ɛ-Coupling and Singular Perturbation Approaches to the Control of Interconnected Distributed Parameter Systems

Abstract

Many distributed-parameter systems consist of interconnected subsystems involving fast and slow physical phenomena or reducing to a number of independent subsystems when a scalar parameter ε is zero. The purpose of this paper is to treat the optimal control of such systems by invoking the ε-coupling and singular perturbation approaches developed by Kokotovic and his associates for lumped-parameter large-scale systems. In the case of ε-coupled distributed-parameter systems it is shown that the optimal state feedback matrix can be approximated by a Volterra-MacLaurin series with coefficients determined by solving two lower-order decoupled Riccati and linear equations. By using an mth-order approximation of the optimal feedback matrix, one obtains a (2m+l)th order approximation of the optimal performance function. In the singular perturbation approach the result is that for an O(ε2) suboptimal control one must solve two decoupled Riccati equations, one for the fast and one for the slow subsystem and then construct appropriately the composite control law. By using only the Riccati equation for the slow subsystem, one obtains an 0(ε) suboptimal control. The singular perturbation technique can also be used to treat interconnected distributed-parameter systems involving many strongly coupled slow subsystems and weakly coupled fast subsystems.