Robust 𝐻_∞ Filter Design for Itô Stochastic Pantograph Systems

Abstract

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 the 𝐻_∞ filter is derived. Furthermore, the explicit expression of the desired filter parameters is characterized. Finally, an example is given to illustrate the results.

1. Introduction

Stochastic pantograph system which is treated as a special class of time-delay systems has also attracted more and more researchers [1–5]. 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 𝜃-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 𝑝th mean and almost sure sense. Reference [3] studied the 𝛼th 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 𝐻_∞ 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 𝐻_∞ filtering design. 𝐻_∞ filtering requires one to design a filter such that the 𝐿₂ gain from the external disturbance to the estimation error is below a prescribed level 𝛾>0. Reference [10] studied the problem of 𝐻_∞ filtering for general continuous-time linear stochastic systems and gave a necessary and sufficient condition for the existence of 𝐻_∞ filter and furthermore designed 𝐻₂/𝐻_∞ filter. Reference [11] gave a necessary and sufficient condition for reduced-order 𝐻_∞ filter of linear continuous and discrete-time stochastic systems. Reference [12] investigated the robust 𝐻_∞ filtering problem for nonlinear stochastic systems and gave a sufficient condition for the existence of 𝐻_∞ filter. Reference [13] studied the mixed 𝐻₂/𝐻_∞ filtering for a class of nonlinear stochastic systems. Reference [14] considered the finite-time 𝐻_∞ filter design for a class of nonlinear stochastic systems. Nevertheless, to the best of our knowledge, the issue on the 𝐻_∞ 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 𝐻_∞ filtering of stochastic pantograph systems. Moreover, the 𝐻_∞ filter design is investigated and a sufficient condition for the existence of 𝐻_∞ 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 𝐻_∞ 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: 𝑑𝑥(𝑡)=𝐴𝑥(𝑡)+𝐴₁𝑥𝑞𝑡𝑑𝑡+𝐶𝑥(𝑡)+𝐶₁𝑥(𝑡)𝑥(0)𝑞𝑡𝑑𝑤=𝑥₀, (1) where 𝑥(𝑡)∈𝑅^𝑛 is system state; 0<𝑞<1; 𝑤(𝑡) is a one-dimension standard Wiener process defined on a complete probability space (Ω,𝐹,𝐹_𝑡,𝑃) with 𝐹_𝑡=𝜎(𝑤(𝑠)∶0≤𝑠≤𝑡); 𝐴, 𝐴₁, 𝐶, 𝐶₁ are all constant matrices of 𝑅^𝑛×𝑛. For initial value 𝑥₀∈𝑅^𝑛 and 𝑇>0, there exists a unique solution 𝑥(𝑡)∈𝐿²_𝐹(𝑅_𝑇,𝑅^𝑛) [1].

Definition 1. The stochastic pantograph system (1) is said to be asymptotically mean square stable if for any initial value 𝑥₀, the corresponding state satisfies lim_𝑡→∞𝔼‖𝑥(𝑡)‖²=0. (2) 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 𝑘₁>0,𝑘₂>0,  and  𝑘₃>0 and a Lyapunov function 𝑉(𝑡,𝑥) satisfying 𝑘₁‖𝑥(𝑡)‖²≤𝑉(𝑡,𝑥)≤𝑘₂‖𝑥‖²,𝐿𝑉(𝑡,𝑥)≤−𝑘₃‖𝑥‖², (3)(4) where ()=()𝐿𝑉𝑡,𝑥𝜕𝑉𝑡,𝑥+𝜕𝑡𝜕𝑉^′(𝑡,𝑥)(𝑡)𝜕𝑥𝐴𝑥+𝐴₁𝑥+1𝑞𝑡2𝐶𝑥(𝑡)+𝐶₁𝑥𝑞𝑡^′𝜕²𝑉(𝑡,𝑥)𝜕𝑥²𝐶𝑥(𝑡)+𝐶₁𝑥𝑞𝑡. (5)

