# Finite-TimeH∞ Filtering for Discrete-Time Singular Markovian Jump Systems with Time Delay and Input Saturation

Authors: Wei Guan; Lei Fu; Yuechao Ma
Journal: Complexity
 (2019)

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## Abstract

The paper is discussed with the problem of finite-timeH∞ filtering for discrete-time singular Markovian jump systems (SMJSs). The systems under consideration consist of time-varying delay, actuator saturation and partly unknown transition probabilities. We pay attention to the design of a H∞ filtering which ensures the filtering error systems to be singular stochastic finite-time boundedness. By employing an adequate stochastic Lyapunov functional together with a class of linear matrix inequalities (LMIs), a sufficient condition is firstly established, which guarantees the systems to achieve our goal and satisfy a prescribed H∞ attenuation level in the given finite-time interval. Considering the above conditions, a distinct presentation for the requested H∞ filter is given. Finally, two numerical examples add to a dynamical Leontief model of economic systems are presented to illustrate the validity of the developed theoretical results.


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## Body

## 1. Introduction

Over the past decades, signal estimation has received remarkable attention in the field of control, as an elementary problem in signal processing. As is known to all, the traditional Kalman filtering [1] is the most popular ways to deal with the signal estimation. However, the celebrated Kalman filtering scheme is no longer shaped while a priori information on the external noises are not accurately known. Some optimal estimation approaches are introduced to overcome this problem. One of them H∞ filtering has a wider range of application value. Compared with the traditional kalman filtering, the main advantage of the H∞ filtering is without assuming the statistics of the noise signal. Therefore, the H∞ filtering problem has been investigated and widely introduced, see [2–8] as well as the references therein. In practice, to guarantee the error system to be asymptotically stable and satisfy a performance indicator is the main intention of H∞ filtering design. Moreover, for the purpose of applying to the practical applications, it is also necessary to be bounded for the state of the error system.As a special class of stochastic hybrid systems, Markovian jump systems have the advantage of better describing physical systems with sudden variations and put into use in manufacturing systems, communication systems [9–11], economic systems and networked control systems. In the past years, a mass of attention has been paid to the study of singular systems, which have been greatly attracted for the reason that singular systems have better than the state-space ones when to describe physical systems. However, many researchers of singular systems, often ignore the influence of Markovian jump, describing the actual systems with some limitations. With in-depth study, the researchers found that combined the singular systems with MJSs to form a new system, which not only can make up for the limitations of description, also can accurately describe more practical applications. This kind of combined singular systems and MJSs systems have been known as singular Markovian jump system (SMJSs) [12–14]. Meanwhile, many of the subsequent researches found that the actual environment mutation and singular system parameters or the internal structure change, all can use SMJSs description, so it is mostly important to study the SMJSs.On the other side, as a precondition, complete knowledge of the mode transition rates was asked to analyse and synthesis MJSs in lots of the studies. Therefore, it is necessary and significant to further study more universal jump systems together with partly unknown transition rates, rather than spending a lot of time to measure or estimate all the transitions rates. Many attractive problems have been solved and investigated for these systems, such as stability analysis,H∞ filtering [15–17] and stabilization [18, 19].It is well known that the Lyapunov stability describes the systems ranging on the dynamic behavior of the trajectories in infinite time space [20], which does not reflect the transient performance of the systems. To copie with this transient performance of control dynamics, short-time stability or finite-time stability were presented by some early researches in [21–24]. Varieties of significant results were provided for discrete-time or continuous-time by applying linear matrix inequality (LMIs) techniques or Lyapunov function approach, these field including singular systems [25, 26], network systems [27, 28], linear and nonlinear systems [29–33], switching systems [34–36], and so forth. In [37–41], the authors studied discrete finite-time stability, finite-time boundedness or finite-time H∞ control. Compared with the above referred, we can find that the definition of finite-time stability have some difference in [42]. When performing the controller, what is noteworthy is that the discrete-time MJSs could be more significant than continuous-time in many practical applications. However, as far as we know, the problem about discrete finite-time H∞ filtering for stochastic systems has less been researched, so that it promotes the main purpose of our study.To the best of our knowledge, time-delay phenomena are usually the main causes for performance deterioration and instability of systems and happened in many practical systems, such as neural networks, economics, chemical processes and mechanics [43, 44]. On the other hand, if we do not consider the effect of actuator saturation when designing the controller, and it can lead to poor performance and unstable of closed-loop system. Consequently, the study of control systems with actuator saturation is challenging not only theory and research value, but also in practice has important research significance and broad application prospects. Recently, Ma and Chen [45] presented singular T-S fuzzy time-varying delay systems with actuator saturation for the problem of memory dissipative control. In [46], the author talked about passive control problem for singular time-delay systems with nonlinearity and actuator saturation. In the paper, we investigates the finite-time H∞ filter design for discrete-time SMJSs with time-varying delay, partly unknown transition probabilities and actuator saturation. Our results are a little different from those previous ones which are in [47, 48] on finite-time H∞ filtering have been presented for singular Markovian jump systems. Our primary purpose is designing a H∞ filtering that can guarantee the filtering error discrete-time SMJSs to be stochastic finite-time stability and boundedness with a prescribed H∞ performance index in the supplied finite-time interval. Enough conditions are supplied for the solvability of the problem, which can be worked out by making use of LMIs techniques. What is more, we consider to design filters for the systems with partly or completely known transition probabilities. Contributions from this paper are as follows(i)
A class of more general systems are investigated, where we take time-varying delay, input saturation, outside disturbance, Markov switching phenomena into consideration at the same time.(ii)
The singular filtering is used in this article, which makes the filtering problem more comprehensive and general.(iii)
The Markovian jump system takes into account the fact that the transfer matrix is not completely known.(iv)
The finite-timeH∞ filtering is investigated in this paper.In this paper, the structure of the arrangement is as follows. In Section2 we present the problem statement as well as preliminaries. Discussed the problem on stochastic finite-time H∞ filtering for discrete-time SMJSs with time-varying delay, actuator saturation and partly known transition probabilities in Section 3. Moreover, the estimation of the largest domain of attraction is solved. Section 4 gives some examples to demonstrate the accuracy of the proposed approaches. Finally, we draw the conclusions and the direction of future research in Section 5.Notations. In this paper, the following notations that will be used: Rn,Rn×m denote the sets of n-dimensional space, n×m real matrices. For any symmetric matrix P, λmin(P) and λmax(P) denote the minimum and maximum eigenvalues of matrix P, respectively. ε. denotes the expectation operator in the case of some probability measure. We use T stands for matrix transposition or vector and ∗ stands for the transposed elements in the symmetric positions of a matrix.


