# Finite-TimeH∞ Filtering for Singular Stochastic Markovian Jump Systems with Time-Varying Delays

Authors: Bin Yan; Xiaojia Zhou; Jun Cheng; Fangnian Lang
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

The issue of finite-timeH∞ filtering for singular stochastic Markovian jump systems with time-varying delays is concerned in
this paper. H∞ filtering is designed for underlying closed-loop singular Markovian jump system and system state does not exceed a given bound over some finite-time interval. Considering the full information of underlying Markov process, sufficient conditions are obtained to guarantee that the described system is finite-time stability and H∞ filtering finite-time boundedness. By establishing the results of stochastic character and finite-time boundedness, the closed-loop singular Markovian jump system trajectory stays within the given bound. At last, a numerical example is supplied to show the efficiency of the proposed method.


---

## Body

## 1. Introduction

As the fundamental problem in the field of control application and output measurement is used, theH∞ filtering problem has been attracting much attention [1, 2]. Among the existing robust control approaches, the traditional Kalman filtering is a well-known effective way to deal with the state estimation problems, which requires the exact knowledge of the statics of system model and external noise signal. The disturbances of stationary Gaussian noises with known statistics and system model are exactly known under the consideration. Over the past two decades, because of their applications in a variety of areas, the H∞ filtering problems have been widely studied in practical engineering. As mentioned above, H∞ filtering technique is employed in most applications to describe the input or output behavior of the controlled system. It is important to investigate H∞ filtering problem, and many results have been established [3–8].On the other hand, as an important kind of hybrid systems, Markovian jump systems have a strong practical background in physical systems, which are subjected to abrupt variations in their structures. Moreover, Markovian chain in a finite mode set always determines the transitions models, which can describe a set of linear Markovian jump systems. In the past two years, Markovian jump systems have received much attention because they continue being a hot research area of mathematical models to represent many physical systems. It should be noted that, the existence of abrupt changes on the modes of operation in many practical systems, and the class of systems known as Markov jump systems has been proved to be useful to model a great number of them. Recently, a large number of results on the stability, estimation, and control problems related to such systems have been reported [9–16].In many engineering systems, singular systems have the extensive applications in power systems, electrical circuits, and other fields [17, 18]. Singular systems are famous for its better performance in physical systems when compared with state-space ones. Recently, the H∞ filtering problems with theory of state-space ones have also found their wide application in the singular systems. Furthermore, as one special family of stochastic systems associated with time delays, the transition probabilities of Markovian jump systems are better to decide the singular system performance. Up to now, many results on H∞ filtering problem for singular Markovian jump systems with or without time delay have been reported [19–24], while there are some related issues that need to be solved.It should be pointed out that the classical filtering problems concern the asymptotic performance over the infinite-time interval. However, the transient behavior of systems is always considered in practical problems and traditional asymptotical stability is not applicable; the finite-time stability was proposed [25]. In the finite-time interval, finite-time stability is investigated to address these transient behaviors of control systems when compared with classical asymptotical stability. Recently, the concept of finite-time stability has been revisited in the light of linear matrix inequalities (LMIs) and theory of Lyapunov function; many results are reported subject to finite-time stability and finite-time boundedness [25–33]. It should be noted that there are still some related issues to be solved; to the best of our knowledge, the finite-time H∞ filtering problems for singular Markovian jump systems have not been fully studied. The topic remains interesting and challenging, which motivates the present study.This paper is concerned with the finite-time boundedness ofH∞ filtering for singular Markovian jump systems. Finite-time stability can be affected by switching behavior significantly, which is an independent concept from traditional Lyapunov stability; thus, it deserves our investigation. In this paper, we choose the appropriate Lyapunov-Krasovskii functional; making full information of the underlying Markov process, the sufficient conditions are derived to guarantee the finite-time boundedness of the systems. The finite-time boundedness (FTB) criteria can be tackled in the form of LMIs and optimization algorithms. At last, an example is given to illustrate the effectiveness of the developed method.


