# RobustH∞ Filtering for a Class of Uncertain Markovian Jump Systems with Time Delays

Authors: Yi Yang; Junwei Lu
Journal: Mathematical Problems in Engineering
 (2013)

---

## Abstract

This paper studies the problem of robustH∞ filtering for a class of uncertain time-delay systems with
Markovian jumping parameters. The system under consideration is subject to norm-bounded time-varying
parameter uncertainties. The problem to be addressed is the design of a Markovian jump filter such that
the filter error dynamics are stochastically stable and a prescribed bound on the ℒ2-induced gain from the
noise signals to the filter error is guaranteed for all admissible uncertainties. A sufficient condition for
the existence of the desired robust H∞ filter is given in terms of two sets of coupled algebraic Riccati
inequalities. When these algebraic Riccati inequalities are feasible, the expression of a desired H∞ filter is
also presented. Finally, an illustrative numerical example is provided.


---

## Body

## 1. Introduction

In the past decades, much attention has been focused on the celebrated Kalman filtering which seems to be one of the most popular estimation approaches; see, for example, [1]. Such a kind of filtering approach assumes that the system is subjected to stationary Gaussian noises with known statistics. However, in practical applications, the statistics of the noise sources may not be exactly known. To deal with this problem, an alternative approach named H∞ filtering was proposed, and a great number of results on this topic have been reported in the literature; see, for example, [2–5]. Note that in the H∞ filtering setting, the exogenous input signals are assumed to belong toℒ2[0,∞); furthermore, no exact statistics are required to be known. When parameter uncertainties appear not only in the exogenous input but also in the system model, the problem of robust H∞  filtering has been investigated and algebraic Riccati equation approach and linear matrix inequality (LMI) approach have been adopted to solve this problem; see [6–9] and the references therein.Systems with Markovian jumping parameters have been extensively studied due to their both practical and theoretical importance; many issues, such as filtering, stability analysis, and synthesis, on this kind of systems have been studied [10–13]. The problem ofH∞filtering for Markovian jumping systems has received much attention. Sufficient conditions for the solvability of this problem were presented in [14], and a design methodology was also proposed based on LMI approach. These results were further extended to uncertain Markovian jump linear systems in [15–18].In this paper, we consider the problem of robustH∞filtering for a class of continuous-time uncertain systems with Markovian jump parameters in all system matrices and time delays in the state variables. We consider uncertain systems with norm-bounded time-varying parameter uncertainties. The objective of this paper is to design a Markovian jump linear filter which guarantees both the stochastic stability and a prescribedH∞performance of the filtering error dynamics for all admissible uncertainties. An algebraic Riccati inequalities approach is developed to solve the previous problem and the desired H∞filter can be constructed by solving two sets of coupled algebraic Riccati inequalities.Notation. Throughout this paper, for symmetric matrices X and Y, the notation X≥Y (resp., X>Y) means that the matrix X-Y is positive semidefinite (resp., positive definite);Iis the identity matrix with appropriate dimension. The notation MT represents the transpose of the matrixM;ℰ{·}denotes the expectation operator with respect to some probability measure 𝒫; L2[0,∞)is the space of square-integrable vector functions over[0,∞); |·|refers to the Euclidean vector norm;∥·∥2stands for the usualL2[0,∞)norm, while∥·∥E2denotes the norm inL2((Ω,ℱ,𝒫),[0,∞));(Ω,ℱ,𝒫) is a probability space;λmin(M)is used to denote the minimum eigenvalue of the matrix M.Matrices, if not explicitly stated, are assumed to have compatible dimensions.


