# H

Authors: Hui Dong; Juntong Chen
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper is concerned with theH∞ filtering for a class of networked Markovian jump systems with multiple communication delays. Due to the existence of communication constraints, the measurement signal cannot arrive at the filter completely on time, and the stochastic communication delays are considered in the filter design. Firstly, a set of stochastic variables is introduced to model the occurrence probabilities of the delays. Then based on the stochastic system approach, a sufficient condition is obtained such that the filtering error system is stable in the mean-square sense and with a prescribed H∞ disturbance attenuation level. The optimal filter gain parameters can be determined by solving a convex optimization problem. Finally, a simulation example is given to show the effectiveness of the proposed filter design method.


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## Body

## 1. Introduction

Analysis and synthesis of the networked systems has received much research attention in the last decade due to the wide applications of networked systems into the industrials, such as process control system, chemical systems, and remote surgery. Despite the advantages brought by the networked systems, various challenging problems occur, such as packet dropouts, communication delays, and medium access constraints. In the last few years, the state estimation for networks systems has become a hot research topic. Basically, there are state estimation algorithms: one is the Kalman filtering and the other one is theH∞ filtering. For example, the authors in [1] studied the remote state estimation for the networked systems with missing observations. They addressed this problem under the discrete Kalman filtering formulation and also gave the statistical convergence properties of the estimation error covariance, showing the existence of a critical value for the arrival rate of the observations. Many new results have also been reported based on the results in [1]; see [2–5] and the references therein. It is well known that the Kalman filter is only applicable to the systems with a certain well-posed model, and the external noise is required to be a stochastic process with known statistical property. In many applications, these requirements may not hold. In this scenario, an alternative filtering algorithm has been proposed, that is, the H∞ filtering algorithm. The recent advances on the H∞ filtering for networked systems with various networked issues are referred to in [6–10].On another research front line, Markovian jump systems have attracted significant attention from control society for many decades. This is due to the fact that such systems are shown to be appropriate and convenient to model a large number of practical systems that are subjected to abrupt variations in their structures, owing to sudden environmental disturbances, abrupt variations in the operating point of a nonlinear plant, and so on. Many works have been reported on the analysis andH∞ filtering for Markovian jump systems [11–16]. To mention a few, the H∞ filtering for a class of Markovian jump systems with time-varying transition probabilities has been considered in [16]; the authors showed that the filter gain parameter can be determined by solving a set of linear matrix inequalities. Recently, the H∞ filtering for Markovian jump systems with missing measurement and signal quantization is investigated in [17]; the corresponding filter synthesis problem is transformed into a convex optimization problem that can be efficiently solved by using standard software packages. It should be noted that the communication delay has not been considered in their design of the filter. The communication delay is an important issue that occurs in the networked control systems and it cannot be ignored. Though the H∞ filtering for networked systems with stochastic delays was studied in [18], the authors assumed that the measurement signal is transmitted to the remote filter within a single packet. In reality, the networked systems may distribute in a large area, and incorporating the measurement into one packet is impossible. Hence, the H∞ filtering for networked systems with multiple communication delays deserves investigation, which motivates the present study.In this paper, theH∞ filtering problem is addressed for a class of networked Markovian jump systems with multiple communication delays. Firstly, a set of stochastic variables are introduced to model the communication delays. Based on the Laypunov stability theory and the stochastic system approach, a sufficient condition is obtained such that the filtering error system is asymptotically stable in the mean-square sense and achieves a prescribed H∞ performance level. Finally, a simulation study is given to show the effectiveness of the proposed design.Notations.
The notations used in this paper are standard. TheRn denotes the n-dimensional Euclidean space, and l2[0,+∞) is the space of summable sequences. AT represents the transpose of the matrix A.