## 2. Problem Statement and Preliminaries

Consider the following discrete-time singular Markovian jump systems (SMJSs) with time-varying delay and actuator saturation:(1)Exk+1=Arkxk+Adrkxk-dk+Brksatuk+Bwrkwkzk=Crkxk+Cdrkx(k-dk+Drkwkyk=Cyrkxk+Dyrkwkxk=ϕk,k=-d2,…,-1,0where x(k)∈Rn is the state variable, z(k)∈Rq is the signal to be estimated, u(k)∈Rp is the control input variable and sat:Rp→Rp is the standard saturation function defined as follows:(2)satuk=satu1k,satu2k,…,satupkTWhere without loss of generality,sat(ui(k))=signuikmin⁡1,ui(k). Here the notation of sat(.) is abused to denote the scalar values and the variable valued saturation functions. w(k)∈Rq is the disturbance input which belongs to(3)ε∑k=0∞wTkwk≤d2,d≥0y(k)∈Rl is the measured output variable, ϕ(k) is a given initial condition, A(rk),Ad(rk),B(rk),Bw(rk),C(rk), Cd(rk),D(rk) and Cy(rk) are real constant matrices with appropriate dimensions. d(k) denotes the time-varying delay satisfying 1≤d1≤d(k)≤d2, where d1 and d2 are known positive integers. The matrix E∈Rn×n may be singular and we assume that rank(E)=r≤n. The process rk,k≥0 is a discrete-time homogeneous Markovian chain taking values in a finite set w=1,2,…,s with mode transition probabilities(4)Prrk+1=j/rk=i=πijwhere πij≥0, ∀i,j∈w, and ∑j=1sπij=1 for rk=i,i∈w.In this paper, the transition probabilities of the Markovian chain are considered to be partially available, while some elements in matrixπ=πij∣i,j∈w are unknown. For notation clarity, ∀i∈w, we denote w=wki+wuki with(5)wki=j:πijisknownwuki=j:πijisunknownMoreover, ifwki≠ϕ, we denote wki as the total number of elements in wki and wki(m) as the mth element in wki, obviously, 1≤m≤wki, 1≤wki≤s.In order to estimatez(k), we are interested in designing a filter of the following structure:(6)Ex~k+1=Afrkx~k+Bfrkykz~k=Cfrkx~k+Dfrkykuk=Krkx~kwhere x~(k)∈Rn is the filter state variable and z~(k)∈Rq is the estimated signal, Af(rk), Bf(rk), Cf(rk), Df(rk) are appropriately dimensioned filter matrices to be determined.For notational simplicity, in the sequel, a matrixN(rk) will be denoted by Ni, for instance, A(rk) will be denoted by Ai, Bf(rk) by Bfi, and so on.Definition 1 (see [14]).
For a matrixH∈Rp×n denote the lth row of H as hl0, and define ħ(H) as:(7)ħH=xk∈Rn:hl0Txk≤1,l0=1,2,…,pLet P∈Rn×n is a positive-symmetric matrix and ETPE≥0,τ>0 be a scalar and denote (8)ℵETPE,τ=xk∈Rn:xTkETPExk≤τ
LetD be the set of p×p diagonal matrices whose diagonal elements are either 1 or 0. Suppose each element of D is labeled as Dl and denote Dl-=1-Dl, Clearly, if Dl∈D, then Dl-∈D.
We denotex(k,ϕ) for the initial value of the state trajectory in (1), which the initial value is x(k)=ϕ(k)∈Cn,d2[-d2,0], and the domain of attraction is followed as: (9)Γ≜ϕk,k=-d2,-d2+1,…,0:limk→∞⁡xk,ϕ=0
However, it is difficult to calculate accurately the domain of attraction in the actual production, so the estimates of the attraction domain is presented by the following set:(10)Bν=ϕk∈Cn,d2-d2,0:max-d2,0⁡ϕk≤νLemma 2 (see [46]).
LetKi,Hi∈Rp×n, then for any x(k)∈ħ(Hi),(11)satKixk=∑l=12pαlkDlKi+Dl-Hixkwhere αl for l=1,2,…,2p are some scalars which satisfy 0≤αl≤1, ∑l=12pαl=1Definex¯T(k)=xT(k)xT(k)-x~T(k), e(k)=z(k)-z~(k), from Lemma 2, for any x¯k∈ħHi-Hi, x~k∈ħHi, we have(12)satKix~k=satKi-Kix¯k=∑l=12pαlkDlKi-Ki+Dl-Hi-Hix¯kThen we can obtain the filtering error discrete-time SMJSs as follows:(13)E¯x¯k+1=∑l=12pαlkA¯ilx¯k+A¯dix¯k-dk+B¯wiwkek=C¯ix¯k+C¯dix¯k-dk+D¯iwkxk=ϕk,k=-d2,-d2+1,…,-1,0where (14)E¯=E00E,A¯il=Ail-BilAi-Afi-BfiCyiAfi,A¯di=Adi0Adi0,B¯wi=Bwi0Bwi-BfiDyi0,C¯i=Ci-DfiCyi-CfiCfi,C¯di=Cdi0,D¯i=Di-DfiDyi.Ail=Ai+BiDlKi+Dl-Hi,Bil=BiDlKi+Dl-Hi.Definition 3 (see [16]).
SystemEx(k+1)=A(rk)x(k) (or pair E,A(rk)) is said to be:
(1) regular if,det⁡(zE-A(rk)) is not identically zero for any rk=i,i∈w;
(2) causal if, degreedet⁡zE-Ark=rank(E) for any rk=i,i∈w.Definition 4 (see [30], singular stochastic finite-time boundedness (SSFTB)).
The discrete-time singular jumping system (20) is said to be SSFTB with respect to (θ,δ,Ri,N,d) with 0<θ<δ,Ri>0 if(15)εxT0ETRiEx0≤θ2⇒εxTkETRiExk<δ2,∀i∈w,k∈1,2,…,NRemark 5.
In many papers, the authors often use the definition of singular stochastic finite-time stability. However, it should be pointed out that SSFTB is more close to practical application and used widely, and SSFTS can be regarded as a particular case of SSFTB by lettingw(t)=0.Definition 6 (see [30], singular stochastic H∞ filtering finite-time boundedness (SSH∞FTB)).
The filtering error discrete-time MJS (20) is said to be SSH∞FTB with respect to (θ,δ,Ri,N,d,γ), where 0<θ<δ,Ri>0,γ>0 and N∈Zk≥0, and under the zero-initial condition the filtering error z(k) satisfies:(16)ε∑k=0NzTkzk<γ2ε∑k=0NwTkwk,∀k∈1,2,…,NLemma 7 (see [45]).
Given matricesX,Y,Z with appropriate dimensions and Y is symmetric positive definite then the following inequality holds: (17)-XTZ-ZTX-ZTYZ≤XTY-1XLemma 8 (see [22]).
Given matricesX,Y,Z with appropriate dimensions and Y is symmetric positive definite, then there exists scalar λ≥0, such that λI+Y>0 and(18)-XTZ-ZTX-ZTYZ≤XTλI+Y-1X+λZTZ


