Volterra and Functional Differential Equations

FUNCTIONAL EQUATIONS AS CONTROL CANONICAL FORMS FOR DISTRIBUTED PARAMETER CONTROL SYSTEMS AND A STATE SPACE THEORY FOR CERTAIN DIFFERENTIAL EQUATIONS OF INFINITE ORDER

David L. Russell

Department of Mathematics University of Wisconsin Madison, Wisconsin

DOI: 10.1201/9781003420026-9

1 FINITE DIMENSIONAL DISCRETE SYSTEMS

Let us begin by discussing the canonical form theory as it is manifested in the control of finite dimensional discrete linear control systems with scalar input:

$${\text{x}}_{\text{k}+1}={\text{Ax}}_{\text{k}}+{\text{bu}}_{\text{k}}+\text{k}=0,1,2,\dots$$ (1.1)

$${\text{y}}_{\text{k}}=\left({\text{x}}_{\text{k}},\text{h}\right)\equiv {\text{h}}^{\text{*}}{\text{x}}_{\text{k}},$$ (1.2)

Wherein A is an n× n matrix, x, b and h are in Eⁿ and u, y are scalar. The objective is to replace (1.1) by an equivalent system, called the control ...