## 2. Problem Formulation

The structure of the considered filtering system is shown in Figure1. The physical plant under the consideration is(1)xk+1=Aσkxk+Bσkwk,zk=Lσkxk,where x(k)∈Rnx is the state, z(k)∈Rnz is the signal to be estimated, and w(k)∈Rnw is the unknown disturbance belonging to l2[0,+∞). Aσ(k), Bσ(k), and Lσ(k) are the constant matrices with appropriate dimensions. σ(k)∈Γ={1,2,…,S} is a discrete-time homogeneous Markov chain with following probabilistic transfer matrix Π=[σij]:(2)σij=Probσ(k+1)=j∣σ(k)=i≥0,∑j=1Sσij=1,∀i,j∈Γ.Figure 1
The structure of networked systems.Suppose there arem distributed sensors for the system and the pth sensor produces its measurement as(3)ypk=Cpxk+Dpwkp=1,2,…,m,where yp(k)∈Rnp; Cp and Dp are the known constant matrix with appropriate dimensions. Due to the existence of the communication delay, the filter may not be able to use the current measurement signal to update its state. In this paper, we assume that the largest delay in the pth channel is Np time step. Then, at each time instant k, the filter input, y-p(tk), will be the most recent member of the transmitted subset of {yp(k),yp(k-1),…,yp(k-Np)}.In the real networked systems, the delay may occur in a stochastic way; hence, a set of stochastic variables,αp,s(k)∈{0,1}, s=0,1,…,Np, is introduced such that αp,s(k)=1 if yp(k-s) is the value used as the filter input at k. Due to the fact that at each time instant only one case may happen in our system we have ∑s=0Npαp,s(k)=1. In this paper, the probabilities E{αp,s(k)=1}=α-p,s are assumed to be known, which may be obtained by some stochastic analysis method. Clearly, ∑s=0Npα-p,s=1. The input of the filter can be described as(4)y-pk=αp,0kypk+αp,1kypk-1+⋯+αp,Npkypk-Np.Let N=max⁡p=1,2,…,m{Np},(5)Xk=xTkxTk-1⋯xTk-NT,Wk=wTkwTk-1⋯wTk-NT.We have(6)y-pk=∑s=0Npαp,skCpEspXk+DpHspWk,where Esp=[0⋯0I0⋯0], and Hsp=[0⋯0I0⋯0], in which all elements are zeros except that the (s+1)th block is an identity matrix.Letαp(k)=[αp,0(k),αp,1(k),…,αp,Np(k)]. Then we have (7)αp(k)∈1,0,0,…,00,1,0,…,0⋮0,0,0,…,1.Define α(k)=[α1(k),α2(k),…,αm(k)]. The total number of possible realizations of α(k) is N=N1×N2×⋯×Nm. Let (8)E-α(k)=E1,α1kTE2,α2kT⋯Em,αmkTT,H-α(k)=H1,α1kTH2,α2kT⋯Hm,αmkTT.One particular realization of α(k) means one sequence of αp(k), which specifies one particular case of (E-α(k),H-α(k)). For presentation simplicity, we now define a new variable βl(k)∈{1,0}, l∈Γ to model each possible case for all the measurement signals, where Γ={1,2,…,N-}, and N- is the possible numbers. For example, β1(k)=1 if and only if αp(k)=[1,0,…,0], p=1,2,…,m and β2(k)=1 if and only if αp(k)=[1,0,…,0], p=1,2,…,m-1, αm(k)=[0,1,…,0] and so on. It is easy to see that ∑l=1N-βl(k)=1 and ∑l=1N-β-l=1.Based on the above discussions, the filter input signal is(9)y-k=∑l=1N-βlkC-E-lXk+D-H-lWk,where C-=diag{C1,C2,…,Cm} and D-=diag{D1,D2,…,Dm}. We propose the following filter:(10)xfk+1=Afxfk+Bfy-k,zfk=Cfxfk,where xf(k)∈Rnx is the state of the filter and zf(k)∈Rnz is the estimate of z(k). Af, Bf, and Cf are the filter parameters to be designed.In order to derive the filtering error system, we rewrite state equation in (1) and the estimation equation in (9) as(11)Xk+1=A-σkXk+B-σkWk,zk=L-σkXk,where (12)A-σ(k)=Aσ(k)00⋯0In00⋯00In0⋯0⋮⋮⋮⋱⋮00⋯In0,B-σ(k)=Bσ(k)00⋯0000⋯0000⋯0⋮⋮⋮⋱⋮00⋯00,L-σ(k)=Lσ(k)00⋯0.Defineη(k)=[XT(k)xfT(k)]T and the estimation error as e(k)=z(k)-zf(k). Then the filtering error system is given by(13)ηk+1=A~σ(k)η(k)+B~σ(k)W(k)+∑l=1N-βlk-β-lA^lηk+B^lWk,ek=L~σkηk,where(14)A~σ(k)=A-σ(k)0BfC-E-Af,B~σ(k)=B-σ(k)BfD-H-,L~σ(k)=L-σ(k)-Cf,A^l=00BfC-E-l0,B^l=0BfD-H-l,E-=∑l=1N-β-lE-l,H-=∑l=1N-β-lH-l.System (13) is a complex stochastic system; we need the following definitions before presenting our main results.Definition 1 (see [12]).
System (13), with w(k)=0, is said to be stochastically stable if the following inequality(15)E∑k=0∞ηk2∣φ(0)<∞holds for any initial condition φ(0)={η(t0),σ(0)}.Definition 2.
For a given scalarγ>0, system (13) is said to be asymptotically stable in the mean-square sense and achieves a prescribed H∞ performance γ, if it is asymptotically stable and under zero initial condition, ∑s=0+∞E{eT(s)e(s)}≤∑s=0+∞γ2wT(s)w(s) holds for all nonzero w(k)∈l2[0,∞).


