# Robust Estimation for Discrete Markov System with Time-Varying Delay and Missing Measurements

Authors: Jia You; Shen Yin; Hamid Reza Karimi
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper addresses theℋ∞ filtering problem for time-delayed Markov jump systems (MJSs) with intermittent measurements. Within network environment, missing measurements are taken into account, since the communication channel is supposed to be imperfect. A Bernoulli process is utilized to describe the phenomenon of the missing measurements. The original system is transformed into an input-output form consisting of two interconnected subsystems. Based on scaled small gain (SSG) theorem and proposed Lyapunov-Krasovskii functional (LKF), the scaled small gains of the subsystems are analyzed, respectively. New conditions for the existence of the ℋ∞ filters are established, and the corresponding ℋ∞ filter design scheme is proposed. Finally, a simulation example is provided to demonstrate the effectiveness of the proposed approach.


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## Body

## 1. Introduction

In recent years, networked control systems (NCSs) have found a great deal of applications in a range of engineering areas [1–6]. Compared to traditional control systems, the NCSs hold many magnificent advantages, such as reduced power and weight requirements, low cost, improved flexibility and suppleness, high reliability, and facilitated system diagnosis and maintenance. Consequently, NCSs have drawn extensive investigation, and a great deal of significant results have been reported [7–12].On the other hand, some unexpected phenomena appear in the networked environment, since the digital communication channels have been introduced, on which all measured signals are transmitted with limited capacity. Intermittent measurements and time-delay are widely emerged in many kinds of NCSs, which could always lead to poor performance even instability of the systems. Some research issues in networked environment, such as filtering problems have become much more complicated since most traditional approaches are proposed on the assumption that the communication channel between the physical plant and filter is perfect. In this sense, a great attention has been attracted to the filtering problems with limited capacity and led to many important results. To mention them, [13] addresses the filtering problem with communication delay, [14] investigates the H∞ filtering problem for nonlinear stochastic systems, and robust filtering problems subject to stochastic missing measurements which described by different methods are introduced in [15–17].Based on the direct Lyapunov method, a great deal of stability criteria have been reported [9, 18]. Meanwhile, an input-output approach has been introduced to analyze the stability of the systems by transforming the original system into a form of interconnection of two subsystems via employing a two-term approximation for the delayed variables [19]. Recently, Observing that the resulting performance largely depends on the error between the delayed variable and the approximation, an approach for estimating the delay by the upper and the lower delay bounds was introduced [20], of which the approximation error is much smaller than the error obtained by other one-term approaches [21, 22]. However, to the best of the authors’ knowledge, corresponding results for time-delay MJSs with limited capacity channel have been relatively few, which motivates the research in this paper.In this paper, we investigate theℋ∞ filtering problem for time-delayed MJSs with intermittent measurements. The communication channel between the plant and the filter is supposed to be imperfect. And the phenomenon of missing measurements is modeled by employing a stochastic variable satisfying Bernoulli random binary distribution. By using a new two-term approximation to the delay variables, the system is transformed into an interconnected systems comprised of two subsystems. Based on the scaled small gain (SSG) theorem and proposed Lyapunov-Krasovskii functional (LKF), sufficient conditions for the existence of the ℋ∞ filters and the corresponding ℋ∞ filter design method are proposed. Finally, a numerical example is provided to show effectiveness of the approach.The remainder of this paper is organized as follows. In Section2, the problem of ℋ∞ filtering is formulated for time-delayed MJSs with intermittent measurements; Section 3 presents the results for ℋ∞ performance analysis, and ℋ∞ filter design problem is solved in Section 4; a numerical example is provided to demonstrate the effectiveness of the design filtering technique in Section 5, and we conclude the paper with Section 6.Notation. Throughout this paper, ℝn represents the n-dimensional Euclidean space, ℝn×m is the set of all n×m real matrices, the superscripts “−1” and “T,” respectively, stand for the matrix inverse and matrix transpose. Sym{A} is the shorten notation for A+AT, and the notation P>0 (resp., P≥0), for P∈ℝn×n means that P is real symmetric and positive definite (resp., semidefinite). The symmetric elements of the symmetric matrix is represented by an asterisk (*), and the block-diagonal matrices are denoted by diag{⋯}. G1∘G2 means the series connection of mapping G1 and G2. E{·} denotes the expectation operator with respect to probability measure, and for vector x(k), ∥x∥E2=E{∑n=0∞‍∥x(n)∥2}1/2.