## 2. Problem Formulation

Fix a complete probability space(Ω,ℱ,𝒫) and consider the following class of uncertain stochastic linear systems with Markovian jumping parameters and time delay:
(1)Σ:{x˙(t)=[A(r(t))+ΔA(t,r(t))]x(t)+[Ad(r(t))+ΔAd(t,r(t))]×x(t-d)+B(r(t))ω(t),y(t)=[C(r(t))+ΔC(t,r(t))]×x(t)+D(r(t))ω(t),z(t)=L(r(t))x(t),x(t)=ϕ(t),∀t∈[-d,0],d>0,r(0)=r0,
where x(t)∈ℝnis the system state; y(t)∈ℝris the measurement;ω(t)∈ℝmis the noise signal which belongs toL2[0,∞);z(t)∈ℝlis a linear combination of state variables to be estimated;d>0is the time delay of the system; andϕ(t)∈ℝnis the continuous initial-value function. The parameter r(t)represents a continuous discrete-state Markov process taking values in a finite set𝒮={1,2,…,s}with transition probability matrix Π≜{πij}given by
(2)Pr{r(t+h)=j∣r(t)=i}={πijh+o(h),i≠j,1+πiih+o(h),i=j,
whereh>0,limh→∞(o(h)/h)=0, and πij≥0is the transition rate from modeiat time t to modejat timet+h and
(3)πii=-∑j=1,j≠isπij.
In  system Σ, A(r(t)),ΔA(t,r(t)),Ad(r(t)),ΔAd(t,r(t)),B(r(t)),C(r(t)),ΔC(t,r(t),D(r(t)),  and L(r(t)) are appropriately dimensioned real-valued matrix functions of r(t). For simplicity of notations, in the sequel, for each possible r(t)=i, i∈𝒮, we will denote the matrices associated with theith mode by
(4)Ai≜A(r(t)),ΔAi(t)≜ΔA(t,r(t)),Adi≜Ad(r(t)),ΔAdi(t)≜ΔAd(t,r(t)),Bi≜B(r(t)),Ci≜C(r(t)),ΔCi(t)≜ΔC(t,r(t)),Di≜D(r(t)),(5)Li≜L(r(t)),
where Ai, Adi, Bi, Ci,  Di, and Li, are known constant matrices representing the nominal system for each i∈𝒮 and ΔAi(t),  ΔAdi(t), and ΔCi(t) are unknown matrices representing time-varying parameter uncertainties and are assumed to be of the form
(6)ΔAi(t)=MAiFAi(t)NAi,ΔAdi(t)=MdiFdi(t)Ndi,ΔCi(t)=MCiFCi(t)NCi,
where MAi, NAi, Mdi, Ndi, MCi, and NCi, for each i∈𝒮, are known constant matrices and FAi(t), Fdi(t), and FCi(t), for each i∈𝒮, are the uncertain time-varying matrices satisfying
(7)FAi(t)TFAi(t)≤I,Fdi(t)TFdi(t)≤I,FCi(t)TFCi(t)≤I,∀i∈𝒮.
It is assumed that all the elements of FAi(t), Fdi(t), and FCi(t) are Lebesgue measurable. ΔAi(t), ΔAdi(t), and ΔCi(t), for each i∈𝒮, are said to be admissible if both (6) and (7) hold.Throughout the paper we will use the following concept of stochastic stability.Definition 1.
Consider the following stochastic jump system:(8)x˙(t)=A(r(t))x(t)+Ad(r(t))x(t-d),x(t)=ϕ(t),∀t∈[-d,0],d>0,r(0)=r0.
The system (8) is said to be stochastically stable, if, for finite ϕ(t)∈ℝn defined on [-d,0] and r0∈𝒮, there exists a scalar c>0such that
(9)limt→∞ℰ{∫0t‍xT(τ,ϕ,r0)x(τ,ϕ,r0)dτ∣ϕ,r0}≤csup-d≤τ≤0|ϕ(τ)|2,
where x(t,ϕ,r0) denotes the solution of system (8) at time t under the initial conditions ϕ(t) and r0.Now, for eachi∈𝒮, consider a Markovian filter of the form
(10)ΣF:{x^.(t)=A^ix^(t)+K^iy(t),z^(t)=L^ix^(t),
where x^(t)∈ℝn is the estimator state and the matrices A^i,  K^i, and L^i are to be chosen. The filtering error is defined by
(11)e(t)≜z(t)-z^(t).Then the robustH∞ filtering problem to be dealt with in this paper can be formulated as determining a filter of the form (10) such that, for all admissible uncertainties ΔAi(t),  ΔAdi(t), and ΔCi(t), i∈𝒮, the following requirements are satisfied:(R1)
the augmented system from systemΣ and the filter ΣF is stochastically stable;(R2)
with zero initial conditions, the following holds:(12)∥e(t)∥E2<γ∥ω(t)∥2,
for all nonzero ω(t)∈L2[0,∞), where γ>0 is a prescribed scalar.Before concluding this section, we introduce the following lemma which will be used in the proof of our main results in the next section.Lemma 2 (see [19, 20]).
LetA, D, E, F, and P be real matrices of appropriate dimensions with P>0 and F satisfying FTF≤I. Then one has the following:(1)
for any scalarϵ>0,
(13)DFE+(DFE)T≤ϵ-1DDT+ϵETE,(2)
for any scalarϵ>0such that P-ϵETE>0,
(14)(A+DFE)P-1(A+DFE)T≤A(P-ϵETE)-1AT+ϵ-1DDT,(3)
AD+(AD)T≤APAT+DTP-1D.