## 3. Main Results

A sufficient condition is firstly presented to guarantee the stability of the filtering error system.Theorem 3.
For given scalarsβ-i and τ, if there exists a positive-definite matrix Pi such that the following inequalities(16)Ξ1Ξ2β-1Φ1⋯β-NΦNΞ3∗-P-i0000∗∗-P-i000∗∗∗⋱00∗∗∗∗-P-i0∗∗∗∗∗-I<0hold for all i∈Γ, then the filtering error system is asymptotically stable in the mean-square sense and achieves a prescribed H∞ performance level γ=τN+1, where(17)Ξ1=-Pi00-τ2I,Ξ2=A~iB~iT,Φl=A^lB^lT,Ξ3=L~i0T,P-i=∑j=1sσijPj-1.Proof.
We first consider the stability of the filtering error system (13) with w(k)=0. To do this, we choose the following Lyapunov function for system (13):(18)Vk=ηTkPσkηk.Then, one sees that(19)EV(k+1)-V(k)=A~iη(k)T∑j=1sσijPjA~iη(k)-ηT(k)Piη(k)+E∑l=1N-ϕl(k)A^lη(k)T∑j=1sσijPjmmmmmmm∑l=1N-T·∑l=1N-ϕlkA^lηk,where ϕi(k)=βi(k)-β-i. On the other hand, we have(20)E∑l=1N-ϕl(k)A^lη(k)T∑j=1sσijPj∑l=1N-ϕl(k)A^lη(k)=E∑l=1N-ϕl(k)2A^lη(k)T∑j=1sσijPjA^lη(k)mmmmmm+∑l=1N∑t=1,t≠lNϕl(k)ϕt(k)A^lη(k)Tmmmmmmmmmmmm·∑j=1sσijPjA^tη(k)≤∑l=1N-β-lA^lηkT∑j=1sσijPjA^lηk.Hence,(21)EVk+1-Vk≤A~iη(k)T∑j=1sσijPjA~iη(k)-ηT(k)Piη(k)+∑l=1N-β-lA^lηkT∑j=1sσijPjA^lηk.
It is seen that the right hand side of (21) is negative under (16) and then E[Vk+1∣φ(k)]<V(k). It is not difficult to find a scalar 0<μ<1, such that E[Vk+1∣φ(k)]<μV(k) holds; then by deduction, we have E[Vk∣φ(0)]<μkV(0), and, consequently, (22)E∑k=0hVk∣φ0≤1+μ+⋯+μhV(k0)=1-μh+11-μVk0.Let h→∞; the following inequality is obtained: (23)E∑k=0∞η(k)2∣φ(0)≤11-μλ-Vk0<∞,where λ-=min⁡i∈Γλmin⁡(Pi). According to Definition 1, system (13) is stochastically stable.
Now we consider theH∞ performance of the filtering error system (13). It follows from the above analysis method that(24)EV(k+1)-V(k)+eT(k)e(k)-τ2WT(k)W(k)=A~iη(k)+B~iW(k)T∑j=1sσijPjA~iη(k)+B~iW(k)-ηT(k)Piη(k)+E∑l=1N-βl(k)-β-lA^lη(k)+B^lW(k)Tmmmmmmm×∑j=1sσijPj∑l=1N-βl(k)-β-lmmmmmmmmmmmmmmmmm∑l=1N-T·A^lη(k)+B^lW(k)∑l=1N-βl(k)-β-l+L~iη(k)TL~iη(k)-τ2WT(k)W(k)≤η-T(k)Ξ1+Ξ2∑j=1sσijPjΞ2Tmmmmmmmm+∑i=1Nβ-iΦi∑j=1sσijPjΦiT+Ξ3Ξ3Tη-(k),where