Perturbations of closed-loop systems in robust feedback stabilization

Abstract

The largest robust stability radius γ(P₀) of a system P₀ is defined as the radius of the largest ball B_max in the gap metric centred at P₀ which can be stabilized by one single controller. Any controller stabilizing B_max is called an optimally robust controller of P₀ . A controller, regarded as a system, should have its own largest robust stability radius also. In this note it is first shown that the largest robust stability radius of any optimally robust controller of P₀ is larger than or equal to γ(Po)- The main result of this paper is the estimate of the variations (in the L∞-norm) of the closed-loop transfer matrix caused by the perturbations of the system or of the optimally robust controller. Finally, the schemes of designing finite-dimensional controllers are presented via the largest robust stability radius. These schemes guarantee that the designed finite-dimensional controllers will stabilize the original infinite-dimensional systems. Moreover, the closed-loop transfer matrices can be estimated.