ME 360: Modeling, Analysis, and Control of Dynamic Systems
Professor Tilbury, University of Michigan, Winter 2009
Linearization Handout
September 10, 2009
Basic idea: The
local
behavior of a nonlinear system can be
approximated
by its linearization about an
operating point.
1
Taylor series approximation
Let
f
(
x
)
be a nonlinear function of
x
, and let
x
o
be the operating point. Then
x
=
x
o
+
δx
, where
δx
is the deviation
(assumed to be small) from the operating point.
Take the Taylor series expansion:
f
(
x
o
+
δx
) =
f
(
x
o
) +
∂f
∂x
vextendsingle
vextendsingle
vextendsingle
vextendsingle
x
=
x
o
·
δx
+
higher order terms
≈
f
(
x
o
) +
kδx
where
k
=
∂f
∂x
vextendsingle
vextendsingle
vextendsingle
x
=
x
o
is a constant. Note that
k
depends on the operating point
x
o
. Also keep in mind that this is only
an approximate relationship because we have kept only the constant and linear terms, and ignored the higher order
terms.
Nonlinear models can arise from experiments or empirical observations, or through first principles (e.g. geometry).
Sometimes a nonlinear function is available; other times, there is only a data set or “lookup table” relating the variables
of interest.
1.1
Nonlinear spring
Assume that the force/displacement curve for the spring is obtained through experimentation, and plotted as shown in
the figure.
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 Fall '10
 Gillespie
 Derivative, Taylor Series, Nonlinear system, operating point, XO

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