η-(k)=[ηT(k)WT(k)]T. By using the schur complement [19], it is easy to see (25)EV(k+1)-V(k)+eT(k)e(k)-τ2WT(k)W(k)<0.Summing both sides of (21) from k=0 to k=T gives(26)EVk+1-V(0)+∑k=0TeT(k)e(k)-τ2WT(k)W(k)<0,which implies, by the zero initial condition and positiveness of V(k+1), that(27)E∑k=0TeT(k)e(k)-τ2WT(k)W(k)<0.Let T→+∞; then(28)∑k=0+∞EeT(k)e(k)≤τ2WTkWk=γ2wTkwk.Hence, system (13) achieves a prescribed H∞ performance level as well. This completes the proof.It should be pointed out that Theorem3 cannot be used to determine the filter gain directly due to the coexistence of P and P-1. In the following theorem, we present the filter gain design.Theorem 4.
For given scalarsβ-i and τ, if there exist a positive-definite matrix Pi and a matrix G with appropriate dimensions, such that the following inequalities(29)Ξ1Ξ-2β-1Φ-1⋯β-N-Φ-N-Ξ-3∗P^i0000∗∗P^i000∗∗∗⋱00∗∗∗∗P^i0∗∗∗∗∗-I<0hold, then the filtering problem is solvable. Moreover, the filter gains are determined by Af=G3-TAF, Bf=G3-TBF, Cf=CF, where(30)Ξ-2=Ξ-21Ξ-22,Φ-i=Φ-i1Φ-i2,Ξ-3=Ξ-310,P^i=∑j=1sσijPj-G-GT,with(31)Ξ-21=A-iTG1+ETBFTMA-iTG2+ETBFTAFTMAFT,Ξ-22=B-iTG1+H-TBFTMB-iTG2+H-TBFT,Φ-i1=E-iTC-TBFTME-iTC-TBFT00,Φ-i2=H-iTD-TBFTMH-iTD-TBFT,Ξ-31=L-T-CFT,Pi=P1iP2i∗P3i,G=G1G2G3MG3,M=I⋯I.Proof.
By pre- and postmultiplying (16) with diag{I,GT,GT,⋯GT︸N,I} and its transpose, respectively, (16) is equivalent to(32)Ξ1Ξ2Gβ-1Φ1G⋯β-N-ΦN-GΞ3∗P~i0000∗∗P~i000∗∗∗⋱00∗∗∗∗P~i0∗∗∗∗∗-I<0,where P~i=-GT(∑j=1sσijPj)-1G. On the other hand, it is easy to see that -GT(∑j=1sσijPj)-1GT≤(∑j=1sσijPj)-G-GT always holds for any matrix G. Then, (32) holds if (33)Ξ1Ξ2Gβ-1Φ1G⋯β-N-ΞN-GΞ3∗P^i0000∗∗P^i000∗∗∗⋱00∗∗∗∗P^i0∗∗∗∗∗-I<0.The above inequality is the same as (29) with Pi=P1iP2i∗P3i, G=G1G2G3MG3, AF=G3TAf, BF=G3TBf, and CF=Cf. This completes the proof.Remark 5.
In order to obtain the minimumH∞ performance γ∗, one can solve the following optimization problem(34)min⁡νsubjectto29withν=τ2and find the minimum H∞ performance γ∗ by γ∗=ν∗(N+1).


