Robust H ∞ Filter Design for Itô Stochastic Pantograph Systems

The problem of robust H ∞ 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 H ∞ filtering problem for this kind of systems is studied and a sufficient condition for the existence of theH ∞ filter is derived. Furthermore, the explicit expression of the desired filter parameters is characterized. Finally, an example is given to illustrate the results.


Introduction
Stochastic pantograph system which is treated as a special class of time-delay systems has also attracted more and more researchers [1][2][3][4][5].Reference [1] 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 [2] 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 [3] studied the th moment stability for stochastic pantograph equation by using Razumikhin technique.Reference [4] 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 [5] 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][7][8][9][10][11][12][13][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  2 gain from the external disturbance to the estimation error is below a prescribed level  > 0. Reference [10] 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  2 / ∞ filter.Reference [11] gave a necessary and sufficient condition for reduced-order  ∞ filter of linear continuous and discrete-time stochastic systems.Reference [12] investigated the robust  ∞ filtering problem for nonlinear stochastic systems and gave a sufficient condition for the existence of  ∞ filter.Reference [13] studied the mixed  2 / ∞ filtering for a class of nonlinear stochastic systems.Reference [14] 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.
Definition 1.The stochastic pantograph system (1) is said to be asymptotically mean square stable if for any initial value  0 , the corresponding state satisfies lim Next, a test criterion for asymptotically mean-square stable of stochastic pantograph systems is given.
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  > 0, then stochastic pantograph system (1) is asymptotically mean-square stable.
Remark 4. Inequality ( 10) is a linear matrix inequality, which provides more convenience to test the asymptotic meansquare stability of stochastic pantograph system (1).

Robust 𝐻 ∞ 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 () ∈   , () ∈    , V() ∈   V , and () ∈    are the system state, the exogenous disturbance signal, the measurement output, and the state combination to be estimated, respectively.,  1 ,  2 , ,  1 , ,  1 ,  2 , and  are constant matrices of suitable dimension.Here we suppose V() ∈  2  ( + ,   V ), which guarantees that the system (20) has a unique solution () ∈  2  (  ,   ) for any  > 0. The so-called  ∞ filtering problem is to design an estimator to estimate the unknown state combination () via output measurement (), which guarantees the  2 gain (from the external disturbance to estimation error) to be less than a prescribe level  > 0, and the extended system is internally stable.Here we construct the following linear pantograph filter via output measurement for the estimation of (): where x() ∈   ,   ∈  × ,   ∈  × ,   ∈  ×  , and   ∈    × are constant matrices to be determined subsequently.Let () = (  () x ())  , z() = ()−ẑ(); then the extended system is where For a given disturbance attenuation level  > 0 and V() ∈  2  ( + ,   V ), define the associated  ∞ filtering performance of (22) as As in [10], 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  > 0, then system (22) with V() = 0 is asymptotically mean-square stable, and  ∞ < 0 holds for any Proof.When V() = 0, from (25) we obtain so system ( 22) is asymptotical mean-square stable according to Theorem 3. Next, we prove  ∞ < 0 for any nonzero V() ∈  2  ( + ,   V ) with (0) = 0, taking the Lyapunov function () =   , where  > 0 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  > 0, where If  < 0, then there exists  > 0, such that By Schur Complement,  < 0 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.
Proof.By Schur Complement, (25) is equivalent to Taking  = [  11 0 0  22 ] and substituting (23) into (31), after a series computations, we have turns out to be (31).Therefore, In the proof of Theorem 8, the matrix  is chosen as diag{ 11 ,  22 } for simplicity.In order to reduce the conservatism of the conditions, the matrix  can also be chosen as [ 22 ].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.( Then the minimum value of optimal  ∞ performance  * is given by  * = (min ) 1/2 .(OP2): The minimum value of  corresponding to the different values of  in the interval (0, 1) can be found.Algorithm I. Consider the following steps.
Step 1.By simple search algorithms, if we find a series of   ( = 1, . . ., ) to make (31) have feasible solutions, then go to Step 2. Otherwise, go to Step 6.
Step 3. Solving the following optimization problem OP1.
Remark 11.The smallest  may be obtained by Algorithm I.

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 The initial condition in the simulation is assumed to be (0) = [0.5 − 0.2 − 0.5 0.2].Figures 1 and 2 show the trajectories of  1 (), x1 (),  2 (), and x2 () by using the proposed  ∞ filter.Figure 3 shows the response of real state () and its estimation ẑ().Figure 4 is the simulation result of the estimation error response of z() = () − ẑ(), which demonstrates that the estimation error is asymptotically mean-square stable.
By the OP2, the minimum value of  can be given by  = 0.534.Figure 5 shows the minimum value of  corresponding to different  in the interval (0, 1).From Figure 5, we see that (31) has no feasible solution when  is in (0, 0.534).In order to see the relationship between  and  more clearly, Figure 6 gives the minimum value of  corresponding to different  in the interval (0.55, 1).

Conclusion
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  2 ,  2 / ∞ control/filtering for linear/nonlinear stochastic pantograph systems.Identify matrix E(⋅):

Notations
The mathematical expectation operator  2  (  ,   ): Th es p a c eo fn o n a n t i c i p a t i v es q u a r e integrable stochastic processes () ∈  2 (Ω,   ) with respect to an increasing -algebra satisfying   -measurable and E ∫  0 ‖ ⋅ ‖ 2  < ∞  1,2 ( + ×   ;  + ): The family of all nonnegative functions (, ) on which are continuously once differentiable in  and twice differentiable in .