# RobustH2/H∞ Filter Design for a Class of Nonlinear Stochastic Systems with State-Dependent Noise

Authors: Weihai Zhang; Bor-Sen Chen; Li Sheng; Ming Gao
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

This paper investigates the problem of robust filter design for a class of nonlinear stochastic systems with state-dependent noise. The state and measurement are corrupted by stochastic uncertain exogenous disturbance and the dynamic system is modeled by Itô-type stochastic differential equations. For this class of nonlinear stochastic systems, the robustH∞ filter can be designed by solving linear matrix inequalities (LMIs). Moreover, a mixed H2/H∞ filtering problem is also solved by minimizing the total estimation error energy when the worst-case disturbance is considered in the design procedure. A numerical example is provided to illustrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

Over the past decades, the robustH∞ filtering problem has been investigated extensively since it is very useful in signal processing and engineering applications [1–5]. The so-called H∞ filtering problem is to design an estimator to estimate the unknown state combination via measurement output, which guarantees the ℒ2 gain (from the external disturbance to the estimation error) to be less than a prescribed level γ>0. In contrast to classical Kalman filter, it is not necessary to know the exact statistic information about the external disturbance in the H∞ filter design. Obviously, there may be more than one solution to H∞ filtering problem with a desired robustness. Since the H2 performance is appealing for engineering, it naturally leads to the mixed H2/H∞ filtering problem [6–8]. Compared with the sole H∞ filter, the mixed H2/H∞ filter is more attractive in engineering practice, since the former is a worst-case design which tends to be conservative whereas the latter minimizes the average performance with a guaranteed worst-case performance. The robust H2/H∞ filtering problem for linear perturbed systems with steady-state error variance constraints was investigated in [6], and the mixed H2/H∞ filter for polytopic discrete-time systems was discussed in [7].On the other hand, stochasticH∞ control and filtering problems for systems expressed by stochastic Itô-type differential equations have attracted a great deal of attention [9–13, 23]. A bounded real lemma was proposed for linear continuous-time stochastic systems [11], according to which full- and reduced-order robust H∞ problems for linear stochastic systems were investigated by [12, 13], respectively. Most of the aforementioned works were limited to linear stochastic systems. Recently, the H∞ filtering problem for nonlinear stochastic systems has become another popular research topic [14–20]. Wang et al. [14] studied the robust H∞ filtering problem for a class of uncertain time-delay stochastic systems with sector-bounded nonlinearities. For general nonlinear stochastic systems, Zhang et al. [15] found that the H∞ filter can be obtained by solving a second-order Hamilton-Jacobi inequality (HJI). Considering that it is difficult to solve the HJI, Tseng [17] designed the H∞ fuzzy filter for nonlinear stochastic systems via solving LMIs instead of an HJI. However, there is little work dealing with the H2/H∞ filtering problem for nonlinear stochastic systems.In this paper, we will deal with the robust filtering problem for a class of nonlinear stochastic systems. The state is corrupted not only by white noise but also by exogenous disturbance signal, and the measurement equation also includes noises. Our goal in this paper is to construct an asymptotically stable observer that leads to a mean square stable estimation error process whoseℒ2 gain with respect to disturbance signal is less than a prescribed level. Moreover, a stochastic H2/H∞ filtering is designed for the nonlinear stochastic systems. Our main results are expressed in linear matrix inequalities (LMIs), which are more easily computed in practical application.This paper is organized as follows: in Section2, some definitions and notations are introduced; Section 3 treats with the H∞ and mixed H2/H∞ filtering problems, and the main outcomes of this section are Theorems 3.2 and 3.6; a numerical example is presented to illustrate the effectiveness of the proposed filtering method in Section 4; Section 5 concludes this paper.Notations. For convenience, we adopt the following notations. 𝒮n: the set of all n×n symmetric matrices; its components may be complex. A′: the transpose of the corresponding matrix A. A≥0  (A>0): A is positive semidefinite (positive definite) symmetric matrix. |x|:=(∑i=1n‍xi2)1/2, that is, |x| denotes the Euclidean 2-norm of x, where x=(x1,x2,…,xn)′∈ℛn. ℒ2(ℛ+,ℛl): the space of nonanticipative stochastic processes y(t) with respect to filter ℱt satisfying ∥y(t)∥L22:=E∫0∞‍|y(t)|2dt<∞. C20({t>0}×U): class of functions V(t,x) twice continuously differential with respect to x∈U and once continuously differential with respect to t>0 except possibly at the point x=0.