## 2. Preliminaries

A probability space(M,F,P) is given, where M, F, and P, respectively, represent the sample space, the algebra of events, and the probability measure defined on F. Let the random form process {rt,t≥0} be the Markov stochastic process taking values on a finite set N={1,2,…,N} with transition rate matrix Ω={μij},  i,j∈R. Define the following transition probability from mode i at time t to mode j at time t+Δt as(1)Pij=Prt+Δt=j∣rt=i=μijΔt+oΔt,ifi≠j,1+μijΔt+oΔt,ifi=j,where Pr{(r0=i)}>0, ∀i∈R, Δt>0, and limΔt→0(oΔt/Δt)→0 with transition probability rates μij≥0 for i,j∈R, i≠j, and ∑j=1N‍μj≠i=-μii.In this paper, we consider the singular systems as follows:(2)Ex˙t=Artxt+Aτrtxt-τt+Brtωt,yt=Crtxt+Drtωt,zt=Lrtxt,where x(t)∈Rn is the state vector of the system, y(t)∈Rl is the measured output, z(t)∈Rm is the controlled output, and E is a singular matrix with rank(E)=r<n. Art, Brt, Crt, Drt, and Lrt are known mode-dependent constant matrices with appropriate dimension. ω(t)∈L2q[0,∞] is the external disturbance vector which satisfies(3)∫0T‍ω⊺tωt<d,d≥0,and time-varying delay τ(t) is the continuous function satisfying(4)0≤τ0≤τt≤τM<∞,(5)0≤d1≤τ˙t≤d2.For system (2), we are interested in designing the filter described by(6)x˙t=Afrtxft+Afτrtxft-τt+Bfrtyt,zft=Cfrtxft,where xf(t)∈Rn is the filter state, zf(t)∈Rq is the estimation of z(t) in system (2), and the matrices Afr(t), Bfr(t), and Cfr(t) are the unknown filter parameters to be designed.Augmenting the model of (2) to include filter (6), we obtain the following filtering error system:(7)E¯η˙t=A¯rtηt+A¯τrtηt-τt+B¯rtωt,et=C¯rtηt,where the state estimate error η(t)=[x¯⊺t,xf⊺(t)]⊺ and the output error e(t)=zf(t)-z(t) and(8)E¯=E00I,A¯rt=Art0BfrtCrtAfrt,A¯τrt=Aτrt00Afτrt,B¯rt=BrtBfrtDrt,C¯rt=-Lrt,Cfrt.For brevity, in the sequel,Ar(t)=Ai, Afr(t)=Afi, and Afτr(t)=Afτi, for every r(t)∈N, and the other symbols are similarly denoted.Some definitions and lemmas should be introduced to facilitate the following discussion.Definition 1 (see [21]).
(i) The singular system (7) is said to be regular if rank(sE-Ai) is not identically zero for every r(t)=i∈R.
(ii) The singular system (7) is said to be impulse-free if deg(det(sEi-Ai))=rank(E) for every r(t)=i∈R.Definition 2 (see [21]).
The closed-loop continuous-time singular Markovian jump system (SMJS) (7) satisfying (3) is said to be singular stochastic finite-time boundedness (SSFTB) with respect to (c1,c2,T,Rr(t),d) with 0≤c1<c2 and R>0, if the stochastic system is regular and impulse-free in time interval [0,T] and(9)Ex⊺0E⊺REx0≤c1⟹Ex⊺tE⊺RExt<c2,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhih∀t∈0,T.Definition 3 (see [21]).
The closed-loop continuous-time singular Markovian jump system (SMJS) (7) satisfying (3) is said to be singular stochastic H∞ finite-time boundedness with respect to (c1,c2,T,R,γ,d), if the closed-loop continuous-time SMJS (7) is with respect to (c1,c2,T,R,d) and under the zero-initial condition the controlled output z(t) satisfies(10)E∫0T‍e⊺tetdt<γ2E∫0T‍ω⊺tωtdtfor any nonzero ω(t) which satisfies (3), where γ is a prescribed positive scalar.Definition 4 (see [16]).
ConsiderV(xt,rt,t>0) as the stochastic Lyapunov function of the resulting system (7); its weak infinitesimal operator is defined as(11)£Vet,rt=limΔt→01ΔtEVxt+Δt,rt+Δt∣xt=xt,rt=i-Vxt,ihhhhhhhhhhh-Vxt,i=∂∂tVxt,i+∂∂xVxt,ix˙t,i+∑j=1N‍μijVxt,j.Definition 5 (see [27]).
The jump rates of the visited modes from a given modei are assumed to satisfy (12)0<minμi≤μij≤maxμi,∀i,j∈R,i≠j,where minμi and maxμi are known parameters for a given mode i and represent the lower and upper bounds when all the jump rates are known; that is, 0<minμi=min{μij≠0,i≠j,j∈R} and minμi≤maxμi. Meanwhile, the number of the visited modes from a given mode i is denoted by Ni including the mode itself.Lemma 6 (see [4]).
For the positive matrices(13)X=X11X12X13∗X22X23∗∗X33≥0,the following integral inequality holds:(14)-∫t-τtt‍x˙s⊺E⊺X33Ex˙sds≤∫t-τtt‍xtxt-τtEx˙s⊺·X11X12X13∗X22X23∗∗0xtxt-τtEx˙sds.


