Invertibility of linear time-invariant dynamical systems

Abstract

The question of inverting linear time-invariant dynamical systems has been of interest to control engineers for many years. An example of the nonstate-variable approach and application of this concept is the very well known work of Bode and Shannon in 1950. With the integration of state-variable descriptions and techniques into systems problems, the question has recently reappeared in a somewhat more complicated guise. Basically, the problem of inverting such dynamical systems is to determine the existence of the inverse, its properties, and its construction in terms of the matrices which characterize its state description. The first general study of existence appears to be due indirectly to Brockett and Mesarovic in 1965, and the first general construction seems to have been proposed by Youla and Dorato in 1966. Neither of these works was intended to develop a substantial insight into the properties of the inverse. The present work introduces the concept of the inherent integration associated with a dynamical system, i.e., the number of integrations which no inverse dynamical system can remove unless ideal differentiators are introduced. The existence of the inverse is discussed in terms of a determination of the inherent integration, and the construction which realizes this minimum number of integrations is given. The existence tests introduced are at worst one-half as complex as that of Brockett and Mesarovic and the construction offers a substantial improvement in conceptual simplicity over that of Youla and Dorato. The results are made possible by recognizing an essential equivalence with an associated problem in real sequential circuits and appear to be applicable to related problems in sensitivity, estimation, and game theory.