Discrete time convolution control systems

In this paper, we study a general class of linear time invariant discrete time distributed systems. We consider both single-input-single-output (SISO) and multi-input-multi-output (MIMO) systems, and study design procedures. We develop a commutative algebra of transfer function, b(p0), for a general class of SISO discrete time convolution systems, which covers sampled distributed systems and, of course, lumped systems as a special case. Each element of b(p0) is formulated as a ratio of two elements in an algebra l1−(p0) of causal p0-stable transfer functions. We demonstrate that l1−(p0) indeed a euclidean ring, give necessary and sufficient conditions for coprimeness between elements in l1−(p0) and characterize poles and zeros for elements in b(p0). In contrast to the algebra l1 the algebra b(p0) includes both stable and unstable systems; furthermore since p0<1 this formulation allows us to study the dominant poles inside the unit disc of the complex plane. We study next MIMO systems whose transfer functions are matrices with elements in b(p0). We establish the matrix fraction representation theory and use it to develop : the dynamic interpretation of poles and transmission zeros, the feedback interconnection of such MIMO systems, and the problem of controller design to achieve stabilization (analogous to arbitrary closed-loop eigenvalue assignment), asymptotic tracking and disturbance rejection ; finally, for the case of stable square plants, we show how to achieve complete decoupling with detailed pole assignment and finite settling time, subject to, of course, the limitations imposed by the plant transmission zeros outside the open unit disc.