## 3. Main Results

This section provides stochastic finite-timeH∞ boundedness analysis for the filtering error discrete-time SMJSs (13) under the given filter gains (6). LMI conditions are established to show that the filtering error discrete-time SMJSs (13) is SSFTB and the filtering error e(k) and disturbance w(k) satisfies the constraint condition (16).Theorem 9.
LetKi be the state feedback controller gain matrix, the filtering error discrete-time SMJSs (13) is SSFTB with respect to θ,δ,N,Ri,d, if there exist a scalar μ≥1,ε1>0,ε2>0, set of symmetric positive-definite matrix Pi>0,Qi>0,i=1,2,3 and nonsingular matrices Gi,Si, such that(19)Π1=Π11′Π1200Π15G¯iTA¯ilTI∗Π22′00Π25A¯diTε1I∗∗-Q20000∗∗∗-Q3000∗∗∗∗Π55′B¯wiTε2I∗∗∗∗∗-P¯j-10∗∗∗∗∗∗-Sj<0,∀j∈wuki,l=1,2,…,2p(20)Π2=Π11′′Π1200Π15G¯iTA¯ilTρkiI∗Π22′′00Π25A¯diTρkiε1I∗∗-ρkiQ20000∗∗∗-ρkiQ3000∗∗∗∗Π55′′B¯wiTρkiε2I∗∗∗∗∗-Pki-10∗∗∗∗∗∗-Ski<0,∀j∈wki,l=1,2,…,2p(21)ℵE¯TP¯iE¯,γ2μ⊂ħH¯(22)θ2supi∈w⁡λmaxp^i+θ2d2+d2+d1-1d2-d12supi∈w⁡λmaxQ^1+θ2d1supi∈w⁡λmaxQ^2+θ2d2supi∈w⁡λmaxQ^3+d2supi∈w⁡λmaxQi≤δ2μ-Ninfi∈w⁡λminP^iwhere(23)Π11′=Π11+G¯iTA¯ilTRT+RA¯ilG¯i,Π12=ε1G¯iTA¯ilTRT+RA¯di,Π11′′=ρkiΠ11+G¯iTA¯ilTRT+RA¯ilG¯iΠ11=d12+1G¯iTQ1G¯i+G¯iTQ2G¯i+G¯iTQ3G¯i-μG¯iE¯P¯iE¯G¯i,Π22′=ε1A¯diTRT+ε1RA¯di-Q1Π22′′=ε1A¯diTRT+ε1RA¯di-ρkiQ1,Π15=ε2G¯iTA¯ilTRT+RB¯wi,Π25=ε2A¯diTRT+ε1RB¯wiΠ55′′=ε2B¯wiTRT+ε2RB¯wi-ρkiQi,Π55′=ε2B¯wiTRT+ε2RB¯wi-Qi,P^i=Ri-1/2P¯iRi-1/2Q^1=Ri-1/2Q1Ri-1/2,Q^2=Ri-1/2Q2Ri-1/2,Q^3=Ri-1/2Q3Ri-1/2,d12=d2-d1ρki=∑j∈wkiπij,Ski=∑j∈wkiπijSj,Pki=∑j∈wkiπijP¯jλmax(.),λmin(.) respectively stand for the maximum and minimum eigenvalues of matrices, R∈R2n×2n is any constant matrix satisfying: RE¯=0, rank(R)=2(n-r). Then system (13) is regular, causal and finite-time bounded inside ℵ(E¯TP¯iE¯,γ2μ) with respect to θ,δ,R,N,d. Moreover, the estimates of the attraction domain is given as the following form:(24)νmax=γ2μ1-N-supi∈w⁡λmaxQid2Υwhere(25)Υ=supi∈w⁡λmaxE¯TP¯iE¯+d2λmaxQ1+d2+d1-1d2-d12λmaxQ1+d1λmaxQ2+d2λmaxQ3Proof.
Here we consider the transition probability of the unknown situation, and use the same method we can get the transfer probability is known. Now, we prove that the system (13) is regular and causal within ℵ(E¯TP¯iE¯,γ2μ). When i∈w,j∈wuki, it follows from the condition (24), we can easily see that Π11′<0, then G¯i is nonsingular. Let T=diagG¯i,I,I,I,I,I,I, then TTΠ1T<0, which is equivalent to:(26)Π¯1=Π¯11′Π¯1200Π¯15G¯iTA¯ilTLiT∗Π22′00Π25A¯diTε1I∗∗-Q20000∗∗∗-Q3000∗∗∗∗Π55B¯wiTε2I∗∗∗∗∗-P¯j-10∗∗∗∗∗∗-Sj<0,l=1,2,…,2pwhere(27)Π¯11′=1+d12Q1+Q2+Q3-μE¯P¯iE¯+A¯ilTRTLi+LiTRA¯ilΠ¯12=ε1A¯ilTRT+LiTRA¯di,Π¯15=ε2A¯ilTRT+LiTRB¯wi,Li=Gi-1then for ∑l=12pαl(k)=1 the following is true:(28)∑l=12pαlkΠ¯1<0
Applying Schur complement lemma. Eq.(28) is equivalent to(29)Φ1=1+d12Q1+Q2+Q3-μE¯TP¯iE¯0000∗-Q1000∗∗-Q200∗∗∗-Q30∗∗∗∗-Qi+ηTkP¯jηk+ηTkRTτk+τTkRηk+τTkSj-1τk<0where(30)ηk=∑l=12pαlkA¯il∑l=12pαlkA¯di00∑l=12pαlkB¯wi,τk=Liε1I00ε2I
From Lemma7 and Sj>0, we know that(31)-ηTkRTSjRηk≤ηTkRTτk+τTkRηk+τTkSj-1τkconsidering (29) and (31), it is obtained that(32)Φ¯1=1+d12Q1+Q2+Q3-μE¯TP¯iE¯0000∗-Q1000∗∗-Q200∗∗∗-Q30∗∗∗∗-Qi+ηTkP¯jηk-ηTkRTSjRηk<Φ1<0using the similar method of the above, when i∈w,j∈wki, we can obtain:(33)Φ¯2=1+d12Q1+Q2+Q3-μE¯TP¯iE¯ρki0000∗-ρkiQ1000∗∗-ρkiQ200∗∗∗-ρkiQ30∗∗∗∗-ρkiQi+ηTkPkiηk-ηTkRTSkiRηk<0
Then we take the unknown transition probability for an example to prove the system is regular and causal.
From (32), we can obtain that(34)1+d12Q1+Q2+Q3-μE¯TP¯iE¯+∑l=12pαlkA¯ilTP¯j-RTSjR∑l=12pαlkA¯il<0since Qi>0,i=1,2,3 and P¯i>0, then we have(35)∑l=12pαlkA¯ilTP¯j-RTSjR∑l=12pαlkA¯il-μE¯TP¯iE¯<0since E¯ is singular and rank(E¯)=r≤n, there exist two nonsingular matrices M,N∈Rn×n, such that(36)ME¯N=Ir000,M∑l=12pαlkA¯ilN=A¯i1A¯i2A¯i3A¯i4M-TP¯iM-1=Pi1Pi2Pi3Pi4,RM-1=R1R2
Noting thatRE¯=0 and rank(R)=2(n-r), we can obtain that R1=0 and rank(R2)=2(n-r), R2∈R2n×2(n-r), that is RM-1=0R2(37)∗∗∗ϖi<0where ∗ represents matrices that are not relevant in the following discussion, and (38)ϖi=A¯i2TP¯j1A¯i2+A¯i4TP¯j3A¯i2+A¯i2TP¯j2A¯i4+A¯i4TP¯j4A¯i4+A¯i4TR2TSjR2A¯i4from Eq.(37), it is obtained that(39)ϖi<0