Proof. Expressing the difference 𝑉(𝑡,𝑥(𝑡))−𝑉(0,𝑥₀) by means of Itô formula [15], calculating expectations, we get 𝔼𝑉(𝑡,𝑥(𝑡))−𝑉0,𝑥₀=^𝑡₀𝔼𝐿𝑉(𝑠,𝑥(𝑠))𝑑𝑠. (6) Differentiating this equality with respect to 𝑡 and using (3), (4), we see that 𝑑((𝑡𝑘𝑑𝑡𝔼𝑉𝑡,𝑥))≤−³𝑘₂𝔼𝑉(𝑡,𝑥(𝑡)). (7) This implies the estimate 𝔼𝑉(𝑡,𝑥(𝑡))≤𝑉0,𝑥₀−𝑘exp³𝑘₂𝑡. (8) Together with (3), this estimate yields 𝔼‖𝑥(𝑡)‖²≤1𝑘₁𝑉0,𝑥₀−𝑘exp³𝑘₂𝑡. (9) Let 𝑡→∞; 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 𝐻_∞ filtering of stochastic pantograph systems.

Theorem 3. If the following linear matrix inequality 𝑃𝐴+𝐴^′𝑃+𝑃+2𝐶^′1𝑃𝐶+𝑞𝐴^′₁𝑃𝐴₁+2𝐶^′₁𝑃𝐶₁<0 (10) has a solution 𝑃>0, then stochastic pantograph system (1) is asymptotically mean-square stable.

Proof. Take a Lyapunov function 𝑉(𝑡,𝑥)=𝑥^′𝑃𝑥, where 𝑃>0 is the solution of (10). Applying Itô formula and by Cauchy inequality 𝑋^′𝑃𝑌+𝑌^′𝑃𝑋≤𝑋^′𝑃𝑋+𝑌^′𝑃𝑌, we obtain (𝑥)=𝑥𝑑𝑉_′𝑃𝐴+𝐴^′𝑃+𝐶^′𝑃𝐶𝑥+𝑥^′𝑞𝑡𝐴^′₁𝑃𝑥+𝑥^′𝑃𝐴₁𝑥𝑞𝑡+𝑥^′𝑞𝑡𝐶^′₁𝑃𝐶𝑥+𝑥^′𝐶^′𝑃𝐶₁𝑥𝑞𝑡+𝑥^′𝑞𝑡𝐶^′₁𝑃𝐶₁𝑥+𝑞𝑡𝑑𝑡𝐶𝑥+𝐶₁𝑥𝑞𝑡^′𝑃𝑥+𝑥^′𝑃𝐶𝑥+𝐶₁𝑥(𝑡)≤𝑥𝑞𝑡𝑑𝑤_′𝑃𝐴+𝐴^′𝑃+𝐶^′𝑃𝐶𝑥+𝑥^′𝑞𝑡𝐴^′₁𝑃𝐴₁𝑥𝑞𝑡+𝑥^′𝑃𝑥+𝑥^′𝑞𝑡𝐶^′₁𝑃𝐶₁𝑥𝑞𝑡+𝑥^′𝐶^′𝑃𝐶𝑥+𝑥^′𝑞𝑡𝐶^′₁𝑃𝐶₁𝑥+𝑞𝑡𝑑𝑡𝐶𝑥+𝐶₁𝑥𝑞𝑡^′𝑃𝑥+𝑥^′𝑃𝐶𝑥+𝐶₁𝑥(𝑡)≤𝑥𝑞𝑡𝑑𝑤_′𝑃𝐴+𝐴^′𝑃+𝑃+2𝐶^′𝑃𝐶𝑥+𝑥^′+𝑞𝑡𝑄𝑥𝑞𝑡𝑑𝑡𝐶𝑥+𝐶₁𝑥𝑞𝑡^′𝑃𝑥+𝑥^′𝑃𝐶𝑥+𝐶₁𝑥𝑞𝑡𝑑𝑤(𝑡), (11) where 𝑄=(𝐴^′₁𝑃𝐴₁+2𝐶^′₁𝑃𝐶₁)≥0 due to 𝑃>0. So for any 0≤𝑠≤𝑡, taking integral from 𝑠 to 𝑡, we have 𝔼𝑉(𝑥(𝑡))−𝔼𝑉(𝑥(𝑠))≤𝔼^𝑡_𝑠𝑥^′(𝑢)𝑃𝐴+𝐴^′𝑃+𝑃+2𝐶^′𝑃𝐶𝑥(𝑢)𝑑𝑢+𝔼^𝑡_𝑠𝑥^′𝑞𝑢𝑄𝑥𝑞𝑢𝑑𝑢, (12) where 𝔼^𝑡_𝑠𝑥^′=1𝑞𝑢𝑄𝑥𝑞𝑢𝑑𝑢𝑞𝔼_𝑠^𝑞𝑡𝑥^′(𝑢)(𝑢)1𝑄𝑥𝑑𝑢≤𝑞𝔼^𝑡_𝑠𝑥^′(𝑢)𝑄𝑥(𝑢)𝑑𝑢. (13) The above last inequality is valid because of 𝑄≥0 and 0<𝑞<1, so 𝔼𝑉(𝑥(𝑡))−𝔼𝑉(𝑥(𝑠))≤𝔼^𝑡_𝑠𝑥^′(𝑢)𝑃𝐴+𝐴^′𝑃+𝑃+2𝐶^′1𝑃𝐶+𝑞𝑄𝑥(𝑢)𝑑𝑢. (14) Multiplying 1/(𝑡−𝑠) by both sides simultaneously and letting 𝑡→𝑠, we obtain 𝔼𝑑𝑉(𝑥(𝑡))≤𝑥^′(𝑡)𝑃𝐴+𝐴^′𝑃+𝑃+2𝐶^′1𝑃𝐶+𝑞𝑄𝑥(𝑡)𝑑𝑡. (15) Therefore, the infinitesimal generator of stochastic pantograph system (1) satisfies 𝐿𝑉(𝑥(𝑡))≤𝑥^′(𝑡)𝑃𝐴+𝐴^′𝑃+𝑃+2𝐶^′1𝑃𝐶+𝑞𝑄𝑥(𝑡)≤−𝑘₃‖𝑥‖², (16) where 𝑃𝐴+𝐴^′𝑃+𝑃+2𝐶^′𝑃𝐶+(1/𝑞)𝑄≤−𝑘₃<0 for some 𝑘₃>0. By Lemma 2, the asymptotic mean-square stability of (1) is derived, which completes the proof.

