NLCS PhasePlane LS

8/13/2019 NLCS PhasePlane LS

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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



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



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



Construction Methods: The analytical method (contd.)

The control law to firethe thrusters is:

u(t) =

U, if >0;

U, if



Construction Methods: The analytical method (contd.)



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 )



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)



Construction Methods: The method of isoclines (contd.)Isoclines for a mass-spring system (remember: no damping)



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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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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The results are as follows:


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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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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 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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