## 2. Problem Setting

Consider the following nonlinear stochastic system governed by Itô differential equation:(2.1)dx(t)=(f(x(t))+B0w(t))dt+σ(x(t))dw0(t),
with the following measurement equation:
(2.2)dy(t)=(A1x(t)+B1w(t))dt+C1x(t)dw1(t),
and the controlled output
(2.3)z(t)=Dx(t).
In the above, x(t)∈ℛn is called the system state, y(t)∈ℛr is the measurement output, z(t) is the state combination to be estimated. w0(t),w1(t) are the standard Wiener processes defined on the probability space (Ω,ℱ,𝒫) related to an increasing family (ℱt)t∈ℛ+ of σ-algebras ℱt⊂ℱ. Without loss of generality, we can suppose w0(t),w1(t) are one-dimensional, mutually uncorrelated. B0,A1,B1,C1,D are constant matrices of suitable dimensions, w∈ℒ2(ℛ+,ℛq) represents the exogenous disturbance signal. Under very general conditions on f and σ, stochastic systems (2.1)-(2.2) have, respectively, a unique strong solution xs,ξ(t) for any t≥s≥0 and initial state x(s)=ξ∈ℛn; see [21].Now, we first introduce the following definitions.Definition 2.1 (see [9]).
We say that the equilibrium pointx≡0 of system
(2.4)dx(t)=f(x(t))dt+σ(x(t))dw0(t)
is exponentially mean square stable, if for some positive constants ρ,ϱ,
(2.5)E|x(t)|2≤ρ|x(0)|2exp(-ϱt),t≥0.Remark 2.2.
It is well known that for stochastic linear time-invariant systems, the exponential mean square stability is equivalent to asymptotical mean square stability [9].Definition 2.3.
Nonlinear stochastic uncertain system (2.1) is said to be internally stable at the origin, if (2.1) with w=0 is exponentially mean square stable.Lemma 2.4 (see [9]).
The trivial solution of (2.4) is exponentially mean square stable for t≥0 if there exists V(t,x)∈C20({t>0}×ℛn) such that
(2.6)k1|x|2≤V(t,x)≤k2|x|2,ℒV(t,x)≤-k3|x|2
for some positive constants k1,k2,k3, where ℒ is the so-called an infinitesimal generator of (2.4).
Now, supposef(x) and σ(x) can be linearized, respectively, as
(2.7)f(x)=Ax+F0(x),F0(0)=0,σ(x)=Cx+F1(x),F1(0)=0,
then the linearized stochastic system of (2.1) becomes
(2.8)dx=(Ax+B0w+F0(x))dt+(Cx+F1(x))dw0,
where A and C are constant matrices.
Consider the following filter for the estimation ofz(t):
(2.9)dx^=Afx^dt+Bfdy,x^(0)=x^0,z^=Dx^,
where x^∈ℛn. Let ξ'=[x′x′-x^′], z~=z-z^, then
(2.10)dξ=A~ξdt+D~1ξdw0+D~2ξdw1+F~1dt+F~2dw0+F~3wdt,
where
(2.11)A~=[A0A-BfA1-Af-Af],D~1=[C0C0],D~2=[00-BfC10],F~1=[F0(x)F0(x)],F~2=[F1(x)F1(x)],F~3=[B0B0-BfB1].
For any given disturbance attenuation level γ>0, one wants to find Af,Bf, such that
(2.12)∥z~(t)∥L22<γ2∥w(t)∥L22
holds for any w∈ℒ2(ℛ+,ℛq). Define the H∞ performance index as
(2.13)Js=∥z~(t)∥L22-γ2∥w(t)∥L22.
Obviously, (2.12) holds iff Js<0. As in [12], H∞ and mixed H2/H∞-based robust state estimation problems are formulated as follows.(i)
StochasticH∞ filtering problem: given γ>0, find an estimator x^ of the form (2.9) leading (2.10) to being internally stable; Moreover, Js<0 for all nonzero w∈ℒ2(ℛ+,ℛn) with ξ(0)=0.(ii)
StochasticH2/H∞ filtering problem: of all the H∞ filter of (i), one finds the one that minimizes the steady error variance
(2.14)limt→∞E[z~′(t)z~(t)],  where in this case, w(t)=η˙, η is taken as a standard Wiener process, independent of w0(t) and w1(t), so w(t) is a white noise. (2.2) and (2.8) can be written as (see, e.g., [22])
(2.15)dy(t)=A1x(t)dt+B1dη(t)+C1x(t)dw1(t),dx(t)=(Ax(t)+F0(x(t)))dt+(Cx(t)+F1(x(t)))dw0(t)+B0dη(t),
respectively.


## 3. StochasticH∞ and Mixed H2/H∞ Filter Design

In this section, we will discuss, respectively, stochasticH∞ and mixed H2/H∞ filtering problems.