Remark 4. Inequality (10) is a linear matrix inequality, which provides more convenience to test the asymptotic mean-square stability of stochastic pantograph system (1).

Remark 5. When 𝐴₁=𝐶₁=0, the pantograph system (1) becomes normal stochastic linear system 𝑑𝑥(𝑡)=𝐴𝑥(𝑡)𝑑𝑡+𝐶𝑥(𝑡)𝑑𝑤(𝑡), (17) and (10) is simplified by 𝑃𝐴+𝐴^′𝑃+𝑃+2𝐶^′𝑃𝐶<0, (18) which implies 𝑃𝐴+𝐴^′𝑃+𝑃+𝐶^′𝑃𝐶<0, so (18) can also guarantee the asymptotic mean-square stability of (17) [15].

Remark 6. Let 𝜏(𝑡)=𝑡−𝑞𝑡; system (1) becomes 𝑑𝑥(𝑡)=𝐴𝑥(𝑡)+𝐴₁𝑥((𝑡+(𝑡)𝑡−𝜏))𝑑𝑡𝐶𝑥+𝐶₁𝑥(𝑡−𝜏(𝑡))𝑑𝑤(𝑡). (19) System (19) is a time-vary delay system. Condition (10) guarantees that system (19) is asymptotically mean-square stable.

3. 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 𝐻_∞ filtering problem for stochastic pantograph system.

Consider the following stochastic linear perturbed pantograph system with measurement output: 𝑑𝑥(𝑡)=𝐴𝑥(𝑡)+𝐴₁𝑥(𝑡)+(𝑡)𝑞𝑡+𝐵𝑣𝑑𝑡𝐶𝑥+𝐶₁𝑥(𝑡)(𝑡)=𝐴𝑞𝑡𝑑𝑤𝑑𝑦₂𝑥(𝑡)+𝐵₁𝑣(𝑡)𝑑𝑡+𝐶₂𝑥(𝑡)(𝑡)𝑧(𝑡)(𝑡),𝑑𝑤=𝑀𝑥 (20) where 𝑥(𝑡)∈𝑅^𝑛,𝑦(𝑡)∈𝑅^𝑛𝑦,𝑣(𝑡)∈𝑅^𝑛𝑣, and 𝑧(𝑡)∈𝑅^𝑛𝑧 are the system state, the exogenous disturbance signal, the measurement output, and the state combination to be estimated, respectively. 𝐴,𝐴₁,𝐴₂,𝐵,𝐵₁,𝐶,𝐶₁,𝐶₂, and 𝑀 are constant matrices of suitable dimension. Here we suppose 𝑣(𝑡)∈𝐿²_𝐹(𝑅₊,𝑅^𝑛𝑣), which guarantees that the system (20) has a unique solution 𝑥(𝑡)∈𝐿²_𝐹(𝑅_𝑇,𝑅^𝑛) for any 𝑇>0.

