IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 3, MARCH 2011 979 On a Rational Transfer Function-Based Approach to Filtering Design for Time-Delay Linear Systems Rubens H. Korogui, Member, IEEE, André R. Fioravanti, Member, IEEE, and José C. Geromel, Member, IEEE Abstract—This paper introduces a new procedure for filter design of time-delay linear systems. A finite-order LTI system, called comparison system, is defined in such a way that its norm is proven to be strongly related to the one of the time-delay system. Differently from what can be viewed as a common feature of filter design methods available in the literature to date, the one presented here treats the filtering design of time-delay systems with classical numerical routines based on Riccati equation and theory of LTI systems. The proposed algorithm is simple, ef- ficient and easy to implement. Illustrative examples are solved and discussed in order to put in evidence the most relevant properties of the theoretical results. Furthermore, a practical application is presented. Index Terms— filtering, linear systems, time-delay systems. I. INTRODUCTION T HE existence of delays generally induces instability, os- cillation, and poor performance leading to an increasing effort to develop efficient design techniques of control and fil- tering for such class of dynamic systems. The books [1] and [2], the survey paper [3] and the references therein contain a large variety of relevant results in this topic. In this context, the design of filters has naturally received great attention, specially in the direction of taking into account the time delays in the design process [4]–[6]. The two main objectives of filter design for time-delay systems are either to determine a filter such that the estimation error is stable under a maximum allowed delay bound given a fixed performance value, or to achieve a min- imum performance level for a prespecified delay bound [6]. In the literature, these problems are addressed by the Riccati equation when dealing exclusively with delayed output, or for more general models including time-varying delays by the Lya- punov-Krasovskii functional approach, leading to LMI-based formulations [4], [5], [7], and [8]. For interesting practical ap- plications involving time-delay systems in the context of fault Manuscript received August 19, 2010; revised October 25, 2010; accepted October 26, 2010. Date of publication November 09, 2010; date of current ver- sion February 09, 2011. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Bogdan Dumitrescu. This work was supported by grants from “Fundação de Amparo à Pesquisa do Es- tado de São Paulo-FAPESP” and by “Conselho Nacional de Desenvolvimento Científico e Tecnológico-CNPq”, Brazil. R. H. Korogui and J. C. Geromel are with the DSCE/School of Electrical and Computer Engineering, UNICAMP, 13083-852, Campinas, SP, Brazil (e-mail: korogui@dsce.fee.unicamp.br; geromel@dsce.fee.unicamp.br). A. R. Fioravanti is with the INRIA Saclay Ilê-de-France, Supélec, 91192, Gif-sur-Yvette, France (e-mail: andre.fioravanti@inria.fr). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2010.2090877 detection and communication channels see [9] and [10], respec- tively. Recently, the paper [11] has proposed an useful technique for stability analysis of time-delay systems. In fact, as commented by the authors, they have successfully generalized the previous results by [12] and [13], among others, for stability margin com- putation of time-delay systems using the Rekasius substitution [13] and the well-known Routh-Hurwitz criterion. For stability analysis it is also important to recall the frequency domain ap- proaches based on the variation of the argument, leading to the Nyquist criterion, [14], [15]. With similar purposes, the results of [16] and [17] are relevant. It is important to notice that all these methods for stability analysis have as a common feature: the necessity to explicitly calculate and handle the character- istic equation of the time-delay system. Finally, in our opinion, the paper [18] provides one of the most important results for stability analysis and norm calculation. Indeed, adopting a comparison system approach, the well known Padé approx- imation is used to determine linear time invariant systems of increasing but finite order, allowing the direct determination of stability margin and bounds for the norm performance of the time-delay system. It is shown that the quality of the re- sults is better whenever the order of the Padé approximation in- creases. This paper follows the same stream as that proposed in [18]. A linear time invariant comparison system of order twice the number of state variables of the time-delay system, built from the Rekasius substitution, is introduced, and the relationship be- tween the stability of the comparison system and the time-delay system is established. This is accomplished by the Nyquist cri- terion applied to some specific characteristic equations, related to the comparison and to the time-delay systems, respectively, see also [15] and [19]. Moreover, it is shown that the norm of the comparison system provides a precise and useful lower bound to the norm of the time-delay system. This property is used for delay-dependent linear filter design and it is shown how lower and upper bounds on the norm of the estima- tion error transfer function are imposed. The results are illus- trated by means of several examples borrowed from the litera- ture. Comparisons including the norm lower bound given in [18] show that our method is simpler to implement and provides good and precise results. In our opinion, the main contributions of this paper are: • The statement of a rational comparison system of finite and constant order that is well adapted to deal with single time-delay systems. However, the generalization to cope with multiple delays is not simple. 1053-587X/$26.00 © 2010 IEEE