## 2. Problem Formulation

Consider the following Markov jump system with the time-varying delay:(1)xk+1=A(rk)xk+Ad(rk)xk-τk+B(rk)ωk,yk=C(rk)xk+Cd(rk)xk-τk+D(rk)ωk,zk=L(rk)xk,xk=ψk,k=-d2,…,-1,0,
where xk∈ℝn denotes system state; yk∈ℝm is the measured output; ωk∈ℝq is the measurement noise which belongs L2[0,∞); zk∈ℝp is the signal to be estimated; and ψk is a given real-valued initial condition. Let {rk,k∈ℤ} be a Markov chain that takes values in a given finite set 𝒮={1,2,…,s} with transition probability pij=Pr{rk+1=j∣rk=i}, where pij≥0 for i,j∈𝒮 and ∑j=1s‍pij=1. And τk is a positive integer, denoting the time-varying delay satisfying
(2)d1≤τ(k)≤d2,
where d1≥0 and d2>0 are the minimum and the maximum delay, respectively. The delay interval is defined as dt=d2-d1.Then, we consider the following mode-independent filter with full order:(3)ℱ:{x^k+1=AFx^k+BFyF(k),z^k=LFx^k,
where x^k∈ℝn is the filter state vector and z^k∈ℝp is the output of the filter; AF∈ℝn×n, BF∈ℝn×m, and CF∈ℝp×n are the filter gains to be determined.In the network the data transmitted in the digital channel may be lost at a certain probability as shown in Figure1. As a result, the output of the physical plant yk is no more equivalent to the input of the filter, which is denoted by yF(k). In this paper, this phenomenon is modeled via a Bernoulli process:
(4)yF(k)=αkyk,
where αk is Bernoulli process, which models the intermittent transmission between the plant and the filter. When α(k)=1, the data is transmitted to the filter successfully. While when α(k)=0, the transmission fails and the data is lost. It can be described as:
(5)Prob{αk=1}=E{αk}=α¯,Prob{αk=0}=1-α¯.Figure 1
Filtering problem with intermittent measurements.Combining (3) and (5), we have
(6)ℱ:{x^k+1=AFx^k+BFαkyk,z^k=LFx^k.
Defining α~=αk-α¯ yields E{α~}=0 and E{(α~)2}=α¯(1-α¯).Then, denote(7)ξk=[xkTx^kT]T,ek=zk-z^k.
By connecting (1) with (3), the filtering error dynamic can be described by
(8)𝒮:{ξk+1=A¯(i)ξk+A¯d(i)Kξk-τk+B¯(i)ωk,ek=L¯(i)ξk,
where
(9)A¯(i)=A¯1(i)+α~A¯2(i),A¯d(i)=A¯d1(i)+α~A¯d2(i),B¯(i)=B¯1(i)+α~B¯2(i),K=[I0],A¯1(i)=[A(i)0α¯BFC(i)AF],A¯2(i)=[00BFC(i)0],A¯d1(i)=[Ad(i)α¯BFCd(i)],A¯d2(i)=[0BFCd(i)],B¯1(i)=[B(i)α¯BFD(i)],B¯2(i)=[0BFD(i)],L¯(i)=[L(i)-LF].Moreover, we introduce the definition of stochastic stability which is vital and important in this paper.Definition 1 (see [23]).
For the system in (1) with ωk=0, the system is stochastically stable if for any ψk∈ℝn,
(10)E{∑k=0∞‍xkTxk∣ψ(0)}<∞.Definition 2 (see [23]).
A mappingG:uk→yk is input-output stable in mean-square if there exists γ≥0 such that
(11)∥yk∥E2=∥G(u)∥E2≤γ∥uk∥E2.Then, theℋ∞ filtering design for MJSs with intermittent measurements to be addressed in this paper can be formulated as follows.Consider system (1) with intermittent measurements modeled by (5). Given positive integers d2>d1 and a prescribed scalar γ>0, the filter gains AF, BF, and LF of filter (3) are designed such that for any time-delay τk satisfying (2), the filter error system (8) is stochastically stable, and ∥e∥E2≤γ∥ω∥E2 holds under zero initial conditions.


## 3. Main Results

In this section, by developing SSG Theorem to stochastic systems and a new model transformation to system (8), we will focus on the filter performance analysis. Sufficient conditions are presented, under which system ℰ is stochastically stable with a prescribed ℋ∞ performance index.