The so-called 𝐻_∞ filtering problem is to design an estimator to estimate the unknown state combination 𝑧(𝑡) via output measurement 𝑦(𝑡), which guarantees the 𝐿₂ gain (from the external disturbance to estimation error) to be less than a prescribe level 𝛾>0, and the extended system is internally stable. Here we construct the following linear pantograph filter via output measurement for the estimation of 𝑧(𝑡): 𝑑𝑥(𝑡)=𝐴_𝑓𝑥(𝑡)+𝐵_𝑓𝑥𝑞𝑡𝑑𝑡+𝐶_𝑓(𝑡)𝑥(0)=𝑥𝑑𝑦₀𝑧(𝑡)=𝑀_𝑓𝑥(𝑡), (21) where 𝑥(𝑡)∈𝑅^𝑛,𝐴_𝑓∈𝑅^𝑛×𝑛,𝐵_𝑓∈𝑅^𝑛×𝑛,𝐶_𝑓∈𝑅^{𝑛×𝑛𝑦},and𝑀_𝑓∈𝑅^{𝑛𝑧×𝑛} are constant matrices to be determined subsequently. Let 𝜂(𝑡)=(𝑥^′(𝑡)𝑥^′(𝑡))^′,𝑧(𝑡)=𝑧(𝑡)−𝑧(𝑡); then the extended system is 𝑑𝜂(𝑡)=𝐴𝜂(𝑡)+𝐴₁𝜂+(𝑡+(𝑡)+𝐶𝑞𝑡𝐵𝑣)𝑑𝑡𝐶𝜂₁𝜂(𝑡),𝑧(𝑡)=(𝑡),𝑞𝑡𝑑𝑤𝑀𝜂 (22) where 𝐶𝐴=𝐴0_𝑓𝐴₂𝐴_𝑓,𝐴₁=𝐴₁00𝐵_𝑓,𝐵𝐶𝐵=_𝑓𝐵₁,𝐶𝐶=𝐶0_𝑓𝐶₂0,𝐶₁=𝐶₁0,00𝑀=0−𝑀_𝑓. (23) For a given disturbance attenuation level 𝛾>0 and 𝑣(𝑡)∈𝐿²_𝐹(𝑅₊,𝑅^𝑛𝑣), define the associated 𝐻_∞ filtering performance of (22) as 𝐽_∞=𝔼^∞₀‖𝑧(𝑡)‖²𝑑𝑡−𝛾²𝔼^∞₀‖𝑣(𝑡)‖²𝑑𝑡. (24) As in [10], the 𝐻_∞ filtering problem is formulated as follows.

Stochastic 𝐻_∞ Filtering Problem. Given 𝛾>0, find an estimator 𝑥 of the form (21) leading (22) to being internally stable. Moreover, for any 𝑣(𝑡)≠0,𝑣(𝑡)∈𝐿²_𝐹(𝑅₊,𝑅^𝑛𝑣) with 𝜂(0)=0, there always  is  𝐽_∞<0.

In what follows, we will give the main result of 𝐻_∞ filtering problem and provide a technique to determine matrices 𝐴_𝑓,𝐵_𝑓,𝐶_𝑓,𝑀_𝑓 of filter (22).

Theorem 7. If the following matrix inequality 𝑃𝐴+𝐴^′𝑃+𝑃+2𝐶^′𝑃1𝐶+𝑞𝐴^′₁𝑃𝐴₁+2𝐶^′𝑃𝐶+𝑀^′𝑀+𝛾⁻²𝑃𝐵𝐵^′𝑃<0 (25) has a solution 𝑃>0, then system (22) with 𝑣(𝑡)=0 is asymptotically mean-square stable, and 𝐽_∞<0 holds for any 𝑣(𝑡)≠0, 𝑣(𝑡)∈𝐿²_𝐹(𝑅₊,𝑅^𝑛𝑣) when 𝜂(0)=0.