### 3.1. StochasticH∞ Filter Design

In this section, some sufficient conditions are given forH∞ filter design; our main results are as follows.Theorem 3.1.
Suppose there exists a scalarλ>0, such that
(3.1)|Fi(x)|≤λ|x|,i=0,1,∀x∈ℛn.
If the following matrix inequalities
(3.2)PA~+A~′P+2D~1′PD~1+D~2′PD~2+P+6λ2αI+Q+1γ2PF~3F~3′P<0,(3.3)0<P≤αI
have a solution P>0, α>0, then (2.10) is internally stable and H∞ filtering performance Js<0, where Q=(0D)′(0D).Proof.
We first show (2.10) to be internally stable, that is, the following system
(3.4)dξ=A~ξdt+D~1ξdw0+D~2ξdw1+F~1dt+F~2dw0(t)
is asymptotically mean square stable. Let ℒξ be the infinitesimal operator of (3.4), V(ξ)=ξ'Pξ with αI≥P>0 to be determined. According to Lemma 2.4, in order to show (3.4) to be internally stable, we only need to show
(3.5)ℒξV(ξ)≤-k3|ξ|2
for some k3>0. Note that
(3.6)ℒξV(ξ)=∂V′(ξ)∂ξ(A~ξ+F~1)+12(D~1ξ+F~2)′∂2V(ξ)∂ξ2(D~1ξ+F~2)+12(D~2ξ)′∂2V(ξ)∂ξ2(D~2ξ)=ξ′(PA~+A~′P+D~1′PD~1+D~2′PD~2)ξ+2F~1′Pξ+F~2′PF~2+2ξ'D~1′PF~2.
By condition (3.1), we have
(3.7)2F~1′Pξ≤ξ'Pξ+F~1′PF~1≤ξ'Pξ+αF~1′F~1=ξ′Pξ+2αF0′F0≤ξ′Pξ+2α|F0|2≤ξ′Pξ+2αλ2|ξ|2.
Similarly,
(3.8)2ξ′D~1′PF~2≤ξ′D~1′PD~1ξ+2αλ2|ξ|2,F~2′PF~2≤2αλ2|ξ|2.
Substituting (3.7), (3.8) into (3.6) and considering (3.2), it follows
(3.9)ℒξV(ξ)≤ξ′(PA~+A~′P+2D~1′PD~1+D~2′PD~2+P+6αλ2I)ξ<-ξ′(Q+1γ2PF~3F~3′P)ξ≤0.
By Lemma 2.4, the internal stability of (2.10) is proved.
Secondly, we further show theH∞ filtering performance Js<0. Let ℒξ,w be the infinitesimal generator of (2.10). For V(ξ)=ξ'Pξ, it is easy to show that
(3.10)ℒξ,wV(ξ)=ℒξV(ξ)+2ξ′PF~3w.
For any T>0 and ξ(0)=0, we have
(3.11)Js(T):=E∫0T‍[|z~(t)|2-γ2|w(t)|2]dt=E∫0T‍{[|z~(t)|2-γ2|w(t)|2]dt+d(ξ′Pξ)}-E[ξ(T)Pξ(T)]≤E∫0T‍[|z~(t)|2-γ2|w(t)|2+ℒξ,wV(ξ)]dt.
Note that
(3.12)ℒξ,wV(ξ)≤ξ′(PA~+A~′P+2D~1′PD~1+D~2′PD~2+P+6λ2αI)ξ+2ξ′PF~3w,|z~(t)|2=ξ′Qξ.
So
(3.13)|z~(t)|2-γ2|w(t)|2+ℒξ,wV(ξ)≤[ξw]′[Λ11PF~3F~3′P-γ2I][ξw]<0,
where
(3.14)Λ11:=PA~+A~′P+2D~1′PD~1+D~2′PD~2+P+6λ2αI+Q.
By the well-known Schur’s complement and (3.2), there exists ɛ>0, such that
(3.15)[Λ11PF~3F~3′P-γ2I]<-ɛI.
Summarizing the above analysis, (3.11) yields
(3.16)Js(T)≤-ɛE∫0T‍(|ξ(t)|2+w(t)|2)dt≤-ɛE∫0T‍|w(t)|2dt.
So for any T>0, E∫0T‍|z~(t)|2dt≤(γ2-ɛ)E∫0T‍|w(t)|2dt.
LetT→∞, then
(3.17)∥z~(t)∥L22≤(γ2-ɛ)∥w(t)∥L22
which yields Js<0. This theorem is proved.Theorem3.1 only has theoretical sense, because it is difficult to be used in designing H∞ filter. The following result is of more important in practice.Theorem 3.2.
Under the condition of Theorem3.1, if the following LMIs
(3.18)[P11-αI00P22-αI]<0,(3.19)[a11A'P22-A1′Z1′-Z′2C'P112C'P22-C1′Z1′P11B0P22A-Z1A1-Za22000P22B0-Z1B12P11C0-P110002P22C00-P2200-Z1C1000-P220B0′P11B0′P22-B1′Z1′000-γ2I]<0
have solutions P11>0,P22>0,α>0,Z1∈ℛn×r, Z∈ℛn×n, then (2.10) is internally stable and Js<0.
Moreover,(3.20)dx^=P22-1Zx^dt+P22-1Z1dy
is the corresponding H∞ filter. In (3.19), a11=P11A+A'P11+6λ2αI+P11, a22=-Z-Z'+6λ2αI+D'D+P22.Proof.
By Schur’s complement, (3.2) is equivalent to
(3.21)[PA~+A~′P+P+6λ2αI+Q2D~1′PD~2′PPF~32PD~1-P00PD~20-P0F~3′P00-γ2I]<0.
Taking P=diag(P11,P22) and substituting (2.11) into (3.21), we have
(3.22)[Ψ11Ψ12′Ψ13′ϕ14′Ψ12Ψ2200Ψ130Ψ330Ψ1400Ψ44]<0,
where
(3.23)Ψ11=[P11A+A′P11+6λ2αI+P11(A-BfA1-Af)′P22P22(A-BfA1-Af)-P22Af-Af′P22+6λ2αI+P22+D′D],Ψ22=Ψ33=-P=[-P1100-P22],Ψ44=-γ2I,Ψ12′=[2C′P112C′P2200],Ψ13′=[0-C1′Bf′P2200],Ψ14′=[P11B0P22(B0-BfB1)].
(3.22) is equivalent to
(3.24)[a¯11(A-BfA1-Af)'P222C'P112C'P22-C1′Bf′P22P11B0P22(A-BfA1-Af)a¯22000P22(B0-BfB1)2P11C0-P110002P22C00-P2200-P22BfC1000-P220B0′P11(B0-BfB1)'P22000-γ2I]<0,
where a¯11=P11A+A′P11+6λ2αI+P11, a¯22=-P22Af-A'fP22+6λ2αI+P11. Let P22Af=Z,P22Bf=Z1, then (3.22) becomes (3.19). From our assumption, Af=P22-1Z,Bf=P22-1Z1, so an H∞ filtering equation is constructed as in the form of (3.20). Theorem 3.2 is proved.


