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Design of Filter for a Class of Switched Linear Neutral Systems
This paper is concerned with the filtering problem for a class of switched linear neutral systems with time-varying delays. The time-varying delays appear not only in the state but also in the state derivatives. Based on the average dwell time approach and the piecewise Lyapunov functional technique, sufficient conditions are proposed for the exponential stability of the filtering error dynamic system. Then, the corresponding solvability condition for a desired filter satisfying a weighted performance is established. All the conditions obtained are delay-dependent. Finally, two numerical examples are given to illustrate the effectiveness of the proposed theory.
Switched time-delay systems have been attracting considerable attention during the recent years [1–9], due to the significance both in theory development and practical applications. However, it is worth noting that only the state time delay is considered, and the time delay in the state derivatives is largely ignored in the existing literature. If each subsystem of a switched system has time delay in the state derivatives, then the switched system is called switched neutral system . Switched neutral systems exist widely in engineering and social systems, many physical plants can be modeled as switched neutral systems, such as distributed networks and heat exchanges. For example, in , a switched neutral type delay equation with nonlinear perturbations was exploited to model the drilling system. Compared with the switched systems with state time delay, switched neutral systems are much more complicated [12–15]. As effective tools, the common Lyapunov function method, dwell time approaches, and average dwell time approaches have been extended to study the switched neutral systems, and many valuable results have been obtained for switched neutral systems. On the research of stability analysis for switched neutral systems, the asymptotically stable problem of switched neutral systems was considered in . If there exists a Hurwitz linear convex combination of state matrices, and subsystems are not necessarily stable, switching rules can be designed to guarantee the asymptotical stability of the switched neutral system. The method of Lyapunov-Metzler linear matrix inequalities in  was extended to switched neutral systems , and some less conservative stability results were obtained.
In contrast with the traditional Kalman filtering, the filtering does not require the exact knowledge of the statistics of the external noise signals, and it is insensitive to the uncertainties both in the exogenous signal statistics and in dynamic models [19, 20]. Because of these advantages, the filtering has attracted much attention in the past decade for nonswitched neutral systems [21–24]. In , some sufficient conditions for the existence of an filter of a Luenberger observer type have been provided. However, to the authors' best knowledge, the filtering for switched neutral systems has been rarely investigated and still remains challenging. This motivates our research.
The contribution of this paper lies in three aspects. First, we address the delay-dependent filtering problem for switched linear neutral systems with time-varying delays, which appear not only in the state, but also in the state derivatives. The resulting filter is of the Luenberger-observer type. Second, by using average dwell time approach and the piecewise Lyapunov function technique, we derive a delay-dependent sufficient condition, which guarantees exponential stability of the filtering error system. Then, the corresponding solvability condition for a desired filter satisfying a weighted performance is established. Here, to reduce the conservatism of the delay-dependent condition, we introduce some slack matrix variables and a new integral inequality recently proposed in . Finally, we succeed in transforming the filter design problem into the feasibility problem of some linear matrix inequalities. To show the efficiency of the obtained results, we present two relevant examples.
The remainder of this paper is organized as follows. The filtering problem for switched neutral systems is formulated in Section 2. Section 3 presents our main results. Numerical examples are given in Section 4, and we conclude this paper in Section 5.
Notation. Throughout this paper, denotes -dimensional Euclidean space; is the set of all real matrices; means that is positive definite; denotes the space of square integrable vector functions on with norm , where denotes the Euclidean vector norm; is the identity matrix with appropriate dimensions; the symmetric terms in a symmetric matrix are denoted by as for example
2. Problem Statement
Consider the following switched linear neutral system: where is the state vector; is the measurements vector; is the noise signal vector, which belongs to ; is the signal to be estimated; is the initial vector function that is continuously differentiable on ; is a piecewise constant function of time called switching signal. Corresponding to the switching signal , we have the switching sequence , which means that the th subsystem is active when . The system coefficient matrices , , , , , , , and are known real constant matrices of appropriate dimensions. and are time-varying delays satisfying
The objective of this paper is to design a family of filters of Luenberger observer type as follows: where are the filter parameters, which are to be determined.
Now, we introduce the estimation errors: , .
The following definitions are introduced, which will play key roles in deriving our main results.
Definition 1 (see ). The equilibrium of the filtering error system (5) is said to be exponentially stable under if the solution of system (5) with satisfies , for all for constants and , where denotes the Euclidean norm, and .
Definition 2 (see ). For any , let denote the number of switching of over . If holds for , , then is called average dwell time. As commonly used in the literature, we choose .
The filtering problem addressed in this paper is to seek for suitable filter gain such that the filtering error system (5) for any switching signal with average dwell time has a prescribed performance ; that is,(1)the error system (5) with is exponentially stable;(2)under the zero initial conditions, that is, , for all , the weighted performance is guaranteed for all nonzero and a prescribed level of noise attenuation .
