Robust 𝐻₂/𝐻_∞ Filter Design for a Class of Nonlinear Stochastic Systems with State-Dependent Noise

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 robust 𝐻_∞ 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.

1. Introduction

Over the past decades, the robust 𝐻_∞ 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, stochastic 𝐻_∞ 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 whose ℒ₂ 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 Section 2, 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. 𝒮_𝑛: 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|∶=(∑^𝑛_𝑖=1𝑥²_𝑖)^1/2, that is, |𝑥| denotes the Euclidean 2-norm of 𝑥, where 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^′∈ℛ^𝑛. ℒ²(ℛ₊,ℛ^𝑙): the space of nonanticipative stochastic processes 𝑦(𝑡) with respect to filter ℱ_𝑡 satisfying ‖𝑦(𝑡)‖²_𝐿2∶=𝐸∫^∞₀|𝑦(𝑡)|²𝑑𝑡<∞. 𝐶⁰₂({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: 𝑑𝑥(𝑡)=𝑓(𝑥(𝑡))+𝐵₀𝑤(𝑡)𝑑𝑡+𝜎(𝑥(𝑡))𝑑𝑤₀(𝑡), (2.1) with the following measurement equation: 𝑑𝑦(𝑡)=𝐴₁𝑥(𝑡)+𝐵₁𝑤(𝑡)𝑑𝑡+𝐶₁𝑥(𝑡)𝑑𝑤₁(𝑡), (2.2) and the controlled output 𝑧(𝑡)=𝐷𝑥(𝑡). (2.3) In the above, 𝑥(𝑡)∈ℛ^𝑛 is called the system state, 𝑦(𝑡)∈ℛ^𝑟 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 (ℱ_𝑡)_𝑡∈ℛ+ of 𝜎-algebras ℱ_𝑡⊂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₊,ℛ^𝑞) 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 𝑥_𝑠,𝜉(𝑡) for any 𝑡≥𝑠≥0 and initial state 𝑥(𝑠)=𝜉∈ℛ^𝑛; see [21].

Now, we first introduce the following definitions.

Definition 2.1 (see [9]). We say that the equilibrium point 𝑥≡0 of system 𝑑𝑥(𝑡)=𝑓(𝑥(𝑡))𝑑𝑡+𝜎(𝑥(𝑡))𝑑𝑤₀(𝑡) (2.4) is exponentially mean square stable, if for some positive constants 𝜌,𝜚, 𝐸|𝑥(𝑡)|²≤𝜌|𝑥(0)|²exp−𝜚𝑡,𝑡≥0. (2.5)

