Introduction

The cheap control regulator for time-invariant systems is studied with respect to uniform L^2-boundedness of the state trajectories in the case where the small parameter “\epsilon” tends to zero. By using the geometric approach, state-space conditions for L^2-boundedness are found and related to the concept of zero dynamics.

In this thesis we study the cheap control regulator in the limiting case where the small parameter “\epsilon” tends to zero. Francis and Glover [1] have given conditions in the frequency domain for uniform L^2-boundedness of the state trajectories in this case.

However, restating these conditions in the time domain (state-space) gives a more intuitive picture of the dynamic behavior of the closed-loop system and is essential for the generalization of these conditions to nonlinear systems.

The thesis is organized as follows. In chapter 2 are presented the main mathematical tools used in the analysis of chapter 3. Section 2.1 consists of a brief survey of the concept of zero dynamics. In section 2.2, the main features of singular perturbation methods are described, methods to be used in solving the general linear cheap control problem in chapter 3.

The frequency domain conditions for L^2-bounded peaking by Francis and Glover [1] are stated in section 2.3. In chapter 3, we first investigate a special case in section 3.1 to illustrate the method of solving the general linear case and from which we draw some important conclusions about the behavior of the closed-loop system.

The general linear case is treated in section 3.2 and is decomposed into four subspaces, from which we get an intuitive picture of the stability properties of the corresponding closed-loop eigenvalues as the parameter “\epsilon” tends to zero.

From the solution of the general linear time-invariant cheap control problem we obtain two sufficient conditions for bounded peaking in the time domain (state-space) which are stated in the end of the section.