Now, we assume that the matrixA¯i4 is singular for one i∈w, then there exists a vector ςi∈Rn-r and ςi≠0 such that A¯i4ςi=0. Pre-multiplying and post-multiplying by ςT and ς result in ςiTA¯i2TP¯j1A¯i2ς<0 which contradicts P¯j1>0. Thus, A¯i4 is nonsingular.
On the other hand, for the case ofi∈w,j∈wki, by using the similar method of above, from (20), we also can obtain that A¯i4 is nonsingular. Therefore, by Definition 3, system (13) is regular and causal.
At this step we will show that system (13) is SSFTB with respect to θ,δ,N,Rrk.
Consider the following Lyapunon-Krasovskii functional:(40)Vx¯k,rk=∑i=13Vix¯k,rkwhere(41)V1x¯k,rk=x¯TkE¯TP¯rkE¯x¯kV2x¯k,rk=∑m=k-dkk-1x¯TmQ1x¯m+∑θ=-d2+1-d1∑m=k+θk-1x¯TmQ1x¯mV3x¯k,rk=∑m=k-d1k-1x¯TmQ2x¯m+∑m=k-d2k-1x¯TmQ3x¯mthen(42)εV1x¯k+1,rk+1-V1x¯k,rk=x¯Tk+1E¯T∑j∈wπijP¯jE¯x¯k+1-x¯TkE¯TP¯iE¯x¯k=x¯Tk+1E¯T∑j∈wkiπijP¯j+∑j∈wukiπijP¯jE¯x¯k+1-x¯TkE¯T∑j∈wkiπij+∑j∈wukiπijP¯iE¯x¯k=x¯Tk+1E¯TPki+∑j∈wukiπijP¯jE¯x¯k+1-x¯TkE¯Tρki+∑j∈wukiπijP¯iE¯x¯k(43)εV2x¯k+1,rk+1-V2x¯k,rk≤x¯TkQ1x¯k-x¯Tk-dkQ1x¯k-dk+d12x¯TkQ1x¯k(44)εV3x¯k+1,rk+1-V3x¯k,rk=x¯TkQ2x¯k+x¯TkQ3x¯k-x¯Tk-d1Q2x¯k-d1-x¯Tk-d2Q3x¯k-d2
FromRE¯=0, the following equation holds for any symmetric matrix Si with appropriate dimensions and rk=i.(45)0=-∑j∈wπijx¯Tk+1E¯TRTSjRE¯x¯k+1=-x¯Tk+1E¯TRT∑j∈wkiπij+∑j∈wukiπijSjRE¯x¯k+1=-x¯Tk+1E¯TRTSkiRE¯x¯k+1-∑j∈wukiπijx¯Tk+1E¯TRTSjRE¯x¯k+1from (42)-(45), we have(46)εVk+1-Vk≤ξTk-(E¯TP¯iE¯+1+d12Q1+Q2+Q3)ρki-ρkiQ1-ρkiQ2-ρkiQ30+ηTkPki-RTSkiRηkξk+∑j∈wukiπijξTk-(E¯TP¯iE¯+1+d12Q1+Q2+Q3)-Q1-Q2-Q30+ηTkP¯j-RTSjRηkξk=ξTk-(E¯TP¯iE¯+1+d12Q1+Q2+Q3)-Q1-Q2-Q30+ηTkP^i-RTS^iRηkξkwhere(47)ξk=x¯Tkx¯Tk-dkx¯Tk-d1x¯Tk-d2wTkTP^i=∑j∈wkiπijP¯j+∑j∈wukiπijP¯j=Pki+∑j∈wukiπijP¯j,S^i=∑j∈wkiπijSj+∑j∈wukiπijSj=Ski+∑j∈wukiπijSjfrom Eq.(32) and (33), we have(48)εVk+1-Vk<μ-1xTkE¯TP¯iE¯xk+wTkQiwk≤μ-1Vk+supi∈w⁡λmaxQiwTkwktherefore, we have(49)εVk+1≤μεVk+supi∈w⁡λmaxQiεwTkwknoting that μ≥1, it follows from Eq.(49) that(50)εVk≤μkεV0+supi∈w⁡λmaxQiε∑j=0k-1μk-j-1wTkwk≤μkεV0+supi∈w⁡λmaxQiμkd2
LettingP^i=Ri-1/2P¯iRi-1/2, Q^1=Ri-1/2Q1Ri-1/2, Q^2=Ri-1/2Q2Ri-1/2, Q^3=Ri-1/2Q3Ri-1/2 and noting that εxT0E¯TRiE¯x(0)≤θ2, we have(51)εV0=εxT0Ri1/2P^Ri1/2x0+ε∑m=-d0-1xTmRi1/2Q^1Ri1/2xm+ε∑θ=-d2+1-d1∑m=θ-1xTmRi1/2Q^1Ri1/2xm+ε∑m=-d1-1xTmRi1/2Q^2Ri1/2xm+ε∑m=-d2-1xTmRi1/2Q^3Ri1/2xm≤supi∈w⁡λmaxP^iθ2+d2supi∈w⁡λmaxQ^1θ2+d2+d1-1d2-d12supi∈w⁡λmaxQ^1θ2+d1supi∈w⁡λmaxQ^2θ2+d2supi∈w⁡λmaxQ^3θ2
On the other hand, for alli∈w, the following is true:(52)εVk≥εxTkE¯P¯iE¯xk=εxTkE¯Ri1/2P^iRi1/2E¯xk≥infi∈w⁡λminP^iεxTkE¯RiE¯xk(53)εxTkE¯RiE¯xk≤μkθ2supi∈w⁡λmaxP^i+μkθ2d2+d2+d1-1d2-d1/2supi∈w⁡λmaxQ^1infi∈w⁡λminP^i+d1μkθ2supi∈w⁡λmaxQ^2+d2μkθ2supi∈w⁡λmaxQ^3+μkd2supi∈w⁡λmaxQiinfi∈w⁡λminP^i
Noticing that condition (22), it follows from Eq. (53) that εxT(k)E¯TRiE¯xk<δ2. This completes the proof of this theorem.Moreover, according to (50), we can obtain(54)xTkE¯TP¯iE¯Txk≤Vk≤μkV0+μksupi∈w⁡λmaxQid2≤γ2μ(55)ϕk2≤γ2μ1-N-supi∈w⁡λmaxQid2Υ≜IϕRemark 10.
For allϕ∈Iϕ, we have Iϕ≤γ2μ and from (54), we also obtain that for all ϕ∈Iϕ, we have xT(k)E¯TP¯iE¯Txk≤γ2μ. This situation shows that began in the trajectory of Iϕ state of x(k) would have been in attracting field.Remark 11.
Though the problem ofH∞ filtering for discrete-time SMJS subject to time-varying delay was considered in [16]. Compared with it, the finite-time and actuator saturation have been taken into the above Theorem, which is more practical and can guarantee the large values of states could not be accepted in the presence of saturations.Base on Theorem9, the filtering error system (13) can be proved that SSFTB, following we will propose a H∞ finite-time sufficient condition for SMJS.Theorem 12.
