REALIZATION OF LINEAR DYNAMICAL SYSTEMS AND NETWORKS

Abstract

With the increasing use of state-space approach in control systems and network theory, considerable interest has been shown in the problem of realization of linear systems, ^fhis thesis is concerned with the state-space realization of linear dynamical systems and its application to networks. la particular, both minimal and n on-minimal realization techniques have been developed and their application to problems in network theory have been sought with a view to obtain better insight and to improve upon the existing techniques in network and system theory, *"•"" The problem of state model realization of a symmetric, positive real matrix for passive RLC networks without the use of gyrators has been investigated and a new minimal realization technique based on the moments of impulse response matrix has been proposed. The method is especially preferable for the cases where the data is contaminated with noise. The algorithms for the realization problem of linear dynamical systems proposed up til now appear compu tationally rather cumbersome, A simplified technique for obtaining a non-minimal state-model of a transfer function matrix has been proposed. In order to determine the Vll dimension of the realization, mode matrices Mand M are defined for the multiple pole case. Roveda and Schmid [91 j have proposed a procedure for obtaining on upper bound on the dimension of a minimal realization. Their method is applicable under the assump tion that no elenent of the transfer-function matrix Ii(s) has multiple poles. Here, a generalized algorithm is developed to obtain a n on-minimad. realization for the case of H(s) having simple as well as multiple poles. The realization results is a still lower dimension, compared with the other methods. Because of a change from transfer-function desc ription of a dynamical system to a more general statespace characterization, it is quite important to establish a communication link between state-space characterization and frequency domain methods. Some work has already been initiated in this direction. A technique for determining the state-model and the impedance matrix Z(s) of order n from given U(s) = Z(s) + Zr(-s) is presented, which is simpler than the one proposed earlier [61 J , It avoids the cumbersome spectral factorization and the determination of a symmetric positive definite matrix P, which gets unwieldy in the case of existing methods especially when the order n of U(s) is largo. Z(s) obtained thus is a minimum reactance matrix. An algorithm is also proposed for obtaining state-space realization and the impedance matrix Z(s) when V(s) = Z(s) - Z'(-s) is given. The method is applicable to V(s) of any order n. Further, a state-space interpretation of the Foster synthesis method for driving point immittance functions of LC networks is presented. A method for determining transfer-function matrix from .a knowledge of its moments is presented. It is shown that at the most (n+1) moments of the impulse response matrix are required in the process, where n is the order of the state matrix. Also, a method is given for deter mining the resolvent matrix (sI-A)"1 and its higher powers, where the given matrix A is in Jordan canonical, form. Further, when A is in the companion form, an algorithm is proposed to compute A~k , k = 1,2,... . These results may be employed to find the moments of the impulse-response matrix. A method is given to construct a transformation N(t) which transforms a time-varying autonomous system to the companion form. In some cases the transformation could be made a constant matrix. Finally, some suggestions are given for further work in this field.