### 3.1. Filtering Performance Analysis

Lemma 3 (see [24]).
One considers the following system consisting of two subsystems as shown in Figure2:
(12)𝒴1:ζk=G(uk-δk),𝒴2:δk=Δζk.
The closed loop system is input-output stable in mean-square for all the subsystemsG and Δ, if ∥Tζ∘G∘Tδ∥<1 holds for some nonsingular matrices Tζ and Tδ with ∥Tδ∘Δ∘Tζ∥≤1.Figure 2
The interconnection subsystems.Proof.
According to (12) and the condition in Lemma 3, by defining Δuk=uk-δk, and without loss of generality, it can be assumed that
(13)∥G(Δuk)∥E2≤k1∥Δuk∥E2,∥Δ(ζk)∥E2≤k2∥ζk∥E2,0<k1<1,0<k2≤1.
Obviously, we have(14)∥Tyk∥E2≤k1∥TΔuk∥E2=k1∥T(uk-Δ(yk))∥E2≤k1(∥Tuk∥E2+k2∥Tyk∥E2),∥Tyk∥E2≤k11-k1k2∥Tuk∥E2.
Sincek1k2<1, system (12) is input-output stable in mean-square, which completes the proof.Then, to separate the uncertaintiesτk from the filtering error system, the time-delay system 𝒮 in (8) is transformed into the interconnection of two subsystems introduced in Lemma 3.Notice that the time-varying delay can be written as(15)xk-τk=12xk-d1+12xk-d2+d122δk,
where (1/2)(xk-d1+xk-d2) is the approximation of the time-varying delay xk-τk, and (d12/2)δk is the approximation error. Then, we can describe system 𝒮 as
(16)𝒮:{ξk+1=A¯(i)ξk+12A¯d(i)Kξk-d1+12A¯d(i)Kξk-d2+d122A¯d(i)δk+B¯(i)ωk,ek=L¯(i)ξk.Definingζk=xk+1-xk and
(17)β(i)≜{1,wheni⩽k-τk-1,-1,wheni>k-τk-1,
we can obtain
(18)δk=2d12[xk-τk-12xk-d1-12xk-d2]=1d12[∑i=k-d2k-τk-1‍ζi-∑i=k-τkk-d1-1‍ζi]=1d12[∑i=k-d2k-d1-1‍β(i)ζi].From (18), we know that mapping Δ:ζk→δk includes the uncertainty in τk leaving only two constant delays d1 and d2 in (16). Then the system in (16) can be described as
(19)𝒮1:[ξk+1ζkek]=[Φ1d122A¯d(i)B¯(i)Φ2d122Ad(i)B¯(i)Φ300]︸G[ϕkδkωk],𝒮2:δk=Δζk,
where
(20)ϕk=[ξkTξk-d1TKTξk-d2TKT]T,Φ1=[A¯(i)12A¯d(i)12A¯d(i)],Φ2=[(A(i)-I)K12Ad(i)12Ad(i)],Φ3=[L¯00].Furthermore, the following lemma shows that the scaled small gain of the mappingΔ has an upper bound.Lemma 4.
γ
(
Δ
T
) has an upper bound:
(21)γ(ΔT)=sup∥Tδk∥E2∥Tζk∥E2≤1.Proof.
By using Jensen's inequality, it can be derived under zero initial condition according to the fact that|β(i)|=1,
(22)∥Tδk∥E22=E{∑j=0∞‍[δjTZδj]}≤2d122E{∑i=0∞‍[d12∑j=i-d2i-d1-1‍β2(j)δjTδj]}=1d12E{∑j=-d2-d1-1‍∑i=0∞‍δi+jTδi+j}≤1d12E{∑j=-d2-d1-1‍∑i=0∞‍δiTδi}≤E{∑j=0∞‍[ζjTZζj]}=∥Tζk∥E22,
where Z=TTT, which implies γ(ΔT)≤1. The proof is completed.Now, we are in the position to propose the sufficient conditions under which the filtering error system (8) is stochastically stable with a prescribed ℋ∞ performance attention level γ. First we assume the filter matrices AF, BF, and LF are fixed and the corresponding conditions are established in the following theorem.Theorem 5.
Consider system (1), and assume the filter matrices AF, BF, and LF are fixed. Given integers d2>d1≥0 and a constant γ>0, the filtering error system (16) is input-output stable in mean-square with an ℋ∞ performance attention level γ for any time-varying delay satisfying (2) if there exist matrices P>0, Qi>0(i=1,2), Ri>0(i=1,2), Z>0 satisfying