## 3. Main Results

In this section, a Riccati-like inequality approach is proposed to design a robustH∞ filter for uncertain stochastic linear systems with Markovian jumping parameters and time delay. Our main result is presented in the next theorem.Theorem 3.
Consider the uncertain jump linear systemΣ. If there exist scalars ε1i>0, ε2i>0, ε3i>0, and ε4i>0 and matrices P1i>0, P2i>0, i=1,…,s, Q1>0, and Q2>0, such that Q1-ε3iNdiTNdi>0, i=1,…,s, and the following coupled Riccati matrix inequalities hold:
(15)AiTP1i+P1iAi+∑j=1sπijP1j+P1iWiP1i+ε1iNAiTNAi+ε2iNCiTNCi+Q1<0,A^iTP2i+P2iA^i+∑j=1sπijP2j+P2iUiP2i+LiTLi+Q2<0,
where
(16)Wi=ε1i-1MAiMAiT+ε3i-1MdiMdiT+Adi(Q1-ε3iNdiTNdi)-1×AdiT+γ-2BiBiT,Ui=ε1i-1MAiMAiT+ε3i-1MdiMdiT+Adi(Q1-ε3iNdiTNdi)-1AdiT+γ-2Bi(I-DiTHi-1Di)BiT,A^i=Ai+[Adi(Q1-ε3iNdiTNdi)-1AdiTε1i-1MAiMAiT+ε3i-1MdiMdiT+Adi(Q1-ε3iNdiTNdi)-1AdiT]P1i+γ-2Bi(I-DiTHi-1Di)BiTP1i-BiDiTHi-1Ci,Hi=DiDiT+γ2ε2i-1MCiMCiT+ε4iI,
for i=1,…,s, then the robust H∞   filtering problem is solvable. In this case, a suitable robust H∞  filter ΣF is given in form (10) with parameters as follows:
(17)K^i=BiDiTHi-1,L^i=Li,i∈𝒮.Proof.
Let the mode at timet be i, that is; r(t)=i, i∈𝒮, and define
(18)x-(t)=x(t)-x^(t),
then from system Σ and filter ΣF, it is easy to show that
(19)x-.(t)=A^ix-(t)+[Ai+ΔAi(t)-A^i-K^i(Ci+ΔCi(t))]x(t)+[Adi+ΔAdi(t)]x(t-d)+(Bi-K^iDi)ω(t).
Therefore, by defining x~(t)T≜[x(t)T,x-(t)T]T, we obtain the augmented system from Σ and ΣF as
(20)Σaug:{x~.(t)=A~i(t)x~(t)+A~di(t)x~(t-d)+B~iω(t),e(t)=C~ix~(t),x~(t)=φ(t),∀t∈[-d,0],d>0,r(0)=r0,
where
(21)A~i(t)=A~i+ΔA~i(t),A~di(t)=A~di+ΔA~di(t),A~i=[Ai0Ai-A^i-K^iCiA^i],ΔA~i(t)=[ΔAi(t)0ΔAi(t)-K^iΔCi(t)0],A~di=[Adi0Adi0],ΔA~di(t)=[ΔAdi(t)0ΔAdi(t)0],B~i=[BiBi-K^iDi],C~i=[0Li].
For r(t)=i, i∈𝒮, we define a matrix Pi>0 by
(22)Pi=[P1i00P2i].
Next, we will show that under the conditions of the theorem, the following matrix inequality
(23)Zi(t)≔A~i(t)TPi+PiA~i(t)+∑j=1sπijPj+PiA~di(t)Q-1A~di(t)TPi+Q+C~iTC~i+γ-2PiB~iB~iTPi≤[V1i00V2i]
holds, when r(t)=i, i∈𝒮, where
(24)Q=[Q100Q2]>0,V1i=AiTP1i+P1iAi+∑j=1sπijP1j+P1iWiP1i+ε1iNAiTNAi+ε2iNCiTNCi+Q1,V2i=A^iTP2i+P2iA^i+∑j=1sπijP2j+P2iUiP2i+LiTLi+Q2.
To this end, we use Lemma 2 to obtain the following matrix inequalities:
(25)PiΔA~i(t)+ΔA~i(t)TPi≤ε1i-1Pi[MAiMAi][MAiTMAiT]Pi+ε1i[NAiT0][NAi0]+ε2i-1Pi[0-K^iMCi][0-MCiTK^iT]Pi+ε21i[NCiT0][NCi0]=[H1iε1i-1P1iMAiMAiTP2iε1i-1P2iMAiMAiTP1iH2i],PiA~di(t)Q-1A~di(t)TPi≤PiA~di(Q-ε3i[NdiT0][Ndi0])-1×A~diTPi+ε3i-1Pi[MdiMdi][MdiTMdiT]Pi=[G1iG2iTG2iG3i],
where (7) and
(26)Q1-ε3iNdiTNdi>0,i=1,…,s,
are used and
