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Phase Plane Analysis [1,2]
Phase-plane:
The plane having state variables as coordinates
Exact Method
includes transient response
Simple graphical construction methods
limited to 2nd order systems with simple inputs
Phase-portrait:
A family of phase plane trajectories that correspond to various
initial conditions.
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Given
x1=f1(x1, x2)
x2=f2(x1, x2)
the slope of phase trajectories:
dx2
dx1 =
f2(x1, x2)
f1(x1, x2)
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The equilibrium:
xi= 0
It implies that the slope of phase trajectories is indeterminate at
equilibrium points.
It is for the same reason the equilibrium points are sometimes called
singular points.
f1(x1eq, x2eq) =f2(x1eq, x2eq) = 0
Nonlinear algebraic equations, possible multiple solutions.
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Example: A First Order System
x= 4x+x3
Three singular points
The arrows denote the direction of motion
The slope at a certain point determines the direction
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Construction MethodsThe analytical method
One way is to use an analytical method. Reader is referred to the text book
[2] for further details. One example follows to illustrates the analyticalmethod.
The method of isoclines
The other method to construct a phase portrait is that of isoclines. Anisocline is defined as a locus of points with a given tangent slope. An
example is included to illustrate the method
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Construction Methods: The analytical method
Example: Satellite control system
The systems can be modeled mathematically as under:
=u
Where u is the torque provided by the thrusters and is the satellite
angle. See the following figures
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Construction Methods: The analytical method (contd.)
The control law to firethe thrusters is:
u(t) =
U, if >0;
U, if
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Construction Methods: The analytical method (contd.)
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Construction Methods: The method of isoclines
In order to construct a phase plane plot using the method of isoclines,
follow the following guidelines.
Draw the axis to represent the state variables (x1andx2) (black lines
in the next figure)
Pick several constant x1 or x2 and evaluate dx2
dx1;dx1
dx2(blue lines)
Pick a radial line x2=x1 and evaluatedx2
dx1;dx1
dx2(green line)
Make dx2
dx1=c (a constant) and solve for x2 = f2(x1) or x1 = f1(x2)
(red line)
Start from one pair of selected initial condition and follow the directions
indicated by the isoclines drawn in aforementioned steps.
Choose different initial conditions to draw another line
Repeat
C M h d Th h d f l ( d )
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Construction Methods: The method of isoclines (contd.)
The following figure illustrates the steps to construct a phase-plane
trajectory using the method of isoclines.
1x
2x
1x
2x
1x
2x
1x
2x
cx 2
02 x
01 x cx
1
12 xx
cdx
dx
1
2
trajectory
(a) (b)
(c) (d)
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Construction Methods: The method of isoclines (contd.)Isoclines for a mass-spring system (remember: no damping)
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Construction Methods: The method of isoclines (contd.)
Phase-plane trajectories for van-der-pol system using the method of
isoclines
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Phase Portraits: Example
The system differential equation for a second order nonlinear system isgiven by:
x= (x a)2 + x3
The state variables are defined as under:
x1=x
x2= x
It implies that:
x1=x2=f1(x1, x2)
x2= (x1 a)2 +x2
3 =f2(x1, x2)
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Phase Portraits: Example (contd.)
Now the equilibrium point xeq:
x1eq = 0 =x2eq
x2eq = 0 = (x1eq a)2 +x2eq
3
It implies that:
x1eq =a
x2eq = 0
and
dx2
dx1=
(x1 a)2 +x2
3
x2
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Phase Portraits: Example (contd.)
The previous expression can be plotted using the method of isoclines. The
results are as follows:
@ x2= 0, dx1dx2
= 0; dx2dx1
=
@ x1=a, dx2dx1
=x22
dx2dx1
= 0 @ (x1 a)2 +x3
2= 0, x2=(x1 a)
2
3
@ x2=x1, dx2dx1
= x31+x2
12ax1+a
2
x1
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Phase Portraits: Example (contd.)
The results are as follows:
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Phase Portraits: Example (contd.)
The closeup:
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Phase plane analysis of linear systems
Next, Lets investigate the stability about the singular (equilibrium) points.
Let us linearizethe system (using perturbation method)
x1=x1eq +x1x2=x2eq +x2
Which implies that:
x1x2
=
f1x1 f1x2f2x1
f2x2
eq
x1
x2
x= A(xeq)x
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Phase plane analysis of linear systems (contd.)
The characteristic equation:
|I A|= 0
and the solution of linearized system
x =et
Coming back to the characteristic equation, there can be a number of
possible stability cases for its roots.
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Phase plane analysis of linear systems (contd.)
Possible Cases: (also see the subsequent figures)1, 2 R
Stable Node
1, 2 R+ Unstable Node
1, 2 have opposite signsSaddle Point1, 2 C, Re(i)0 Unstable Focus
1, 2 C, Re(i) = 0 Center
1= 0, 20 Center (Unstable)
1= 0, 2= 0 Center
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Phase plane analysis of linear systems (contd.)
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Phase plane analysis of linear systems (contd.)
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Phase plane analysis of nonlinear systems
Local behavior of nonlinear systems:
The local behavior of the nonlinear systems can be approximated
with the results of linear systems.
Limit Cycles:
Periodic
An isolated closed curve
Types:
1. Stable limit cycles
2. Unstable limit cycles
3. Semi-stable limit cycles
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Phase plane analysis of nonlinear systems (contd.)
Limit cycles:
Existence of limit cycles:Poincare
Poincare-Bendixson
Bendixson
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References
[1] Benito R. Fernandez. Nonlinear control systems: Class notes. UTAustin, 2010.
[2] J.J.E. Slotine and W. Li. Applied nonlinear control. Prentice Hall,
1991.
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Questions
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