Proof. When 𝑣(𝑡)=0, from (25) we obtain 𝑃𝐴+𝐴^′𝑃+𝑃+2𝐶^′𝑃1𝐶+𝑞𝐴^′₁𝑃𝐴₁+2𝐶^′𝑃𝐶<−𝑀^′𝑀−𝛾⁻²𝑃𝐵𝐵^′𝑃<0, (26) so system (22) is asymptotical mean-square stable according to Theorem 3.
Next, we prove 𝐽_∞<0 for any nonzero 𝑣(𝑡)∈𝐿²_𝐹(𝑅₊,𝑅^𝑛𝑣) with 𝜂(0)=0, taking the Lyapunov function 𝑉(𝜂)=𝜂^′𝑃𝜂, where 𝑃>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 𝐿𝑉𝜂≤𝜂^′𝑃𝐴+𝐴^′𝑃+𝑃+2𝐶^′𝑃𝐶+1𝑞𝐴^′₁𝑃𝐴₁+2𝐶^′𝑃𝐶𝜂+𝑣^′𝐵^′𝑃𝜂+𝜂^′𝑃𝐵𝑣. (27) Note that for 𝑇>0, 𝐽_𝑇𝜂,𝑣=𝔼^𝑇₀‖𝑧(𝑡)‖²𝑑𝑡−𝛾²𝔼^𝑇₀‖𝑣(𝑡)‖²𝑑𝑡=𝔼^𝑇₀‖𝑧(𝑡)‖²𝑑𝑡−𝛾²𝔼^𝑇₀‖𝑣(𝑡)‖²𝑑𝑡+𝑑𝜂^′𝑃𝜂−𝑑𝜂^′𝑃𝜂=−𝔼𝜂^′(𝑇)𝑃𝜂(𝑇)+𝔼^𝑇₀𝜂^′𝑀^′𝑀−𝛾²𝑣^′𝑣+𝐿𝑉𝜂𝑑𝑡≤𝔼^𝑇₀𝜂𝑀^′𝑀−𝛾²𝑣^′𝑣+𝐿𝑉𝜂𝑑𝑡≤𝔼^𝑇₀𝜂𝑣^′𝑁𝜂𝑣𝑑𝑡, (28) where 𝑃𝐴𝑁=𝐴+^′𝑃+𝑃+2𝐶^′𝑃𝐶+1𝑞𝐴^′₁𝑃𝐴₁+2𝐶^′𝑃𝐶+𝑀^′𝑀𝑃𝐵𝐵^′𝑃−𝛾⁻²𝐼. (29)
If 𝑁<0, then there exists 𝜖>0, such that 𝐽_𝑇𝜂,𝑣≤−𝜖𝔼^𝑇₀𝜂²+‖𝑣‖²𝑑𝑡≤−𝜖𝔼^𝑇₀‖𝑣‖²𝑑𝑡. (30)
Let 𝑇→∞; then 𝐽_∞(𝜂,𝑣)≤−𝜖𝔼∫^∞₀‖𝑣‖²𝑑𝑡<0. By Schur Complement, 𝑁<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.

Theorem 8. If the following LMI^{𝑃11𝐴+𝐴′𝑃11+𝑃11+2𝑞𝐶′1𝑃11𝐶1𝐴′2𝑍′12𝐶′𝑃112𝐶′1𝑍′1𝑀′𝑃11𝐵𝐴′1𝑃110∗𝑍3+𝑍′3+𝑃2200−𝑀′𝑓𝑍1𝐵10𝑍′2∗∗−2𝑃1100000∗∗∗−2𝑃220000}_{∗∗∗∗−𝐼000∗∗∗∗∗−𝛾2𝐼00∗∗∗∗∗∗−𝑞𝑃110∗∗∗∗∗∗∗−𝑞𝑃22}<0 (31)has solutions 𝑃₁₁>0, 𝑃₂₂>0, 𝑍₁∈𝑅^{𝑛×𝑛𝑦}, 𝑍₂∈𝑅^𝑛×𝑛, 𝑍₃∈𝑅^𝑛×𝑛, 𝑀_𝑓, then system (22) is internally asymptotically mean-square stable, and filtering performance 𝐽_∞<0 holds for any 𝑣(𝑡)≠0, 𝑣(𝑡)∈𝐿²_𝐹(𝑅₊,𝑅^𝑛𝑣) with 𝜂(0)=0. The corresponding 𝐻_∞ filter (21) can be formulated by 𝑑𝑥(𝑡)=𝑃⁻¹₂₂𝑍₃𝑥(𝑡)+𝑃⁻¹₂₂𝑍₂𝑥𝑞𝑡𝑑𝑡+𝑃⁻¹₂₂𝑍₁(𝑡),𝑥(0)=𝑥𝑑𝑦₀,𝑧(𝑡)=𝑀_𝑓𝑥(𝑡). (32)