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)∈𝐶⁰₂({𝑡>0}×ℛ^𝑛) such that 𝑘₁|𝑥|²≤𝑉(𝑡,𝑥)≤𝑘₂|𝑥|²,ℒ𝑉(𝑡,𝑥)≤−𝑘₃|𝑥|² (2.6) for some positive constants 𝑘₁,𝑘₂,𝑘₃, where ℒ is the so-called an infinitesimal generator of (2.4).
Now, suppose 𝑓(𝑥) and 𝜎(𝑥) can be linearized, respectively, as 𝑓(𝑥)=𝐴𝑥+𝐹₀(𝑥),𝐹₀(0)𝜎(𝑥)=0,=𝐶𝑥+𝐹₁(𝑥),𝐹₁(0)=0, (2.7) then the linearized stochastic system of (2.1) becomes 𝑑𝑥=𝐴𝑥+𝐵₀𝑤+𝐹₀(𝑥)𝑑𝑡+𝐶𝑥+𝐹₁(𝑥)𝑑𝑤₀, (2.8) where 𝐴 and 𝐶 are constant matrices.
Consider the following filter for the estimation of 𝑧(𝑡): 𝑑𝑥=𝐴_𝑓𝑥𝑑𝑡+𝐵_𝑓𝑑𝑦,𝑥(0)=𝑥₀,𝑧=𝐷𝑥, (2.9) where 𝑥∈ℛ^𝑛. Let 𝜉′=[𝑥^′𝑥^′−𝑥^′], 𝑧=𝑧−𝑧, then 𝑑𝜉=𝐴𝜉𝑑𝑡+𝐷₁𝜉𝑑𝑤₀+𝐷₂𝜉𝑑𝑤₁+𝐹₁𝑑𝑡+𝐹₂𝑑𝑤₀+𝐹₃𝑤𝑑𝑡, (2.10) where 𝐴=𝐴0𝐴−𝐵_𝑓𝐴₁−𝐴_𝑓−𝐴_𝑓,𝐷₁=,𝐷𝐶0𝐶0²=00−𝐵_f𝐶₁0,𝐹₁=𝐹₀(𝑥)𝐹₀(𝑥),𝐹₂=𝐹₁(𝑥)𝐹₁(𝑥),𝐹₃=𝐵₀𝐵₀−𝐵_𝑓𝐵₁. (2.11) For any given disturbance attenuation level 𝛾>0, one wants to find 𝐴_𝑓,𝐵_𝑓, such that ‖𝑧(𝑡)‖²_𝐿2<𝛾²‖𝑤(𝑡)‖²_𝐿2 (2.12) holds for any 𝑤∈ℒ²(ℛ₊,ℛ^𝑞). Define the 𝐻_∞ performance index as 𝐽_𝑠=‖𝑧(𝑡)‖²_𝐿2−𝛾²‖𝑤(𝑡)‖²_𝐿2. (2.13) Obviously, (2.12) holds iff 𝐽_𝑠<0. As in [12], 𝐻_∞ and mixed 𝐻₂/𝐻_∞-based robust state estimation problems are formulated as follows.(i)Stochastic 𝐻_∞ filtering problem: given 𝛾>0, find an estimator 𝑥 of the form (2.9) leading (2.10) to being internally stable; Moreover, 𝐽_𝑠<0 for all nonzero 𝑤∈ℒ²(ℛ₊,ℛ^𝑛) with 𝜉(0)=0.(ii)Stochastic 𝐻₂/𝐻_∞ filtering problem: of all the 𝐻_∞ filter of (i), one finds the one that minimizes the steady error variance lim_𝑡→∞𝐸𝑧^′(𝑡)𝑧(𝑡), (2.14) where in this case, 𝑤(𝑡)=𝜂̇, 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]) 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), (2.15) respectively.

3. Stochastic 𝐻_∞ and Mixed 𝐻₂/𝐻_∞ Filter Design

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

3.1. Stochastic 𝐻_∞ Filter Design

In this section, some sufficient conditions are given for 𝐻_∞ filter design; our main results are as follows.

Theorem 3.1. Suppose there exists a scalar 𝜆>0, such that 𝐹_𝑖(𝑥)≤𝜆|𝑥|,𝑖=0,1,∀𝑥∈ℛ^𝑛. (3.1) If the following matrix inequalities 𝑃𝐴+𝐴^′𝑃+2𝐷^′₁𝑃𝐷₁+𝐷^′₂𝑃𝐷₂+𝑃+6𝜆²1𝛼𝐼+𝑄+𝛾²𝑃𝐹₃𝐹^′₃𝑃<0,0<𝑃≤𝛼𝐼 (3.2)(3.3) have a solution 𝑃>0, 𝛼>0, then (2.10) is internally stable and 𝐻_∞ filtering performance 𝐽_𝑠<0, where 𝑄=(0𝐷)^′(0𝐷).