The filtering error system (13) is SSH∞FTB with respect to θ,δ,N,γ,Rrk, if there exist scalars μ≥1,ε1>0,ε1>0,δ>0,γ>0, symmetric definite matrices P¯i>0,Q1,Q2,Q3>0, and nonsingular matrices G¯i,Si,such that(56)Π¯1=Π11′Π1200Π15G¯iTA¯ilTIG¯iTC¯iT∗Π22′00Π25A¯diTε1IC¯diT∗∗-Q200000∗∗∗-Q30000∗∗∗∗Π¯55′B¯wiTε2ID¯iT∗∗∗∗∗-P¯j-100∗∗∗∗∗∗-Sj0∗∗∗∗∗∗∗-I<0,∀j∈wuki(57)Π¯2=Π11′′Π1200Π15G¯iTA¯ilTWiρkiIρkiG¯iTC¯iT∗Π22′′00Π25A¯diTWiρkiε1IρkiC¯diT∗∗-ρkiQ200000∗∗∗-ρkiQ30000∗∗∗∗Π¯55′′B¯wiTWiρkiε2IρkiDiT∗∗∗∗∗-P¯-100∗∗∗∗∗∗-Ski0∗∗∗∗∗∗∗-ρkiI<0,∀j∈wki(58)θ2supi∈w⁡λmaxp^i+θ2d2+d2+d1-1d2-d12supi∈w⁡λmaxQ^1+θ2d1supi∈w⁡λmaxQ^2+θ2d2supi∈w⁡λmaxQ^3+μ-Nγ2d2≤δ2μ-Ninfi∈w⁡λminP^i(59)ℵE¯TP¯iE¯,γ2μ⊂ħH¯where(60)Π¯55′=-γ2μ-NI+ε2B¯wiTRT+ε2RB¯wi,Π¯55′′=-ρkiγ2μ-NI+ε2B¯wiTRT+ε2RB¯wi,Wi=πi1I,πi2I,…,πisI,P¯-1=diag⁡P¯1-1,P¯2-1,…,P¯s-1Proof.
Using Schur complement lemma, it follows from Eq.(56) that the following inequality holds:(61)Π^1=Π11′Π1200Π15G¯iTA¯ilTI∗Π22′00Π25A¯diTε1I∗∗-Q20000∗∗∗-Q3000∗∗∗∗Π¯55′B¯wiTε2I∗∗∗∗∗-P¯j-10∗∗∗∗∗∗-Sj<0,∀j∈wuki,l=1,2,…,2p
LetQi=μ-Nγ2I, then conditions (58) and (61) can guarantee that the filtering error SMJS(13) is SSFTB with respect to θ,δ,N,R(rk),d according to Theorem 9. Thus, we only need to prove that the constraint relation (16) holds under zero-value initial condition. Applying Schur complement to (56) we have(62)Π^1′=Π11′+G¯iTC¯iTC¯iG¯iΠ12+G¯iTC¯iTC¯di00Π15+G¯iTC¯iTD¯iG¯iTA¯ilTI∗Π22′+C¯diTC¯di00Π25+C¯diTD¯iA¯diTε1I∗∗-Q20000∗∗∗-Q3000∗∗∗∗Π¯55′+D¯iTD¯iB¯wiTε2I∗∗∗∗∗-P¯j-10∗∗∗∗∗∗-Sj<0,∀j∈wuki
Using the same Lyapunov-Krasovskii functional as Theorem9, it follows that:(63)εVk+1-μVk+eTkek-γ2μ-NwTkwk<0then we have (64)εVk+1≤μεVk+εeTkek-γ2μ-NεwTkwkaccording to Eq.(64), we can deduce(65)εVk≤μkεV0+∑j=0k-1μk-j-1εeTjej+γ2μ-Nε∑j=0k-1μk-j-1wTjwjunder the zero-value initial condition and noting that Vk≥0 for all k∈1,2,…,N we have(66)∑j=0k-1μk-j-1εeTjej<γ2μ-Nε∑j=0k-1μk-j-1wTjwj
Notice thatμ≥1, we have(67)ε∑k=0NeTkek=∑k=0NεeTkek≤∑k=0NεμN-keTkek<γ2μ-Nε∑k=0NμN-kwTkwk<γ2ε∑k=0NwTkwk
Therefore, condition (16) holds. In addition, taking into account (57) and using the similar method of the above, (16) can be worked out. This proof is completed.To solve Theorem12, letting P¯i=diagPi,Pi, Ri=diagR¯i,R¯i, the following theorem provides LMI conditions to ensure stochastic H∞ finite-time boundedness of the filtering errors SMJS (13).Theorem 13.
The filtering error discrete-time SMJS (13) with saturating actuators within ℵ(E¯TP¯iE¯,γ2μ) is stochastically H∞ finite-time bounded with respect to θ,δ,N,γ,Rrk,d, if there exist scalars μ≥1,γ>0,ε1>0,ε2>0,δe>0,ρ>0,η1>0,η2>0, a set of symmetric positive-definite matrices Pi>0,Q1,Q2,Q3>0, and sets of nonsingular matrices Gi,Si,Zi, sets of matrices Ui,Vi,Xi,Fi,Yi,Mi,Ti, for all i∈w, such that(68)CyiGi=ZiCyi(69)-1γ2μh¯l0Gi-h¯l0Gi∗EGi+GiTET-2I+Pi0∗∗EGi+GiTET-2I+Pi≤0,l0=1,2,…,p(70)Π~1=Ω1Ω2∗Ω3<0,∀j∈wuki,l=1,2,…,2p(71)Π~2=Ω′1ρkiΩ2∗Ω3<0,∀j∈wki,l=1,2,…,2p(72)R¯i<Pi<δeR¯i,η1R¯i<Q~1,Q~2,Q~3<η2R¯i(73)δe+d2+d1-1d2-d12+d1+2d2η2θ2+μ-Nγ2d2≤δ2μ-Nwhere(74)Ω1=Π~11′Π~1200Π~13Π14′IΠ15∗Π~22′00Π~23Π24′ε1IΠ25∗∗-Q~200000∗∗∗-Q~30000∗∗∗∗Π~55′Π56′ε2IΠ57∗∗∗∗∗Π^1′00∗∗∗∗∗∗Π^20∗∗∗∗∗∗∗Π^3,Ω2=Π16Π17Π18Π19Π2000000000000000000000000000000000000Ω′1=Π~11′′Π~1200Π~13Π14′′ρkiIρkiΠ15∗Π~22′′00Π~23Π24′′ρkiε1IρkiΠ25∗∗-ρkiQ~200000∗∗∗-ρkiQ~30000∗∗∗∗Π~55′′Π56′′ρkiε2IρkiΠ57∗∗∗∗∗Π^1′′00∗∗∗∗∗∗Π^20∗∗∗∗∗∗∗Π^3,Ω3=Π^40000∗Π^5000∗∗Π^600∗∗∗Π^70∗∗∗∗Π^8Π~11′′=Λ1+ρkiμGiTET+EGiΛ2∗Λ3+μρkiGiTET+EGi,Π~12=ε1Λ4+R1+R2Adiε1Λ5Λ6Λ7Π~11′=Λ1+μGiTET+EGiΛ2∗Λ3+μGiTET+EGi,Π~13=Λ8ε2Λ5Λ10Λ11,Pi14′′=Π14′WiΠ14′=2Λ¯1T+Λ¯2T-XiT-FiCyiT2Λ¯1T+Λ¯2T-XiT-FiCyiTXiT-Λ¯2TXiT-Λ¯2T,Π57=DiT-DyiTDfiT0Π15=Λ12-YiT,Π16=1+d12GiT001+d12GiT,Q~2=Q21Q22∗Q23Π17=Π18=GiT00GiT,Π19=μρGiTETμI00,Π^7=Π^8=-μρI00-μρI+PiΠ~22′′=Λ13-ρkiQ11Λ14-ρkiQ12∗-ρkiQ13,Π~22′=Λ13-Q11Λ14-Q12∗-Q13,Π24′′=Π24′WiΠ~23=Λ15ε2AdiTR3T+R4Tε1R3Bwi+R4Bwi-R4Ni0,Π24′=AdiTAdiT00Π56′=BwiTBwiT-NiT00,Π56′′=Π56′Wi,Π25=CdiT0,Π20=00μρGiTETμIQ~3=Q31Q32∗Q33,Q~1=Q11Q12∗Q13,Π~55′′=Λ16-ρkiγ2μ-NIε2BwiTR3T+R4T-ε2NiTR4T∗-ρkiγ2μ-NIΠ^1′=-2I+Pj00-2I+Pj,Π^2=-Ski00-Ski,Π^3=-ρkiI0,Π^4=-2I+Q11Q12∗-2I+Q13Π^5=-2I+Q21Q22∗-2I+Q23,Π^6=-2I+Q31Q32∗-2I+Q33,Π^1′′=-2I+P00-2I+PΛ1=symR1Λ¯1+R1Λ¯2-R2FiCyi+R2Λ¯1-R2Xi,Λ¯1=AiGiΛ2=Λ¯1T+Λ¯2TR3T-R4FiCyiT+R2Xi-R1Λ¯2+Λ¯1TR4T-XiTR4TΛ¯2=BiDlUi+Dl-Vi,Λ3=XiTR4T+R4Xi-R3Λ¯2-R3Λ¯2TΛ4=Λ¯1T+Λ¯2TR1T-FiCyiTR2T+Λ¯1TR2T-XiTR2TΛ5=Λ¯1T+Λ¯2TR3T-FiCyiTR4T+Λ¯1TR4T-XiTR4TΛ6=ε1XiTR2T+R3+R4Adi-R1Λ¯2TΛ7=ε1XiTR4T-R3Λ¯2T,Λ8=ε2Λ4+R1+R2Bwi-R2NiΛ10=ε2XiTR2T+R3+R4Bwi-R1Λ¯2T-R4NiΛ11=ε2XiTR4T-R3Λ¯2T,Λ12=CiGiT-MiCyiT-YiTΛ13=ε1symR1+R2Adi,Λ16=symε2R1Bwi+ε2R2Bwi-ε2R2N2Λ14=ε1AdiTR3T+R4T,Λ15=ε2AdiTR1T+R2T+ε1R1+R2Bwi-ε1RiNiIn this case, the desired filtering parameters and filtering controller gain can be chosen by:(75)Ki=UiGi-1,Hi=ViGi-1,Afi=XiGi-1,Bfi=FiZi-1,Cfi=YiGi-1,Dfi=MiTi-1Proof.