(23)Ψ≜[Ξ1[d1Γ3TR1d2Γ3TR2Γ3TZΓ4TΓ1TPfΓ2TP]*diag{-R1,-R2,-Z,-Ip,-P,-P}]<0,
where
(24)Ξ1=[Ξ11[KTR1KTR200]*diag{Ξ2,Ξ3,-Z,-γ2Iq}],Γ1=[A¯1(i)12A¯d1(i)12A¯d1(i)12d12A¯d1(i)B¯1(i)],Γ2=[A¯2(i)12A¯d2(i)12A¯d2(i)12d12A¯d2(i)B¯2(i)],Γ3=[(A(i)-I)K12Ad(i)12Ad(i)12d12Ad(i)B(i)],Γ4=[L¯(i)0p×(3n+q)],f=η¯(1-η¯),Ξ11=-P(i)+KT(Q1+Q2-R1-R2)K,Ξ2=-Q1-R1,Ξ3=-Q2-R2.Proof.
Choose the following Lyapunov-Krasovskii functional:(25)V(k)=V1(k)+V2(k)+V3(k),
with
(26)V1(k)=ξkTP(i)ξk,V2(k)=∑i=k-d1k-1‍ξiTKTQ1Kξi+∑i=k-d2k-1‍ξiTKTQ2Kξi,V3(k)=d1∑i=-d1-1‍∑j=k+ik-1‍ζjTR1ζj+d2∑i=-d2-1‍∑j=k+ik-1‍ζjTR2ζj,
where P>0 and Qi>0 are the matrices to be determined.
DefiningΔV(k)=E{V(k+1)}-V(k), we have
(27)ΔV1(k)=E{ξk+1TP(i+)ξk+1}-ξkP(i)ξk=φkT(Γ1TP(i+)Γ1+f2Γ2TP(i+)Γ2)φk-ξkP(i)ξk,ΔV2(k)=ξkTKTQ1Kξk-ξk-d1TKTQ1Kξk-d1+ξkTKTQ2Kξk-ξk-d2TKTQ2Kξk-d2,ΔV3(k)=d12ζkTR1ζk+d22ζkTR2ζk-d1∑j=k-d1k-1‍ζjTR1ζj-d2∑j=k-d2k-1‍ζjTR2ζj,
where φk=[ξkTξk-d1TKTξk-d2TKTδkTωkT]T.
Moreover, define(28)Js=E{∑k=0∞‍(ykTZyk-δkTZδk+ekTek-γ2ωkTωk)}.
Together with (25), and according to V(0)=0 and V(∞)≥0, we have
(29)Js≤Js+V(∞)-V(0)=E{∑k=0∞‍(ΔV(k)+ζkTZζk-δkTZδk+ekTek-γ2ωkTωk)∑k=0∞‍}=E{∑k=0∞‍φkT(Ξ1+Γ3T(d12R1+d22R2+Z)Γ3+Γ4TΓ4+Γ1TP(i+)Γ1+fΓ2TP(i+)Γ2)φk∑k=0∞‍},
It is obvious thatΔV(k)+ζkTZζk-δkTZδk+ekTek-γ2ωkTωk<0, if
(30)Ξ1+Γ3T(d12R1+d22R2+Z)Γ3+Γ4TΓ4+Γ1TP(i+)Γ1+fΓ2TP(i+)Γ2<0,
which, by Schur complement, is equivalent to (23). Therefore, defining Z=TTT, we have Js<0, if (23) holds, which means
(31)∥Tζk∥E22+∥ek∥E22≤∥Tδk∥E22+γ2∥ωk∥E22,
if Theorem 5 holds, which means the SSG of subsystem 𝒮1 satisfying γ(GT)<1. By Lemmas 3 and 4, the filtering error system (16) is input-output stable in mean-square. Furthermore, we can obtain ∥ek∥E22≤γ2∥ωk∥E22 from (31) together with (22). The proof is completed.Theorem5 guarantees the input-output stability in mean-square of system in (19). Through the following research, it can be proved that the system in (19) is stochastically stable by Lyapunov-Krasovskii functional method.Theorem 6.
For the forward system𝒮1 in (19) with ωk=0, if there exists an LKF V(k) and Z>0 satisfying ΔV(k)+ζkTZζk-δkTZδk<0, then the system in (19) is stochastically stable by the following LKF:
(32)Vc(k)=V(k)+1d12∑i=-d2-d1-1‍∑j=k+ik-1‍ζkTZζk.Proof.
DefiningΔVc(k)=Vc(k+1)-Vc(k); we have
(33)ΔVc(k)=ΔV(k)+1d12∑i=-d2-d1-1‍[ζkTZζk-ζk+iTZζk+i]=ΔV(k)+ζkTZζk-1d12∑i=k-d2k-d1-1‍[β(i)ζiTZβ(i)ζi],≤ΔV(k)+ζkTZζk-1d122[∑i=k-d2k-d1-1‍β(i)ζi]TZ[∑i=k-d2k-d1-1‍β(i)ζi]=ΔV(k)+ζkTZζk-δkTZδk<0,
which is implied by (23). Let λ>0, we have
(34)ΔVc(k)=E{Vc(k+1)}-Vc(k)≤-θxkTxk<0.
By summing up the inequality on both sides fromk=0,…,R, it can be obtained that
(35)E{Vc(R)}-Vc(0)≤-θ∑k=0R‍xkTxk<0.
WhenR→∞, we have
(36)θ∥xkTxk∥E22≤Vc(0)-E{Vc(∞)}≤Vc(0)<∞,
which implies that the systems in (19) are stochastically stable. The proof is completed.