Proof. We first show (2.10) to be internally stable, that is, the following system 𝑑𝜉=𝐴𝜉𝑑𝑡+𝐷₁𝜉𝑑𝑤₀+𝐷₂𝜉𝑑𝑤₁+𝐹₁𝑑𝑡+𝐹₂𝑑𝑤₀(𝑡) (3.4) is asymptotically mean square stable. Let ℒ_𝜉 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 ℒ_𝜉𝑉(𝜉)≤−𝑘₃𝜉² (3.5) for some 𝑘₃>0. Note that ℒ_𝜉𝑉(𝜉)=𝜕𝑉^′(𝜉)𝐹𝜕𝜉𝐴𝜉+₁+12𝐷₁𝜉+𝐹₂^′𝜕²𝑉(𝜉)𝜕𝜉²𝐷₁𝜉+𝐹₂+12𝐷₂𝜉^′𝜕²𝑉(𝜉)𝜕𝜉²𝐷₂𝜉=𝜉^′𝑃𝐴+𝐴^′𝑃+𝐷^′₁𝑃𝐷₁+𝐷^′₂𝑃𝐷₂𝜉+2𝐹^′₁𝑃𝜉+𝐹^′₂𝑃𝐹₂+2𝜉′𝐷^′₁𝑃𝐹₂. (3.6) By condition (3.1), we have 2𝐹^′₁𝑃𝜉≤𝜉′𝑃𝜉+𝐹^′₁𝑃𝐹₁≤𝜉′𝑃𝜉+𝛼𝐹^′₁𝐹₁=𝜉^′𝑃𝜉+2𝛼𝐹^′₀𝐹₀≤𝜉^′𝑃𝜉+2𝛼𝐹₀²≤𝜉^′𝑃𝜉+2𝛼𝜆²𝜉². (3.7) Similarly, 2𝜉^′𝐷^′₁𝑃𝐹₂≤𝜉^′𝐷^′₁𝑃𝐷₁𝜉+2𝛼𝜆²𝜉²,𝐹^′₂𝑃𝐹₂≤2𝛼𝜆²𝜉². (3.8) Substituting (3.7), (3.8) into (3.6) and considering (3.2), it follows ℒ_𝜉𝑉(𝜉)≤𝜉^′𝑃𝐴+𝐴^′𝑃+2𝐷^′₁𝑃𝐷₁+𝐷^′₂𝑃𝐷₂+𝑃+6𝛼𝜆²𝐼𝜉<−𝜉^′1𝑄+𝛾²𝑃𝐹₃𝐹^′₃𝑃𝜉≤0. (3.9) By Lemma 2.4, the internal stability of (2.10) is proved.
Secondly, we further show the 𝐻_∞ filtering performance 𝐽_𝑠<0. Let ℒ_𝜉,𝑤 be the infinitesimal generator of (2.10). For 𝑉(𝜉)=𝜉′𝑃𝜉, it is easy to show that ℒ_𝜉,𝑤𝑉(𝜉)=ℒ_𝜉𝑉(𝜉)+2𝜉^′𝑃𝐹₃𝑤. (3.10) For any 𝑇>0 and 𝜉(0)=0, we have 𝐽_𝑠(𝑇)∶=𝐸^𝑇₀|𝑧(𝑡)|²−𝛾²|𝑤(𝑡)|²𝑑𝑡=𝐸^𝑇₀|𝑧(𝑡)|²−𝛾²|𝑤(𝑡)|²𝑑𝑡+𝑑𝜉^′𝑃𝜉−𝐸𝜉(𝑇)𝑃𝜉(𝑇)≤𝐸^𝑇₀|𝑧(𝑡)|²−𝛾²|𝑤(𝑡)|²+ℒ_𝜉,𝑤𝑉(𝜉)𝑑𝑡. (3.11) Note that ℒ_𝜉,𝑤𝑉(𝜉)≤𝜉^′𝑃𝐴+𝐴^′𝑃+2𝐷^′₁𝑃𝐷₁+𝐷^′₂𝑃𝐷₂+𝑃+6𝜆²𝛼𝐼𝜉+2𝜉^′𝑃𝐹₃|𝑧(𝑡𝑤,)|²=𝜉^′𝑄𝜉. (3.12) So |𝑧(𝑡)|²−𝛾²|𝑤(𝑡)|²+ℒ_𝜉,𝑤𝑉(𝜉)≤𝜉𝑤^′Λ₁₁𝑃𝐹₃𝐹^′₃𝑃−𝛾²𝐼𝜉𝑤<0, (3.13) where Λ₁₁∶=𝑃𝐴+𝐴^′𝑃+2𝐷^′₁𝑃𝐷₁+𝐷^′₂𝑃𝐷₂+𝑃+6𝜆²𝛼𝐼+𝑄. (3.14) By the well-known Schur’s complement and (3.2), there exists 𝜀>0, such that Λ₁₁𝑃𝐹₃𝐹^′₃𝑃−𝛾²𝐼<−𝜀𝐼. (3.15) Summarizing the above analysis, (3.11) yields 𝐽_𝑠(𝑇)≤−𝜀𝐸^𝑇₀𝜉(𝑡)²+𝑤(t)|²𝑑𝑡≤−𝜀𝐸^𝑇₀|𝑤(𝑡)|²𝑑𝑡. (3.16) So for any 𝑇>0, 𝐸∫^𝑇₀|𝑧(𝑡)|²𝑑𝑡≤(𝛾²−𝜀)𝐸∫^𝑇₀|𝑤(𝑡)|²𝑑𝑡.
Let 𝑇→∞, then ‖𝑧(𝑡)‖²_𝐿2≤𝛾²−𝜀‖𝑤(𝑡)‖²_𝐿2 (3.17) which yields 𝐽_𝑠<0. This theorem is proved.

