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22 Introduction to Control Theory

model of the robot dynamics is obtained as described in Chapter 2, the
automatic control concepts presented in the current chapter may be used to
modify the actions and reactions of the robot to different stimuli. Subsequent
chapters will therefore deal with the application of control principles to the
robot equations. The particular controller used will depend on the complexity
of the mathematical model, the application at hand, the available resources,
and a host of other criteria.
We begin the chapter in section 2.2 with a review of the state-space
description for linear, continuous- and discrete-time systems. A similar review
of nonlinear systems is presented in section 2.3. The Equilibria of nonlinear
systems is reviewed in section 2.4, while concepts of vector spaces is presented
in section 2.5. Stability theory is presented in section 2.6, which constitutes
the bulk of the chapter. In Section 2.7, Lyapunov stability results are presented
while input-output stability concepts are presented in section 2.8. Advanced
stability concepts are compiled to make later developments more concise in
section 2.9. In section 2.10 we review some useful theorems and lemmas. In
section 2.11 we the basic linear controller designs from a state-space point
of view, and the chapter is concluded in Section 2.12.

2.2 Linear State-Variable Systems

Many physical systems such as the robots considered in this book are
described by differential or difference equations. These describing equations,
which are usually obtained from fundamental physical laws, provide the
starting point for the analysis and control of systems. There are, of course,
some systems which are so complicated that describing differential (or
difference) equations are not available. We do not consider those systems in
this book.
In this section we study the state-space model of physical systems that are
linear. We limit ourselves to systems described by ordinary differential
equations which will lead to a finite-dimensional state space. Partial
differential equations, leading to infinite-dimensional systems, are needed to
study flexible robotic manipulators, but those are not considered in this
textbook. We stress that the material of this chapter is intended as a quick
introduction to these topics and will not be comprehensive. The readers are
referred to [Kailath 1980], [Antsaklis and Michel 1997] for more rigorous
presentations of linear control systems.

Continuous-Time Systems

A continuous-time system is said to be linear if it obeys the principle of
superposition] that is, if the output y 1(t) results from the input u 1(t) and the
output y 2(t) results from the input u 2(t), then the output resulting from

Copyright © 2004 by Marcel Dekker, Inc.