Phase Plane Analysis

Phase plane analysis is a graphical method for studying

second-order systems.

The basic idea is to generate a two-dimensional plane

(called the phase plane) in the state space of a second

order dynamical system.

Motion trajectories corresponding to various initial

conditions are examined to obtain the qualitative

features like stability and motion patterns of the

system.

ADVANTAGES:

Visualising behaviour of a nonlinear system,

starting from various initial conditions without

having to solve the nonlinear equations

analytically.

Not restricted to small or smooth nonlinearities,

but equally applied to hard nonlinearities.

DISADVATGES:

Restricted to only second order or first order

systems.

Concepts of Phase Plane Analysis

Phase Portraits:

Graphical representation of 2nd order autonomous

systems i.e.

( )

( )

The state space of this system is a plane having

( ) as coordinates and is called the phase plane.

The family of phase plane trajectories corresponding to various initial conditions is called a phase portrait of a system.

Lets take an example of spring mass system having dynamical equation ( )

The nature of the system response corresponding to various initial conditions is directly displayed on the phase plane.

A major class of 2nd order systems can be represented as

( ) Can you tell me some examples of 2nd order systems

in electrical systems and mechanical systems?

Singular Points:

A singular point is an equilibrium point in the phase

plane, which means

( ) ( )

Stability is uniquely characterised by the nature of their singular points.

Lets take some examples to find the singular points: o For the 2nd order linear system

o For the 2nd order nonlinear system

Singular points are (0,0) and (-3,0).

For the 1st order nonlinear system

Singular points are at 0, 2, -2.

Phase Plane analysis of Linear systems

The phase portrait of

This system has two stable poles at -1 and -2. The phase portrait of

The system has one unstable and one stable pole at 1 and -2 respectively.

The phase portrait of

The system has two complex stable poles at

Phase Plane analysis of nonlinear systems

Local behaviour of nonlinear system can be

approximated by the behaviour of a linear system.

Nonlinear systems can display much complicated

patterns such as multiple equilibrium points and

limit cycles.

A limit cycle is an isolated closed curve in the phase

plane.

a) Stable limit cycle

b) Unstable limit cycle

c) Semi-stable limit cycle

Consider a Satellite control system having

mathematical model as (rotational unit

inertia )

( ) {

i.e. the thrusters push in the counter-clockwise

direction if is positive.

Control input and thus the phase trajectories are switched on the vertical switching line.