Theorem 3.1 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 Theorem 3.1, if the following LMIs 𝑃₁₁−𝛼𝐼00𝑃₂₂−𝛼𝐼<0,𝑎₁₁𝐴^′𝑃22−𝐴^′₁𝑍^′₁−𝑍^′√2𝐶^′𝑃11√2𝐶^′𝑃22−𝐶^′₁𝑍^′₁𝑃₁₁𝐵₀𝑃₂₂𝐴−𝑍₁𝐴₁−𝑍𝑎₂₂000𝑃₂₂𝐵₀−𝑍₁𝐵₁√2𝑃₁₁𝐶0−𝑃₁₁√0002𝑃₂₂𝐶00−𝑃₂₂00−𝑍₁𝐶₁000−𝑃₂₂0𝐵^′₀𝑃₁₁𝐵^′₀𝑃₂₂−𝐵^′₁𝑍^′₁000−𝛾²𝐼<0 (3.18)(3.19) have solutions 𝑃₁₁>0,𝑃₂₂>0,𝛼>0,𝑍₁∈ℛ^𝑛×𝑟, 𝑍∈ℛ^𝑛×𝑛, then (2.10) is internally stable and 𝐽_𝑠<0.
Moreover, 𝑑𝑥=𝑃⁻¹₂₂𝑍𝑥𝑑𝑡+𝑃⁻¹₂₂𝑍₁𝑑𝑦 (3.20) 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 𝑃𝐴+𝐴^′𝑃+𝑃+6𝜆²𝛼𝐼+𝑄√2𝐷^′₁𝑃𝐷^′₂𝑃𝑃𝐹₃√2𝑃𝐷₁𝑃𝐷−𝑃00₂𝐹0−𝑃0_′3𝑃00−𝛾²𝐼<0. (3.21) Taking 𝑃=diag(𝑃₁₁,𝑃₂₂) and substituting (2.11) into (3.21), we have Ψ₁₁Ψ^′₁₂Ψ^′₁₃𝜙^′₁₄Ψ₁₂Ψ₂₂Ψ00₁₃0Ψ₃₃0Ψ₁₄00Ψ₄₄<0, (3.22) where Ψ₁₁=𝑃₁₁𝐴+𝐴^′𝑃₁₁+6𝜆²𝛼𝐼+𝑃₁₁𝐴−𝐵_𝑓𝐴₁−𝐴_𝑓^′𝑃₂₂𝑃₂₂𝐴−𝐵_𝑓𝐴₁−𝐴_𝑓−𝑃₂₂𝐴_𝑓−𝐴^′_𝑓𝑃₂₂+6𝜆²𝛼𝐼+𝑃₂₂+𝐷^′𝐷,Ψ₂₂=Ψ₃₃=−𝑃=−𝑃₁₁00−𝑃₂₂,Ψ₄₄=−𝛾²Ψ𝐼,_′12=√2𝐶^′𝑃₁₁√2𝐶^′𝑃₂₂00,Ψ^′₁₃=0−𝐶^′₁𝐵^′_𝑓𝑃₂₂00,Ψ^′₁₄=𝑃₁₁𝐵₀𝑃₂₂𝐵₀−𝐵_𝑓𝐵₁. (3.23) (3.22) is equivalent to 𝑎₁₁𝐴−𝐵_𝑓𝐴₁−𝐴_𝑓′𝑃₂₂√2𝐶′𝑃₁₁√2𝐶′𝑃₂₂−𝐶^′₁𝐵^′_𝑓𝑃₂₂𝑃₁₁𝐵₀𝑃₂₂𝐴−𝐵_𝑓𝐴₁−𝐴_𝑓𝑎₂₂000𝑃₂₂𝐵₀−𝐵_𝑓𝐵₁√2𝑃₁₁𝐶0−𝑃₁₁√0002𝑃₂₂𝐶00−𝑃₂₂00−𝑃₂₂𝐵_𝑓𝐶₁000−𝑃₂₂0𝐵^′₀𝑃₁₁𝐵₀−𝐵_𝑓𝐵₁′𝑃₂₂000−𝛾²𝐼<0, (3.24) where 𝑎₁₁=𝑃₁₁𝐴+𝐴^′𝑃₁₁+6𝜆²𝛼𝐼+𝑃₁₁, 𝑎₂₂=−𝑃₂₂𝐴_𝑓−𝐴′_𝑓𝑃₂₂+6𝜆²𝛼𝐼+𝑃₁₁. Let 𝑃₂₂𝐴_𝑓=𝑍,𝑃₂₂𝐵_𝑓=𝑍₁, then (3.22) becomes (3.19). From our assumption, 𝐴_𝑓=𝑃⁻¹₂₂𝑍,𝐵_𝑓=𝑃⁻¹₂₂𝑍₁, so an 𝐻_∞ filtering equation is constructed as in the form of (3.20). Theorem 3.2 is proved.

