Robust H2 / H [infinity] 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 [subscript] H ∞[/subscript] filter can be designed by solving linear matrix inequalities (LMIs). Moreover, a mixed [subscript] H2[/subscript] / [subscript] H ∞[/subscript] 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.

(ProQuest: ... denotes non-US-ASCII text omitted.)

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

Recommended by Jun Hu

1, College of Information and Electrical Engineering, Shandong University of Science and Technology, Qingdao 266590, China
2, Department of Electrical Engineering, National Tsing Hua University, Hsinchu 30013, Taiwan
3, College of Information and Control Engineering, China University of Petroleum (East China), Qingdao 266555, China

Received 17 June 2012; Accepted 24 August 2012

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.

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 _{[Lagrangian (script capital L)] 2} 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 _{H 2} performance is appealing for engineering, it naturally leads to the mixed _{H 2} / _{H ∞} filtering problem [ 6- 8]. Compared with the sole _{H ∞} filter, the mixed _{H 2} / _{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 _{H 2} / _{H ∞} filtering problem for linear perturbed systems with steady-state error variance constraints was investigated in [ 6], and the mixed _{H 2} / _{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 _{H 2} / _{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 _{[Lagrangian (script capital L)] 2} gain with respect to disturbance signal is less than a prescribed level. Moreover, a stochastic _{H 2} / _{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 3treats with the _{H ∞} and mixed _{H 2} / _{H ∞} filtering problems, and the main outcomes of this section are Theorems 3.2and 3.6; a numerical example is presented to illustrate the effectiveness of the proposed filtering method in Section 4; Section 5concludes this paper.

Notations. For convenience, we adopt the following notations. _{...AE; n} : the set of all n ×n symmetric matrices; its components may be complex. ^{A [variant prime]} : the transpose of the corresponding matrix A . A ...5;0 (A >0 ) : A is positive semidefinite (positive definite) symmetric matrix. |x | : = ( ^{∑ i =1 n} ... ^{x i 2 ) 1 /2} , that is, |x | denotes the Euclidean 2-norm of x , where x = ^{( _{x 1} , _{x 2} , ... , _{x n} ) [variant prime]} ∈ ^{... n} . ^{[Lagrangian (script capital L)] 2} ( _{... +} , ^{... l} ) : the space of nonanticipative stochastic processes y (t ) with respect to filter _{... t} satisfying ||y (t ) ^{|| _{L 2} 2} : =E ^{∫ 0 ∞} ... |y (t ) ^{| 2} dt < ∞ . ^{C 2 0} ( {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: [figure omitted; refer to PDF] with the following measurement equation: [figure omitted; refer to PDF] and the controlled output [figure omitted; refer to PDF] In the above, x (t ) ∈ ^{... n} is called the system state, y (t ) ∈ ^{... r} is the measurement output, z (t ) is the state combination to be estimated. _{w 0} (t ) , _{w 1} (t ) are the standard Wiener processes defined on the probability space ( Ω , ... , ...AB; ) related to an increasing family ( _{... t ) t ∈ ... +} of σ -algebras _{... t} ⊂ ... . Without loss of generality, we can suppose _{w 0} (t ) , _{w 1} (t ) are one-dimensional, mutually uncorrelated. _{B 0} , _{A 1} , _{B 1} , _{C 1} ,D are constant matrices of suitable dimensions, w ∈ ^{[Lagrangian (script capital L)] 2} ( _{... +} , ^{... 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 ...5;s ...5;0 and initial state x (s ) = ξ ∈ ^{... n} ; see [ 21].

Now, we first introduce the following definitions.

Definition 2.1 (see [ 9]).

We say that the equilibrium point x ...1;0 of system [figure omitted; refer to PDF] is exponentially mean square stable, if for some positive constants ρ , [varrho] , [figure omitted; refer to PDF]

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 ...5;0 if there exists V (t ,x ) ∈ ^{C 2 0} ( {t >0 } × ^{... n} ) such that [figure omitted; refer to PDF] for some positive constants _{k 1} , _{k 2} , _{k 3} , where [Lagrangian (script capital L)] is the so-called an infinitesimal generator of ( 2.4).

