# H

Authors: Ying Zhou; Shuming Yang; Qiang Zang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper is concerned withH∞ filter design problem for large-scale systems with missing measurements. The occurrence of missing measurements is assumed to be a Bernoulli distributed sequence with known probability. The new full-dimensional filter is designed to make the filter error system exponentially mean-square stable and achieve a prescribed H∞ performance. Sufficient conditions are derived in terms of linear matrix inequality (LMI) for the existence of the filter, and the parameters of filter are obtained by solving the LMI. Finally, the numerical simulation results illustrate the effectiveness of the proposed scheme.


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## Body

## 1. Introduction

In many practical applications, due to the limitations imposed by the network, missing measurements often occur due to the network link transmission errors, network congestion, and so forth. Currently, the research of filter and controller design for systems with missing measurements has attracted more attention [1–7]. In [1], the robust control problem with missing measurements was investigated, where the missing measurements were described by a binary switch sequence satisfied conditional probability distribution. The similar model was employed in [2–4], where the filtering problem was investigated in [2, 3], and the distributed state estimation problem was studied in [4]. In [5], the quantized H∞ control problem is investigated for a class of nonlinear stochastic time-delay network-based systems with probabilistic data missing. In [6], the filtering problem with packet loss was considered using Markov chains to describe probabilistic losses. The problem of robust H∞ filtering for discrete-time switched systems with missing measurements under asynchronous switching is considered in [7].Most of the existing research is focused on general linear or nonlinear discrete system. However, many actual systems are large-scale systems which are composed of interconnected subsystems. Although ideas of decentralized control of large-scale systems have attracted much attention in the literature during the past two decades, the research about large-scale systems with missing measurements is seldom. In [8], a decentralized H∞ controller design for a class of large-scale systems with missing measurements is considered. In [9], a state feedback H∞ controller is designed for a class of linear discrete-time large-scale system with both measurement data and control data missing simultaneously.In this paper,H∞ filter is considered for a class of large-scale systems with missing measurements. We apply Bernoulli distributed sequence to describe the occurrence of missing measurements, and the linear discrete-time large-scale system is modeled as interconnection of N subsystems with missing measurements. Then, we design a new decentralized filter. Sufficient conditions are derived in terms of linear matrix inequality (LMI) which is easy to be solved by using MATLAB LMI Toolbox for the decentralized stabilization of this class of large-scale system.


## 2. Problem Formulation

Consider the linear discrete-time large-scale system comprisingN subsystems Ξi, i=1,2,…,N; the dynamics of the ith subsystem is described by (1)Ξi:{xi(k+1)=Aixi(k)+Biwi(k)+∑j=1j≠iNGijxj(k)zi(k)=Cixi(k)+Diwi(k)yi(k)=xi(k),
where xi(k)∈Rni is the state vector of the ith subsystem at time k, yi(k)∈Rqi is the measurement output, zi(k)∈Rpi is the controlled output, wi(k)∈Rri is the disturbance vector belonging to l2[0,∞), Ai, Bi, Ci, and Di are known real constant matrices with appropriate dimensions, and Gij∈Rni×nj is the interconnection matrix of the subsystem of j and i.The measurement with missing data can be characterized by(2)x-i(k)=αi(k)xi(k),
where x-i(k)∈Rmi is the actual measured state, the stochastic variable αi(k)∈R is a Bernoulli distributed white noise sequence taking the values of 0 and 1 with certain probability
(3)prob{αi(k)=1}=E{αi(k)}:=α-i,prob{αi(k)=0}=1-E{αi(k)}:=1-α-i,
and 0<α-i<1 is a known positive constant.In order to observe the states of the system (1), we consider the following filter of order n described by
(4)x^i(k+1)=Aix^i(k)+∑j=1j≠iNGijx^j(k)+Ki(α-ix^i(k)-x-i(k)),z^i(k)=Cix^i(k),y^i(k)=x^i(k),
where x^i(k)∈Rni is the state estimate of system (1) and Ki is the observer gain to be determined later.Define the state estimation error by(5)εi(k)=xi(k)-x^i(k),
and the filter error output is denoted by
(6)ei(k)=zi(k)-z^i(k).
Then it follows from (1), (2), and (4) that
(7)εi(k+1)=(Ai+α-iKi)εi(k)+∑j=1j≠iNGijεj(k)+(αi(k)-α-i)Kixi(k)+Biwi(k),(8)ei(k)=zi(k)-z^i(k)=Ciεi(k)+Diwi(k).
Since it contains stochastic quantities αi(k), the filter error system (8) is actually a stochastic parameter system. Then we use the following definition.Definition 1 (see [10]).
The filter error system (8) is said to be exponentially mean-square asymptotically stable if with w(k)=0, there exist constants κ>0 and 0<τ<1,  such that
(9)E{∥e(k)∥2}<κτkE{∥e(0)∥2},∀e(k)≠0,
where e(k)=[e1T(k)⋯eNT(k)]Tand w(k)=[w1T(k)⋯wNT(k)]T.
With this definition, our objective is to design the full-order filter of form (4), such that(1)
the filter error system (8) is exponentially mean-square asymptotically stable with w(k)=0;(2)
under zero-initial condition, the filter errore(k) satisfies
(10)∑k=0∞E{∥e(k)∥2}<γ2∑k=0∞E{∥w(k)∥2},where γ is a given positive constant.