3.2. Mixed 𝐻₂/𝐻_∞ Filtering

To design the mixed stochastic 𝐻₂/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 𝐽₂∶=lim_𝑡→∞𝐸𝑧^′(𝑡)𝑧(𝑡)=lim_𝑡→∞𝐸𝜉^′(𝑡)(0𝐼)^′𝐷^′𝐷(0𝐼)𝜉(𝑡)=lim_𝑡→∞Tr𝐷(0𝐼)𝐸𝜉(𝑡)𝜉^′(𝑡)(0𝐼)^′𝐷^′. (3.25) Two performances 𝐽_𝑠 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 𝐽_𝑠 and minimize 𝐽₂. Consider the following linear stochastic constant system 𝑑𝜉=𝐴₁₁𝜉𝑑𝑡+^𝑙_𝑖=1𝐵_𝑖𝑖𝜉𝑑𝑤_𝑖, (3.26) where {𝑤_𝑖,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 𝑃𝐴₁₁+𝐴^′₁₁𝑃+^𝑙_𝑖=1𝐵^′_𝑖𝑖𝑃𝐵_𝑖𝑖=−𝑅 (3.27) has a unique positive definite solution 𝑃>0.
In the next, for simplicity, when (3.26) is exponentially stable, one also says (𝐴₁₁,𝐵₁₁,…,𝐵_𝑙𝑙) is stable.
As we have pointed out before, at this stage, we assume 𝑤(𝑡)=𝜂̇(𝑡); (2.10) accordingly becomes 𝑑𝜉=𝐴𝜉𝑑𝑡+𝐷₁𝜉𝑑𝑤₀+𝐷₂𝜉𝑑𝑤₁+𝐹₁𝑑𝑡+𝐹₂𝑑𝑤₀+𝐹₃𝑑𝜂. (3.28) Let 𝑋(𝑡)=𝐸[𝜉(𝑡)𝜉′(𝑡)] in (3.28), then by Itô’s formula, we have 𝑋̇(𝑡)=𝐴𝑋(𝑡)+𝑋(𝑡)𝐴^′+𝐸𝐹₁𝜉^′+𝜉𝐹^′₁+𝐷₁𝑋𝐷^′₁+𝐸𝐷₁𝜉𝐹^′₂+𝐹₂𝜉′𝐷^′₁+𝐸𝐹₂𝐹^′₂+𝐷₂𝑋(𝑡)𝐷^′₂+𝐹₃𝐹^′₃. (3.29) By means of 