## 3.1. Filtering Performance Analysis

Lemma 3 (see [24]).
One considers the following system consisting of two subsystems as shown in Figure2:
(12)𝒴1:ζk=G(uk-δk),𝒴2:δk=Δζk.
The closed loop system is input-output stable in mean-square for all the subsystemsG and Δ, if ∥Tζ∘G∘Tδ∥<1 holds for some nonsingular matrices Tζ and Tδ with ∥Tδ∘Δ∘Tζ∥≤1.Figure 2
The interconnection subsystems.Proof.
According to (12) and the condition in Lemma 3, by defining Δuk=uk-δk, and without loss of generality, it can be assumed that
(13)∥G(Δuk)∥E2≤k1∥Δuk∥E2,∥Δ(ζk)∥E2≤k2∥ζk∥E2,0<k1<1,0<k2≤1.
Obviously, we have(14)∥Tyk∥E2≤k1∥TΔuk∥E2=k1∥T(uk-Δ(yk))∥E2≤k1(∥Tuk∥E2+k2∥Tyk∥E2),∥Tyk∥E2≤k11-k1k2∥Tuk∥E2.
Sincek1k2<1, system (12) is input-output stable in mean-square, which completes the proof.Then, to separate the uncertaintiesτk from the filtering error system, the time-delay system 𝒮 in (8) is transformed into the interconnection of two subsystems introduced in Lemma 3.Notice that the time-varying delay can be written as(15)xk-τk=12xk-d1+12xk-d2+d122δk,
where (1/2)(xk-d1+xk-d2) is the approximation of the time-varying delay xk-τk, and (d12/2)δk is the approximation error. Then, we can describe system 𝒮 as
(16)𝒮:{ξk+1=A¯(i)ξk+12A¯d(i)Kξk-d1+12A¯d(i)Kξk-d2+d122A¯d(i)δk+B¯(i)ωk,ek=L¯(i)ξk.Definingζk=xk+1-xk and
(17)β(i)≜{1,wheni⩽k-τk-1,-1,wheni>k-τk-1,
we can obtain
(18)δk=2d12[xk-τk-12xk-d1-12xk-d2]=1d12[∑i=k-d2k-τk-1‍ζi-∑i=k-τkk-d1-1‍ζi]=1d12[∑i=k-d2k-d1-1‍β(i)ζi].From (18), we know that mapping Δ:ζk→δk includes the uncertainty in τk leaving only two constant delays d1 and d2 in (16). Then the system in (16) can be described as
(19)𝒮1:[ξk+1ζkek]=[Φ1d122A¯d(i)B¯(i)Φ2d122Ad(i)B¯(i)Φ300]︸G[ϕkδkωk],𝒮2:δk=Δζk,
where
(20)ϕk=[ξkTξk-d1TKTξk-d2TKT]T,Φ1=[A¯(i)12A¯d(i)12A¯d(i)],Φ2=[(A(i)-I)K12Ad(i)12Ad(i)],Φ3=[L¯00].Furthermore, the following lemma shows that the scaled small gain of the mappingΔ has an upper bound.Lemma 4.
γ
(
Δ
T
) has an upper bound:
(21)γ(ΔT)=sup∥Tδk∥E2∥Tζk∥E2≤1.Proof.
By using Jensen's inequality, it can be derived under zero initial condition according to the fact that|β(i)|=1,
(22)∥Tδk∥E22=E{∑j=0∞‍[δjTZδj]}≤2d122E{∑i=0∞‍[d12∑j=i-d2i-d1-1‍β2(j)δjTδj]}=1d12E{∑j=-d2-d1-1‍∑i=0∞‍δi+jTδi+j}≤1d12E{∑j=-d2-d1-1‍∑i=0∞‍δiTδi}≤E{∑j=0∞‍[ζjTZζj]}=∥Tζk∥E22,
where Z=TTT, which implies γ(ΔT)≤1. The proof is completed.Now, we are in the position to propose the sufficient conditions under which the filtering error system (8) is stochastically stable with a prescribed ℋ∞ performance attention level γ. First we assume the filter matrices AF, BF, and LF are fixed and the corresponding conditions are established in the following theorem.Theorem 5.
Consider system (1), and assume the filter matrices AF, BF, and LF are fixed. Given integers d2>d1≥0 and a constant γ>0, the filtering error system (16) is input-output stable in mean-square with an ℋ∞ performance attention level γ for any time-varying delay satisfying (2) if there exist matrices P>0, Qi>0(i=1,2), Ri>0(i=1,2), Z>0 satisfying
