Non Linear Systems
61
Nonlinear System Analysis Lyapunov Based Approach Lecture 4 Module 1 Dr. Laxmidhar Behera Department of Electrical Engineering, Indian Institute of Technology, Kanpur. January 4, 2003 Intelligent Control Lecture Series Page 1
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Transcript of Non Linear Systems
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Nonlinear System AnalysisLyapunov Based Approach
Lecture 4 Module 1
Dr. Laxmidhar Behera
Department of Electrical Engineering,Indian Institute of Technology, Kanpur.
January 4, 2003 Intelligent Control Lecture Series Page 1
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Overview
Non-linear Systems : An Introduction
Linearization
Lyapunov Stability Theory
Examples
Summary
January 4, 2003 Intelligent Control Lecture Series Page 2
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Nonlinear Systems: An Introduction
Model:
Properties:
Dont follow the principle of superposition,i.e,
January 4, 2003 Intelligent Control Lecture Series Page 3
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Multiple equilibrium points
Limit cycles : oscillations of constantamplitude and frequency
Subharmonic, harmonic oscillations forconstant frequency inputs
Chaos: randomness, complicated steady statebehaviours
Multiple modes of behaviour
January 4, 2003 Intelligent Control Lecture Series Page 4
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Nonlinear Systems: An Introduction
Autonomous Systems: the nonlinear function doesnot explicitly depend on time
.
Affine System:
Unforced System: input
,
January 4, 2003 Intelligent Control Lecture Series Page 5
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Nonlinear Systems: An Introduction
Example:Pendulum:
The state equations are
January 4, 2003 Intelligent Control Lecture Series Page 6
dellSticky Note.khedde/
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Nonlinear Systems: An Introduction
Exercise
Identify the category to which the followingdifferential equations belong to? Why?1.
2.
3.
where
is an external input.
4.
5.
January 4, 2003 Intelligent Control Lecture Series Page 7
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Linearization
Concept of Equilibrium Point: Consider a system
where functions
!
are continuouslydifferentiable. The equilibrium point
"
"
forthis system is defined as
"
"
January 4, 2003 Intelligent Control Lecture Series Page 8
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What is linearization?Linearization is the process of replacing thenonlinear system model by its linear counterpartin a small region about its equilibrium point.
Why do we need it?We have well stablished tools to analyze andstabilize linear systems.
January 4, 2003 Intelligent Control Lecture Series Page 9
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Linearization
The method:Let us write the the general form of nonlinearsystem
as:
#
#
$
%
#
#
$
%
(1)
...
#
$
#
$
$
%
January 4, 2003 Intelligent Control Lecture Series Page 10
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Linearization
Let
"
&
"
"
% "
'( be a constant input
that forces the system
to settle into aconstant equilibrium state
"
&
"
"
$ "
'
( such that
"
"
holds true.
January 4, 2003 Intelligent Control Lecture Series Page 11
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Linearization
We now perturb the equilibrium state by allowing:
)
)
"
)
and*
*
"
*
. Taylorsexpansion yields
#
)
#
)
"
)
*
"
*
)
"
*
"
+
+
)
)
"
*
"
)
++
*
)
"
*
"
*
January 4, 2003 Intelligent Control Lecture Series Page 12
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Linearization
where+
+
)
)
"
*
"
,,
,-
./
-0
/
1 1 1
-
.
/
-
0
2
... ...
-
.
2
-0
/
1 1 1
-
.
2
-
0
2
33
3
44
44
44
44
4
0
5
687
5
++
*
)
"
*
"
,,
,-
./
-
7
/
1 1 1
-
./
-
7
9
... ...
-
.
2
-
7
/
1 1 1
-
.
2
-
7
9
33
3
44
44
44
44
4
0
5
687
5
are the Jacobian matrices ofJanuary 4, 2003 Intelligent Control Lecture Series Page 13
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Linearization
Note that
#
)
#
#
)
"
#
#
)
#
#
)
#
because
)
"
is constant. Furthermore,
)
"
*
"
.Let
+
+
)
)
"
*
"
and +
+
*
)
"
*
"
Neglecting higher order terms, we arrive at thelinear approximation
#
)
#
) *
January 4, 2003 Intelligent Control Lecture Series Page 14
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Linearization
Similarly, if the outputs of the nonlinear systemmodel are of the form
$
%
$
%
...
:
:
$
%
or in vector notation
;