We firstly verify that conditions (68) and (70) are equivalent to condition (56). Letting(76)P¯i=Pi00Pi,Ri=R¯i00R¯i,G¯i=Gi00Gi,R=R1R2R3R4,Qi=Qi1Qi2∗Qi3and R satisfies RE¯=0.
From Lemma8, we know that:(77)-μGiTETPiEGi≤μGiTET+μEGi+μρI+Pi-1+μρGiTETEGi(78)-μρkiGiTETPiEGi≤μρkiGiTET+μρkiEGi+μρkiρI+Pi-1+μρρkiGiTETEGinoting that the fact,(79)-Pi-1≤-2I+Pi,Qi-1≤-2I+Qi,i=1,2,3let(80)Ui=KiGi,Vi=HiGi,Xi=AfiGi,Yi=CfiGi,BfiCyiGi=FiCyi,Fi=BfiZi,DfiCyiGi=MiCyi,Mi=DfiZi,Ni=BfiDyi
Applying (77), (79), (80) and by Schur complement, it is easily obtained that Eq.(68) and (70) is a sufficient condition for Eq.(56).
On the other hand, we denoteP~i=R¯-1/2PiR¯-1/2, Q~i=R¯-1/2QiR¯-1/2,i=1,2,3. So condition (58) is equivalent to(81)θ2supi∈w⁡λmaxp~i+d2+d2+d1-1d2-d12supi∈w⁡λmaxQ~1+d1supi∈w⁡λmaxQ~2d2supi∈w⁡λmaxQ~3+μ-Nγ2d2≤δ2μ-Ninfi∈w⁡λminP~i
LettingR¯i<Pi<δeR¯i, η1R¯i<Q1, Q2,Q3<η2R¯i, it is easy to check that condition (58) can guarantee that condition (81) holds. When j∈wki, we consider (78), (79), (80) and use the same method as above. Here, the proof is omitted.Remark 14.
The condition (68) may be difficult to solve using the LMI toolbox of Matlab. To overcome this, we can replace the condition by the following one that may approximate this constraint:(82)CyiGi-ZiCyiTCyiGi-ZiCyi<αIwhere α is a given sufficiently small positive scalar. Applying Schur complement, the constrained condition (82) is equivalent to the following LMI:(83)-αI∗CyiGi-ZiCyi-I<0Remark 15.
Using the same method in [46], condition ℵE¯TP¯iE¯,γ2μ⊂ħHi is equivalent to:(84)-1γ2μh¯j-h¯jG¯i∗-E¯TP¯iE¯≤0h¯j is the j-th row of Hi, pre-multiplying and post-multiplying inequality (84) by diag⁡1,G¯T and diag⁡1,G¯ respectively and then from Lemma 8 it is obtained:(85)-1γ2μh¯jGi-h¯jGi∗EGi+GiTET-2I+Pi0∗∗EGi+GiTET-2I+Pi≤0Then the condition ℵE¯TP¯iE¯,γ2μ⊂ħHi holds.Remark 16.
According to the above discussion, we can obtain that the feasibility of conditions stated in Theorem13 can be turned into the following LMI based on feasibility problem with a parameter μ, respectively:(86)
m
i
n
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2
+
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2
U
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V
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X
i
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Y
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F
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M
i
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N
i
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e
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i
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i
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G
i
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η
1
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η
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t
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L
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69
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70
.
71
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72
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73
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83
a
n
d
85Remark 17.
For all the ellipsoid invariant set of Theorem9, we hope to get the biggest estimation of attraction domain. According to (86), the problem can be solved by the following method:(87)minκs.t.λ1I-E¯TPiE¯≥0,λ2I-Q1≥0,λ3I-Q2≥0,λ4I-Q3≥0where κ=λ1+d2+(d2+d1-1)(d2-d1)/2λ2+d1λ3+d2λ4, then the biggest estimation of attraction domain as following:(88)νmax=γ2μ1-N-γ2μ-Nd2κmin