### 3.2. MixedH2/H∞ Filtering

To design the mixed stochasticH2/H∞ filter, we need to choose the one from the set of all H∞ filters, which also minimizes the estimation error variance, or concretely speaking, minimizes the H2 performance
(3.25)J2:=limt→∞E{z~′(t)z~(t)}=limt→∞E{ξ′(t)(0I)′D′D(0I)ξ(t)}=limt→∞Tr{D(0I)Eξ(t)ξ′(t)(0I)′D′}.
Two performances Js in (2.13) and J2 in (3.25) associated with H∞ robustness and H2 optimization have constructed, respectively. Now, we need to design the mixed H2/H∞ filter to maximize Js and minimize J2. Consider the following linear stochastic constant system
(3.26)dξ=A11ξdt+∑i=1l‍Biiξdwi,
where {wi,i=1,…,l} are independent, standard Wiener processes. The following lemma will be used in this section.Lemma 3.3 (see [23]).
System (3.26) is exponentially mean square stable iff for any R>0, the following Lyapunov-type equation
(3.27)PA11+A11′P+∑i=1l‍Bii′PBii=-R
has a unique positive definite solution P>0.
In the next, for simplicity, when (3.26) is exponentially stable, one also says (A11,B11,…,Bll) is stable.
As we have pointed out before, at this stage, we assumew(t)=η˙(t); (2.10) accordingly becomes
(3.28)dξ=A~ξdt+D~1ξdw0+D~2ξdw1+F~1dt+F~2dw0+F~3dη.
Let X(t)=E[ξ(t)ξ'(t)] in (3.28), then by Itô’s formula, we have
(3.29)X˙(t)=A~X(t)+X(t)A~′+E[F~1ξ′+ξF~1′]+D~1XD~1′+E[D~1ξF~2′+F~2ξ'D~1′]+E[F~2F~2′]+D~2X(t)D~2′+F~3F~3′.
By means of
(3.30)E[F~1ξ'+ξF~1′]≤E[F~1F~1′]+X(t),E[D~1ξF~2′+F~2ξ'D~1′]≤D~1XD~1′+E[F~2F~2′],
we have
(3.31)X˙(t)≤A~X(t)+X(t)A~'+2D~1X(t)D~1′+D~2X(t)D~2′+X(t)+2E[F~2F~2′]+E[F~1F~1′]+F~3F~3′.
Now, we suppose Fi(x)(i=0,1) satisfy
(3.32)Fi(x)Fi′(x)≤Gixx′Gi′,i=0,1,∀x∈ℛn,
where G1,G2 are constant matrices of suitable dimensions. At this stage,
(3.33)F~iF~i′=[I0II][FiFi′000][II0I]≤[I0II][Gixx'Gi′000][II0I]=[I0II][Gi000]ξξ'[Gi′000][II0I]=[Gi0Gi0]ξξ'[Gi′Gi′00]:=G~iξξ′G~i′,i=0,1,
where
(3.34)G~i=[Gi0Gi0].
So (3.31) becomes
(3.35)X˙(t)≤A~X(t)+X(t)A~'+2D~1X(t)D~1′+D~2X(t)D~2′+X(t)+2G~2X(t)G~2′+G~1X(t)G~1′+F~3F~3′.
In addition, if X1(t) solves
(3.36)X˙1(t)=A~X1(t)+X1(t)A~'+2D~1X1(t)D~1′+D~2X1(t)D~2′+X1(t)+2G~2X1(t)G~2′+G~1X1(t)G~1′+F~3F~3′X1(0)=X(0)
then it is easy to prove that X(t)≤X1(t). Denoting X¯1:=limt→∞X1(t), where X¯1 satisfies
(3.37)A~X¯1+X¯1A~'+2D~1X¯1D~1′+D~2X¯1D~2′+2G~2X¯1G~2′+G~1X¯1G~1′+X¯1+F~3F~3′=0.
Obviously, limt→∞X(t)≤X¯1, accordingly,
