A simple definition of hidden modes, poles and zeros

This paper should be read in conjunction with [8], where basic structural properties, such as controllability, observability and state-variable representation, of constant or time-varying generalized linear systems were examined in the context of module theory. Our aim here is to illustrate this approach by giving a simple and transparent definition of hidden modes, poles and zeros of constant multivariable linear systems. The algebraic interpretation of poles and zeros is of course easy via transfer matrices (see, e.g., Kailath [10]). The need has nevertheless been felt by several authors (cf. Francis and Wonham [9], MacFarlane and Karkanias [12], Wonham [15], Wyman and Sain [16], Conte and Perdon [3]) for other types of developments. The first attempt with multivariable hidden modes is due to Rosenbrock [13] (see also Blomberg and Ylinen [1] and Callier and Desoer [2]). The fact that this field of research is still active is demonstrated by a recent and well documented survey due to Schrader and Sain [14]. Elementary module theory permits to define a system without reference to a peculiar choice of equations [8]. It is therefore possible to circumvent some of the difficulties which were previously encountered. Hidden modes, poles and zeros are characterized as the eigenvalues of the derivation d/dt acting on appropriate finitely generated torsion modules.