## 3. Main Results

For investigating the stability conditions of the filter error system (8), the following lemma is needed.Lemma 2 (see [10]).
LetV(η(k)) be a Lyapunov functional. If there exist constants λ≥0, μ>0, v>0, and 0<ψ<1 such that
(11)μ∥η(k)∥2≤V(η(k))≤v∥η(k)∥2,E{V(η(k+1)∣η(k))}-V(η(k))≤λ-ψV(η(k)),
then the sequence η(k) satisfies
(12)E{∥η(k)2∥}≤vμ∥η(0)∥2(1-ψ)k+λμψ.
The main results are concluded into the following theorems.Theorem 3.
Given0<α-i<1 and w(k)=0, the filter error system (8) is exponentially mean-square asymptotically stable if there exist positive definite matrices P1i=P1iT, P2i=P2iT and gain matrix Ki,i=1,2,…,N, satisfying
(13)[-P10Q1T+α-K000-P20Q1TKQ1+α-K0-P1-1000Q10-P2-100K00-β2P1-1]<0,
where
(14)Q1=[A1G12⋯G1NG21A2⋯G2N⋮⋮⋱⋮GN1GN2⋯AN],P1=diag{P11,P12,…,P1N},P2=diag{P21,P22,…,P2N},K=diag{K1,K2,…,KN},β=diag{β1,β2,…,βN},βi=((1-α-i)α-i)1/2,i=1,2,…,N,α-=diag{α-1,α-2,…,α-N}.Proof.
Whenw(k)=0, define the Lyapunov functional
(15)V(k)=∑i=1NεiT(k)P1iεi(k)+∑i=1NxiT(k)P2ixi(k),
where P1i=P1iT, P2i=P2iT are positive definite matrices. It follows from (8) that
(16)V(k+1)-V(k)=∑i=1N(∑j=1j≠iNGijεj(k)(Ai+α-iKi)εi(k)+∑j=1j≠iNGijεj(k)+(αi(k)-α-i)Kixi(k))TP1i×(∑j=1j≠iNGijεj(k)(Ai+α-iKi)εi(k)+∑j=1j≠iNGijεj(k)+(αi(k)-α-i)Kixi(k))+∑i=1N(Aixi(k)+∑j=1j≠iNGijxj(k))TP2i×(Aixi(k)+∑j=1j≠iNGijxj(k))-∑i=1NεiT(k)P1iεi(k)-∑i=1NxiT(k)P2ixi(k).
Noting that E{αi(k)-α-i}=0, E{(αi(k)-α-i)2}=(1-α-i)α-i≜βi2, we have
(17)E{V(k+1)}-V(k)=∑i=1N((Ai+α-iKi)εi(k)+∑j=1j≠iNGijεj(k))TP1i×((Ai+α-iKi)εi(k)+∑j=1j≠iNGijεj(k))+∑i=1N(Aixi(k)+∑j=1j≠iNGijxj(k))TP2i×(Aixi(k)+∑j=1j≠iNGijxj(k))+∑i=1Nβi2(Kixi(k))TP1i(Kixi(k))-∑i=1NεiT(k)P1iεi(k)-∑i=1NxiT(k)P2ixi(k)=ηT(k)θ1η(k),
where x(k)=[x1T(k)⋯xNT(k)]T, ε(k)=[ε1T(k)⋯εNT(k)]T, and η(k)=[εT(k),xT(k)]T.
By Schur complement, inequality (13) implies θ1<0. Then we have
(18)E{V(k+1)}-V(k)=ηT(k)θ1η(k)≤-λmin(-θ1)ηT(k)η(k)<-γηT(k)η(k),
where 0<γ<min{λmin(-θ1),σ}, and then from (18), we get
(19)E{V(k+1)}-V(k)<-γηT(k)η(k)<-γσV(η(k))=-ψV(η(k)),
where ψ=γ/σ∈(0,1).
From Definition1 and Lemma 2, we can conclude that the filter error system (8) is exponentially mean-square asymptotically stable. This completes the proof.In the sequel, we further provide method for solving matrix inequality (13) that is not a linear matrix inequality (LMI).Theorem 4.
Given0<α-i<1 if there exist positive definite matrices M1=M1T, M2=M2T and gain matrices N1, N2 that satisfy linear matrix inequality
(20)[-M10M1Q1T+α-N1000-M20M2Q1TβN2Q1M1+α-N10-M1000Q1M20-M200βN200-M1]<0
and equation
(21)N1P1=N2P2=K,
where M1=P1-1, M2=P2-1, N1=KP1-1, and N2=KP2-1, then the error system (8) is exponentially mean-square asymptotically stable.Proof.
Through left- and right-multiplying (13) by diag{P1-1,P2-1,I,I,βI}, we have
(22)[-P1-10P1-1(Q1+α-K)T000-P2-10P2-1Q1TβP2-1K(Q1+α-K)P1-10-P1-1000Q1P2-10-P2-100βKP2-100-P1-1]<0.
