outline home work phase plane analysis phase portraits symmetry in phase plane portraits
DESCRIPTION
Nonlinear Controls ( 3 Credits, Spring 2009 ) Lecture 3: Equilibrium Points, Phase Plane Analysis March 31, 2009 Instructor: M Junaid Khan. Outline Home Work Phase Plane Analysis Phase Portraits Symmetry in Phase Plane Portraits Constructing Phase Portraits - PowerPoint PPT PresentationTRANSCRIPT
Nonlinear ControlsNonlinear Controls (3 Credits, Spring 2009)
Lecture 3: Equilibrium Points, Phase Plane
Analysis
March 31, 2009
Instructor: M Junaid Khan
Outline•Home Work
•Phase Plane Analysis
•Phase Portraits
•Symmetry in Phase Plane Portraits
•Constructing Phase Portraits
•Phase Plane Analysis of Linear Systems
•Phase Plane Analysis of Nonlinear Systems
•Local Behavior of Nonlinear Systems
Phase Plane Analysis•Introduced in the end of 19th century by Henry Poincare
•Phase Plane analysis is a graphical method of studying second order nonlinear systems
•Basic Idea is to solve 2nd order Diff Eqn graphically
•The result is a family of system motion trajectories on 2D plane, called phase plane
•Only applicable where 2nd order approximation is possible
•Give intuitive insights to nonlinear effects
•Applies equally well to the analysis of hard nonlinearities
•Fundamental disadvantage is application to 2nd order systems
Phase Portraits•Phase Plane method is concerned with graphical study of 2nd order systems described by:
1 2 and are the coordinates of the plane, this plane is
called the phase plane
x x
Phase Portraits
+ 0x x
Example
Solution0
0
( ) cos
( ) sin
x t x t
x t x t
22 2
0x x x
Phase Portraits
+ ( , ) 0x f x x
A major class of nonlinear systems
can be described by:
1 2
2 1 2( , )
x x
x f x x
In the state space form
Singular Points
2+0.6 3 0x x x x
A singular point is an equilibrium point in the phase plane
1 1 2
2 1 2
( , ) 0
( , ) 0
f x x
f x x
For linear systems, there is usually only one singular point, while nonlinear systems often have more than one isolated singular point
Example
This systems has two equilibrium points
(0,0) and ( 3,0)
Phase Plane Method can also be applied to the analysis of first order systems
+ ( ) 0x f x
34 x x x
Example
There are three singular points
0, 2 and 2x
Symmetry in Phase Plane Portraits
+ ( , ) 0x f x x
1 2
2 1 2( , )
x x
x f x x
Symmetry in Phase Plane Portraits
Constructing Phase Portraits
Two methods:
Analytical Method and Isocline Method
Analytical Method requires analytical solution of the differential equations describing the system
Isocline Method is a graphical method, applied to those systems which cannot be solved analytically
Constructing Phase Portraits
Analytical Method:
Refer to slide 5 for the example
Constructing Phase Portraits
Analytical Method:
Constructing Phase Portraits
Analytical Method:
Remark
Constructing Phase Portraits
Analytical Method:
u
Constructing Phase Portraits
Analytical Method:
u
U
d Ud
Constructing Phase Portraits
Analytical Method:u
U
d Ud
2
12U c
Constructing Phase Portraits
Analytical Method:
Constructing Phase Portraits
The method of Isoclines:
An isocline is defined to be the locus of the points with a given tangent slope:
1 2( , )x xAt a point in the phase plane, the slope of the tangent to the trajectory can be given by:
Constructing Phase Portraits
The method of Isoclines:
+ 0x x
Example
The slope of the trajectories is:
Constructing Phase Portraits
The method of Isoclines:
+ 0x x
Example
The slope of the trajectories is:
Constructing Phase Portraits
The method of Isoclines:
Example
Therefore all the points on the curve:
will have slope
Constructing Phase PortraitsThe method of Isoclines:
Phase Plane Analysis of Linear Systems
Differentiation of first equation and substitution in 2nd
Phase Plane Analysis of Linear Systems
Phase Plane Analysis of Linear Systems
Phase Plane Analysis of Linear Systems
Phase Plane Analysis of Linear Systems
Phase Plane Analysis of Linear Systems
Phase Plane Analysis of Linear Systems
