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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 747890, 8 pages
Robust Filter Design for Itô Stochastic Pantograph Systems
1School of Electrical Engineering and Automation, Shandong Polytechnic University, Jinan 250353, China
2School of Science, Shandong Polytechnic University, Jinan 250353, China
Received 20 January 2013; Accepted 18 February 2013
Academic Editor: Weihai Zhang
Copyright © 2013 Zhiguo Yan and Yulin Huang. 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.
The problem of robust filter design is investigated for stochastic pantograph systems governed by linear Itô differential equation. First, a sufficient condition for asymptotic mean-square stability of stochastic pantograph systems is presented by means of Lyapunov approach. Then, based on matrix inequalities, the filtering problem for this kind of systems is studied and a sufficient condition for the existence of the filter is derived. Furthermore, the explicit expression of the desired filter parameters is characterized. Finally, an example is given to illustrate the results.
Stochastic pantograph system which is treated as a special class of time-delay systems has also attracted more and more researchers [1–5]. Reference  gave the necessary analytical theory for the existence and uniqueness of a strong solution of the linear stochastic pantograph differential equation and presented the strong approximations to the solution obtained by a continuous extension of -Euler scheme. Reference  investigated the asymptotic growth and delay properties of solution of linear stochastic pantograph equation and gave the sufficient conditions on parameters when the solution grows at a polynomial rate in th mean and almost sure sense. Reference  studied the th moment stability for stochastic pantograph equation by using Razumikhin technique. Reference  investigated the convergence of the Euler method of stochastic pantograph equations and proved that the Euler approximation solution converges to the analytic solution in probability under weaker conditions. Reference  studied the almost surely asymptotic stability of the nonlinear stochastic pantograph differential equations with Markovian switching under the weakened linear growth condition. At present, most literatures on stochastic pantograph equation focus on the existence, uniqueness, and convergence of the numerical solution produced by kinds of approximate methods.
On the other hand, due to great many applications of robust control and filtering in real world, the problems on these two have been studied extensively [6–14]. Compared with classical Kalman filter, one does not need to know the exact statistic information about the external disturbance in the filtering design. filtering requires one to design a filter such that the gain from the external disturbance to the estimation error is below a prescribed level . Reference  studied the problem of filtering for general continuous-time linear stochastic systems and gave a necessary and sufficient condition for the existence of filter and furthermore designed filter. Reference  gave a necessary and sufficient condition for reduced-order filter of linear continuous and discrete-time stochastic systems. Reference  investigated the robust filtering problem for nonlinear stochastic systems and gave a sufficient condition for the existence of filter. Reference  studied the mixed filtering for a class of nonlinear stochastic systems. Reference  considered the finite-time filter design for a class of nonlinear stochastic systems. Nevertheless, to the best of our knowledge, the issue on the filtering for stochastic linear pantograph systems with state-dependent noise has not been investigated in previous literatures.
In this paper, we first consider the problem on the asymptotic mean-square stability and give a test criterion for stochastic pantograph systems by the Lyapunov approach. On this basis, a sufficient condition of the asymptotic mean square stability is obtained, which can be available for studying the filtering of stochastic pantograph systems. Moreover, the filter design is investigated and a sufficient condition for the existence of filter is obtained in the form of linear matrix inequality. Finally, an example is given to illustrate our proposed methods.
This paper is organized as follows. Section 2 discusses the asymptotic mean-square stability of stochastic pantograph systems and presents a sufficient condition of stability by means of the Lyapunov approach. The filtering problem of stochastic pantograph systems is investigated in Section 3. Section 4 provides a numerical example to demonstrate the effectiveness and applicability of the proposed methods. Section 5 concludes this paper.
2. Asymptotic Mean-Square Stability
Consider the following linear stochastic pantograph system: where is system state; ; is a one-dimension standard Wiener process defined on a complete probability space with ; , , , are all constant matrices of . For initial value and , there exists a unique solution .
Definition 1. The stochastic pantograph system (1) is said to be asymptotically mean square stable if for any initial value , the corresponding state satisfies Next, a test criterion for asymptotically mean-square stable of stochastic pantograph systems is given.
Lemma 2. Stochastic pantograph system (1) is asymptotically mean-square stable if there exist some positive constant scalars , and and a Lyapunov function satisfying where
Proof. Expressing the difference by means of Itô formula , calculating expectations, we get Differentiating this equality with respect to and using (3), (4), we see that This implies the estimate Together with (3), this estimate yields Let ; then (2) is obtained. This proof is complete.
On the basis of Lemma 2, the following theorem gives a sufficient condition of the asymptotic mean-square stability is obtained, which can be available for studying the filtering of stochastic pantograph systems.
Theorem 3. If the following linear matrix inequality has a solution , then stochastic pantograph system (1) is asymptotically mean-square stable.
Proof. Take a Lyapunov function , where is the solution of (10). Applying Itô formula and by Cauchy inequality , we obtain where due to . So for any , taking integral from to , we have where The above last inequality is valid because of and , so Multiplying by both sides simultaneously and letting , we obtain Therefore, the infinitesimal generator of stochastic pantograph system (1) satisfies where for some . By Lemma 2, the asymptotic mean-square stability of (1) is derived, which completes the proof.
