Pergamon Automotica zyxwvutsrqponmlkjihgfedcba Vol. 34. No. zyxwvutsrqponmlkjihgfe 1. DD. 111-124. 1998 PII: sooo5-1098(97)00187-8 o 1998 zyxwvutsrqponmlkjihgfedcbaZ Else&r S&z Li bjl tightsn?scrwd Printed in Gmt Britain cmos-1098/98 s19.00 + 0.00 Brief Paper Mappings of the Finite and Infinite Zero Structures and Invertibility Structures of General Linear Multivariable Systems Under the Bilinear Transformation* BEN M. CHENt and STEVEN R. WELLER1 Key Words-Linear system theory; continuous-time systems; discrete-time systems; bilinear trans- formation. Abstract-This paper presents a comprehensive picture of the mapping of structural properties associated with general linear multivariable systems under bilinear transformation. While the mapping of poles of linear multivariable systems under such a transformation is well known, the question of how the struc- tural invariant properties of a given system are mapped remains unanswered. This paper addresses this question. More specifi- cally, we investigate how the finite and infinite zero structures, as well as invertibility structures, of a general continuous-time (discrete-time) linear time-invariant multivariable system are mapped to those of its discrete-time (continuous-time) counter- part under the bilinear (inverse bilinear) transformation. We demonstrate that the structural invariant indices lists Yx and .Ysof Morse remain invariant under the bilinear transformation, while the structural invariant indices lists 4, and X4 of Morse are, in general, changed. 0 1998 Elsevier Science Ltd. All rights reserved. 1. INTRODUCTION AND PROBLEM STATEMENT The need to perform continuous-time to discrete- time model conversions arises in a range of engin- eering contexts, including sampled-data control system design, and digital signal processing. As a consequence, numerous discretization procedures exist, including zero- and first-order hold input approximation, impulse invariant transformation, !nd bilinear transformation (see, for example Astrijm et zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA al., 1984; Franklin et al., 1980). Despite of the widespread use of the bilinear transform, however, a comprehensive treatment is lacking which details how key structural properties of con- tinuous-time systems, such as the finite and infinite zero structures, and invertibility properties, are in- herited by their discrete-time counterparts. Given *Received 10 July 1996; revised 17 January 1997; received in final form 3 September 1997. This paper was recommended for tmbhcation in revised form bv Editor H. Kwakernaak. Cor&sponding author Ben M. Chen. Tel. 0065 874 2289; Fax 0065 779 1103; E-mail bmchen@nus.edu.sg. TDepartment of Electrical Engineering, National University of Singapore, Singapore 119260, Republic of Singapore. $Department of Electrical and Computer Engineering, Uni- versity of Newcastle, Callaghan, NSW 2308, Australia. the important role played by the infinite and finite zero structures in control system design, a clear understanding of the zero structures under bilinear transformation would be useful in the design of sampled-data control systems, and would comp- lement existing results on the mapping of finite and infinite zero structures under zero-order hold sampling (see, for example, Astriim et al., 1984; Grizzle and Shor, 1988). In this paper, we present a comprehensive study of how the structures, i.e. the finite and infinite zero structures, as well as invertibility structures, of a general continuous- time (discrete-time) linear time-invariant system are mapped to those of its discrete-time (continuous- time) counterpart under the well-known bilinear (inverse bilinear) transformations z-l ( > a+s s=a - z+l and z = -, (1) a-s respectively. Since the bilinear and inverse bilinear transformations find widespread use in digital con- trol and signal processing, the results obtained have immediate applications. In particular, some of the results have already been applied to solve discrete- time algebraic Riccati equations (see for example Chen et al., 1994), and in the solution of certain H, control problems (see for example, Chen, 1996; and Chen et al., 1996). We consider in this paper general linear time- invariant systems characterized by ~ *I 6x = Ax + Bu, . y= cx +Du, (2) where x E: R”, y E RP, u E R” and A, B, C and D are matrices of appropriate dimensions. In equation (2), 6 is an operator defined as follows: 6x = i if X is a continuous-time system, and 6x = x(k + 1) if X is a discrete-time system. Without loss of any general- ity, we assume throughout this paper that both matrices [C D] and [B’ D’] are of full rank. 111