(27)H1i=ε1i-1P1iMAiMAiTP1i+ε1iNAiTNAi+ε2iNCiTNCi,H2i=P2i(ε1i-1MAiMAiT+ε2i-1K^iMCiMCiTK^iT)P2i,G1i=P1i[ε3i-1MdiMdiT+Adi(Q1-ε3iNdiTNdi)-1AdiT]P1i,G2i=P2i[Adi(Q1-ε3iNdiTNdi)-1AdiT+ε3i-1MdiMdiT]P1i,G3i=P2i[Adi(Q1-ε3iNdiTNdi)-1AdiT+ε3i-1MdiMdiT]P2i.
Hence
(28)Zi(t)≤A~iTPi+PiA~i+∑j=1sπijPj+Q+C~iTC~i+γ-2PiB~iB~iTPi+[H1i+G1iε1i-1P1iMAiMAiTP2i+G2iTε1i-1P2iMAiMAiTP1i+G2iH2i+G3i].
Substituting (16) and (17) into the right-hand side of the previous inequality and using algebraic manipulations we have that (23) holds.
Now, we will show the stochastic stability of the augmented systemΣaug when ω(t)=0. In the following we will simply use x~(t) to stand for the solution of the system Σaug at time t with the initial conditions φ(t) and r0.
Introduce the following stochastic Lyapunov functional candidate for systemΣaug:
(29)V(x~(t),r(t))=x~(t)TP(r(t))x~(t)+∫t-dt‍x~(τ)TQx~(τ)dτ.
It can then be shown that the weak infinitesimal generator 𝒜 of the random process {x~(t),r(t)} is given by
(30)𝒜V(x~(t),i)=limh⟶∞1h[ℰ{V(x~(t+h),r(t+h))∣x~(t),r(t)=iVV}-V(x~(t),r(t)=i)ℰV]=x~(t)T[∑j=1sπijPj+Q∑A~i(t)TPi+PiA~i(t)+∑j=1sπijPj+Q]×x~(t)+x~(t)TPiA~di(t)x~(t-d)+x~(t-d)TA~di(t)TPix~(t)-x~(t-d)Qx~(t-d),
for any r(t)=i,  i∈𝒮. Then, from Lemma 2, it follows that
(31)𝒜V(x~(t),i)≤x~(t)T[∑j=1sπijPjA~i(t)TPi+PiA~i(t)≤x~(t)T15+∑j=1sπijPj+PiA~di(t)Q-1A~di(t)TPi+Q∑j=1sπijPj]x~(t).
Using (23), we have
(32)𝒜V(x~(t),i)≤x~(t)T[V1i00V2i]x~(t)≤-β1x~(t)Tx~(t),
where β1=mini∈𝒮{λmin(-V1i),λmin(-V2i)}. From (15), it is easy to see that β1>0.
Now using Dynkin’s formula, we have, for eachr(t)=i, i∈𝒮, t>0,
(33)ℰ{V(x~(t),i)}-V(x~0,r0)=ℰ{∫0t‍𝒜V(x~(τ),r(τ))dτ}≤-β1∫0t‍ℰ{x~(τ)Tx~(τ)}dτ.
On the other hand, for each r(t)=i, i∈𝒮, we can show that
(34)ℰ{V(x~(t),i)}=ℰ{x~(t)TPix~(t)}+ℰ{∫t-dt‍x~(τ)TQx~(τ)dτ}≥β2ℰ{x~(t)Tx~(t)},
where β2=mini∈𝒮{λmin(Pi)} and β2>0. The previous inequality and (33) imply that
(35)ℰ{x~(t)Tx~(t)}≤-λ1∫0t‍ℰ{x~(τ)Tx~(τ)}dτ+λ2V(x~0,r0),
where λ1=β1β2-1>0 and λ2=β2-1>0. Then, it can be verified that
(36)ℰ{x~(t)Tx~(t)}≤λ2exp(-λ1t)V(x~0,r0).
Therefore,
(37)ℰ{∫0t‍x~(τ)Tx~(τ)dτ∣φ,r0}≤λ1-1λ2[1-exp(-λ1t)]V(x~0,r0).
Taking limit as t→∞, we have
(38)limt→∞ℰ{∫0t‍x~(τ)Tx~(τ)dτ∣φ,r0}≤λ1-1λ2V(x~0,r0).
Note that there always exists a scalar c>0 such that
(39)λ1-1λ2V(x~0,r0)≤csup-d≤τ≤0|φ(τ)|2.
Considering (38), we have that the augmented system Σaug is stochastically stable. Next we will show that
(40)∥e(t)∥E2<γ∥ω(t)∥2,
for all nonzero ω(t)∈L2[0,∞). To this end, we introduce
(41)J=ℰ{∫0∞‍[e(τ)Te(τ)-γ2ω(τ)Tω(τ)]dτ}.
For any r(t)=i, i∈𝒮, ω(t)≠0, we have
