Control of Infinite Dimensional Systems Using Finite Dimensional Techniques: A Systematic Approach
MASSACHUSETTS INST OF TECH CAMBRIDGE DEPT OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE
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In this thesis, the problem of designing finite dimensional controllers for infinite dimensional single-input single-output systems is addressed. More specifically, it is shown how to systematically obtain near-optimal finite dimensional compensators for a large class of scalar infinite dimensional plants. The criteria used to determine optimality are standard Hinfinity and H2 weighted sensitivity and mixed-sensitivity measures. Unlike other approaches which appear in the literature, the approach taken here avoids solving an infinite dimensional optimization problem to get an infinite dimensional compensator and then approximating to get an appropriate finite dimensional compensator. Rather than this Design Approximate approach, we take an ApproximateDesign approach. In this approach one starts with a good finite dimensional approximant for the infinite dimensional plant and then solves a finite dimensional optimization problem to get a suitable finite dimensional compensator. Traditionally, however, this approach has not come with any guarantees. The key difficulties which have arisen can be attributed to the fact that these measures are sometimes not continuous with respect to plant perturbations, even when the uniform topology is imposed. Moreover, even if they were, it is a known fact that many interesting infinite dimensional plants can not be approximated in the uniform topology on Hinfinity e.g. a delay. Also, it must be noted that the concept of a good approximant, in the context of feedback design, has never been rigorously formulated. The goal and main contribution of this research endeavour has been to resolve these difficulties. It is shown that given a suitable finite dimensional approximant for an infinite dimensional plant, one can solve a natural finite dimensional problem in order to obtain a near-optimal finite dimensional compensator. Moreover, very weak conditions are presented to indicate what a good approximant is.
- Numerical Mathematics