## 4. Numerical Examples

Two numerical examples and a dynamical Leontief model of economic systems are shown to demonstrate the usefulness and flexibility of our theory in this section. The first example adopted to account for conservatism of the conditions shown less than [2]. The second example which is employed to apply the filter we designed is feasible and effective. Finally, as the third example, we use the dynamical Leontief model of economic systems to elucidatory the practical application in our life.Example 1.
In order to show the advantage of the method proposed, consider the systems shown in Example 1 of [2].
Mode 1 (89) A 1 = - 2 0 1.5 1 , B 1 = 1 1 0 1 , B w 1 = 1 0 0 1 , C 1 = 1 1 0.5 1 , D 1 = 0.2 0.1 0 0.1 , C y 1 = 1 0 0.8 1 .
Mode 2 (90) A 1 = - 2 0 1.5 1 , B 2 = 1 0 1 1 , B w 2 = 1 1 0 1 , C 2 = 0.5 1 0.8 1 , D 2 = 0.1 0.2 0 0.2 , C y 2 = 1.2 0 0.5 1 . R 1 = R 2 = I 2 , θ = 1 , N = 5 , α = 1 0 - 10 , μ = 2.5 , ρ = 1.6 , ε 1 = 0.05 , ε 2 = 0.05
In addition, the transition probability matrix is given by(91)Pr=0.70.30.40.6.
Using the method in [2], the optimal values γ=54.8648, while using the method in this paper, the optimal values γ=44.2536. We can obtain that the method in this paper is better than it in [2]. The corresponding filter gains aregiven as follows: (92)Af1=-25.8409-1.0119-1.0856-20.1545,Af2=-23.2328-2.5570-2.5630-24.0083,Bf1=-47.2815-1.9114-9.4202-39.5987Bf2=-39.5178-5.5231-2.3040-45.5163,Cf1=-37.3258-1.4616-1.5681-29.1121,Cf2=-32.5259-3.5798-3.5882-33.6117Df1=-30.5939-1.2368-6.0954-25.6227,Df2=-26.3452-3.6820-1.5360-30.3442K1=1.75050.5009-0.7510-0.5002,K2=-0.5719-0.47840.1980-0.0094
Then, we consider the systems shown in Example 2 of [2]. Without input feedback, we consider the two-mode MJS with the same matrices and transition matrix given in Example 1. The optimization result of this paper is γ=115.2842, which is better than the value of γ=209.4661 of [2]. The corresponding filter gains aregiven as follows:(93)Af1=-0.82870.14550.1641-0.6967,Af2=-0.8435-0.1198-0.1159-0.8002Bf1=-0.16670.26850.2379-0.4011,Bf2=-0.41800.36110.3554-0.3213Cf1=-52.4816-2.9693-3.1335-47.9769,Cf2=-47.4428-6.0609-6.0976-47.0605Remark 18.
The problem of finite-timeH∞ filtering for discrete-time Markovian jump systems were investigated in [2]. However, the time-delay and actuator saturation have not been considered, which have a great impact on the performance and stability of the system. Thus, we consider the singular Markovian jump systems with actuator saturation, time-varying delay and partly unknown transition probabilities. Moreover, the examples above have presented that our methods is feasible and less conservative than [2].Example 2.
Consider a three-mode SMJSs (1) with the following parameters:
Mode 1 (94) A 1 = 1 0.2 1.1 2 , B 1 = 0.1 0.2 , A d 1 = 0.02 - 0.09 0.02 - 0.02 , B w 1 = 0.02 0 - 0.04 0.1 , C 1 = - 0.12 - 0.2 1 0 C d 1 = 0.04 0.5 0.1 0.2 , D 1 = 0.3 1 1 0.1 , C y 1 = 1 0.3 0 0.2 , D y 1 = 0.01 0 0 0.01
Mode 2 (95) A 2 = - 0.4 1 - 2 4 , A d 2 = 0.01 0 - 0.02 - 0.01 , B 2 = 0.3 0.2 , B w 2 = 0.1 0.02 0.1 0.3 , C 2 = - 0.02 - 0.2 1 1 C d 2 = 0.3 0.1 0.3 0.2 , D 2 = 0.1 0 0 0.2 , C y 2 = 1 3 0 1 , D y 2 = - 0.01 0 0 - 0.01
Mode 3 (96) A 3 = - 0.3 1.5 - 3 4 , A d 3 = 0.02 0 - 0.03 - 0.01 , B 3 = 0.3 0.2 , B w 3 = 0.1 0.02 0.1 0.3 , C 3 = - 0.02 - 0.2 1 1 C d 3 = 0.3 0.1 0.3 0.2 , D 3 = 0.1 0 0 0.2 , C y 3 = 1 3 0 1 , D y 3 = - 0.01 0 0 - 0.01
In addition, the singular matrix and the transition matrix is given as follows:(97)E=1000,Pr=0.30.50.20.1???0.3?.Then, we choose N=3,d=0.7,d1=1,d2=3,ε1=0.05,ε2=0.05,ρ=2.65,θ=0.00001,α=10-10, and(98)R=R1R2R3R4,where (99)R1=0000.1,R2=0001.7,R3=0000.1,R4=000-0.2,R¯1=R¯2=R¯3=I2.
According to Theorem13, the optimal bound with value of δ2+γ2 depend on the parameter μ.
