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
•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:
x1 and x2 are the coordinates of the plane, this plane is
called the phase plane
Phase Portraits
Example
Solution

x+ x  0
x(t )  x0 cos t

x(t )   x0 sin t
2
x 2  x  x0 2
Phase Portraits
A major class of nonlinear systems
can be described by:


x +f ( x, x)  0
In the state space form

x1  x2

x 2   f ( x1 , x2 )
Singular Points
A singular point is an equilibrium point in the
phase plane
f1 ( x1 , x2 )  0
f 2 ( x1 , x2 )  0
For linear systems, there is usually only one
singular point, while nonlinear systems often
have more than one isolated singular point
Example


x +0.6 x  3x  x 2  0
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

x +f ( x)  0
Example

x  4 x  x3
There are three singular points
x  0, 2 and 2
Symmetry in Phase Plane Portraits


x +f ( x, x)  0

x1  x2

x 2   f ( x1 , x2 )
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


 d   Ud
2
  2U  c1

 u
Constructing Phase Portraits
Analytical Method:
Constructing Phase Portraits
The method of Isoclines:
At a point ( x1 , x2 ) in the phase plane, the
slope of the tangent to the trajectory can
be given by:
An isocline is defined to be the locus of
the points with a given tangent slope:
Constructing Phase Portraits
The method of Isoclines:
Example

x+ x  0
The slope of the trajectories is:
Constructing Phase Portraits
The method of Isoclines:
Example

x+ x  0
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 Portraits
The 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