Before concluding this section, we introduce three lemmas which are essential for the development of the results.
Lemma 3 (see ). Let be a vector-valued function with first-order continuous-derivative entries. Then, the following integral inequality holds for any matrices , , and , and a scalar , where .
Lemma 4 (see ). For any constant matrix , scalar , vector function such that the integrations concerned are well defined; then, .
Lemma 5 (Schur complement). For given , where and , the following is equivalent:
3. Main Results
In this section, we first present a sufficient condition for exponential stability of the filtering error system (5) with . Then, it is applied to formulate an approach to design the desired filters for switched neutral system (2).
3.1. Stability Analysis
Theorem 6. Given , , for all . If there exist matrices , , , , , and , , () of appropriate dimensions, and , such that for all ,where then the error dynamic system (5) with is exponentially stable for any switching signal with average dwell time satisfying .
Proof. Define the piecewise Lyapunov-Krasovskii functional candidate
Now, taking the derivative of , with respect to along the trajectory of the error system (5) with , according to (3) and Lemma 4, we have
From Lemma 3, it holds
Define where .
By some algebraic manipulations, it is easy to show that where where
From Lemma 5, is equivalent to where
In addition, the following is true from (5) with : where .
Then, (19) along with (21) gives , which yields ; thus, Combining (9) with (22), for any , we have
According to (11), we have where
Considering (23) and (24), it holds .
Therefore, if , that is , then error dynamic system (5) is exponentially stable.
Remark 7. When , we have , which means that the switching signal can be arbitrary. In this case, condition (9) implies that there exists a common Lyapunov functional for all subsystems. Moreover, setting in (8) gives asymptotic stability of the filtering error system (5) under arbitrary switching.
Remark 8. The filters of Luenberger observer type has been adopted in the literatures, see . The Luenberger-type observer can produce an approximation to the system state that is independent of the system trajectory, and it only depends on the initial value of the system state.
Remark 9. The condition guarantees that Lipschitz constant for the right hand of (2) with respect to is less than one.
3.2. Filter Design
Now, we design the desired filter for the switched neutral system (2).
Theorem 10. Given , if , for all , and if there exists matrices , , , , , and , , , of appropriate dimensions, and , such that, for any ,where
then the filter problem for the system (2) is solvable for any switching signal with average dwell time satisfying .
Moreover, the filter gain are given by .
Proof. Consider the piecewise Lyapunov-krasovskii functional candidate as (11) and introduce the vector , where is defined in (21). Then, replace (21) with the following
Let and , similar to the proof of Theorem 6, we have
If , it has
Integrating both sides of (31) from to , for any , gives
Therefore, similar to the proof method of Theory 2 in , we have
Under the zero initial condition, (33) gives
Multiplying both sides of (34) by yields
Notice that and , we have . Thus, (35) implies that
Integrating both sides of (36) from to , we have
Then, we can obtain
Obviously, it follows from (38) that
Remark 11. Theorem 10 provides a sufficient condition for the solvability of the filtering problem for switched neutral system with time-varying delay. If the condition is satisfied, then matrices are nonsingular.
4. Numerical Examples
In this section, we present two numerical examples to illustrate the effectiveness of the results presented previously.
Example 1. Consider the switched neutral system (2) with two subsystems.
By using the LMI toolbox, it can be checked that the conditions given in Theorem 10 are satisfied. Therefore, the previously switched neutral system has the given performance , when (here, the allowable minimum of is ).
Setting and solving LMIs (26) using the LMI Toolbox in MATLAB, it follows that the minimized feasible is , , and the corresponding filter parameters are computed as
In the following, we illustrate the effectiveness of the designed filter through simulation. Let the initial condition be , . Figures 1 and 2 are, respectively, the simulation results on and ; we can see that the filtering error dynamic system with is stable. and of the filtering error dynamic system with are given in Figures 3 and 4. Figure 5 shows .
Example 2. Consider the switched neutral system in Example 1 with constant delays; that is, , , , and . We calculate the admissible maximum value of , of , which ensures that the resulting filtering error system is exponentially stable with a prescribed level of noise attenuation. For the different values, , the obtained , , and the filter gain are listed in Table 1.
We have addressed the filtering problem for a class of switched neutral systems with time-varying delays which appear in both the state and the state derivatives. For switched neutral systems with average dwell time scheme, we have provided a condition, in terms of upper bounds on the delays and in terms of a lower bound on the average dwell time, for the solvability of the filtering problem. The piecewise Lyapunov functional technique has been used, which makes the proposed conditions are both delay-dependent and neutral delay-dependent. The design of filters for switched neutral systems is a difficult issue that is far from being well explored. Since multiple Lyapunov functional approach is commonly considered less conservative, the design of filters for switching neutral system with an appropriate switching law using multiple Lyapunov functionals is of great significance which deserves further study.
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Copyright © 2013 Caiyun Wu and Yue-E. Wang. 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.