Research Article  Open Access
Chengming Yang, Zhandong Yu, Pinchao Wang, Zhen Yu, Hamid Reza Karimi, Zhiguang Feng, "Robust  Filtering for DiscreteTime Delay Systems", Mathematical Problems in Engineering, vol. 2013, Article ID 408941, 10 pages, 2013. https://doi.org/10.1155/2013/408941
Robust  Filtering for DiscreteTime Delay Systems
Abstract
The problem of robust  filtering for discretetime system with interval timevarying delay and uncertainty is investigated, where the time delay and uncertainty considered are varying in a given interval and normbounded, respectively. The filtering problem based on the  performance is to design a filter such that the filtering error system is asymptotically stable with minimizing the peak value of the estimation error for all possible bounded energy disturbances. Firstly, sufficient  performance analysis condition is established in terms of linear matrix inequalities (LMIs) for discretetime delay systems by utilizing reciprocally convex approach. Then a less conservative result is obtained by introducing some variables to decouple the Lyapunov matrices and the filtering error system matrices. Moreover, the robust  filter is designed for systems with timevarying delay and uncertainty. Finally, a numerical example is given to demonstrate the effectiveness of the filter design method.
1. Introduction
The uncertainty is unavoidable in practical engineering due to the parameter drafting, modeling error, and component aging. The controllers or filtering obtained based on nominal systems cannot be employed to get the desired performance. Therefore, more and more researchers are devoted to robust control or robust filtering problems; see, for instance, [1â€“4]. On the other hand, timedelay often exists in the practical engineering systems and is the main reason of the instability and poor performance of the systems. Timedelay systems have been widely studied during the past two decades [5â€“7]. In order to get less conservative results, more and more approaches have been proposed to develop delaydependent conditions for discretetime system with timevarying delay. For examples, Jensen's inequality is proposed in [8]; delaypartitioning method is utilized in [9]; improved results are obtained by using convex combination approach in [10].
In some practical applications, the peak value of the estimation error is required to be within a certain range and the aim of the  (energytopeak) filtering is to minimize the peak values of the filtering error for any bounded energy disturbance, which has received many attention. By using a parameterdependent approach, the robust energytopeak filtering problem is considered in [11]. An improved robust energytopeak filtering condition is proposed by increasing the flexible dimensions in the solution space in [12]. The robust  filtering for stochastic systems and the exponential  filtering for Markovian jump systems are investigated in [13, 14], respectively. Compared with the corresponding continuoustime systems, discretetime systems with timevarying delay have more stronger application background [15]. For discretetime Markovian jumping systems, the reducedorder filter is designed in [16] such that the filtering error system satisfies an energytopeak performance. When timedelay appears, the robust energytopeak filtering problem for networked systems is tackled in [17]. For discretetime switched systems with timevarying delay, an improved robust energytopeak filtering design method is proposed in [18].
In this paper we consider the problem of robust  filtering for uncertain discretetime systems with timevarying delay. The filter is designed by employing the reciprocally convex approach proposed in [19] such that the filtering error system is asymptotically stable with an  performance. Firstly, a sufficient condition of the  performance analysis for nominal systems is obtained in terms of LMIs for systems with timevarying delay and uncertainty. Based on this criterion, by introducing some slack matrices, a less conservative result is obtained. Moreover, the desired filter for nominal systems with timevarying delay is obtained by solving a set of LMIs. Then the result is extended to the uncertain systems. A numerical example is given to illustrate the effectiveness of the presented results.
Notation. The notation used throughout the paper is given as follows. is the dimensional Euclidean space and (=0) denotes that matrix is real symmetric and positive definite (semidefinite); and present the identity matrix and zero matrix with compatible dimensions, respectively; means the symmetric terms in a symmetric matrix and stands for ; means the space of square summable infinite vector sequences; for any real function , we define and ; refer to the Euclidean vector norm. Matrices are assumed to be compatible for algebraic operations if their dimensions are not explicitly stated.
2. Problem Statement
Consider a class of uncertain discretetime systems with timevarying delay described by where is the state vector; is the measured output; represents the signal to be estimated; is assumed to be an arbitrary noise belonging to and is a given initial condition sequence; is a timevarying delay satisfying , , , , , , , , and are system matrices and satisfy Matrices , , , , , , , , and are unknown timeinvariant matrix representing the uncertainty of the system satisfying the following conditions: where and is a compact set in . The system in (1) is assumed to be asymptotically stable. Our purpose is to design a full order linear filter for the estimate of : where , , , and are filter gains to be determined.
Let the augmented state vector and . Then the filtering error system is described as where and
The nominal system of (6) is system (6) without uncertainty; that is, , , , , , , , , and .
The following lemmas and definition will be utilized in the derivation of the main results.
Lemma 1 (see [20]). For any matrices and , the following inequality holds:
Lemma 2 (see [19]). Let have positive values in a subset of . Then, the reciprocally convex combination of over satisfies subject to
Lemma 3. For any constant matrix , integers , vector function : , then
Lemma 4. Given a symmetric matrix and matrices , with appropriate dimensions, then for all , if and only if there exists a scalar such that
Definition 5. Given a scalar , the filtering error in (6) is said to satisfy the  disturbance attenuation level under zero initial state, and the following condition is satisfied:
Our aim is to design a filter in the form of (5) such that the filtering error system in (6) is asymptotically stable and satisfies the  performance defined in Definition 5.