𝐸𝐹₁𝜉′+𝜉𝐹^′₁≤𝐸𝐹₁𝐹^′₁(𝑡),𝐸𝐷+𝑋₁𝜉𝐹^′₂+𝐹₂𝜉′𝐷^′₁≤𝐷₁𝑋𝐷^′₁+𝐸𝐹₂𝐹^′₂, (3.30) we have 𝑋̇(𝑡)≤𝐴𝑋(𝑡)+𝑋(𝑡)𝐴′+2𝐷₁𝑋(𝑡)𝐷^′₁+𝐷₂𝑋(𝑡)𝐷^′₂+𝑋(𝑡)+2𝐸𝐹₂𝐹^′₂+𝐸𝐹₁𝐹^′₁+𝐹₃𝐹^′₃. (3.31) Now, we suppose 𝐹_𝑖(𝑥)(𝑖=0,1) satisfy 𝐹_𝑖(𝑥)𝐹^′_𝑖(𝑥)≤𝐺_𝑖𝑥𝑥^′𝐺^′_𝑖,𝑖=0,1,∀𝑥∈ℛ^𝑛, (3.32) where 𝐺₁,𝐺₂ are constant matrices of suitable dimensions. At this stage, 𝐹_𝑖𝐹^′_𝑖=𝐼0𝐹𝐼𝐼^𝑖𝐹^′_𝑖0≤𝐺00𝐼𝐼0𝐼𝐼0𝐼𝐼^𝑖𝑥𝑥′𝐺^′_𝑖0=𝐺00𝐼𝐼0𝐼𝐼0𝐼𝐼^𝑖0𝐺00𝜉𝜉′^′𝑖0=𝐺00𝐼𝐼0𝐼_𝑖0𝐺_𝑖0𝐺𝜉𝜉′^′𝑖𝐺^′_𝑖𝐺00∶=_𝑖𝜉𝜉^′𝐺^′_𝑖,𝑖=0,1, (3.33) where 𝐺_𝑖=𝐺_𝑖0𝐺_𝑖0. (3.34) So (3.31) becomes 𝑋̇(𝑡)≤𝐴𝑋(𝑡)+𝑋(𝑡)𝐴′+2𝐷₁𝑋(𝑡)𝐷^′₁+𝐷₂𝑋(𝑡)𝐷^′₂+𝑋(𝑡)+2𝐺₂𝑋(𝑡)𝐺^′₂+𝐺₁𝑋(𝑡)𝐺^′₁+𝐹₃𝐹^′₃. (3.35) In addition, if 𝑋₁(𝑡) solves 𝑋̇₁(𝑡)=𝐴𝑋₁(𝑡)+𝑋₁(𝑡)𝐴′+2𝐷₁𝑋₁(𝑡)𝐷^′₁+𝐷₂𝑋₁(𝑡)𝐷^′₂+𝑋₁(𝑡)+2𝐺₂𝑋₁(𝑡)𝐺^′₂+𝐺₁𝑋₁(𝑡)𝐺^′₁+𝐹₃𝐹^′₃𝑋₁(0)=𝑋(0) (3.36) then it is easy to prove that 𝑋(𝑡)≤𝑋₁(𝑡). Denoting 𝑋₁∶=lim_𝑡→∞𝑋₁(𝑡), where 𝑋₁ satisfies 𝐴𝑋₁+𝑋₁𝐴′+2𝐷₁𝑋₁𝐷^′₁+𝐷₂𝑋₁𝐷^′₂+2𝐺₂𝑋₁𝐺^′₂+𝐺₁𝑋₁𝐺^′₁+𝑋₁+𝐹₃𝐹^′₃=0. (3.37) Obviously, lim_𝑡→∞𝑋(𝑡)≤𝑋₁, accordingly, 𝐽₂≤Tr𝐷(0𝐼)𝑋₁(0𝐼)^′𝐷^′=Tr𝑋₁𝑄. (3.38) As in [12, 24], it is easily seen the following fact.