## 3. Finite-TimeH∞ Performance Analysis

Theorem 7.
Consider the closed-loop continuous-time SMJS (7); there exist positive scalars c1 and c2 with 0≤c1<c2, for all admissible ω(t) subject to condition (3), if there exist symmetric positive definite matrices X=X=X11X12X13∗X22X23∗∗X33, Pi, i∈N, and Qji>0, j=1,2,3, such that, for all i∈R, the following linear matrix inequalities(15)PiE=E⊺Pi⊺≥0,(16)Ξi=Ξ11iPiA¯τi00PiB¯ieατ0τM-τ0AiKi∗Ξ22iΞ23iΞ24i0eατ0τM-τ0AτiKi∗∗Ξ33i000∗∗∗Ξ44i00∗∗∗∗-αQieατ0τM-τ0BiKi∗∗∗∗∗-Ki<0,(17)PiE=E⊺R1/2S1iR1/2E,Qji=R1/2TjiR1/2,j=1,2,3,Ki=R1/2UiR1/2,c1λ2+τ0ατ0λ3+τMατMλ4+λ5+12τM-τ0eατMλ6+αdλmaxQi1-e-αT<λ1c2e-αT,where(18)Ξ11i=PiA¯i+A¯i⊺Pi-1-Niminμi+αPiE+maxμi∑j=1,j≠iN‍PjE+Q1i+Q2i+Q3i,Ξ22i=-eατ0Q2i+eατMQ3i-2eατ0E⊺Ki-X33iE+eατ0τM-τ0X22i-X23iE-E⊺X23i⊺+eατ0τM-τ0X11i+X13iE+E⊺X13i⊺,Ξ23i=-eατ0E⊺Y1⊺E+eατ0τM-τ0X12i⊺+E⊺X13i⊺+X23iE,Ξ24i=-eατ0E⊺Y2⊺E+eατ0τM-τ0X12i⊺+E⊺X13i⊺+X23iE,Ξ33i=-eατ0Q1i-Q2i-eατ0E⊺Ki-X33i⊺E+eατ0τM-τ0X11i+X13iE+E⊺X13i⊺,Ξ44i=-eατMQ3i-eατ0E⊺Ki⊺E+eατ0τM-τ0X22i-X23iE-E⊺X23i⊺,λ1=maxi∈NλminR-1/2E⊺SiR-1/2,λ2=maxi∈NλmaxR-1/2E⊺SiR-1/2,λ3=maxi∈NλmaxR-1/2T1iR-1/2,λ4=maxi∈NλmaxR-1/2T2iR-1/2,λ5=maxi∈NλmaxR-1/2T3iR-1/2,λ6=maxi∈NλmaxR-1/2UiR-1/2hold; the closed-loop SMJS (7) is SSFTB with respect to (c1,c2,T,R,d).Proof.
At first, we will prove that the singular Markovian jump system (7) is regular and impulse-free.
Noting (16), we can have that(19)A¯iPi⊺+PiA¯i⊺+1-Niminμi-αEPi⊺+Wi<-maxμi∑j=1,j≠iN‍Pj-1E≤0.
Since0<rankEi=ri<n, there must exist two invertible matrices Mi and Ni such that(20)E~i=MiENi=Iri000.
We denote(21)MiA¯iNi=A¯11iA¯12iA¯21iA¯22i,MiPiNi-⊺=P¯11iP¯12iP¯21iP¯22i.
According to (15), it can be obtained that(22)P¯11i=P¯11i⊺≥0,P¯12i=0.
Pre- and postmultiplying (22) by M and M⊺, we can easily obtain A¯22iP¯22i⊺+P¯22iA¯22i⊺<0. Therefore, A¯22i is nonsingular. Otherwise, supposing A¯22i is singular, there exists a nonzero vector ςi∈Rn-ri which ensures that A¯22iςi=0. And then we can conclude that ςi⊺A¯22iP¯22i⊺+P¯22iA¯22i⊺ςi=0, and this contradicts A¯22iP¯22i⊺+P¯22iA¯22i⊺<0.