(3.38)J2≤Tr{D(0I)X¯1(0I)′D′}=Tr{X¯1Q}.
As in [12, 24], it is easily seen the following fact.Lemma 3.4.
IfP^ is a solution of
(3.39)A~'P^+P^A~+2D~1′P^D~1+D~2′P^D~2+2G~2′P^G~2+G~1′P^G~1+Q+P^=0
then Tr(X¯1Q)=Tr(P^(F~3F~3′)).
Secondly, supposeP>0 satisfies
(3.40)A~′P+PA~+2D~1′PD~1+D~2′PD~2+Q+P+2G~2′PG~2+G~1′PG~1<0.
By means of Lemma 3.3, one can show P>P^. So we have the following lemma.Lemma 3.5.
P
>
P
^, where P and P^ stand for the positive definite solutions of (3.40) and (3.39), respectively.
From Lemmas3.4–3.5, it gives
(3.41)J2=limt→∞Tr{D(0I)X(t)(0I)′D′}≤limt→∞Tr{D(0I)X1(t)(0I)′D′}=Tr{D(0I)X¯1(0I)′D′}=Tr{X¯1Q}=Tr(P^F~3F~3′)=Tr(F~3′P^F~3)≤Tr(F~3′PF~3):=J^2.
Hence, to solve the mixed stochastic H2/H∞ filtering problem, we seek to minimize an upper-bound on J^2 subject to (3.2), (3.3), and
(3.42)PA~+A~′P+2D~1′PD~1+D~2′PD~2′+P+2G~2′PG~2+G~1′PG~1+Q<0.
(3.42) having a positive definite solution P>0 is equivalent to
(3.43)[PA~+A~'P+P+Q2D~1′PD~2′PG~′1P2G~2′P2PD~1-P000PD~20-P00PG~100-P02PG~2000-P]<0.
A suboptimal H2/H∞ filtering can be obtained by minimizing Tr(H) subject to (3.2), (3.3), (3.43), and
(3.44)H-F~3′PF~3>0.
(3.44) is equivalent to
(3.45)[HF~3′PPF~3P]>0.
We still take P=diag(P11,P22)>0, P22Bf=Z1, P22Af=Z, then (3.3), (3.2), (3.43), and (3.45) become, respectively, as (3.18), (3.19),
(3.46)[γ11γ122C'P112C'P22-C1′Z1′G1′P11G1′P22G2′P11G2′P22γ21γ2200000002P11C0-P110000002P22C00-P2200000-Z1C1000-P220000P11G10000-P11000P22G100000-P2200P11G2000000-P110P22G20000000-P22]<0,[HB0′P11B0′P22-B1′Z1′P11B0P110P22B0-Z1B10P22]>0,
where γ11=P11A+A′P11+P11, γ12=A′P22-A1′Z1′-Z′, γ21=P22A-Z1A1-Z, γ22=-Z-Z′+D′D+P22. Therefore, we have the following theorem.Theorem 3.6.
Under the conditions of Theorem3.2 and assumption (3.32), if there exists a solution (P11>0,P22>0,Z,Z1,α>0) to (3.18), (3.19), (3.46), then a suboptimal mixed stochastic H2/H∞ filtering is obtained by solving P11 and P22 from the following convex optimization problem: minP11,P22,Z,Z1,αTr(H) subject to (3.18),(3.19), (3.46), and the corresponding filter is given by (3.20).Remark 3.7.
In the proof of Theorems3.2 and 3.6, the matrix P is chosen as diag(P11,P22) for simplicity. In order to reduce the conservatism of the conditions, the matrix P can also be chosen as [P11P12P12′P22]. However, this case will increase the complexity of computation.


