Nonlinear Control Sy.. - Free

Chapter 1
Introduction
This first chapter serves as an introduction to the rest of the book. We present several
simple examples of dynamical systems, showing the evolution from linear time-invariant to
nonlinear. We also define the several classes of systems to be considered throughout the rest
of the book. Phase-plane analysis is used to show some of the elements that characterize
nonlinear behavior.
1.1 Linear Time-Invariant Systems
In this book we are interested in nonlinear dynamical systems. The reader is assumed to
be familiar with the basic concepts of state space analysis for linear time-invariant (LTI)
systems. We recall that a state space realization of a finite-dimensional LTI system has the
following form
x = Ax + Bu (1.1)
y=Cx+Du (1.2)
where A, B, C, and D are (real) constant matrices of appropriate dimensions. Equation
(1.1) determines the dynamics of the response. Equation (1.2) is often called the read out
equation and gives the desired output as a linear combination of the states.
Example 1.1 Consider the mass-spring system shown in the Figure 1.1. Using Newton's
second law, we obtain the equilibrium equation
1