A behavioural approach to the zero structure of multidimensional linear systems
We use the behavioural approach and commutative algebra to define and characterize poles and zeros of multidimensional (nD) linear systems. In the case of a system with a standard input output structure we provide new definitions and characterizations of system, controllable and uncontrollable zeros and demonstrate strong relationships between the controllable poles and zeros and properties of the system transfer matrix, and we show that the uncontrollable zeros are in fact uncontrollable poles. We also show that we can regard the zero as a form of pole with respect to an additional form of input output structure imposed on the zero output sub-behaviour. In the case when the behaviour has a latent variable description we make a further distinction of the zeros into several other classes including observable, unobservable and invariant zeros. In addition we also introduce their corresponding controllable and uncontrollable zeros, such as the observable controllable, unobservable controllable, invariant controllable, observable uncontrollable, unobservable uncontrollable and invariant uncontrollable etc. We again demonstrate strong relationships between these and other types of zeros and provide physical interpretations in terms of exponential and polynomial exponential trajectories. In the 1D case of a state-space model we show that the definitions and characterizations of the observable controllable and invariant zeros correspond to the transmission zeros and the invariant zeros in the classical 1D framework. This then completes the correspondences between the behavioural definitions of poles and zeros and those classical poles and zeros which have an interpretation in nD.