Now, suppose f (x ) and σ (x ) can be linearized, respectively, as [figure omitted; refer to PDF] then the linearized stochastic system of ( 2.1) becomes [figure omitted; refer to PDF] where A and C are constant matrices.

Consider the following filter for the estimation of z (t ) : [figure omitted; refer to PDF] where x ^ ∈ ^{... n} . Let ξ ' = [ ^{x [variant prime] x [variant prime]} - ^{x ^ [variant prime]} ] , z ~ =z - z ^ , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] For any given disturbance attenuation level γ >0 , one wants to find _{A f} , _{B f} , such that [figure omitted; refer to PDF] holds for any w ∈ ^{[Lagrangian (script capital L)] 2} ( _{... +} , ^{... q} ) . Define the _{H ∞} performance index as [figure omitted; refer to PDF] Obviously, ( 2.12) holds iff _{J s} <0 . As in [ 12], _{H ∞} and mixed _{H 2} / _{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 ∈ ^{[Lagrangian (script capital L)] 2} ( _{... +} , ^{... n} ) with ξ (0 ) =0 .

(ii) Stochastic _{H 2} / _{H ∞} filtering problem: of all the _{H ∞} filter of (i), one finds the one that minimizes the steady error variance [figure omitted; refer to PDF]

where in this case, w (t ) = η , η is taken as a standard Wiener process, independent of _{w 0} (t ) and _{w 1} (t ) , so w (t ) is a white noise. ( 2.2) and ( 2.8) can be written as (see, e.g., [ 22]) [figure omitted; refer to PDF] respectively.

3. Stochastic _{H ∞} and Mixed _{H 2} / _{H ∞} Filter Design

In this section, we will discuss, respectively, stochastic _{H ∞} and mixed _{H 2} / _{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 [figure omitted; refer to PDF] If the following matrix inequalities [figure omitted; refer to PDF] have a solution P >0 , α >0 , then ( 2.10) is internally stable and _{H ∞} filtering performance _{J s} <0 , where Q = ^{( 0 D ) [variant prime]} (0 D ) .

Proof.

We first show ( 2.10) to be internally stable, that is, the following system [figure omitted; refer to PDF] is asymptotically mean square stable. Let _{[Lagrangian (script capital L)] ξ} be the infinitesimal operator of ( 3.4), V ( ξ ) = ξ 'P ξ with αI ...5;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 [figure omitted; refer to PDF] for some _{k 3} >0 . Note that [figure omitted; refer to PDF] By condition ( 3.1), we have [figure omitted; refer to PDF] Similarly, [figure omitted; refer to PDF] Substituting ( 3.7), ( 3.8) into ( 3.6) and considering ( 3.2), it follows [figure omitted; refer to PDF] By Lemma 2.4, the internal stability of ( 2.10) is proved.

Secondly, we further show the _{H ∞} filtering performance _{J s} <0 . Let _{[Lagrangian (script capital L)] ξ ,w} be the infinitesimal generator of ( 2.10). For V ( ξ ) = ξ 'P ξ , it is easy to show that [figure omitted; refer to PDF] For any T >0 and ξ (0 ) =0 , we have [figure omitted; refer to PDF] Note that [figure omitted; refer to PDF] So [figure omitted; refer to PDF] where [figure omitted; refer to PDF] By the well-known Schur's complement and ( 3.2), there exists [varepsilon] >0 , such that [figure omitted; refer to PDF] Summarizing the above analysis, ( 3.11) yields [figure omitted; refer to PDF] So for any T >0 , E ^{∫ 0 T} ... | z ~ (t ) ^{| 2} dt ...4; ( ^{γ 2} - [varepsilon] )E ^{∫ 0 T} ... |w (t ) ^{| 2} dt .

Let T [arrow right] ∞ , then [figure omitted; refer to PDF] which yields _{J s} <0 . This theorem is proved.