(23)Ψ≜[Ξ1[d1Γ3TR1d2Γ3TR2Γ3TZΓ4TΓ1TPfΓ2TP]*diag{-R1,-R2,-Z,-Ip,-P,-P}]<0,
where
(24)Ξ1=[Ξ11[KTR1KTR200]*diag{Ξ2,Ξ3,-Z,-γ2Iq}],Γ1=[A¯1(i)12A¯d1(i)12A¯d1(i)12d12A¯d1(i)B¯1(i)],Γ2=[A¯2(i)12A¯d2(i)12A¯d2(i)12d12A¯d2(i)B¯2(i)],Γ3=[(A(i)-I)K12Ad(i)12Ad(i)12d12Ad(i)B(i)],Γ4=[L¯(i)0p×(3n+q)],f=η¯(1-η¯),Ξ11=-P(i)+KT(Q1+Q2-R1-R2)K,Ξ2=-Q1-R1,Ξ3=-Q2-R2.Proof.
Choose the following Lyapunov-Krasovskii functional:(25)V(k)=V1(k)+V2(k)+V3(k),
with
(26)V1(k)=ξkTP(i)ξk,V2(k)=∑i=k-d1k-1‍ξiTKTQ1Kξi+∑i=k-d2k-1‍ξiTKTQ2Kξi,V3(k)=d1∑i=-d1-1‍∑j=k+ik-1‍ζjTR1ζj+d2∑i=-d2-1‍∑j=k+ik-1‍ζjTR2ζj,
where P>0 and Qi>0 are the matrices to be determined.
DefiningΔV(k)=E{V(k+1)}-V(k), we have
(27)ΔV1(k)=E{ξk+1TP(i+)ξk+1}-ξkP(i)ξk=φkT(Γ1TP(i+)Γ1+f2Γ2TP(i+)Γ2)φk-ξkP(i)ξk,ΔV2(k)=ξkTKTQ1Kξk-ξk-d1TKTQ1Kξk-d1+ξkTKTQ2Kξk-ξk-d2TKTQ2Kξk-d2,ΔV3(k)=d12ζkTR1ζk+d22ζkTR2ζk-d1∑j=k-d1k-1‍ζjTR1ζj-d2∑j=k-d2k-1‍ζjTR2ζj,
where φk=[ξkTξk-d1TKTξk-d2TKTδkTωkT]T.
Moreover, define(28)Js=E{∑k=0∞‍(ykTZyk-δkTZδk+ekTek-γ2ωkTωk)}.
Together with (25), and according to V(0)=0 and V(∞)≥0, we have
(29)Js≤Js+V(∞)-V(0)=E{∑k=0∞‍(ΔV(k)+ζkTZζk-δkTZδk+ekTek-γ2ωkTωk)∑k=0∞‍}=E{∑k=0∞‍φkT(Ξ1+Γ3T(d12R1+d22R2+Z)Γ3+Γ4TΓ4+Γ1TP(i+)Γ1+fΓ2TP(i+)Γ2)φk∑k=0∞‍},
It is obvious thatΔV(k)+ζkTZζk-δkTZδk+ekTek-γ2ωkTωk<0, if
(30)Ξ1+Γ3T(d12R1+d22R2+Z)Γ3+Γ4TΓ4+Γ1TP(i+)Γ1+fΓ2TP(i+)Γ2<0,
which, by Schur complement, is equivalent to (23). Therefore, defining Z=TTT, we have Js<0, if (23) holds, which means
(31)∥Tζk∥E22+∥ek∥E22≤∥Tδk∥E22+γ2∥ωk∥E22,
if Theorem 5 holds, which means the SSG of subsystem 𝒮1 satisfying γ(GT)<1. By Lemmas 3 and 4, the filtering error system (16) is input-output stable in mean-square. Furthermore, we can obtain ∥ek∥E22≤γ2∥ωk∥E22 from (31) together with (22). The proof is completed.Theorem5 guarantees the input-output stability in mean-square of system in (19). Through the following research, it can be proved that the system in (19) is stochastically stable by Lyapunov-Krasovskii functional method.Theorem 6.
For the forward system𝒮1 in (19) with ωk=0, if there exists an LKF V(k) and Z>0 satisfying ΔV(k)+ζkTZζk-δkTZδk<0, then the system in (19) is stochastically stable by the following LKF:
(32)Vc(k)=V(k)+1d12∑i=-d2-d1-1‍∑j=k+ik-1‍ζkTZζk.Proof.
DefiningΔVc(k)=Vc(k+1)-Vc(k); we have
(33)ΔVc(k)=ΔV(k)+1d12∑i=-d2-d1-1‍[ζkTZζk-ζk+iTZζk+i]=ΔV(k)+ζkTZζk-1d12∑i=k-d2k-d1-1‍[β(i)ζiTZβ(i)ζi],≤ΔV(k)+ζkTZζk-1d122[∑i=k-d2k-d1-1‍β(i)ζi]TZ[∑i=k-d2k-d1-1‍β(i)ζi]=ΔV(k)+ζkTZζk-δkTZδk<0,
which is implied by (23). Let λ>0, we have
(34)ΔVc(k)=E{Vc(k+1)}-Vc(k)≤-θxkTxk<0.
By summing up the inequality on both sides fromk=0,…,R, it can be obtained that
(35)E{Vc(R)}-Vc(0)≤-θ∑k=0R‍xkTxk<0.
WhenR→∞, we have
(36)θ∥xkTxk∥E22≤Vc(0)-E{Vc(∞)}≤Vc(0)<∞,
which implies that the systems in (19) are stochastically stable. The proof is completed.


