Mathematical Description of Linear Dynamical Systems

TLDR

Linear dynamical systems can be described either by state variables or by input/output relations, with the former corresponding to Newtonian mechanics and the latter determining only the observable and controllable portion of the system. The paper presents controllability‑based algorithms for computing irreducible realizations of impulse‑response matrices and a procedure to determine the minimal state‑variable count for a given transfer‑function matrix.

Abstract

There are two different ways of describing uynamicu systems: (i) bymeans of state variables and (ii) by input/output relations. The first method may be regarded as an axiornatization of Newton’s laws of mechanics and is taken to be the basic definition of a system. It is then shown (in the linear case) that the input/output relations determine only one part of a system, that which is completely observable and completely controllable. Using the theory of controllability and observability, methods are given for calculating irreducible realization of a given impulse-response matrix. In particular, an explicit procedure is given to determine the minimal number of state variables necessary to realize a given transfer-function matrix. Difficulties arising from the use of reducible realizations are discussed briefly.