## 3.1. StochasticH∞ Filter Design

In this section, some sufficient conditions are given forH∞ filter design; our main results are as follows.Theorem 3.1.
Suppose there exists a scalarλ>0, such that
(3.1)|Fi(x)|≤λ|x|,i=0,1,∀x∈ℛn.
If the following matrix inequalities
(3.2)PA~+A~′P+2D~1′PD~1+D~2′PD~2+P+6λ2αI+Q+1γ2PF~3F~3′P<0,(3.3)0<P≤αI
have a solution P>0, α>0, then (2.10) is internally stable and H∞ filtering performance Js<0, where Q=(0D)′(0D).Proof.
We first show (2.10) to be internally stable, that is, the following system
(3.4)dξ=A~ξdt+D~1ξdw0+D~2ξdw1+F~1dt+F~2dw0(t)
is asymptotically mean square stable. Let ℒξ be the infinitesimal operator of (3.4), V(ξ)=ξ'Pξ with αI≥P>0 to be determined. According to Lemma 2.4, in order to show (3.4) to be internally stable, we only need to show
(3.5)ℒξV(ξ)≤-k3|ξ|2
for some k3>0. Note that
(3.6)ℒξV(ξ)=∂V′(ξ)∂ξ(A~ξ+F~1)+12(D~1ξ+F~2)′∂2V(ξ)∂ξ2(D~1ξ+F~2)+12(D~2ξ)′∂2V(ξ)∂ξ2(D~2ξ)=ξ′(PA~+A~′P+D~1′PD~1+D~2′PD~2)ξ+2F~1′Pξ+F~2′PF~2+2ξ'D~1′PF~2.
By condition (3.1), we have
(3.7)2F~1′Pξ≤ξ'Pξ+F~1′PF~1≤ξ'Pξ+αF~1′F~1=ξ′Pξ+2αF0′F0≤ξ′Pξ+2α|F0|2≤ξ′Pξ+2αλ2|ξ|2.
Similarly,
(3.8)2ξ′D~1′PF~2≤ξ′D~1′PD~1ξ+2αλ2|ξ|2,F~2′PF~2≤2αλ2|ξ|2.
Substituting (3.7), (3.8) into (3.6) and considering (3.2), it follows
(3.9)ℒξV(ξ)≤ξ′(PA~+A~′P+2D~1′PD~1+D~2′PD~2+P+6αλ2I)ξ<-ξ′(Q+1γ2PF~3F~3′P)ξ≤0.
By Lemma 2.4, the internal stability of (2.10) is proved.
Secondly, we further show theH∞ filtering performance Js<0. Let ℒξ,w be the infinitesimal generator of (2.10). For V(ξ)=ξ'Pξ, it is easy to show that
(3.10)ℒξ,wV(ξ)=ℒξV(ξ)+2ξ′PF~3w.
For any T>0 and ξ(0)=0, we have
(3.11)Js(T):=E∫0T‍[|z~(t)|2-γ2|w(t)|2]dt=E∫0T‍{[|z~(t)|2-γ2|w(t)|2]dt+d(ξ′Pξ)}-E[ξ(T)Pξ(T)]≤E∫0T‍[|z~(t)|2-γ2|w(t)|2+ℒξ,wV(ξ)]dt.
Note that
(3.12)ℒξ,wV(ξ)≤ξ′(PA~+A~′P+2D~1′PD~1+D~2′PD~2+P+6λ2αI)ξ+2ξ′PF~3w,|z~(t)|2=ξ′Qξ.
So
(3.13)|z~(t)|2-γ2|w(t)|2+ℒξ,wV(ξ)≤[ξw]′[Λ11PF~3F~3′P-γ2I][ξw]<0,
where
(3.14)Λ11:=PA~+A~′P+2D~1′PD~1+D~2′PD~2+P+6λ2αI+Q.
By the well-known Schur’s complement and (3.2), there exists ɛ>0, such that
(3.15)[Λ11PF~3F~3′P-γ2I]<-ɛI.
Summarizing the above analysis, (3.11) yields
(3.16)Js(T)≤-ɛE∫0T‍(|ξ(t)|2+w(t)|2)dt≤-ɛE∫0T‍|w(t)|2dt.
So for any T>0, E∫0T‍|z~(t)|2dt≤(γ2-ɛ)E∫0T‍|w(t)|2dt.
LetT→∞, then
(3.17)∥z~(t)∥L22≤(γ2-ɛ)∥w(t)∥L22
which yields Js<0. This theorem is proved.Theorem3.1 only has theoretical sense, because it is difficult to be used in designing H∞ filter. The following result is of more important in practice.Theorem 3.2.
Under the condition of Theorem3.1, if the following LMIs
(3.18)[P11-αI00P22-αI]<0,(3.19)[a11A'P22-A1′Z1′-Z′2C'P112C'P22-C1′Z1′P11B0P22A-Z1A1-Za22000P22B0-Z1B12P11C0-P110002P22C00-P2200-Z1C1000-P220B0′P11B0′P22-B1′Z1′000-γ2I]<0
have solutions P11>0,P22>0,α>0,Z1∈ℛn×r, Z∈ℛn×n, then (2.10) is internally stable and Js<0.
Moreover,(3.20)dx^=P22-1Zx^dt+P22-1Z1dy
is the corresponding H∞ filter. In (3.19), a11=P11A+A'P11+6λ2αI+P11, a22=-Z-Z'+6λ2αI+D'D+P22.Proof.
By Schur’s complement, (3.2) is equivalent to
(3.21)[PA~+A~′P+P+6λ2αI+Q2D~1′PD~2′PPF~32PD~1-P00PD~20-P0F~3′P00-γ2I]<0.
Taking P=diag(P11,P22) and substituting (2.11) into (3.21), we have
(3.22)[Ψ11Ψ12′Ψ13′ϕ14′Ψ12Ψ2200Ψ130Ψ330Ψ1400Ψ44]<0,
where
(3.23)Ψ11=[P11A+A′P11+6λ2αI+P11(A-BfA1-Af)′P22P22(A-BfA1-Af)-P22Af-Af′P22+6λ2αI+P22+D′D],Ψ22=Ψ33=-P=[-P1100-P22],Ψ44=-γ2I,Ψ12′=[2C′P112C′P2200],Ψ13′=[0-C1′Bf′P2200],Ψ14′=[P11B0P22(B0-BfB1)].
(3.22) is equivalent to
(3.24)[a¯11(A-BfA1-Af)'P222C'P112C'P22-C1′Bf′P22P11B0P22(A-BfA1-Af)a¯22000P22(B0-BfB1)2P11C0-P110002P22C00-P2200-P22BfC1000-P220B0′P11(B0-BfB1)'P22000-γ2I]<0,
where a¯11=P11A+A′P11+6λ2αI+P11, a¯22=-P22Af-A'fP22+6λ2αI+P11. Let P22Af=Z,P22Bf=Z1, then (3.22) becomes (3.19). From our assumption, Af=P22-1Z,Bf=P22-1Z1, so an H∞ filtering equation is constructed as in the form of (3.20). Theorem 3.2 is proved.


