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CS545—Contents VI
Control Theory II
Linear Stability Analysis
Linearization of Nonlinear Systems
Discretization
Reading Assignment for Next Class
See http://wwwclmc.usc.edu/~cs545
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View Full Document Stability Analysis
Given the control system
How can we the show that a particular choice of a
controller generates a stable control system?
In order to get started, consider whether the generic
dynamical system is stable:
˙
x
=
f
x
,
u
( )
or
˙
x
=
Ax
+
Bu
˙
x
=
f
x
( )
or
˙
x
=
Equilibrium Points and
Stability
Definition of an Equilibrium Point
A state x is an equilibrium state (or equilibrium point) of the system if
once x(t) is equal to x, it remains equal to x for all future time.
Mathematically, this means:
Definition of Stability
An equilibrium state x is said to be stable, if, for any R>0, there exists
r>0, such that if x(0)<r,then x(t)<R for all t
≥
0. Otherwise, the
equilibrium point is unstable.
r
R
Equilibrium
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View Full DocumentLinear Stability Analysis
(Local Stability Analysis)
What is needed at the outset:
The system model (linear or nonlinear)
An equilibrium point
The linearization of the system about the equilibrium point
Then, we have the linear(ized) system:
This system is stable if and only if:
The complete stability definitions are:
˜
˙
x
=
A
x
=
x
*
˜
x
where
˜
x
=
x

x
*
( )
REAL eig
(
A
)
( )<
0
continuous system
discrete system
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This note was uploaded on 11/06/2010 for the course CS 101 at Cornell University (Engineering School).
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