Proof. By Schur Complement, (25) is equivalent to 𝑃𝐴+𝐴^′2𝑃+𝑃+𝑞𝐶^′₁𝑃𝐶₁2𝐶^′𝑃𝑀^′𝑃𝐵𝐴^′₁𝑃∗−2𝑃000∗∗−𝐼00∗∗∗−𝛾²𝐼0∗∗∗∗−𝑞𝑃<0. (33)
Taking 𝑃=^𝑃110_0𝑃22 and substituting (23) into (31), after a series computations, we have^{𝑃11𝐴+𝐴′𝑃11+𝑃11+2𝑞𝐶′1𝑃11𝐶1𝐴′2𝐶′𝑓𝑃222𝐶′𝑃112𝐶′1𝐶′𝑓𝑃22𝑀′𝑃11𝐵𝐴′1𝑃110∗𝑃22𝐴𝑓+𝐴′𝑓𝑃22+𝑃2200−𝑀′𝑓𝑃22𝐶𝑓𝐵10𝐵′𝑓𝑃22∗∗−2𝑃1100000∗∗∗−2𝑃220000}_{∗∗∗∗−𝐼000∗∗∗∗∗−𝛾2𝐼00∗∗∗∗∗∗−𝑞𝑃110∗∗∗∗∗∗∗−𝑞𝑃22}<0. (34)
Setting 𝑃₂₂𝐶_𝑓=𝑍₁,𝑃₂₂𝐵_𝑓=𝑍₂,𝑃₂₂𝐴_𝑓=𝑍₃, then (34) turns out to be (31). Therefore, 𝐴_𝑓=𝑃⁻¹₂₂𝑍₃,𝐵_𝑓=𝑃⁻¹₂₂𝑍₂,𝐶_𝑓=𝑃⁻¹₂₂𝑍₁; then the proof is complete.

Remark 9. In the proof of Theorem 8, 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 𝑃₁₁𝑃^′₁₂𝑃₁₂𝑃₂₂. 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 𝐻_∞ 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 𝑞, that is, very large time delay. However, the smallest 𝑞 can be found by numerical algorithm. The results in Theorem 8 suggest the following optimization problems.(OP1): The optimal 𝐻_∞ filtering problem for stochastic pantograph systems is defined by min_{𝑃11>0,𝑃22>0,𝑍1,𝑍2,𝑍3,𝑀𝑓}𝜒subjectto(31)with𝜒=𝛾². (35) Then the minimum value of optimal 𝐻_∞ performance 𝛾^∗ is given by 𝛾^∗=(min𝜒)^1/2.(OP2): The minimum value of 𝛾 corresponding to the different values of 𝑞 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 𝑞_𝑖(𝑖=1,…,𝑛) to make (31) have feasible solutions, then go to Step  2. Otherwise, go to Step  6.
Step  2. Set 𝑖=1, take a 𝑞_𝑖.
Step  3. Solving the following optimization problem OP1.
Step  4. Set 𝑖=𝑖+1, if 𝑖+1>𝑛, then go to Step  5; otherwise 𝑞_𝑖=𝑞_𝑖+1, go to Step  3.
Step  5. (31) has feasible solutions. Stop.
Step  6. (31) has no feasible solutions. Stop.