For the definitions of the matrices M1, M2, N1, and N2, the matrix inequality (13) is equivalent to (20) and (21). This completes the proof.By solving the linear matrix inequality (20) and (21), we have M1, M2 and N1, N2. Moreover, the matrices are given by P1=M1-1, P2=M2-1, and K=N1P1=N2P2.Theorem 5.
Given0<α-i<1 if there exist positive definite matrices P1i=P1iT,P2i=P2iT and gain matrix Ki,i=1,2,…,N, that satisfy the following linear matrix inequality:
(23)[-P1******0-P2*****00-γ2I****Q1+α-K0B-P1-1***0Q1B0-P2-1**0K000-β2P1-1*C0D000-I]<0,
where C=diag{C1,C2,…,CN}, D=diag{D1,D2,…,DN}, B=diag{B1,B2,…,BN}, and Q1, P1, P2, K, α-, βi, and β are the same as (13), then the filter error system (8) is exponentially mean-square asymptotically stable and achieves the prescribed H∞ performance.Proof.
Whenw(k)=0, (23) is equivalent to (13), so the filter error system is exponentially mean-square asymptotically stable.
Whenw(k)≠0, define the Lyapunov functional as
(24)V(k)=∑i=1NεiT(k)P1iεi(k)+∑i=1NxiT(k)P2ixi(k),
then
(25)E{V(k+1)}-V(k)+E{eT(k)e(k)}-γ2E{wT(k)w(k)}=∑i=1N((Ai+α-iKi)εi(k)+∑j=1j≠iNGijεj(k)+Biwi(k))T×P1i×((Ai+α-iKi)εi(k)+∑j=1j≠iNGijεj(k)+Biwi(k))+∑i=1N(Aixi(k)+∑j=1j≠iNGijxj(k)+Biwi(k))TP2i×(Aixi(k)+∑j=1j≠iNGijxj(k)+Biwi(k))+βi2∑i=1N(Kixi(k))TP1i(Kixi(k))+E{∑i=1N(Ciεi(k)+Diwi(k))T×(Ciεi(k)+Diwi(k))∑i=1N(Ciεi(k)+Diwi(k))T}-∑i=1NεiT(k)P1iεi(k)-∑i=1NxiT(k)P2ixi(k)-γ2E{wT(k)w(k)}=η-T(k)θ2η-(k),
where η-(k)=[εT(k),xT(k),wT(k)]T. By the Schur complement, inequality (23) implies θ2<0, and then we have
(26)E{V(k+1)}-V(k)+E{eT(k)e(k)}-γ2E{wT(k)w(k)}<0,∑k=0∞E{eT(k)e(k)}<γ2∑k=0∞E{wT(k)w(k)}+E{V(0)}-E{V(∞)}.
Since the system is exponentially mean-square asymptotically stable, it is straightforward to see that
(27)∑k=0∞E{∥e(k)∥2}<γ2∑k=0∞E{∥w(k)∥2},
under the zero-initial condition. This completes the proof.In the sequel, we further present how to convert the matrix inequality (23) into an LMI with matrix equality constraint.Theorem 6.
Given0<α-i<1 if there exist positive definite matrices M1T=M1, M2T=M2 and gain matrices N1, N2 that satisfy the following linear matrix inequality
(28)[-M1******0-M2*****00-γ2I****Q1M1+α-N10B-M1***0Q1M2B0-M2**0βN2000-M1*CM10D000-I]<0,K=N1P1=N2P2,
then the filter error system (8) is exponentially mean-square asymptotically stable and achieves the prescribed H∞ performance.Proof.
Through left- and right-multiplying (23) by diag{P1-1,P2-1,I,I,I,βI,I}>0, we have
(29)[-P1-1******0-P2-1*****00-γ2I****(Q1+αK)P1-10B-P1-1***0Q1P2-1B0-P2-1**0βKP2-1000-P1-1*CP1-10D000-I]<0.
Similar to the proof of Theorem 4, we define M1=P1-1, M2=P2-1, N1=KP1-1, and N2=KP2-1. Then the matrix inequality (29) is equivalent to (23). From Theorem 5, we can conclude that the filter error system (8) is exponentially mean-square asymptotically stable and achieves the prescribed H∞ performance. The proof is completed.


