On the modal control of distributed systems with distributed feedback

Abstract

This paper presents a solution to the problem of the control of a class of linear distributed systems. The system is described by a linear operator acting on functions of time and distance. It is shown that if the operator separates in time and distance and the distance operator has a real discrete spectrum (self-adjoint and completely continuous), the operator can be represented by an infinite diagonal matrix in which the entries are functions of the Laplace transform variable <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</tex> . In particular, if the problem stems from separable partial differential equations, the entries are rational ratios of polynomials of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</tex> . The system can be compensated with a series of discrete conventional filters using techniques of conventional lumped, single loop control system design. To implement the control system, the assumption is made that the distance dependent part of the output and forcing functions have negligible eigenfunction content beyond the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> th one. If this assumption holds, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> sensors, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> filters, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> manipulators, plus 2 matrix multipliers and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> subtractors provide a synthesis of the feedback control system. An illustrative numerical example is given.