Mathematical Problems in Engineering

Volume 2012 (2012), Article ID 750841, 16 pages

doi:10.1155/2012/750841

Research Article

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

Weihai Zhang,¹ Bor-Sen Chen,² Li Sheng,³ and Ming Gao¹

¹College of Information and Electrical Engineering, Shandong University of Science and Technology, Qingdao 266590, China

²Department of Electrical Engineering, National Tsing Hua University, Hsinchu 30013, Taiwan

³College of Information and Control Engineering, China University of Petroleum (East China), Qingdao 266555, China

Received 17 June 2012; Accepted 24 August 2012

Academic Editor: Jun Hu

Copyright © 2012 Weihai Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 filter can be designed by solving linear matrix inequalities (LMIs). Moreover, a mixed / 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 filtering problem has been investigated extensively since it is very useful in signal processing and engineering applications [1–5]. The so-called filtering problem is to design an estimator to estimate the unknown state combination via measurement output, which guarantees the gain (from the external disturbance to the estimation error) to be less than a prescribed level . In contrast to classical Kalman filter, it is not necessary to know the exact statistic information about the external disturbance in the filter design. Obviously, there may be more than one solution to filtering problem with a desired robustness. Since the performance is appealing for engineering, it naturally leads to the mixed filtering problem [6–8]. Compared with the sole filter, the mixed 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 filtering problem for linear perturbed systems with steady-state error variance constraints was investigated in [6], and the mixed filter for polytopic discrete-time systems was discussed in [7].

On the other hand, stochastic 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 problems for linear stochastic systems were investigated by [12, 13], respectively. Most of the aforementioned works were limited to linear stochastic systems. Recently, the filtering problem for nonlinear stochastic systems has become another popular research topic [14–20]. Wang et al. [14] studied the robust 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 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 fuzzy filter for nonlinear stochastic systems via solving LMIs instead of an HJI. However, there is little work dealing with the 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 gain with respect to disturbance signal is less than a prescribed level. Moreover, a stochastic 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 and mixed 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. : the set of all symmetric matrices; its components may be complex. : the transpose of the corresponding matrix .   : is positive semidefinite (positive definite) symmetric matrix. , that is, denotes the Euclidean 2-norm of , where . : the space of nonanticipative stochastic processes with respect to filter satisfying . : class of functions twice continuously differential with respect to and once continuously differential with respect to except possibly at the point .

2. Problem Setting

Consider the following nonlinear stochastic system governed by Itô differential equation:

with the following measurement equation:

and the controlled output

In the above, is called the system state, is the measurement output, is the state combination to be estimated. are the standard Wiener processes defined on the probability space related to an increasing family of -algebras . Without loss of generality, we can suppose are one-dimensional, mutually uncorrelated. are constant matrices of suitable dimensions, represents the exogenous disturbance signal. Under very general conditions on and , stochastic systems (2.1)-(2.2) have, respectively, a unique strong solution for any and initial state ; see [21].

Now, we first introduce the following definitions.

Definition 2.1 (see [9]). We say that the equilibrium point of system

is exponentially mean square stable, if for some positive constants ,

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 is exponentially mean square stable.

Lemma 2.4 (see [9]). The trivial solution of (2.4) is exponentially mean square stable for if there exists such that

for some positive constants , where is the so-called an infinitesimal generator of (2.4).

Now, suppose and can be linearized, respectively, as

then the linearized stochastic system of (2.1) becomes

where and are constant matrices.

Consider the following filter for the estimation of :

where . Let , , then

where

For any given disturbance attenuation level , one wants to find , such that

holds for any . Define the performance index as

Obviously, (2.12) holds iff . As in [12], and mixed -based robust state estimation problems are formulated as follows.

(i)Stochastic filtering problem: given , find an estimator of the form (2.9) leading (2.10) to being internally stable; Moreover, for all nonzero with .(ii)Stochastic filtering problem: of all the filter of (i), one finds the one that minimizes the steady error variance

where in this case, , is taken as a standard Wiener process, independent of and , so is a white noise. (2.2) and (2.8) can be written as (see, e.g., [22])

respectively.

3. Stochastic and Mixed Filter Design

In this section, we will discuss, respectively, stochastic and mixed filtering problems.

3.1. Stochastic Filter Design

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

Theorem 3.1. Suppose there exists a scalar , such that

If the following matrix inequalities

have a solution , , then (2.10) is internally stable and filtering performance , where .

