outline home work phase plane analysis phase portraits symmetry in phase plane portraits

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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 Presentation

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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

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