Theorem 3.1only 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 [figure omitted; refer to PDF] have solutions _{P 11} >0 , _{P 22} >0 , α >0 , _{Z 1} ∈ ^{... n ×r} , Z ∈ ^{... n ×n} , then ( 2.10) is internally stable and _{J s} <0 .

Moreover, [figure omitted; refer to PDF] is the corresponding _{H ∞} filter. In ( 3.19), _{a 11} = _{P 11} A +A ' _{P 11} +6 ^{λ 2} αI + _{P 11} , _{a 22} = -Z -Z ' +6 ^{λ 2} αI +D 'D + _{P 22} .

Proof.

By Schur's complement, ( 3.2) is equivalent to [figure omitted; refer to PDF] Taking P =diag ( _{P 11} , _{P 22} ) and substituting ( 2.11) into ( 3.21), we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF] ( 3.22) is equivalent to [figure omitted; refer to PDF] where _{a ¯ 11} = _{P 11} A + ^{A [variant prime]} _{P 11} +6 ^{λ 2} αI + _{P 11} , _{a ¯ 22} = - _{P 22 A f} - _{A ' f P 22} +6 ^{λ 2} αI + _{P 11} . Let _{P 22 A f} =Z , _{P 22 B f} = _{Z 1} , then ( 3.22) becomes ( 3.19). From our assumption, _{A f} = ^{P 22 -1} Z , _{B f} = ^{P 22 -1} _{Z 1} , so an _{H ∞} filtering equation is constructed as in the form of ( 3.20). Theorem 3.2is proved.

3.2. Mixed _{H 2} / _{H ∞} Filtering

To design the mixed stochastic _{H 2} / _{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 _{H 2} performance [figure omitted; refer to PDF] Two performances _{J s} in ( 2.13) and _{J 2} in ( 3.25) associated with _{H ∞} robustness and _{H 2} optimization have constructed, respectively. Now, we need to design the mixed _{H 2} / _{H ∞} filter to maximize _{J s} and minimize _{J 2} . Consider the following linear stochastic constant system [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] has a unique positive definite solution P >0 .

In the next, for simplicity, when ( 3.26) is exponentially stable, one also says ( _{A 11} , _{B 11} , ... , _{B ll} ) is stable.

As we have pointed out before, at this stage, we assume w (t ) = η (t ) ; ( 2.10) accordingly becomes [figure omitted; refer to PDF] Let X (t ) =E [ ξ (t ) ξ ' (t ) ] in ( 3.28), then by Itô's formula, we have [figure omitted; refer to PDF] By means of [figure omitted; refer to PDF] we have [figure omitted; refer to PDF] Now, we suppose _{F i} ( x ) (i =0,1 ) satisfy [figure omitted; refer to PDF] where _{G 1} , _{G 2} are constant matrices of suitable dimensions. At this stage, [figure omitted; refer to PDF] where [figure omitted; refer to PDF] So ( 3.31) becomes [figure omitted; refer to PDF] In addition, if _{X 1} (t ) solves [figure omitted; refer to PDF] then it is easy to prove that X (t ) ...4; _{X 1} (t ) . Denoting _{X ¯ 1} : = _{lim t [arrow right] ∞ X 1} (t ) , where _{X ¯ 1} satisfies [figure omitted; refer to PDF] Obviously, _{lim t [arrow right] ∞} X (t ) ...4; _{X ¯ 1} , accordingly, [figure omitted; refer to PDF] As in [ 12, 24], it is easily seen the following fact.

Lemma 3.4.

If P ^ is a solution of [figure omitted; refer to PDF] then Tr ( _{X ¯ 1} Q ) =Tr ( P ^ ( _{F ~ 3} ^{F ~ 3 [variant prime]} ) ) .