## 4. Simulation Example

Consider a satellite yaw angles control system with noise perturbation, which is given by the following state-space representations [20]:(35)1000010000J10000J2θ˙1(t)θ˙2(t)δ˙1(t)δ˙2(t)=00100001-kk-ffk-kf-fθ1(t)θ2(t)δ1(t)δ2(t)+0010ut+00.100.1wt.The satellite yaw angles control system consists of two rigid bodies jointed by a flexible link. This link is modeled as a spring with torque constant k and viscous damping f. θ1(t) and θ2(t) are the yaw angles for the main body and the instrumentation module of the satellite. Moreover, δ1(t)=θ˙1(t) and δ2(t)=θ˙2(t). u(t) is the control torque, while J1 and J2 are the moments of the main body and the instrumentation module, respectively.ChoosingJ1=J2=1, k=0.3, f=0.004 and setting the sampling period T=0.1 s, then with the controller u(k)=103[-0.1591-5.9343-0.0172-2.8604]x(k), we obtain the following discrete-time satellite yaw angles closed-loop system:(36)xk+1=0.2035-29.65800.0142-14.29600.00120.98710.00000.0944-15.9256-592.9813-0.7168-285.83300.0188-0.44510.00070.7980x(k)+0.0000051890.010490.00015690.009843wk,where x(k)=θ1(k)θ2(k)δ1(k)δ2(k)T.In real control systems, controller failure may occur. In this example, we only consider a partial failure case; that is, the real controller is taken asur(k)=ρ(k)u(k) with ρ(k)∈{0.5,1}. The binary variable ρ(k) is assumed to be a two-state Markovian process. The probabilistic transfer matrix is taken as Π=0.60.40.60.4, which will be generated stochastically in the simulation setup. Hence, such a satellite yaw angles control system with partial controller failure can be modeled as the discrete-time Markovian jump system (1).Suppose the remote filter is designed to estimate the signalz(k)=Lx(k) by the measurement yp(k)=Cpx(k)+Dpw(k) of the satellite system, where C1=[1000],C2=[0100],D1=0.3, D2=0.2, and L=[1100]. The two measurements y1 and y2 are transmitted to the remote filter through communication channels with stochastic communication delays. The maximal time delays are assumed to be one time step, and the occurrence probabilities are α-1,0=0.2, α-1,1=0.8, α-2,0=0.4, and α-2,1=0.6. By solving the optimization problem (22), we have γ∗=3.1916.In the simulation setup, we choose the zero initial conditions, and the noise signal is taken asw(k)=exp⁡(-0.1k)∗sin(0.8k). One-sample trajectories of z(k) and zf(k) are shown in Figure 2. The trajectory of the filtering error is shown in Figure 3. By simple calculation, we have ∑k=0100eT(k)e(k)/∑k=0100wT(k)w(k)=1.1622<γ∗.Figure 2
Trajectories ofz(k) and zf(k).Figure 3
Trajectory ofe(k).


## 5. Conclusions

In this paper, theH∞ filtering for a class of networked Markovian jump system with multiple stochastic communication delays has been investigated. A new stochastic delay model is proposed and formulated. Based on the stochastic system approach, a sufficient condition has been obtained such that the filtering error system is asymptotically stable in the mean-square sense and achieves a prescribed H∞ performance level. A simulation study of the satellite yaw control system is given to show the effectiveness of our results.



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