Lemma 3.4. If 𝑃 is a solution of 𝐴′𝑃+𝑃𝐴+2𝐷^′₁𝑃𝐷₁+𝐷^′₂𝑃𝐷₂+2𝐺^′₂𝑃𝐺₂+𝐺^′₁𝑃𝐺₁+𝑄+𝑃=0 (3.39) then Tr(𝑋₁𝑄)=Tr(𝑃(𝐹₃𝐹^′₃)).
Secondly, suppose 𝑃>0 satisfies 𝐴^′𝑃+𝑃𝐴+2𝐷^′₁𝑃𝐷₁+𝐷^′₂𝑃𝐷₂+𝑄+𝑃+2𝐺^′₂𝑃𝐺₂+𝐺^′₁𝑃𝐺₁<0. (3.40) 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 Lemmas 3.4–3.5, it gives 𝐽₂=lim_𝑡→∞Tr𝐷(0𝐼)𝑋(𝑡)(0𝐼)^′𝐷^′≤lim_𝑡→∞Tr𝐷(0𝐼)𝑋₁(𝑡)(0𝐼)^′𝐷^′=Tr𝐷(0𝐼)𝑋₁(0𝐼)^′𝐷^′=Tr𝑋₁𝑄=Tr𝑃𝐹₃𝐹^′₃=Tr𝐹^′₃𝑃𝐹₃≤Tr𝐹^′₃𝑃𝐹₃∶=𝐽₂̂. (3.41) Hence, to solve the mixed stochastic 𝐻₂/𝐻_∞ filtering problem, we seek to minimize an upper-bound on 𝐽₂̂ subject to (3.2), (3.3), and 𝑃𝐴+𝐴^′𝑃+2𝐷^′₁𝑃𝐷₁+𝐷^′₂𝑃𝐷^′₂+𝑃+2𝐺^′₂𝑃𝐺₂+𝐺^′₁𝑃𝐺₁+𝑄<0. (3.42) (3.42) having a positive definite solution 𝑃>0 is equivalent to 𝑃𝐴+𝐴′𝑃+𝑃+𝑄√2𝐷^′₁𝑃𝐷^′₂𝑃𝐺′₁𝑃√2𝐺^′₂𝑃√2𝑃𝐷₁𝑃𝐷−𝑃000₂𝑃𝐺0−𝑃00₁√00−𝑃02𝑃𝐺₂000−𝑃<0. (3.43) A suboptimal 𝐻₂/𝐻_∞ filtering can be obtained by minimizing Tr(𝐻) subject to (3.2), (3.3), (3.43), and 𝐻−𝐹^′₃𝑃𝐹₃>0. (3.44) (3.44) is equivalent to 𝐻𝐹^′₃𝑃𝑃𝐹₃𝑃>0. (3.45) We still take 𝑃=diag(𝑃₁₁,𝑃₂₂)>0, 𝑃₂₂𝐵_𝑓=𝑍₁, 𝑃₂₂𝐴_𝑓=𝑍, then (3.3), (3.2), (3.43), and (3.45) become, respectively, as (3.18), (3.19), 𝛾₁₁𝛾₁₂√2𝐶′𝑃₁₁√2𝐶′𝑃₂₂−𝐶^′₁𝑍^′₁𝐺^′₁𝑃₁₁𝐺^′₁𝑃₂₂𝐺^′₂𝑃₁₁𝐺^′₂𝑃₂₂𝛾₂₁𝛾₂₂√00000002𝑃₁₁𝐶0−𝑃₁₁√0000002𝑃₂₂𝐶00−𝑃₂₂00000−𝑍₁𝐶₁000−𝑃₂₂𝑃0000₁₁𝐺₁0000−𝑃₁₁𝑃000₂₂𝐺₁00000−𝑃₂₂𝑃00₁₁𝐺₂000000−𝑃₁₁0𝑃₂₂𝐺₂0000000−𝑃₂₂<0,𝐻𝐵^′₀𝑃₁₁𝐵^′₀𝑃₂₂−𝐵^′₁𝑍^′₁𝑃₁₁𝐵₀𝑃₁₁0𝑃₂₂𝐵₀−𝑍₁𝐵₁0𝑃₂₂>0, (3.46) where 𝛾₁₁=𝑃₁₁𝐴+𝐴^′𝑃₁₁+𝑃₁₁, 𝛾₁₂=𝐴^′𝑃₂₂−𝐴^′₁𝑍^′₁−𝑍^′, 𝛾₂₁=𝑃₂₂𝐴−𝑍₁𝐴₁−𝑍, 𝛾₂₂=−𝑍−𝑍^′+𝐷^′𝐷+𝑃₂₂. Therefore, we have the following theorem.

