=d0=0. The trajectories of the closed-loop system’s states and the corresponding estimated value are shown in Figures 5 and 6 which indicate that the closed-loop system is stochastically stable.Figure 4
The angular positioning system.Figure 5
Statex1 and its estimated value x^1.Figure 6
Statex2 and its estimated value x^2.Remark 11.
The results in this paper only require a part of transition probabilities while the results in [13, 14] need the full information of the transition probabilities. In view of these, the proposed controller design method is more general than that of [13, 14].


## 5. Conclusion

The observer-based controller design problem for a class of nonlinear networked control systems with random time-delays is investigated in this paper. Two dependent Markov chains are employed to describe S-C delay and C-A delay, respectively. The transition probabilities of S-C delay and C-A delay are both assumed to be partly accessible. Sufficient conditions on the stochastic stability for the closed-loop systems are obtained by the Lyapunov stability theory. The CCL algorithm is employed to calculate the controller and observer gain matrices. Finally, two examples are used to illustrate the effectiveness of the proposed method.



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*Source: 1523825-2017-03-30.xml*