Secondly, suppose P >0 satisfies [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] Hence, to solve the mixed stochastic _{H 2} / _{H ∞} filtering problem, we seek to minimize an upper-bound on _{J ^ 2} subject to ( 3.2), ( 3.3), and [figure omitted; refer to PDF] ( 3.42) having a positive definite solution P >0 is equivalent to [figure omitted; refer to PDF] A suboptimal _{H 2} / _{H ∞} filtering can be obtained by minimizing Tr (H ) subject to ( 3.2), ( 3.3), ( 3.43), and [figure omitted; refer to PDF] ( 3.44) is equivalent to [figure omitted; refer to PDF] We still take P =diag ( _{P 11} , _{P 22} ) >0 , _{P 22 B f} = _{Z 1} , _{P 22 A f} =Z , then ( 3.3), ( 3.2), ( 3.43), and ( 3.45) become, respectively, as ( 3.18), ( 3.19), [figure omitted; refer to PDF] where _{γ 11} = _{P 11} A + ^{A [variant prime]} _{P 11} + _{P 11} , _{γ 12} = ^{A [variant prime]} _{P 22} - ^{A 1 [variant prime] Z 1 [variant prime]} - ^{Z [variant prime]} , _{γ 21} = _{P 22} A - _{Z 1 A 1} -Z , _{γ 22} = -Z - ^{Z [variant prime]} + ^{D [variant prime]} D + _{P 22} . Therefore, we have the following theorem.

Theorem 3.6.

Under the conditions of Theorem 3.2and assumption ( 3.32), if there exists a solution ( _{P 11} >0 , _{P 22} >0 ,Z , _{Z 1} , α >0 ) to ( 3.18), ( 3.19), ( 3.46), then a suboptimal mixed stochastic _{H 2} / _{H ∞} filtering is obtained by solving _{P 11} and _{P 22} from the following convex optimization problem: _{min P 11 , P 22 ,Z , Z 1 , 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 3.2and 3.6, the matrix P is chosen as diag ( _{P 11} , _{P 22} ) for simplicity. In order to reduce the conservatism of the conditions, the matrix P can also be chosen as [ _{P 11 P 12} ^{P 12 [variant prime]} _{P 22} ] . 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 [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Consider the following filter for the estimation of z (t ) : [figure omitted; refer to PDF] Setting γ =0.9 , and using the LMI control toolbox of Matlab, the estimation gains of _{H ∞} filter are derived from Theorem 3.2: [figure omitted; refer to PDF] From Theorem 3.6, the estimation gains of _{H 2} / _{H ∞} filter are obtained as follows: [figure omitted; refer to PDF] The initial condition in the simulation is assumed to be _{ξ 0} = [0.3 0.2 -0.02 -0.05 ] ' . Figures 1and 2show the trajectories of _{x 1} (t ) , _{x ^ 1} (t ) , _{x 2} (t ) , _{x ^ 2} (t ) by using the proposed _{H ∞} and _{H 2} / _{H ∞} filters, respectively. The trajectories of the estimation error z ~ (t ) for _{H ∞} and _{H 2} / _{H ∞} filters are shown in Figures 3and 4, respectively. From Figures 3and 4, it is obvious that the performance of the proposed _{H 2} / _{H ∞} filter is better than that of the _{H ∞} filter.

In [ 15], the _{H ∞} and _{H 2} / _{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 _{H 2} / _{H ∞} filtering problems can be solved via the LMI technique instead of the HJI according to Theorems 3.2and 3.6in this paper. Simulation results show the effectiveness of the proposed method.

Trajectories of _{x 1} (t ) , _{x ^ 1} (t ) and _{x 2} (t ) , _{x ^ 2} (t ) for the proposed _{H ∞} filter.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Trajectories of _{x 1} (t ) , _{x ^ 1} (t ) and _{x 2} (t ) , _{x ^ 2} (t ) for the proposed _{H 2} / _{H ∞} filter.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Figure 3: Trajectory of the estimation error z ~ (t ) for the proposed _{H ∞} filter.

[figure omitted; refer to PDF]

Figure 4: Trajectory of the estimation error z ~ (t ) for the proposed _{H 2} / _{H ∞} filter.

[figure omitted; refer to PDF]

5. Conclusions

In this paper, we have discussed the robust _{H ∞} filtering problem for a class of nonlinear stochastic systems. Meanwhile, the mixed _{H 2} / _{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.

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