Delay-Dependent Robust H Filtering for Uncertain ... - IEEE Xplore

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 53, NO. 12, DECEMBER 2006

H

Delay-Dependent Robust Filtering for Uncertain Discrete-Time Systems With Time-Varying Delay Based on a Finite Sum Inequality Xian-Ming Zhang and Qing-Long Han

Abstract—This brief is concerned with delay-dependent robust filtering for uncertain discrete-time systems with timevarying delay. The uncertainty is of convex polytopic type. By establishing a finite sum inequality based on quadratic terms, a new delay-dependent bounded real lemma (BRL) is derived. In combination with a parameter-dependent Lyapunov–Krasovskii functional, which allows the Lyapunov–Krasovskii matrices to be vertex dependent, the obtained BRL is modified into a new version to suit for convex polytopic uncertainties. Neither model transformation nor bounding technique for cross terms is involved. Based on the new BRL, the designed filter is provided in terms of a linear matrix inequality (LMI), which is easily solved by Matlab LMI toolbox. A numerical example is given to illustrate the effectiveness of the proposed method. filtering, linear Index Terms—Discrete-time linear systems, matrix inequality (LMI), stability, time-varying delay.

I. INTRODUCTION

T

HE topic of filtering for systems has been widely investigated in the past decade (see, for example, [2], [5], [6], [9], [12], and references therein). The aim is to design a suitable norm such that the induced and stable filter to minimize the gain from the noise signal to the estimation error is less than a prescribed level. Some approaches to this issue, such as the algebraic Riccati equations/inequalities, interpolation, and linear matrix inequalities (LMIs), have been developed in the recent years. A great number of practical systems are unavoidably subject to uncertainties and delays, which usually degrade the performance of the systems under consideration. Therefore, the robust filtering for uncertain systems with time delay have been focused on recently. For discrete-time systems with time delay, some sufficient conditions were obtained for the existence of the desired filters [3], [4], [8], [10]. However, these conditions are delay independent, which are conservative, especially for Manuscript received March 21, 2006; revised May 30, 2006. This work was supported in part by Central Queensland University for the Research Advancement Awards Scheme Project “Robust Fault Detection, Filtering and Control for Uncertain Systems With Time-Varying Delay” (January 2006–December 2008). This paper was recommended by Associate Editor Y. Lian. X.-M. Zhang is with the School of Mathematical Science and Computing Technology, Central South University, Changsha 410083, China, and also with the School of Computing Sciences, Faculty of Business and Informatics, Central Queensland University, Rockhampton, Qld. 4702, Australia. Q.-L. Han is with the School of Computing Sciences, Faculty of Business and Informatics, Central Queensland University, Rockhampton, Qld. 4702, Australia (e-mail: [email protected]). Digital Object Identifier 10.1109/TCSII.2006.884116

small delay. In [1], a model transformation approach incorporating with Moon’s inequality to bound some cross terms was employed, and some delay-dependent conditions were derived filter. As is well known, these condito design the desired tions may be also conservative due to the model transformation and bounding technique for cross terms. Moreover, the aforementioned results were established under the assumption that the delay was constant; when the delay is time varying, they are inapplicable. To the best of our knowledge, no delay-dependent filtering result for discrete-time systems with time-varying delay has been reported in the open literature. filter for In this brief, we will design the delay-dependent discrete-time systems with time-varying delay. A new sufficient filter will be obcriterion for the existence of a suitable tained by introducing a new finite sum inequality, which avoids using both model transformation and bounding technique for cross terms. A numerical example will be given to show that the obtained results are less conservative than some existing ones. Notation: The symmetric term in a symmetric matrix is denoted by , e.g.,

. The other notations

are routine ones. II. PROBLEM STATEMENT Consider the following uncertain discrete-time system with time-varying delay:

(1)

where is the state vector, is the measured is the signal to be estimated, output, is assumed to be an arbitrary noise signal in is a known given initial condition sequence, and is a positive integer time-varying delay satisfying (2) Denote . Clearly, means that the time-delay is time invariant. The system matrices are supposed to be uncertain and unknown but belong to a known convex compact set of polytopic type, i.e.,

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(3)

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where

Lemma 1: For any constant matrices , , , , then a positive integer time-varying

with , which denotes the th vertex of the polyhedral domain . Suppose that system (1) is robustly asymptotically stable over the entire polytopic domain . In this case, we will design a fullorder filter with state-space realization of the following form:

with

, and

(7) where (8)

(4)

(9) where constant matrices , , , and are filter parameters to be determined. Defining the augmented state vector and the estimation error , one obtains the following filtering system:

(5)

where

Proof: Let

, then

It follows that

and

(10) Notice that

Rearranging (10) yields (7). The purpose of this brief is to design a robust filter of the form performance (4) such that the system (5) has a prescribed for all uncertainties satisfying (3). is asymptotically stable. 1) System (5) with noise attenua2) System (5) has a prescribed level of tion, i.e., under the zero initial condition, is satisfied for any nonzero . In the following, we introduce two vectors:

III.