Theorem 3.6. Under the conditions of Theorem 3.2 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: min_{𝑃11,𝑃22,𝑍,𝑍1,𝛼}Tr(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 Theorems 3.2 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 𝑃₁₁𝑃₁₂𝑃^′₁₂𝑃₂₂. 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 𝑑𝑥=𝐴𝑥+𝐵₀𝑤+𝐹₀(𝑥)𝑑𝑡+𝐶𝑥+𝐹₁(𝑥)𝑑𝑤₀,𝑑𝑦=𝐴₁+𝐵₁𝑤𝑑𝑡+𝐶₁𝑥𝑑𝑤₁,𝑧=𝐷𝑥, (4.1) where 𝐴=−31/2−1−3,𝐵₀=10,𝐹,𝐶=1000₀(𝑥)=0.3tanh(𝑥),𝐹₁(𝑥)(𝑥),𝐴=0.3sin₁=−111−1,𝐵₁=01,𝐶₁=,0111001𝐷=,𝑤=1+2𝑡,𝑡≥0. (4.2) Consider the following filter for the estimation of 𝑧(𝑡): 𝑑𝑥=𝐴_𝑓𝑥𝑑𝑡+𝐵_𝑓𝑑𝑦,𝑧=𝐷𝑥. (4.3) Setting 𝛾=0.9, and using the LMI control toolbox of Matlab, the estimation gains of 𝐻_∞ filter are derived from Theorem 3.2: 𝐴_𝑓=5.62313.7259−0.16178.2289,𝐵_𝑓=.0.1812−1.8190−0.25250.4635 (4.4) From Theorem 3.6, the estimation gains of 𝐻₂/𝐻_∞ filter are obtained as follows: 𝐴_𝑓=4.14493.4665−0.24696.3382,𝐵_𝑓=.0.5270−1.2388−0.36930.3445 (4.5) 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),  𝑥₁(𝑡),  𝑥₂(𝑡),  𝑥₂(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.

(a)

(b)

Figure 1 

Trajectories of 𝑥1(𝑡),𝑥1(𝑡) and 𝑥2(𝑡),𝑥2(𝑡) for the proposed 𝐻∞ filter.

(a)

(b)

Figure 2 

Trajectories of 𝑥1(𝑡),  𝑥1(𝑡) and 𝑥2(𝑡),  𝑥2(𝑡) for the proposed 𝐻2/𝐻∞ filter.

Figure 3 

Trajectory of the estimation error 𝑧(𝑡) for the proposed 𝐻∞ filter.

Figure 4 

Trajectory of the estimation error 𝑧(𝑡) for the proposed 𝐻2/𝐻∞ filter.

5. Conclusions

In this paper, we have discussed the robust 𝐻_∞ 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.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 61174078, 61203053), Natural Science Foundation of Shandong Province of China (no. ZR2011FL025), Fundamental Research Funds for the Central Universities (no. 11CX04042A), Research Fund for the Taishan Scholar Project of Shandong Province of China, and SDUST Research Fund.