516 Younseok Choo and Jaeho Choi Properties of a Generalized Impulse Response Gramian with Application to Model Reduction Younseok Choo and Jaeho Choi Abstract: In this paper we investigate the properties of a generalized impulse response Gramian. The recursive relationship satisfied by the family of Gramians is established. It is shown that the generalized impulse response Gramian contains information on the characteristic polynomial of a linear time-invariant continuous system. The results are applied to model reduction problem. Keywords: Generalized impulse response Gramian, Lyapunov equation, Markov parameter, model reduction, time-moment. 1. INTRODUCTION In identification or model reduction problems, an important task is the computation of the characteristic polynomial of the original or reduced-order system. Several literature have shown that the characteristic polynomial of a system can be extracted from the information generated by the impulse response data. For continuous systems, the Gram matrix [1,2] and the impulse response Gramian [3] are good examples of information from which the characteristic polynomial of a system can be obtained. For discrete systems, the Hankel matrix [4] and the impulse response Gramian [5] possess the same properties. Recently a new impulse response Gramian was introduced in [6] that can also be utilized to compute the characteristic polynomial of a discrete system. In addition to computing the characteristic polynomial, those impulse response data are useful for the order reduction of linear time-invariant systems [6-10]. In this paper we investigate the properties of a generalized impulse response Gramian [11], which includes the Gram matrix of [1,2] and the impulse response Gramian in [3] as special cases. The recursive relationship satisfied by the family of Gramians is established. It is also shown that the generalized impulse response Gramian contains information on the characteristic polynomial of a linear time-invariant continuous system. The results are applied to model reduction problem. This paper is organized as follows. In Section 2, some preliminaries are presented. The properties of a generalized impulse response Gramian are studied in Section 3. An application to a model reduction problem is considered in Section 4 and the paper is concluded in Section 5. 2. PRELIMINARIES 2.1. Canonical realizations Consider a stable nth-order linear time-invariant system described by the transfer function 1 2 1 2 1 1 1 1 () n n n n n n n n bs bs b s b Hs s as a s a − − − − − + + + + = + + + + " " (1) or by the minimal state-space realization () () () t A t br t = + x x  , (2) () ( ), yt c t = x (3) where () . n t ∈ ℜ x The transfer function () Hs can be expanded into the following two forms 2 3 1 2 3 4 () Hs t ts ts ts = − − − − − " , (4) 3 1 2 2 3 () m m m Hs s s s = + + + " , (5) where i t s and i m s respectively denote the time- moments and Markov parameters of the system, and are computed from the coefficients of () Hs or from __________ Manuscript received June 21, 2003; revised March 17, 2004; accepted October 13, 2004. Recommended by Editorial Board member Young Il Lee under the direction of Editor Chung Choo Chung. This paper was accomplished with the help of a research fund provided by the Korean Council for University Education, support for 2003 Domestic Faculty Exchange. Younseok Choo is with the School of Electronic, Electrical and Computer Engineering, Hongik University, San 34, Sinan- Ri, Jochiwon-Eup, Yeongi-Gun, Chungnam 339-701, Korea (e-mail: yschoo@wow.hongik.ac.kr). Jaeho Choi is with the School of Electrical and Electronic Engineering, Chungbuk National University, San 48, Gaesin- Dong, Heungdeok-Gu, Cheongju, Chungbuk 361-763, Korea (e-mail: choi@power.chungbuk.ac.kr). International Journal of Control, Automation, and Systems, vol. 2, no. 4, pp. 516-522, December 2004