When we choose3.1≤μ≤8.61 that can find feasible solution. Figure 1 shows the optimal value with different value of μ. Moreover, while μ=3.2 we can obtain the optimal bound γ=8.0483, δ=8.0534 and the controller gains: (100)H1=-2.9765-0.1328,K1=-2.96980.0705,Af1=2.253316.79140.74485.5915Bf1=2.198378.46294.3065153.6901,Cf1=0.37830.12370.1386-0.0826,Df1=0.0807-0.5132-0.99232.1256H2=-0.45751.4874,K2=-0.04450.1727,Af2=-5.336017.13960.5732-1.8683Bf2=-2.925918.3000-7.191044.7616,Cf2=0.0599-0.19840.08420.0842,Df2=-0.09350.8072-1.10502.6129H3=1.8040-4.7583,K3=1.6792-4.4952,Af3=-4.41129.38153.0823-6.4983Bf3=-0.98995.0880-16.507884.6279,Cf3=0.0982-0.01110.03400.3148,Df3=-0.49512.6592-1.44704.1678
Under the above optimal solution, chose the jumping mode as Figure2., let the disturbance function w(k)=0.5e-k and assume the given initial condition x1(0)=-0.2,x2(0)=0.4,x~10=0,x~20=0, and the initial mode r0=3, then, the error response of the resulting filtering error systems as depicted in Figure 3. From the figure, it is obvious that the filtering error system is finite-time bounded.Figure 1
The local optimal bound ofδ and γ.Figure 2
The jumping modes.Figure 3
The filtering errore(k).Remark 19.
Using the same matrix variables, whenN=3,d=1,d1=1,d2=2,ε1=0.05,ε2=0.05,ρ=1.5,θ=0.00001,α=10-10,μ=2.8,γ=7.3,δ=7.9, according to (87), we can obtain the biggest estimation of attraction domain νmax=1.3331.Example 3.
Consider the dynamic Leontief model of economic systems, which explains the time pattern of production sectors shown by [49](101)xk=Axk+Exk+1-xk+vkwhere xk is the vector of production levels, A is the input-output matrix and Ax(k) is the amount required as direct input for the current produce, E is the capital coefficient matrix, and we know Exk+1-xk is the amount necessary for capacity expansion to be capable to produce x(k+1) in the next period. v(k) is the vector of final demands, according to [50], it is assumed that vk=-Bisatuk-Bwiwk+Hθk, then (101) can be rewritten as(102)Exk+1=I-A+Exk+Bisatuk+Bwiwk-Hθk
It is assumed that the control law with fault model is given by(103)θLk=LσkK^1xk+K^2xk+1where Lσk=diagf1σk,f2σk,…,fpσk,0≤fpσk≤1,q=1,2,…,p Then, consider a Leontief model described by(104)E=1000,A=2.0410.81,H=-13.05
LetK^1=[-0.010.6], K^2=[0.16760.117], L1=0.3, L2=0.8, L3=1, then we can obtain the following parameters:(105)A1=-0.0430-0.8200-0.7909-0.5490,A2=-0.0480-0.5200-0.7756-1.4640,A3=-0.0500-0.4000-0.7695-1.8300,Ad1=0.05030.0351-0.1534-0.1071,Ad2=0.13410.0936-0.4089-0.2855,Ad3=-0.16760.1170-0.5112-0.3569.
The other parameters are completely similar with Example2. Then, by using the Matlab LMI Toolbox, the desired filter gains can be given as follows:(106)H1=1.42515.8779,H2=-2.1754-12.4704,H3=4.338720.8669,K1=1.00160.4554,K2=-1.0094-5.7993,K3=7.755137.3305,Af1=-1.3413-13.68190.31333.0341,Af2=-12.8384-73.52091.817910.3895,Af3=56.1115270.4071-7.5392-36.3306,Bf1=-0.6284-30.7700-2.0133-98.6463,Bf2=-6.2666-17.1297-17.4107-47.4759,Bf3=25.390646.170481.5071148.2659,Cf1=-0.1222-0.3708-0.0322-0.0553,Cf2=0.68063.89930.79044.5337,Cf3=-5.1008-24.5798-5.6820-27.3749,Df1=0.16064.2467-1.01432.0039,Df2=-1.7920-4.1510-2.7473-2.0614,Df3=11.404521.617310.375223.5226.
For simulation purposes, we assume that the initial conditionsx0=0.2-0.1T, x~0=00T, the exogenous disturbance signal w(k) is the same as Example 2. Consider the jumping mode as Figure 4, and the trajectories of state are shown in Figure 5, from the picture, it is obvious that the filtering error systems is finite-time bounded.Figure 4
The jumping modes.Figure 5
The filtering errore(k).


## 5. Conclusion

The finite-time discreteSMJSsH∞ filtering problems which are considering time-varying delay, partially unknown transition probabilities and actuator saturation. First, we realized our goal, a H∞ filter has been designed while the filtering error system is stochastic boundedness in the finite time and satisfying a prescribed H∞ attenuation level. Following, we enumerates three examples to prove the validity of the theory that we obtained. However, in practical application, the transition probabilities are often time-varying or difficult to available. Based on the above content, lots of attentions would focus on studying Markovian jump systems with transition probabilities unknown or time-varying under actuator saturation in the near future.



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*Source: 4675397-2019-04-02.xml*