The Block form of Linear Systems over Commutative Rings with Applications to Control

This paper deals with a new approach to the study of linear time-invariant discrete-time systems whose coefficients belong to an arbitrary commutative ring. Such systems arise in the study of integer systems, systems depending on parameters, and multi-dimensional systems. The key idea is to consider the representation of systems over a ring in terms of a block-input/block-output form. By time-compressing the block representation, new results are derived on assignability by state feedback control including the construction of deadbeat controllers. A new type of state observer is then considered based on a block-output form for the update term in the state estimate. The results on state observers are combined with the time-compression approach to state feedback control to yield a new type of input/output regulator. In the last section of the paper the results are applied to the problem of state and output tracking of set points.