PERFORMANCE ANALYSIS

Based on Lemma 1, a new delay-dependent condition on performance analysis is derived, which can guarantee that the performance . filtering system (5) has a prescribed , the system (5) with Proposition 1: Given is asymptotically stable with a guaranteed performance if there exist real matrices , , , , , and with appropriate dimensions such that

(11) then

where (6)

where

The following lemma gives the relationship between the vecand , which will play a key role in the delay-detors pendent performance analysis.

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 53, NO. 12, DECEMBER 2006

Proof: Choose a Lyapunov–Krasovskii functional candidate as (12) where

Using Schur complement and considering (18) with yields , which guarantees the asymptotic stability . of (5) with Next, assuming that under zero initial condition, is satisfied for any nonzero . Noting that , rewrite (18) as

Thus, if (11) holds, using Schur complement yields (20) Summing both sides of (20) from 0 to

(21)

with , , and are to be determined. Taking the forward difference along the trajectory of system (5) yields

Noting that

, we obtain

Under zero initial condition,

, one obtains

(13)

(22)

(14)

That is, is true for all nonzero , which completes the proof. Remark 1: From the proof process of Proposition 1, one can clearly see that neither model transformation nor bounding technique for cross terms is involved. Therefore, the obtained result is expected to be less conservative. To exploit parameter-dependent Lyapunov–Krasovskii functionals to handle the polytopic uncertainties, using the idea in and , we can con[7] by introducing two slack variables clude that the matrix (11) is implied by

from (2), we derive (15)

Use Lemma 1 to obtain (16)

(23)

where and are defined in (8) and (9), respectively. Similar to [11], we have where (17) Combining (13), (16), and (17), one obtains

(18) where is defined in (11). We first show that the system (5) with ically stable. In fact, (11) implies

is asymptot-

Then, employing parameter-dependent Lyapunov–Krasovskii functionals yields the following conclusion. , the filtering system (5) with Proposition 2: Given (2) is asymptotically stable with a guaranteed level of noise attenuation for all uncertainties satisfying (3) if there exist real matrices , , , , , , and matrices such that for

(19) (24)

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where

Proof: We are about to prove the conclusion using Propo. sition 2. If the matrix inequality (24) is true, then as Partition

where can deduce that

IV.

. Then, is invertible. Define

, from which we

and let

FILTER DESIGN

In this section, based on Proposition 2, we present the following sufficient condition for the existence of a desired filter of form (4). Proposition 3: Consider the system (1) with uncertainties (3), an admissible robust filter of the form (4) exists if there exist with

matrix ,

, ,

,

, ,

,

, ,

, , and

,

(27)

Then, we have

, and matrices such that

for

(25)

where

(28) where and are defined in (24) and (25), respectively. If , then is true. Therefore, the (25) holds, i.e., performance . filtering system (5) has a prescribed On the other hand, clearly, the filter parameters , , , are implicit in (27) except that can be directly oband , , tained from (25). To solve the other filter parameters , at first, it is obvious that is invertible from (28) if and (25) is feasible, then the filter parameters , , and can be rewritten as

(29)

Since the following systems are algebraically equivalent:

(30)

Moreover, a suitable filter realization is given by (26)

thus a state-space realization as (26) of the desired filter is readily obtained from (30), which completes the proof. Remark 2: Proposition 3 provides a delay-dependent condifilter for uncertain discrete-time tion to design a suitable systems with time-varying delay. This condition depends on the upper bound as well as the lower bound of the time-varying delay. Thus, when these bounds are available, Proposition 3 can achieve less conservative results. Moreover, if the lower bound

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ACHIEVED MINIMUM H

TABLE I =5 LEVEL FOR DIFFERENT h WHEN h

TABLE II OF H NOISE ATTENUATION ACHIEVED MINIMUM H LEVEL  CORRESPONDING TO DELAY UPPER BOUNDS h

is not exactly known, we can replace it with zero; if the lower bound is equal to the upper bound, then the delay is time infiltering variant. In this sense, Proposition 3 can handle the for a large class of discrete-time systems.