(42)𝒜V(x~(t),i)=x~(t)T[A~i(t)TPi+PiA~i(t)+∑j=1sπijPj+Q]×x~(t)+x~(t)TPiA~di(t)x~(t-d)+x~(t-d)TA~di(t)TPix~(t)-x~(t-d)TQx~(t-d)+x~(t)TPiB~iω(t)+ω(t)TB~iTPix~(t).
Thus, under zero initial condition, for any t>0, r(t)=i, i∈𝒮, ω(t)≠0,
(43)J1=ℰ{∫0t‍[e(τ)Te(τ)-γ2ω(τ)Tω(τ)]dτ}=ℰ{∫0t‍[𝒜V(x~(τ),i)e(τ)Te(τ)-γ2ω(τ)Tω(τ)+𝒜V(x~(τ),i)e(τ)Te(τ)]dτ∫0t‍}-ℰ{∫0t‍𝒜V(x~(τ),i)dτ}=ℰ{∫0t[B~iTx~(τ)TZi(τ)x~(τ)-(x~(t-d)T-x~(t)TPiA~di(t)Q-1)×Q(x~(t-d)-Q-1A~di(t)TPix~(t))-γ2(ω(τ)T-γ-2x~(t)TPiB~i)×(ω(τ)-γ-2B~iTPix~(t))]dτ∫0t[x~(τ)TZi(τ)x~(τ)∫}-ℰ{V(x~(t),i)}.
Hence
(44)J≤ℰ{∫0∞‍x~(τ)TZi(τ)x~(τ)dτ}.
Finally, from (23), (40) follows immediately. This completes the proof.Remark 4.
Theorem3 provides a sufficient condition for the solvability of robust H∞ filtering problem for uncertain stochastic time-delay systems with Markovian jumping parameters, and the desired filter can be constructed by solving two sets of coupled algebraic Riccati inequalities.In the case when𝒮={1}, that is, there is only one mode operation, we have πii=0,  i∈𝒮={1}, and system Σ reduces to the following uncertain time-delay system with no jumping parameters:
(45)Σ1:{x˙(t)=[A+ΔA(t)]x(t)+[Ad+ΔAd(t)]x(t-d)+Bω(t),y(t)=[C+ΔC(t)]x(t)+Dω(t),z(t)=Lx(t),x(t)=ϕ(t),∀t∈[-d,0],d>0,r(0)=r0,
where ΔA(t),  ΔAd(t), and ΔC(t) are unknown matrices and are assumed to be of the form
(46)ΔA(t)=MAFA(t)NA,ΔAd(t)=MdFd(t)Nd,ΔC(t)=MCFC(t)NC,
where MA, NAi, Md, Nd, MC, and NC are known constant matrices and FA(t), Fd(t), and FC(t) are the uncertain time-varying matrices satisfying
(47)FA(t)TFA(t)≤I,Fd(t)TFd(t)≤I,FC(t)TFC(t)≤I.
Then, from Theorem 3, we have the following robust H∞ filtering result for the previous system.Corollary 5.
Consider the uncertain time-delay system(Σ1). If there exist scalars εi>0, i=1,…,4, and matrices P1>0, P2>0, Q1>0, and Q2>0, such that Q1-ε3NdTNd>0, and the following coupled Riccati matrix inequalities hold:
(48)ATP1+P1A+P1WP1+ε1NATNA+ε2NCTNC+Q1<0A^TP2+P2A^+P2UP2+LTL+Q2<0,
where
(49)W=ε1-1MAMAT+ε3-1MdMdT+Ad(Q1-ε3NdTNd)-1AdT+γ-2BBT,U=ε1-1MAMAT+ε3-1MdMdT+Ad(Q1-ε3NdTNd)-1AdT+γ-2B(I-DTH-1D)BT,A^=A+[Ad(Q1-ε3NdTNd)-1AdTε1-1MAMAT+ε3-1MdMdT+Ad(Q1-ε3NdTNd)-1AdT]P1+γ-2B(I-DTH-1D)BTP1-BDTH-1C,H=DDT+γ2ε2-1MCMCT+ε4I,
then the robust H∞  filtering problem is solvable. In this case, a suitable robust H∞  filter is given by
(50)x^.(t)=A^x^(t)+K^y(t),z^(t)=L^x^(t),
where
(51)K^=BDTH-1,L^=L.Remark 6.
The solvability for robustH∞ filtering problem for uncertain continuous delay-free systems can be easily derived from Corollary 5, in this case. It can be shown that the result coincides with that proposed in [8].