Remark 11. The smallest 𝑞 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: 𝑑𝑥(𝑡)=𝐴𝑥(𝑡)+𝐴₁𝑥(𝑡)+(𝑡)𝑞𝑡+𝐵𝑣𝑑𝑡𝐶𝑥+𝐶₁𝑥(𝑡)(𝑡)=𝐴𝑞𝑡𝑑𝑤𝑑𝑦₂𝑥(𝑡)+𝐵₁𝑣(𝑡)𝑑𝑡+𝐶₂𝑥(𝑡)(𝑡)𝑧(𝑡)(𝑡),𝑑𝑤=𝑀𝑥 (36) where 𝐴=−500.80.4−10,𝐴₁=,,𝐶21.50.2−2𝐵=0.3−0.5,𝐶=40.531₁=−0.81.20.81,𝐴₂=,𝐵15.61.10.68₁=0.60.2,𝐶₂=,1.6201𝑀=−11,𝑞=0.9. (37)

Consider the following filter for estimation of 𝑧(𝑡): 𝑑𝑥(𝑡)=𝐴_𝑓𝑥(𝑡)+𝐵_𝑓𝑥𝑞𝑡𝑑𝑡+𝐶_𝑓(𝑡),𝑥(0)=𝑥𝑑𝑦₀,𝑧(𝑡)=𝑀_𝑓𝑥(𝑡). (38) According to OP1, the minimum value of 𝛾 is 0.13 and the corresponding estimation gains of 𝐻_∞ filter are derived from theorem 𝐴_𝑓=−1.44140.05180.0512−1.5298,𝐵_𝑓=,𝐶0.5184−0.0095−0.00930.5418_𝑓=0.0610−0.0221−0.0219−0.0915,𝑀_𝑓=−0.38700.7240. (39)

The initial condition in the simulation is assumed to be 𝜂(0)=[0.5−0.2−0.50.2]. Figures 1 and 2 show the trajectories of 𝑥₁(𝑡), 𝑥₁(𝑡), 𝑥₂(𝑡), and 𝑥₂(𝑡) by using the proposed 𝐻_∞ filter. Figure 3 shows the response of real state 𝑧(𝑡) and its estimation 𝑧(𝑡). Figure 4 is the simulation result of the estimation error response of 𝑧(𝑡)=𝑧(𝑡)−𝑧(𝑡), which demonstrates that the estimation error is asymptotically mean-square stable.

Figure 1 

The trajectories of 𝑥1(𝑡) and 𝑥1(𝑡).

Figure 2 

The trajectories of 𝑥2(𝑡) and 𝑥2(𝑡).

Figure 3 

The trajectories of 𝑧(𝑡) and 𝑧(𝑡).

Figure 4 

The trajectories of 𝑧(𝑡).

By the OP2, the minimum value of 𝑞 can be given by 𝑞=0.534. Figure 5 shows the minimum value of 𝛾 corresponding to different 𝑞 in the interval (0,1). From Figure 5, we see that (31) has no feasible solution when 𝑞 is in (0,0.534). In order to see the relationship between 𝛾 and 𝑞 more clearly, Figure 6 gives the minimum value of 𝛾 corresponding to different 𝑞 in the interval (0.55,1).

Figure 5 

The minimum value of 𝛾 versus 𝑞 in (0,1).

Figure 6 

The minimum value of 𝛾 versus 𝑞 in (0.55,1).

5. Conclusion

This paper has discussed infinite horizon 𝐻_∞ 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 𝐻_∞ 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 𝐻₂, 𝐻₂/𝐻_∞ control/filtering for linear/nonlinear stochastic pantograph systems.

Notations

𝐴′:	Thetransposeof𝐴
𝐴>0(𝐴≥0):	𝐴 is positive (nonnegative)
𝑅𝑛:	The 𝑛-dimensional Euclidean space with ‖⋅‖2
𝑅𝑚×𝑛:	The set of all 𝑚×𝑛 matrices
𝑅+	=[0,∞)
𝑅𝑇	=[0,𝑇] for 𝑇>0
𝐼:	Identify matrix
𝔼(⋅):	The mathematical expectation operator
𝐿2𝐹(𝑅𝑟,𝑅𝑘):	The space of nonanticipative square integrable stochastic processes 𝑦(𝑡)∈𝐿2(Ω,𝑅𝑘) with respect to an increasing 𝜎-algebra satisfying 𝐹𝑡-measurable and 𝔼∫𝑇0‖⋅‖2𝑑𝑡<∞
𝐶1,2(𝑅+×𝑅𝑘;𝑅+):	The family of all nonnegative functions 𝑉(𝑡,𝑥) on which are continuously once differentiable in 𝑡 and twice differentiable in 𝑥.

Acknowledgments

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).