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Linearization of Nonlinear Systems
A general form of the state-space equation (for nonlinear systems) is

(
x ˙ t() = gx, u) (23.114)
yt() = hx, u) (23.115)
(
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where g and h can be nonlinear functions. The behavior of nonlinear systems is beyond the scope of
this section; however, a detailed discussion can be found in [10]. The behavior of a nonlinear system can
be approximated by a linear model in a neighborhood of an equilibrium point. Such linearizations
can simplify the analysis and design of nonlinear systems because the tools developed for linear systems
can be applied under certain conditions [10]. Let x 0 and u 0 be the equilibrium point and equilibrium
input, respectively, such that [10, Chapter 1]
gx 0 , u 0 ) = 0 (23.116)
(
(
hx 0 , u 0 ) = y 0 (23.117)
u
x
Consider small perturbations in the equilibrium point x(t) = x 0 + (t), the input u(t) = u 0 (t) + (t),
y
and the output y(t) = y 0 + (t). If the perturbation (t) is small for all t, we obtain the following byx
u
x
expanding (23.114) in Taylor series (neglecting higher order terms of (t) and (t)):
˙
(
x ˙ + x t() = gx 0 + x t(),u 0 + ut())
0
∂g ∂g
x t() = gx 0 , u 0 ) + ------ xt() + ------ ut() (23.118)
(
˙
∂x x=x ∂u x=x
0 0
u=u 0 u=u 0
Recognizing that g(x 0 , x 0 ) = 0 we obtain
˙
x t() = Ax t() + Bu t() (23.119)
where
g ∂
A = ----- g ∂ and B = ------ (23.120)
x ∂ x=x ∂ u x=x
u=u 0 0 u=u 0 0

The matrices A and B are the Jacobians evaluated at x 0 and u 0 . Equation (23.119) is a linear state
equation and is valid for small perturbations about x 0 and u 0 . A similar result can be obtained for the
change (t) in the output from the equilibrium value y 0 as
y
yt() = Cx t() + Du t() (23.121)

where

h ∂
h ∂
C = ----- and D = ------ (23.122)
x ∂ x=x ∂ u x=x
0 0
u=u u=u
0 0
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The MATLAB command ode45 can be used to obtain the numeric solution to the general nonlinear state space
equation.

©2002 CRC Press LLC