Systems and Control
Broad in scope, this text shows the multidisciplinary role of dynamics and control, presents neural networks, fuzzy systems, and genetic algorithms, and provides a self-contained introduction to chaotic systems.
TABLE OF CONTENTS Systems and Control
4.11: Uniform Ultimate Boundedness
4.11 Uniform Ultimate Boundedness
A drawback to the discontinuous controllers, discussed in the previous section, is that rapid switching tends to excite high-frequency modes of the plant, and this may degrade the performance of the controller. These problems can be alleviated by adding a so-called boundary layer to the discontinuous controller resulting in a continuous control strategy. However, there is a price for making a discontinuous controller a continuous one. The closed-loop system driven by a boundary layer controller may only be uniformly ultimately bounded rather than asymptotically stable. The notions we are concerned with in this section are described in the following three definitions.
Definition 4.18
The closed ball B q( 0) = { x : ? x ? ? q} is uniformly stable if for any ? > q, there is a ? = ?( ?) > 0 so that if ? x( t 0) ? ? ?, then ? x( t) ? ? ? for t ? t 0.
For the benefit of the reader, we now rephrase the definition of uniform boundedness from Section 4.8.
Definition 4.19
The solutions of ? = f( t, x) are uniformly bounded if for any given d > 0, there exists b = b( d) < ? so that for each solution x( t) = x( t; t 0
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