Regulator Problem of Partial Differential System

Abstract

This paper deals with the regulator problem of a distributed-parameter system described by a parabolic or hyperbolic linear partial differential equation. The Riccati-type partial differential equation or the infinite-dimensional two point boundary value problem must be solved to obtain the solution u₀ of the regulator problem of a distributed-parameter system. It is very difficult to calculate the solution of such a complex problem. Therefore, it is reasonable to approximate the distributed-parameter system with a lumped-parameter system having n main modes. By solving the regulator problem of the lumped-parameter system, an approximate solution u_n will be obtained instead of u₀. This authors' idea is an extension of Modal Control Method proposed by Gould to the regulator problem of a distributed-parameter system. But if J(u) is the performance index of the regulator problem and n tends to infinity, the convergence of J(u_n) to J(u₀) is equivalent to the convergence of u_n to u₀ in an appropriate topology. Then it must be investigated whether u_n converges to u₀ and in what topology this convergence is assured.
The authors show that u_n converges to u₀ in the mean square, and furthermore it converges uniformly if J(u) contains the derivative of the control input with time.