## 4. Numerical Example

In this section, we will give a numerical example to demonstrate the applicability of the proposed approach.Consider the uncertain time-delay stochastic linear systems with Markovian jumping parameters in form (1) with two modes. For mode 1, the dynamics of the system are described as follows:
(52)A1=[-5200-4100-5],Ad1=[-100.200.510.500],B1=[1-2.3010.50],C1=[-102011],D1=[1001],MC1=[0.5100.5],NC1=[00.510-0.51],MA1=[10010.51],Md1=[00.50.510.50],NA1T=[-0.20.10],Nd1T=[-0.100.2],L1=[0.20000.3-0.1].
For mode 2, the dynamics of the system are described as follows:
(53)A2=[-610.5-1-7000.5-6],Ad2=[0100.5-1-0.1001],B2=[-10.5-3101],C2=[0.512-102],D2=[-10.50.21],MC2=[0.1-0.30.50],NC2=[-10.20.30.50.50],MA2=[0.50.300.50.21],Md2=[-0.60.5010.51],NA2T=[00.10],Nd2T=[0.200],L2=[-0.20.100.300.5].
Let the transition probability matrix be given by
(54)Π=[-0.50.51-1].
In this example, we set γ=2.8. By solving (15), we obtain
(55)P11=[2.15920.3579-0.05890.35793.20180.1337-0.05890.13374.0867],P12=[2.9083-0.0024-0.2047-0.00241.68550.0031-0.20470.00312.0020],Q1=[11.1343-0.4670-0.3949-0.467010.5734-0.6361-0.3949-0.636111.8166],P21=[0.21570.10940.02210.10940.33240.05240.02210.05240.1848],P22=[0.1572-0.01610.0122-0.01610.0989-0.01760.0122-0.01760.1414],Q2=[0.7630-0.0624-0.0257-0.06240.7896-0.0205-0.0257-0.02050.8370],ε11=12.1023,ε21=9.2498,ε31=12.1293,ε41=0.0032,ε12=12.1346,ε22=10.1634,ε32=12.1219,ε42=0.0012.
Therefore, using Theorem 3, a suitable Markovian robust H∞ filter can be constructed with parameters given as follows:
(56)A^1=[-3.55044.44960.33040.0239-3.83660.99410.29270.4860-4.7540],K^1=[0.9411-2.2211-0.18210.88650.2611-0.0911],L^1=[0.20000.3-0.1],A^2=[-5.96950.0301-1.2747-2.8680-9.2528-4.38900.78400.5184-7.3116],K^1=[0.9452-0.01622.7441-0.42870.18260.7602],L^2=[-0.20.100.300.5].


## 5. Conclusions

In this paper, we have studied the problem of robustH∞ filtering for a class of uncertain Markovian jump systems with time-delay and norm-bounded time-varying parameter uncertainties. A Markovian jump filter is designed which guarantees the stochastic stability of the filter error dynamics and a prescribed bound on the ℒ2-induced gain from the noise signals to the filter error irrespective of the parameter uncertainties. It has been shown that the desired robust H∞ filter can be constructed by solving two sets of coupled algebraic Riccati inequalities.



---

*Source: 574571-2013-08-18.xml*