3. Main Results
In this section, the sufficient  performance analysis condition is first derived for nominal filtering error system of (6). Then an equivalent result is obtained by introducing three slack matrices. Based on these results, a desired filter is designed to render the nominal system of (6) asymptotically stable with an  performance. Then the result is extended to the uncertain system in (6).
3.1.  Performance Analysis
In this subsection, we first give the result of  performance analysis for nominal system of (6).
Theorem 6. Given a scalar , the nominal system of (6) is asymptotically stable with an  performance if there exist matrices ,? , , ??, , ??, and such that the following LMIs hold: where
Proof. First, the asymptotic stability of the nominal system of (6) is proved. We denote and the following Lyapunov functional is chosen:
where
Calculating the forward difference of along the trajectories of filtering error system (6) with yields
By using Lemma 3, we have
Since , the following inequality holds:
where and . Then employing Lemma 2, for , we have
Note that when or , it yields or . Hence, the inequality in (24) still holds. Combining the conditions from (21) to (24), we have
where
On the other hand, the following inequality can be obtained from (17):
which is equivalent to . Hence, which implies that the filtering error system in (6) with is asymptotically stable.
Next, we show the  performance of system (6). To this end, we define
Then under the zero initial condition, that is, , ??, it can be shown that for any nonzero
where
By using Schur complement equivalence, the inequality in (17) is equivalent to . Then we have ; that is,
On the other hand, it yields from (16) and (34) that
where
Then, we have by taking the supremum over time . By Definition 5, the filtering error satisfies a given  disturbance attenuation level. This completes the proof.
Remark 7. The advantage of the results benefits from utilizing the reciprocally convex combination approach proposed in [19]. For the extensively used Jensen inequality [8], the integral term with , ?? by the term which is a special case of the following term with with , which is one of the advantages of reciprocally convex combination approach. On the other hand, the delay partitioning method is widely applied to reduce the conservatism of the results [9, 21, 22]. Also, the method can be extended to the problem considered in this paper. However, it will rise significant computation cost with the partitioning number increasing. Therefore, the reciprocally convex method needs less decision variables and can be seen as a tradeoff between the conservatism and the computation cost.
Then, an equivalent condition of LMI (17) is obtained by introducing three slack matrices , , and , which is presented in the following theorem.
Theorem 8. Given a scalar , the nominal system of (6) is asymptotically stable with an  performance if there exist matrices , ?? ,??, , , , , and , such that the following LMIs hold: where , , , , , and are defined in (17).
Proof. On one hand, if (17) holds, then there exist , , and such that (42) holds. On the other hand, if (42) holds, we have the following inequality based on Lemma 1: In addition, matrices , , and are nonsingular due to , , and . Then, pre and promultiplying (43) by and its transpose yields (17). Therefore, the equivalence between (42) and (17) is proved.
3.2. Robust Filter Design
In this subsection, the filter in the form of (5) is firstly designed such that the nominal filtering error system of (6) is asymptotically stable with an  performance. Then the robust filtering problem is solved. Based on the result of Theorem 8, the filter design method for nominal system of (1) is presented in the following theorem.
Theorem 9. Given a scalar , the nominal system of (6) is asymptotically stable with an  performance if there exist matrices , , , , , , , , , diagonal matrix , , and such that the following set of LMIs hold: where Moreover, a suitable  filter is given by
Proof. Firstly, we introduce four matrices , , , and with invertible and define From (45), we have which implies that is nonsingular. Hence, and are nonsingular. The inequality in (46) can be obtained by pre an promultiplying (34) with and , respectively. Noting that we have . On the other hand, because and cannot be obtained from (45), we cannot determine the filters from (49). However, we can construct an equivalent filter transfer function from to : Therefore, the desired filter can be obtained from (48). This completes the proof.
Then the filter design result for uncertain system (6) is presented in the following theorem.
Theorem 10. Given a scalar , the system in (6) with uncertainty is asymptotically stable with an  performance if there exist matrices ,? ?, ?, , , ?, , , , diagonal matrix , , , and scalars ,?? such that the following set of LMIs hold: where is defined in (45) and Moreover, a suitable  filter is given by
Proof. Firstly, replace matrices , , , , , and in (45) with , , ,? , , and , respectively, and the following inequality is obtained: where , are defined in (53). Then by using Lemma 4, the above inequality holds if and only if Then by using Schur complement equivalence, the inequality in (58) is equivalent to (45). Substituting , , and in (46) with ,??, and , respectively, we can get where By following the similar line, the equivalence between (59) and (54) can be proved.
4. Illustrative Example
In this section, the following example is given to demonstrate the effectiveness of the proposed approach.
Example 1. Firstly, consider a nominal discretetime delay system in (1) with the following parameters: For different delay cases, the different minima of can be calculated by solving the LMIs in Theorem 9. When the upper bound of the timevarying delay is 5, that is, , the minima of for a given are listed in Table 1.

Moreover, when , , the corresponding  filter is given as follows: When uncertainty appears in the system, Theorem 10 will be used for the desired filter design. The following uncertainty parameters are considered: Similarly, the allowed minimal values of can be obtained by solving the LMIs in Theorem 10. For , the different minimum allowed are listed in Table 2 for the uncertain system with different