Proof. We first show (2.10) to be internally stable, that is, the following system

is asymptotically mean square stable. Let be the infinitesimal operator of (3.4), with to be determined. According to Lemma 2.4, in order to show (3.4) to be internally stable, we only need to show

for some . Note that

By condition (3.1), we have

Similarly,

Substituting (3.7), (3.8) into (3.6) and considering (3.2), it follows

By Lemma 2.4, the internal stability of (2.10) is proved.

Secondly, we further show the filtering performance . Let be the infinitesimal generator of (2.10). For , it is easy to show that

For any and , we have

Note that

So

where

By the well-known Schur’s complement and (3.2), there exists , such that

Summarizing the above analysis, (3.11) yields

So for any , .

Let , then

which yields . This theorem is proved.

Theorem 3.1 only has theoretical sense, because it is difficult to be used in designing filter. The following result is of more important in practice.

Theorem 3.2. Under the condition of Theorem 3.1, if the following LMIs

have solutions , , then (2.10) is internally stable and .

Moreover,

is the corresponding filter. In (3.19), , .

Proof. By Schur’s complement, (3.2) is equivalent to

Taking and substituting (2.11) into (3.21), we have

where

(3.22) is equivalent to

where , . Let , then (3.22) becomes (3.19). From our assumption, , so an filtering equation is constructed as in the form of (3.20). Theorem 3.2 is proved.

3.2. Mixed Filtering

To design the mixed stochastic filter, we need to choose the one from the set of all filters, which also minimizes the estimation error variance, or concretely speaking, minimizes the performance

Two performances in (2.13) and in (3.25) associated with robustness and optimization have constructed, respectively. Now, we need to design the mixed filter to maximize and minimize . Consider the following linear stochastic constant system

where 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 , the following Lyapunov-type equation

has a unique positive definite solution .

In the next, for simplicity, when (3.26) is exponentially stable, one also says is stable.

As we have pointed out before, at this stage, we assume ; (2.10) accordingly becomes

Let in (3.28), then by Itô’s formula, we have

By means of

we have

Now, we suppose satisfy

where are constant matrices of suitable dimensions. At this stage,

where

So (3.31) becomes

In addition, if solves

then it is easy to prove that . Denoting , where satisfies

Obviously, , accordingly,

As in [12, 24], it is easily seen the following fact.

Lemma 3.4. If is a solution of

then .

Secondly, suppose satisfies

By means of Lemma 3.3, one can show . So we have the following lemma.

Lemma 3.5. , where and stand for the positive definite solutions of (3.40) and (3.39), respectively.

From Lemmas 3.4–3.5, it gives

Hence, to solve the mixed stochastic filtering problem, we seek to minimize an upper-bound on subject to (3.2), (3.3), and

(3.42) having a positive definite solution is equivalent to

A suboptimal filtering can be obtained by minimizing subject to (3.2), (3.3), (3.43), and

(3.44) is equivalent to

We still take , , , then (3.3), (3.2), (3.43), and (3.45) become, respectively, as (3.18), (3.19),

where , , , . 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 to (3.18), (3.19), (3.46), then a suboptimal mixed stochastic filtering is obtained by solving and from the following convex optimization problem: 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 is chosen as for simplicity. In order to reduce the conservatism of the conditions, the matrix can also be chosen as . 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

where

Consider the following filter for the estimation of :

Setting , and using the LMI control toolbox of Matlab, the estimation gains of filter are derived from Theorem 3.2:

From Theorem 3.6, the estimation gains of filter are obtained as follows:

The initial condition in the simulation is assumed to be . Figures 1 and 2 show the trajectories of ,  ,  ,   by using the proposed and filters, respectively. The trajectories of the estimation error for and filters are shown in Figures 3 and 4, respectively. From Figures 3 and 4, it is obvious that the performance of the proposed filter is better than that of the filter.

In [15], the and 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 and 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 and for the proposed filter.

(a)

(b)

Figure 2: Trajectories of ,   and ,   for the proposed filter.

Figure 3: Trajectory of the estimation error for the proposed filter.

Figure 4: Trajectory of the estimation error for the proposed filter.

5. Conclusions

In this paper, we have discussed the robust filtering problem for a class of nonlinear stochastic systems. Meanwhile, the mixed 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.

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