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2.3 Nonlinear State-Variable Systems 31

where

P(z)=C(zI-A) B+D
-1

Note that the Z transform is used in the discrete-time case versus the Laplace
transform in the continuous-time case.
EXAMPLE 2.2–5: Tranfer Function of Discrete-Tiem Double Integrator

The transfer function of the Example 2.2.4 is given by

2.3 Nonlinear State-Variable Systems

In many cases, the underlying physical behavior may not be described using
linear state-variable equations. This is the case of robotic manipulators where
the interaction between the different links is described by nonlinear differential
equations, as shown in Chapter 3. The state-variable formulation is still
capable of handling these systems, while the transfer function and frequency-
domain methods fail. In this section we deal with the nonlinear variant of
the preceding section and stress the classical approach to nonlinear systems
as studied in [Khalil 2001], [Vidyasagar 1992] and in [Verhulst 1997], [LaSale
and Lefschetz 1961], [Hahn 1967].

Continuous-Time Systems

A nonlinear, scalar, continuous-time, time-invariant system is described by a
nonlinear, scalar, constant-coefficient differential equation such as

(2.3.1)

where y(t) is the output and u(t) is the input to the system under consideration.
As with the linear case, we define the state vector x by its components as
follows:

Copyright © 2004 by Marcel Dekker, Inc.