## 4. Numerical Simulations

Consider a linear discrete-time large-scale system which consists of two interconnected subsystems:(30)x1(k+1)=[-130-0.1]x1(k)+[0.20.30.10.2]x2(k)+[1001]w1(k)=A1x1(k)+G12x2(k)+E1w1(k),z1=[10-10.8]x1(k)+[1110]w1(k)=C1x1(k)+D1(k)w1(k),y1(k)=x1(k),x2(k+1)=[-110-0.2]x2(k)+[0.30.10.20.2]x1(k)+[1001]w2(k)=A2x2(k)+G21x1(k)+E2w2(k),z2=[101-0.8]x2(k)+[1101]w2(k)=C2x2(k)+D2(k)w2(k),y2(k)=x2(k).
Choose the disturbance input w1(k)=w2(k)=0.01[sin(100k)sin(100k)]. The initial state values of original system and its observer are x1(0)=[21], x2(0)=[22] and x^1(0)=[00], x2(0)=[-10], respectively. Suppose that stochastic sequence obeys Bernoulli distribution with probability E{α1(k)∣α1(k)=1}=E{α2(k)∣α2(k)=1}=0.996, given γ=1. For simplicity, let M=P1-1=P2-1 and N=KP1-1=KP2-1. We can obtain the following parameters in Theorem 6 by using the MATLAB YALMIP Toolbox:
(31)M1=[1.35490.40140.40140.3819],M2=[1.10580.78810.78811.6449],N1=[0.1755-0.3232-0.32320.0041],N2=[0.2464-0.3847-0.38470.3580].
The Lyapunov function solution matrices and observer parameter are given by
(32)P11=P21=M1-1=[1.0718-1.1264-1.12643.8021],P12=P22=M2-1=[1.3732-0.6579-0.65790.9231],K1=N1P1=[0.5521-1.4264-0.35100.3789],K2=N2P2=[0.5919-0.5172-0.76370.5835].
The simulation results are shown in Figures 1, 2, 3, and 4.Figure 1
Filter errore11(k).Figure 2
Filter errore12(k).Figure 3
Filter errore21(k).Figure 4
Filter errore22(k).It can be verified that∑k=0∞E{∥e(k)∥2}<γ2∑k=0∞{∥w(k)∥2} and the filter error system satisfies the prescribed H∞ performance.


## 5. Conclusion

In this paper, theH∞ filter for a class of linear discrete-time large-scale system has been designed, where the measurements are probably missing. The missing probability is assumed to obey Bernoulli distribution. By employing the Lyapunov stability theory combined with stochastic analysis method, a filter is designed to reconstruct the states of original system such that the filter error system is exponentially stable in the sense of mean square and achieves the prescribed H∞ performance.



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