Then, it can be shown that(23)det⁡sEi-Ai=detMi-1det-A¯22i·detsIri-A¯11i-A¯12iA¯22i-1A¯21idetNi-1which implies that det(sEi-Ai) is not identically zero and deg(det(sEi-Ai))=rank(Ei)=ri. Therefore, by Definition 1, we can obtain that the closed-loop SMJS (7) is regular and impulse-free in time interval [0,T].
We consider the following Lyapunov-Krasovskii functional:(24)Vηt,rt=V1ηt,rt+V2ηt,rt+V3ηt,rt,(25)V1ηt,rt=η⊺tPrtEηt,V2ηt,rt=∫t-τ0t‍η⊺seαt-sQ1rtηsds+∫t-τtt-τ0‍η⊺seαt-sQ2rtηsds+∫t-τMt-τt‍η⊺seαt-sQ3rtηsds,V3ηt,rt=∫-τM-τ0‍∫t+θt‍η˙⊺sE⊺eαt-sKrtEη˙sdsdθ.
Taking the time derivative ofV(ηt,r(t)) along the trajectory of system (7), one has(26)£V1ηt,rt=η⊺tPiA¯i+Wiηt+2η⊺tPiB¯iωt+η⊺t∑j=1NμijPjEηt,£V2ηt,rt-α£V2ηt,rt=η⊺tQ1i+Q2i+Q3iηt-η⊺t-τ0eατ0Q1i-Q2iηt-τ0-η⊺t-τteατ0Q2i-eατMQ3i·ηt-τt-η⊺t-τMeατMQ3iηt-τM,£V3ηt,rt-α£V3ηt,rt=eατ0τM-τ0η˙⊺tE⊺KiEη˙t-∫t-τMt-τ0‍eαt-sη˙⊺sE⊺KiEη˙sds≤eατMτM-τ0η˙⊺tE⊺KiEη˙t-eατ0∫t-τMt-τt‍η˙⊺sE⊺Ki-X33iEη˙sds-eατ0∫t-τtt-τ0‍η˙⊺sE⊺Ki-X33iEη˙sds-eατ0∫t-τMt-τt‍η˙⊺sE⊺X33iEη˙sds-eατ0∫t-τtt-τ0‍η˙⊺sE⊺X33iEη˙sds.
It should be noted that(27)-eατ0∫t-τMt-τt‍η˙⊺sE⊺Ki-X33iEη˙sds-eατ0∫t-τtt-τ0‍η˙⊺sE⊺Ki-X33iEη˙sds=-ηt-τ0ηt-τt⊺·eατ0E⊺Ki-X33iEeατ0E⊺Y1iE∗eατ0E⊺Ki-X33iE·ηt-τ0ηt-τt-ηt-τtηt-τM⊺·eατ0E⊺Ki-X33iEeατ0E⊺Y2iE∗eατ0E⊺Ki-X33iE·ηt-τtηt-τM.
From Lemma6, one has(28)-eατ0∫t-τtt-τ0‍η˙⊺sE⊺X33iEη˙sds≤η⊺t-τ0eατ0τM-τ0X11i+X13iE+E⊺X13i⊺·ηt-τ0+2η⊺t-τ0eατ0·τM-τ0X12i+X13iE+E⊺X23i⊺ηt-τt+η⊺t-τteατ0τM-τ0X22i-X23iE-E⊺X23i⊺·ηt-τt,-eατ0∫t-τMt-τt‍η˙⊺sE⊺X33iEη˙sds≤η⊺t-τteατ0τM-τ0X11i+X13iE+E⊺X13i⊺·ηt-τt+2η⊺t-τteατ0·τM-τ0X12i+X13iE+E⊺X23i⊺ηt-τM+η⊺t-τMeατ0τM-τ0X22i-X23iE-E⊺X23i⊺·ηt-τM.
Moreover, we have(29)∑j=1N‍μijPjE=μiiPiE+∑j=1,j≠iN‍μijPjE=-∑j=1,j≠iN‍μijPiE+∑j=1,j≠iN‍μijPjE≤-Ni-1minμiPi+maxμi∑j=1,j≠iN‍PjE.
Then, from (24)–(29) and Schur complement, one can obtain(30)£Vηt,rt-αVηt,rt≤χ⊺tΞiχt+αω⊺tQiωt,where(31)χ⊺t=η⊺t,η⊺t-τt,η⊺t-τ0,η⊺t-τM,ω⊺t.
From condition (15), we have(32)£Vηt,rt-αVηt,rt<αω⊺tQiωt.
Multiplying (32) by e-αt, it yields(33)£e-αtVηt,rt<αe-αtω⊺tQiωt.