## 3.2. MixedH2/H∞ Filtering

To design the mixed stochasticH2/H∞ filter, we need to choose the one from the set of all H∞ filters, which also minimizes the estimation error variance, or concretely speaking, minimizes the H2 performance
(3.25)J2:=limt→∞E{z~′(t)z~(t)}=limt→∞E{ξ′(t)(0I)′D′D(0I)ξ(t)}=limt→∞Tr{D(0I)Eξ(t)ξ′(t)(0I)′D′}.
Two performances Js in (2.13) and J2 in (3.25) associated with H∞ robustness and H2 optimization have constructed, respectively. Now, we need to design the mixed H2/H∞ filter to maximize Js and minimize J2. Consider the following linear stochastic constant system
(3.26)dξ=A11ξdt+∑i=1l‍Biiξdwi,
where {wi,i=1,…,l} are independent, standard Wiener processes. The following lemma will be used in this section.Lemma 3.3 (see [23]).
System (3.26) is exponentially mean square stable iff for any R>0, the following Lyapunov-type equation
(3.27)PA11+A11′P+∑i=1l‍Bii′PBii=-R
has a unique positive definite solution P>0.
In the next, for simplicity, when (3.26) is exponentially stable, one also says (A11,B11,…,Bll) is stable.
As we have pointed out before, at this stage, we assumew(t)=η˙(t); (2.10) accordingly becomes
(3.28)dξ=A~ξdt+D~1ξdw0+D~2ξdw1+F~1dt+F~2dw0+F~3dη.
Let X(t)=E[ξ(t)ξ'(t)] in (3.28), then by Itô’s formula, we have
(3.29)X˙(t)=A~X(t)+X(t)A~′+E[F~1ξ′+ξF~1′]+D~1XD~1′+E[D~1ξF~2′+F~2ξ'D~1′]+E[F~2F~2′]+D~2X(t)D~2′+F~3F~3′.
By means of
(3.30)E[F~1ξ'+ξF~1′]≤E[F~1F~1′]+X(t),E[D~1ξF~2′+F~2ξ'D~1′]≤D~1XD~1′+E[F~2F~2′],
we have
(3.31)X˙(t)≤A~X(t)+X(t)A~'+2D~1X(t)D~1′+D~2X(t)D~2′+X(t)+2E[F~2F~2′]+E[F~1F~1′]+F~3F~3′.
Now, we suppose Fi(x)(i=0,1) satisfy
(3.32)Fi(x)Fi′(x)≤Gixx′Gi′,i=0,1,∀x∈ℛn,
where G1,G2 are constant matrices of suitable dimensions. At this stage,
(3.33)F~iF~i′=[I0II][FiFi′000][II0I]≤[I0II][Gixx'Gi′000][II0I]=[I0II][Gi000]ξξ'[Gi′000][II0I]=[Gi0Gi0]ξξ'[Gi′Gi′00]:=G~iξξ′G~i′,i=0,1,
where
(3.34)G~i=[Gi0Gi0].
So (3.31) becomes
(3.35)X˙(t)≤A~X(t)+X(t)A~'+2D~1X(t)D~1′+D~2X(t)D~2′+X(t)+2G~2X(t)G~2′+G~1X(t)G~1′+F~3F~3′.
In addition, if X1(t) solves
(3.36)X˙1(t)=A~X1(t)+X1(t)A~'+2D~1X1(t)D~1′+D~2X1(t)D~2′+X1(t)+2G~2X1(t)G~2′+G~1X1(t)G~1′+F~3F~3′X1(0)=X(0)
then it is easy to prove that X(t)≤X1(t). Denoting X¯1:=limt→∞X1(t), where X¯1 satisfies
(3.37)A~X¯1+X¯1A~'+2D~1X¯1D~1′+D~2X¯1D~2′+2G~2X¯1G~2′+G~1X¯1G~1′+X¯1+F~3F~3′=0.
Obviously, limt→∞X(t)≤X¯1, accordingly,
(3.38)J2≤Tr{D(0I)X¯1(0I)′D′}=Tr{X¯1Q}.
As in [12, 24], it is easily seen the following fact.Lemma 3.4.
IfP^ is a solution of
(3.39)A~'P^+P^A~+2D~1′P^D~1+D~2′P^D~2+2G~2′P^G~2+G~1′P^G~1+Q+P^=0
then Tr(X¯1Q)=Tr(P^(F~3F~3′)).
Secondly, supposeP>0 satisfies
(3.40)A~′P+PA~+2D~1′PD~1+D~2′PD~2+Q+P+2G~2′PG~2+G~1′PG~1<0.
By means of Lemma 3.3, one can show P>P^. So we have the following lemma.Lemma 3.5.
P
>
P
^, where P and P^ stand for the positive definite solutions of (3.40) and (3.39), respectively.
From Lemmas3.4–3.5, it gives
(3.41)J2=limt→∞Tr{D(0I)X(t)(0I)′D′}≤limt→∞Tr{D(0I)X1(t)(0I)′D′}=Tr{D(0I)X¯1(0I)′D′}=Tr{X¯1Q}=Tr(P^F~3F~3′)=Tr(F~3′P^F~3)≤Tr(F~3′PF~3):=J^2.
Hence, to solve the mixed stochastic H2/H∞ filtering problem, we seek to minimize an upper-bound on J^2 subject to (3.2), (3.3), and
(3.42)PA~+A~′P+2D~1′PD~1+D~2′PD~2′+P+2G~2′PG~2+G~1′PG~1+Q<0.
(3.42) having a positive definite solution P>0 is equivalent to
(3.43)[PA~+A~'P+P+Q2D~1′PD~2′PG~′1P2G~2′P2PD~1-P000PD~20-P00PG~100-P02PG~2000-P]<0.
A suboptimal H2/H∞ filtering can be obtained by minimizing Tr(H) subject to (3.2), (3.3), (3.43), and
(3.44)H-F~3′PF~3>0.
(3.44) is equivalent to
(3.45)[HF~3′PPF~3P]>0.
We still take P=diag(P11,P22)>0, P22Bf=Z1, P22Af=Z, then (3.3), (3.2), (3.43), and (3.45) become, respectively, as (3.18), (3.19),
(3.46)[γ11γ122C'P112C'P22-C1′Z1′G1′P11G1′P22G2′P11G2′P22γ21γ2200000002P11C0-P110000002P22C00-P2200000-Z1C1000-P220000P11G10000-P11000P22G100000-P2200P11G2000000-P110P22G20000000-P22]<0,[HB0′P11B0′P22-B1′Z1′P11B0P110P22B0-Z1B10P22]>0,
where γ11=P11A+A′P11+P11, γ12=A′P22-A1′Z1′-Z′, γ21=P22A-Z1A1-Z, γ22=-Z-Z′+D′D+P22. Therefore, we have the following theorem.Theorem 3.6.
Under the conditions of Theorem3.2 and assumption (3.32), if there exists a solution (P11>0,P22>0,Z,Z1,α>0) to (3.18), (3.19), (3.46), then a suboptimal mixed stochastic H2/H∞ filtering is obtained by solving P11 and P22 from the following convex optimization problem: minP11,P22,Z,Z1,αTr(H) subject to (3.18),(3.19), (3.46), and the corresponding filter is given by (3.20).Remark 3.7.
In the proof of Theorems3.2 and 3.6, the matrix P is chosen as diag(P11,P22) for simplicity. In order to reduce the conservatism of the conditions, the matrix P can also be chosen as [P11P12P12′P22]. However, this case will increase the complexity of computation.