Fig. 1. Maximum singular value curves of the filtering transfer function at the four vertices.

VI. CONCLUSION V. NUMERICAL EXAMPLE Example : Consider system (1) with

(32)

and . The uncertain parameters satisfy is time varying, the results in [1] are inapplicable When to this case. However, using Proposition 3, the achieved performances of the filtering system are listed in Table I . It is clearly shown for different lower bounds when decreases as increases. from this table that is time invariant. We now suppose that the delay performance of filtering We calculated the achieved system (5) corresponding to different delay upper bounds by using the approaches proposed both in this brief and [1], which are listed in Table II, from which one can clearly see that Proposition 3 obtains much less conservative results than that in [1]. , Proposition 3 yields the minimum Especially, when level , and the corresponding filter parameters are given in (33), whereas the method in [1] fails to make any conclusion, i.e.,

(33) Connecting filter parameters (33) to the filtering systems (5) and (32), we depicted the singular value curves of the transfer functions at four vertices, which are shown in Fig. 1. Clearly, all of the maximum singular values are less than 8.25, which demonstrates the effectiveness of the proposed method in this brief.

filtering for uncertain The delay-dependent robust discrete-time systems with time-varying delay has been investigated. By introducing a new finite sum inequality based on quadratic terms, a new bounded real lemma (BRL) for the filtering system has been obtained in combining with Lyapunov– Krasovskii functional method. Neither model transformation nor bounding technique for cross terms has been employed. The new BRL has been modified to be an LMI, which enables filter. Finally, a numerical example us to easily solve the have been given to illustrate the effectiveness of the proposed approach. REFERENCES [1] H. Gao and C. Wang, “A delay-dependent approach to robust H filtering for uncertain discrete-time state-delayed systems,” IEEE Trans. Signal Process., vol. 52, no. 6, pp. 1631–1640, Jun. 2004. [2] J. C. Geromel and M. C. de Oliveria, “ H and H robust filtering for convex bounded uncertain systems,” IEEE Trans. Autom. Control, vol. 46, no. 1, pp. 100–107, Jan. 2001. [3] N. T. Hoang, H. D. Tuan, P. Apkarian, and S. Hosoe, “Robust filtering for discrete nonlinear fractional transformation systems,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 51, no. 11, pp. 587–592, Nov. 2004. [4] J. H. Kim, S. J. Ahn, and S. Ahn, “Guaranteed cost and H filtering for discrete-time polytopic uncertain systems with time-delay,” J. Franklin Inst., vol. 342, no. 4, pp. 365–378, Jul. 2005. [5] X. Lai, “Projected least-squares algorithms for constrained FIR filter design,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 52, no. 11, pp. 2436–2443, Nov. 2005. [6] H. Liu, F. Sun, K. He, and Z. Sun, “Design of reduced-order H filter for Markovian jumping systems with time delay,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 51, no. 11, pp. 607–612, Nov. 2004. [7] M. C. de Oliverira, J. Bernussou, and C. Geromel, “A new discretetime robust stability condition,” Syst. Control Lett., vol. 37, no. 4, pp. 261–265, Jul. 1999. [8] R. M. Palhares, C. E. de Souza, and P. L. D. Peres, “Robust H filtering for uncertain discrete-time state-delayed systems,” IEEE Trans. Signal Process., vol. 49, no. 8, pp. 1696–1703, Aug. 2001. [9] L. Xie, L. Lu, D. Zhang, and H. Zhang, “Improved robust H and H filtering for uncertain discrete-time systems,” Automatica, vol. 40, no. 5, pp. 873–880, May 2004. [10] S. Xu, “Robust H filtering for a class of discrete-time uncertain nonlinear systems with state delay,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 49, no. 12, pp. 1853–1859, Dec. 2002. [11] S. Xu and T. Chen, “Robust H control for uncertain discrete-time systems with time-varying delays via exponential output feedback controllers,” Syst. Control Lett., vol. 51, no. 3–4, pp. 171–183, Mar. 2004. [12] Z. Wang, D. W. C. Ho, and X. Liu, “Robust filtering under randomly varying sensor delay with variance constraints,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 51, no. 6, pp. 320–326, Jun. 2004.