Integrating (33) from 0 to t, then we can obtain(34)e-αtEVηt,rt<Vx0,r0+α∫0t‍e-αsω⊺sQiωsds.
Noting thatα≥0, t∈[0,T]. Define PiE=E⊺R1/2S1iR1/2E, Qji=R1/2TjiR1/2, j=1,2,3, and Ki=R1/2UiR1/2; by Dynkin’s formula, we have(35)
E
V
η
t
,
r
t
<
e
α
t
V
x
0
,
r
0
+
α
e
α
t
∫
0
t
‍
e
-
α
s
ω
⊺
s
Q
i
ω
s
d
s
≤
e
α
T
c
1
λ
m
a
x
S
1
i
+
τ
0
α
τ
0
λ
m
a
x
T
1
i
∫
0
T
‍
e
-
α
s
d
s
h
h
h
h
h
h
h
h
h
+
τ
M
α
τ
M
λ
m
a
x
T
2
i
+
λ
m
a
x
T
3
i
h
h
h
h
h
h
h
h
h
∫
0
T
‍
e
-
α
s
d
s
+
1
2
τ
M
-
τ
0
e
α
τ
M
λ
m
a
x
U
i
h
h
h
h
h
h
+
α
d
λ
m
a
x
Q
i
∫
0
T
‍
e
-
α
s
d
s
≤
e
α
T
c
1
λ
2
+
τ
0
α
τ
0
λ
3
+
τ
M
α
τ
M
λ
4
+
λ
5
1
2
h
h
h
h
h
h
h
+
1
2
τ
M
-
τ
0
e
α
τ
M
λ
6
h
h
h
h
1
2
+
α
d
λ
m
a
x
Q
i
1
-
e
-
α
T
.
On the other hand, it follows from (24) that(36)EVηt,i=Eη⊺tPiEηt≥λminS1iEη⊺tE⊺RiEηt.
Then, one has(37)Eη⊺tE⊺RiEηt<eαT·c1λ2+τ0ατ0λ3+τMατMλ4+λ512τM-τ0eατMλ6+αdλmaxQi1-e-αThhhhhhhhhhh+12τM-τ0eατMλ6hhhhhhhhhhh12+αdλmaxQi1-e-αT12·λ1-1.
Therefore, from (37) and by Definition 2, we conclude that E{η⊺(t)E⊺RiEη(t)}<c2. This completes the proof.Theorem 8.
For a given constantT>0 and α>0, system (4) is robust finite-time stability with respect to (c1,c2,N,Rr,d), for all admissible ω(t) subject to condition (6), if there exist symmetric positive definite matrices Pi, Qji, j=1,2,3, 1≤i≤N, such that the following linear matrix inequalities hold:(38)Ξi=Ξ11iPiA¯τi00PiB¯ieατ0τM-τ0AiKiC¯i∗Ξ22iΞ22iΞ24i0eατ0τM-τ0AτiKi0∗∗Ξ33i0000∗∗∗Ξ44i000∗∗∗∗-αQieατ0τM-τ0BiKi0∗∗∗∗∗-Ki0∗∗∗∗∗∗-I<0.Proof.
We will show theH∞ performance of system (7); from Theorem 7, we have(39)£Vηt,rt-αVηt,rt+e⊺tet-γ2ω⊺tωt≤χ⊺tΩiχt.
Condition (38) yields(40)£Vηt,rt-αVηt,rt+e⊺tet-γ2ω⊺tωt<0.
Define(41)Jt=e⊺tet-γ2ω⊺tωt.
Multiplying (41) by e-αt, it follows that(42)£e-αtVηt,rt<e-αtJt.
According to condition (42), integrating this inequality during [0,T] yields(43)0≤e-αTEVηt,rt<E∫0T‍e-αtJtdt.Then we have(44)e-αT∫0T‍e⊺tetdt<∫0T‍e-αte⊺tetdt<γ2∫0T‍e-αtω⊺tωtdt<γ2∫0T‍ω⊺tωtdt.
Thus it is concluded by Definition3 that the closed-loop SMJS (7) is SSFTB with H∞ performance γ. The proof is completed.