## 4.ℋ∞ Filter Design

Theorem 7.
Consider the Markov jump system (1). For given integers d2>d1≥0 and a constant γ>0, if there exist matrices Ri, S, Q1>0, Q2>0, R1>0, R2>0, Z>0, A¯F, B¯F, and L¯F such that
(37)Π≜[Ξ~1[d12Γ~3TR1d22Γ~3TR2Γ~3TZΓ~4TΓ~1TfΓ~2T]*diag{-R1,-R2,-Z,-Ip,Θ,Θ}]<0,S-Ri<0,
where
(38)Ξ~1=[Ξ~11[KTR1KTR200]*diag{-Q1-R1,-Q2-R2,-Z,-γ2Iq}],Γ~1=[ΓATR+η¯ΓF1T+ΓF2TΓATS+η¯ΓF1T+ΓF2T]T,Γ~3=[A(i)012Ad(i)12Ad(i)12d12Ad(i)B(i)],Γ~2=[ΓF1TΓF1T]T,Γ~4=[L(i)-LF0p×(3n+q)],ΓA=[A(i)012Ad(i)12Ad(i)d122Ad(i)B(i)],ΓF1=[B¯FC(i)012B¯FCd(i)12B¯FCd(i)d122B¯FCd(i)B¯FD(i)],ΓF2=[0n×nA¯F0n×(3n+q)],Ξ~11=-P~(i)+KT(Q1+Q2-R1-R2)K,Θ=[R~-S-S-S],R~=-∑i=1s‍pijRi,
then, there exists an ℋ∞ filter of the type (3) such that the filtering error system in (8) is stochastically stable with a prescribed ℋ∞ performance level γ. Furthermore, the filter gains are given by
(39)[AFBFLF0]=[S-100I][A¯FB¯FL¯F0].Proof.
Suppose there exist matricesP(i)>0, Qi>0, Ri>0, (i=1,2), Z>0, AF, BF, and LF satisfying (23). Partition P as
(40)P(i)≜[P1iP2P2TP3],
and define matrices J and J1 as follows:
(41)J≜[I00P3-1P2T],J1≜diag{J,I,I,I,I,I,I,J,J}.
According to the construction ofJ and J1 and the fact that P2 and P3 are nonsingular even disturbed by a small perturbation, it is obvious that matrix J is invertible. Let
(42)Ri≜P1i,S≜P2P3-1P2T.
Then, multiply (23) by J1T from the left side and its transpose from the right side, respectively, and define
(43)[A¯FB¯FL¯F0]≜[P200I][AFBFLF0][P3-1P2T00I],
we obtain (37).
On the other hand, suppose that matricesRi>0, S>0, Q1>0, Q2>0, R1>0, R2>0, Z>0, A¯F, B¯F, and L¯F satisfy (37). Introduce P1i≜Ri and nonsingular square matrices P2 and P3 satisfying S=P2P3-1P2T. By Schur complement, P(i)>0 is equivalent to Ri>0 and Ri-S=P1i-P2P3-1P2T>0. Then, without loss of generality, setting P2-TP3=I yields (39). The proof is completed.