## 4. Numerical Example

Example 4.1.
Consider the following nonlinear stochastic system governed by Itô differential equation(4.1)dx=(Ax+B0w+F0(x))dt+(Cx+F1(x))dw0,dy=(A1+B1w)dt+C1xdw1,z=Dx,
where
(4.2)A=[-31/2-1-3],B0=[10],C=[1000],F0(x)=0.3tanh(x),F1(x)=0.3sin(x),A1=[-111-1],B1=[01],C1=[1001],D=[01],w=11+2t,t≥0.
Consider the following filter for the estimation of z(t):
(4.3)dx^=Afx^dt+Bfdy,z^=Dx^.
Setting γ=0.9, and using the LMI control toolbox of Matlab, the estimation gains of H∞ filter are derived from Theorem 3.2:
(4.4)Af=[5.62313.7259-0.16178.2289],Bf=[0.1812-1.8190-0.25250.4635].
From Theorem 3.6, the estimation gains of H2/H∞ filter are obtained as follows:
(4.5)Af=[4.14493.4665-0.24696.3382],Bf=[0.5270-1.2388-0.36930.3445].
The initial condition in the simulation is assumed to be ξ0=[0.30.2-0.02-0.05]'. Figures 1 and 2 show the trajectories of x1(t),  x^1(t),  x2(t),  x^2(t) by using the proposed H∞ and H2/H∞ filters, respectively. The trajectories of the estimation error z~(t) for H∞ and H2/H∞ filters are shown in Figures 3 and 4, respectively. From Figures 3 and 4, it is obvious that the performance of the proposed H2/H∞ filter is better than that of the H∞ filter.
In [15], the H∞ and H2/H∞ filters for general nonlinear stochastic systems were obtained by solving a second-order nonlinear HJI. Generally, it is difficult to solve the HJI. In fact, for the special nonlinear stochastic system (4.1), the H∞ and H2/H∞ filtering problems can be solved via the LMI technique instead of the HJI according to Theorems 3.2 and 3.6 in this paper. Simulation results show the effectiveness of the proposed method.Trajectories ofx1(t),x^1(t) and x2(t),x^2(t) for the proposed H∞ filter.
(a)
(b)Trajectories ofx1(t),  x^1(t) and x2(t),  x^2(t) for the proposed H2/H∞ filter.
(a)
(b)Figure 3
Trajectory of the estimation errorz~(t) for the proposed H∞ filter.Figure 4
Trajectory of the estimation errorz~(t) for the proposed H2/H∞ filter.


## 5. Conclusions

In this paper, we have discussed the robustH∞ filtering problem for a class of nonlinear stochastic systems. Meanwhile, the mixed H2/H∞ filtering analysis is also considered. Since the results can be solved by LMIs, the proposed method has much advantage in practical computation. Although we only demand the state equation to be nonlinear, one can tackle the case that when both the state and measurement equations are nonlinear.



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*Source: 750841-2012-10-02.xml*