Beyond the Givone-Roesser/Fornasini-Marchesini models: A universal input/state/output model for 2-D systems

Regardless of their widespread popularity, the Givone-Roesser and the Fornasini-Marchesini models are known to be incapable of representing a huge class of $ 2 $-D systems [11]. In this paper, we provide a method to construct a state space (input/state/output) representation from a given higher order representation of an arbitrary $ 2 $-D system. A crucial role is played by a normalization process – based on a suitable linear change of coordinates on the indexing set – that changes the system's equations to a useful form. The continuous version of this process is the classical result from algebraic geometry called Noether's normalization lemma; an extension of the same to the discrete case was done in [13]. Using this normalization, called Noether normalization, we first provide a method of obtaining an input/state/output (i/s/o) representation of a given controllable $ 2 $-D system. Then we do the same for autonomous systems. Finally, we use the notion of decomposing a general uncontrollable system as a sum of its controllable part and an autonomous subsystem to obtain an i/s/o representation for the general uncontrollable $ 2 $-D system. A key issue that we solve in this decomposition based approach is that of the existence of a Noether normalization that simultaneously normalizes the controllable and autonomous subsystems.