## 5. Numerical Example

In this section, we provide an example to demonstrate the effectiveness of the filter design method proposed in the preceding sections.Consider system (1) with the following parameters:
(44)A1=[-0.100.01-0.59-0.10.020.1-0.06-0.1],Ad1=[0.010.0200.0100.0100-0.01],B1=[-0.2020.3830.139]T,C1=[-0.02-0.01-0.020.050.020.021],Cd1=[-0.010-0.020.010.020],L1=[0.20.50.1],D1=[0.10.47]T,A2=[-0.10.10.04-0.51-0.210.010.1-0.04-0.1],Ad2=[0.010.0200.0100.01000.01],B2=[-0.1340.2480.117]T,C2=[0.01-0.040.0150.0310.0420.071],Cd2=[0.010-0.010.020-0.01],L2=[0.10.30.2],D2=[0.230.15]T,
and the mode switching is governed by a Markov chain with transition probability:
(45)p11=0.8,p12=0.2;p21=0.7,p22=0.3.Supposed1=0 and d2=1 and the initial condition x(0)=[000]T, xF(0)=[00]T. Then, by Theorem 7, a filter in the form of (3) is designed such that the system (8) is stochastically stable with a guaranteed ℋ∞ performance level γ.Suppose the packet loss parameter isη¯=0.8 as shown in Figure 3. By (37), it can be obtained that the minimum ℋ∞ performance γmin=0.2801, and
(46)S=[0.64170.24090.49630.24090.4799-0.21050.4963-0.21050.9024],R1=[3.54911.14361.42311.14361.1211-0.38081.4231-0.38082.9697],R2=[4.48481.23664.01991.23661.18420.08954.01990.08956.1286],Q11=[0.28610.05670.27080.05670.01320.04260.27080.04260.3200],Q12=[0.65480.10100.83360.10100.01910.11310.83360.11311.1341],Q21=[0.31410.06770.28250.06770.01640.05130.28250.05130.3075],Q22=[0.61090.09340.77520.09340.01740.10470.77520.10471.0582],R11=[0.36300.05160.21460.05160.01160.00780.21460.00780.2992],R12=[0.60510.08880.81730.08880.01960.10330.81730.10331.1513],R21=[0.13210.01830.07250.01830.00350.00490.07250.00490.0786],R22=[0.12880.02110.16790.02110.00580.02140.16790.02140.2356],Z=[0.16420.02330.22290.02330.00850.01930.22290.01930.3438],
and the filter matrices are as follows:
(47)A¯F=[-0.2262-0.0798-0.0365-0.06780.0340-0.0505-0.1747-0.13940.0068],B¯F=[0.0390-0.10990.2651-0.0906-0.1568-0.0028],L¯F=[-0.1391-0.48500.3233].Figure 3
Packet loss parameterη(k).In addition, by (39), we have
(48)AF=[-0.5488-0.05920.00270.20240.0525-0.11570.1555-0.1096-0.0210],BF=[-0.8717-0.44681.25240.15810.59790.2796],L¯F=[-0.1391-0.48500.3233].To illustrate the performance of the filter, it is assumed that the external disturbanceωk=2/(k+10). By calculation, the switching of Markov chain is shown in Figure 4. Figure 5 shows the system state xk and its estimation x^k and signal zk and its estimation z^k. Moreover, it can be obtained that the ∥e(k)∥E2=0.0166 and ∥ω(k)∥E2=0.3754 as shown in Figure 5, that yields γ=0.0626 which is below the minimum γmin=1.7051, demonstrating the effectiveness of the proposed filter design.Figure 4
Switching states of Markov chain.Original signal and estimation signal.
(a)
x
1 and estimation x^1
(b)
x
2 and estimation x^2
(c)
x
3 and estimation x^3
(d)
signalz1-z^1


## 6. Conclusion

This paper addresses theℋ∞ filtering problem for time-delay Markov jump systems subject to intermittent measurements. The communication link failure from the plant to the filter is modeled by a stochastic variable satisfying the Bernoulli random binary distribution. The system is transformed into an input-output form consisted of two interconnected subsystems. Based on the SSG Theorem developed to stochastic systems and the proposed Lyapunov-Krasovskii function, sufficient conditions under which the ℋ∞ filter can be achieved with the prescribed ℋ∞ performance index is established. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed filter design scheme.



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*Source: 321935-2013-10-23.xml*