Remark 5. When , the pantograph system (1) becomes normal stochastic linear system and (10) is simplified by which implies , so (18) can also guarantee the asymptotic mean-square stability of (17) .
3. Robust Filter Design
Based on the asymptotic mean-square stability of pantograph system discussed in the above section, we are in a position to deal with the filtering problem for stochastic pantograph system.
Consider the following stochastic linear perturbed pantograph system with measurement output: where , and are the system state, the exogenous disturbance signal, the measurement output, and the state combination to be estimated, respectively. , and are constant matrices of suitable dimension. Here we suppose , which guarantees that the system (20) has a unique solution for any .
The so-called filtering problem is to design an estimator to estimate the unknown state combination via output measurement , which guarantees the gain (from the external disturbance to estimation error) to be less than a prescribe level , and the extended system is internally stable. Here we construct the following linear pantograph filter via output measurement for the estimation of : where are constant matrices to be determined subsequently. Let ; then the extended system is where For a given disturbance attenuation level and , define the associated filtering performance of (22) as As in , the filtering problem is formulated as follows.
In what follows, we will give the main result of filtering problem and provide a technique to determine matrices of filter (22).
Theorem 7. If the following matrix inequality has a solution , then system (22) with is asymptotically mean-square stable, and holds for any , when .
Proof. When , from (25) we obtain
so system (22) is asymptotical mean-square stable according to Theorem 3.
Next, we prove for any nonzero with , taking the Lyapunov function , where is a solution of (25), and following the outline of the proof in Theorem 3, we obtain that the infinitesimal generator of (22) satisfies Note that for , where
If , then there exists , such that
Let ; then . By Schur Complement, is equivalent to (25), which ends the proof.
It is difficult to solve the inequality (25) because of its nonlinearity, so Theorem 7 cannot be directly available for designing the filter. Next we will give a sufficient condition easy to be solved.
Theorem 8. If the following LMIhas solutions , , , , , , then system (22) is internally asymptotically mean-square stable, and filtering performance holds for any , with . The corresponding filter (21) can be formulated by
Proof. By Schur Complement, (25) is equivalent to
Taking and substituting (23) into (31), after a series computations, we have
Setting , then (34) turns out to be (31). Therefore, ; then the proof is complete.
Remark 9. In the proof of Theorem 8, the matrix is chosen as diag 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.
Remark 10. In many engineering applications, the performance constraint is often specified a priori. In Theorem 8, the filter is designed after performance is prescribed. In fact, we can obtain an improved performance by optimization method. In addition, inequality (31) may be no feasible solution for very small , that is, very large time delay. However, the smallest can be found by numerical algorithm. The results in Theorem 8 suggest the following optimization problems.(OP1): The optimal filtering problem for stochastic pantograph systems is defined by Then the minimum value of optimal performance is given by .(OP2): The minimum value of corresponding to the different values of in the interval can be found.
Algorithm I. Consider the following steps.
Step 1. By simple search algorithms, if we find a series of to make (31) have feasible solutions, then go to Step 2. Otherwise, go to Step 6.
Step 2. Set , take a .
Step 3. Solving the following optimization problem OP1.
Step 4. Set , if , then go to Step 5; otherwise , go to Step 3.
Step 5. (31) has feasible solutions. Stop.
Step 6. (31) has no feasible solutions. Stop.
Remark 11. The smallest may be obtained by Algorithm I.
4. Numerical Example
In this section, a numerical example is provided to demonstrate the effectiveness and applicability of the proposed methods. Consider the following Itô stochastic pantograph system: where
Consider the following filter for estimation of : According to OP1, the minimum value of is and the corresponding estimation gains of filter are derived from theorem
The initial condition in the simulation is assumed to be . Figures 1 and 2 show the trajectories of , , , and by using the proposed filter. Figure 3 shows the response of real state and its estimation . Figure 4 is the simulation result of the estimation error response of , which demonstrates that the estimation error is asymptotically mean-square stable.
By the OP2, the minimum value of can be given by . Figure 5 shows the minimum value of corresponding to different in the interval . From Figure 5, we see that (31) has no feasible solution when is in . In order to see the relationship between and more clearly, Figure 6 gives the minimum value of corresponding to different in the interval .
This paper has discussed infinite horizon filtering for stochastic linear pantograph systems with state-dependent noise, which has not been studied for pantograph system in the previous literatures. A sufficient condition for asymptotic mean-square stability of stochastic linear pantograph systems is presented and a sufficient condition for the existence of the filter is given in the form of linear matrix inequality. The results obtained in this paper may be significant in studying the other control/filtering problem such as , control/filtering for linear/nonlinear stochastic pantograph systems.
|:||is positive (nonnegative)|
|:||The -dimensional Euclidean space with|
|:||The set of all matrices|
|:||The mathematical expectation operator|
|:||The space of nonanticipative square integrable stochastic processes with respect to an increasing -algebra satisfying -measurable and|
|:||The family of all nonnegative functions on which are continuously once differentiable in and twice differentiable in .|
This work is supported by the Starting Research Foundation of Shandong Polytechnic University under Grant 12045501, Outstanding Mid-Young Scientist Prize Foundation of Shandong Province (BS2011DX032), and a Project of Shandong Province Higher Educational Science and Technology Program (J10LG13).
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