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
Chapter 5: Optimal Control
BEFORE me grew the human soul, And after I am dead and gone, Through grades of effort and control The marvellous work shall still go on
Archibald Lampman
If everything seems under control, you're just not going fast enough.
Mario Andretti
Overview
One of the essential elements of the control problem, discussed in Section 1.1, is a means of testing the performance of any proposed control law. Whenever we use the term "best" or "optimal" to describe the effectiveness of a given control strategy, we do so with respect to some numerical index of performance called the performance index. We assume that the value of the performance index decreases as the quality of the given admissible control law increases. The admissible controller that ensures the completion of the system objective and at the same time minimizes the performance index is called an optimal controller for the system. In this chapter, we discuss various methods of constructing optimal controllers.
5.1 Performance Indices
Constructing a performance index that is, choosing a means to measure the system performance can be considered a part of the system modeling. Here, we discuss some typical choices of performance indices.
First, suppose that the objective is to control a dynamical system modeled by the equations
on a fixed interval [ t 0, t f] so that the components of the state vector are "small." A suitable performance index to be minimized would be
Obviously, if J 1 is small, then the state vector norm, ?
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