Properties of a Generalized Impulse Response Gramian with Application to Model Reduction

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.