## 4. Finite-TimeH∞ Filter Design

Theorem 9.
Consider the finite-time switched discrete-time system (2) and a given scalar γ>0. Then there exists a switched filter in the form of (6) such that the filtering error system (7) is finite-time boundedness with H∞ performance level γ, if there exist real matrixes Pi>0, Sij>0, P¯i>0, S¯ij>0, and Gi¯. A¯fi, B¯fi, and C¯fi  (i,j∈N) of appropriate dimensions and real matrices Gi  (i∈N), such that ∀i∈N and(45)Σ1iΣ2i∗Σ3i<0,where(46)Σ1i=Q1i+Q2i+Q3i-Gi-Gi⊺X1i0∗-1-Niminμi+αHiEX2i∗∗Ξ22i,Σ2i=000-LiCfiGi0Σ23iΣ24iX3i000,Σ3i=Ξ33i00000∗Ξ44i0000∗∗-γ2I000∗∗∗-Ki00∗∗∗∗-I0∗∗∗∗∗-Hi,with (47)X1i=G1iAi+ViCiWiG3iAi+ViCiWi,X2i=G1iAτiUiG3iAτiUi,X3i=G1iBi+ViDiG3iBi+ViDi,Gi=maxμiGi⊺,maxμiGi⊺,…,maxμiGi⊺︷N-1,Hi=diagEH1,EH2,…,EHi-1,EHi+1,…,EHN,hold; then the closed-loop SMJS (7) is SSFTB with respect to (c1,c2,T,R,d) with H∞ performance γ.
In this case, desiredH∞ filter parameter matrices are given by(48)Afi=G2i-1Wi,Aτfi=G2i-1Ui,Afi=G2i-1Vi,Cfi=Cfi.Proof.
The Lyapunov matricesPi can be parameterized as follows:(49)Pi=GiHi-1Gi,i=1,2,…,n,where Hi are symmetric positive definite mode-dependent matrices. We also parameterize Gi as(50)Gi=G1iG2iG3iG2i,in which G2i are nonsingular matrices. Note that Pi are 2n×2n matrices and G2i belong to Rn×n. Also we will assume μii≠0 which excludes the existence of a permanent mode in the system.
First, we have(51)∑j=1,j≠iN‍PjE=Gi⊺Hi-1Gi.
According to Schur’s complement, (37) can be rewritten as(52)ΩiG¯i⊺∗-Hi<0,where (53)G¯i=Gi,02nN-2n×nΩi-2n.
With matricesPi, inequalities (51) could be decomposed into(54)Σi⊺ΠjiΣi<0,where(55)Σi=Σ~i00I3n×3n,Σ~i=I2n0Hi-1Gi0,Πi=Π1i0GiA¯τi000GiB¯ieατ0τM-τ0A¯τiKiC¯i0Gi0∗Ξ22iΞ23iΞ24i0000∗∗Ξ33i00000∗∗∗Ξ44i0000∗∗∗∗-γ2Ieατ0τM-τ0BiKi00∗∗∗∗∗-Ki00∗∗∗∗∗∗-I0∗∗∗∗∗∗∗-Hi<0,Π1i=Q1i+Q2i+Q3iA¯i⊺Gi⊺∗-1-Niminμi+αHiE.
Since the equality(56)-GiNi0000Σi=0holds for every Σi, Σi⊺ΠjiΣi<0 is equivalent to(57)Π¯i=Πi+I2n00I2n00-GiHi0000+-Gi⊺0Hi000I2n000I2n0<0.
Also we have(58)GiA¯i=G1iAi+G2iBfiCiG2iAfiG3iAi+G2iBfiCiG2iAfi,GiA¯τi=G1iAτiG2iAfτiG3iAτiG2iAfτi,GiB¯i=G1iBi+G2iBfiDiG3iBi+G2iBfiDi.
TakingG2iAfi=Wi,  G2iBfi=Vi, and  G2iAfτi=Ui, we get (59)X1i=GiA¯i=G1iAi+ViCiWiG3iAi+ViCiWi,X2i=GiA¯τi=G1iAτiUiG3iAτiUi,X3i=GiB¯i=G1iBi+ViDiG3iBi+ViDi.
Therefore, we can get (44) and the proof is complete.


## 5. Illustrative Example

Consider a finite-time stabilization of switched system as follows:(60)A1=0.25970.01110.16990.5738,A2=-0.13320.32110.19040.6603,B1=-1.05991.19970.11830.1451,B2=1.84720.90600.7990-0.6362,C1=0.0540-0.4714-0.67400.1520,C2=0.2616-0.6257-0.16120.4696,D1=-0.59321.2077-0.4571-0.1404,D2=1.28551.75211.02550.7475,L1=0.49920.1807-1.4114-0.4540,L2=0.40281.43350.09170.2119,E=0.02500.0200.030.Initial states satisfyx(0)≤1; the filtering objective is to find feedback filter parameter ensuring that system (5) is finite-time boundedness with minimum value of γmin2+c2min. Choosing c1=4, d=0.01. According to Theorem 9, we can also obtain that feasible solution c2=97.08, α=0.420, and performance level is γ=1.54. Furthermore, Figures 1 and 2 are given to illustrate the optimal value with different value of α. By solving the matrix equalities in Theorem 9, we have the following filter parameters:(61)Af1=1.4870-1.9748-0.98981.7607,Af2=-0.92141.1072-0.4963-0.6799,Bf1=-1.42222.1525-2.15763.0650,Bf2=-0.74841.6955-2.0858-1.2667,Cf1=-0.0279-0.00980.06670.0214,Cf2=-0.0183-0.0649-0.0032-0.0068.Figure 1
The local optimal bound ofc1.Figure 2
The local optimal bound ofc2.Two classes of filters are involved in this example; filter (4) serves in short-time singular Markovian jump systems to guarantee the error state bounded in the prescribed boundary.


## 6.  Conclusion

In this paper,we have examined the problems ofH∞ filtering for short-time singular Markovian jump systems. Based on the analysis result, the static state feedback finite-time boundedness is given. Although the derived result is not in LMIs form, we can turn it into LMIs feasibility problem by fixing some parameters. At last, a numerical example is also given to illustrate the effectiveness of the proposed approach. Furthermore, the mode-dependent singular matrices Ert and uncertain matrices will be studied in the future for the recurrent singular Markovian jump system.



---

*Source: 567394-2015-09-16.xml*