cds 101 precourse phase plane analysis and stabilitymurray/courses/cds101/fa02/p...cds 101 precourse...

CDS 101 PrecoursePhase Plane Analysis and Stability

Melvin Leok

Control and Dynamical SystemsCalifornia Institute of Technology

Pasadena, CA, 26 September, 2002.

[email protected]://www.cds.caltech.edu/˜mleok/

Control and Dynamical Systems

2

Introduction

¥ Overview

• Equilibrium points

• Stability of equilibria

• Tools for analyzing stability

• Phase portraits and visualization of dynamical systems

• Computational tools

3

Equilibrium and Stability

¥ Equilibrium Points

• Consider a pendulum, under the influence of gravity.

• An Equilibrium Point is a state that does not change underthe dynamics.

• The fully down and fully up positions to a pendulum are ex-amples of equilibria.

4


¥ Equilibria of Dynamical Systems

• To understand what is an equilibrium point of a dynamical systems,we consider the equation of motion for a pendulum,

θ +g

Lsin θ = 0,

which is a second-order linear differential equation without damp-ing.

•We can rewrite this as a system of first-order differential equationsby introducing the velocity variable, v.

θ = v,

v = −g

Lsin θ.

5


¥ Equilibria of Dynamical Systems

• The dynamics of the pendulum can then be visualized by plottingthe vector field, (θ, v).

• The equilibrium points correspond to the positions at which thevector field vanishes.

6


¥ Stability of Equilibrium Points

• A point is at equilibrium if when we start the system at exactlythat point, it will stay at that point forever.

• Stability of an equilibrium point asks the question what happensif we start close to the equilibrium point, does it stay close?

• If we start near the fully down position, we will stay near it, so thefully down position is a stable equilibrium.

7


¥ Stability of Equilibrium Points

• If we start near the fully up position, the pendulum will wanderfar away from the equilibrium, and as such, it is an unstable

equilibrium.

8

Types of Stability

¥ Lyapunov Stability

• An equilibrium point is Lyapunov Stable if whenever we startsufficiently close to the equilibrium, we will stay close to the equi-librium.

Examples of Lyapunov stable and unstable behavior

9

Types of Stability

¥ Asymptotic Stability

• An equilibrium point is Asymptotically Stable if it is Lya-punov stable, and for any solution that starts sufficiently closeto the equilibrium point will converge to the equilibrium point.

10

Tools for Analyzing Stability

¥ Potential Energy near the Equilibrium

•When the system only experiences forces that can be expressed interms of a potential energy, looking at the potential energy nearthe equilibrium can give one information about the stability of thatpoint.

Energy minimumStable

Energy maximumUnstable

•More generally, such stability analysis methods are known as Lya-punov Function methods.

11


¥ Eigenvalue Analysis

• An analytic method of analyzing stability is related toEigenvalueAnalysis in linear algebra.

• As an example, consider the following scalar linear differential equa-tion,

x = ax,

Which we readily verify to have the solution,

x(t) = x0eat.

• Notice that the behavior of the equilibrium at the origin, x = 0,depends on the value of the parameter a.

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• If a > 0, we see that the solution diverges from 0, and the originis unstable.

0 0.5 1 1.5 2 2.5 30

5

10

15

20

25

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• If a < 0, we see that the solution converges to 0, and the origin isstable.

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

14



• In general, if we are given a system of coupled first-order lineardifferential equations of the form,

x = Ax,

where x ∈ Rn is a n-vector, and x ∈ R

n×n is a n × n matrix,the stability of an equilibrium can be analyzed by determining theeigenvalues of the matrix A.

15


¥ What about nonlinear systems?

•We can do this analysis for linear systems of differential equations,but what happens in the case of nonlinear systems of differentialequations, which we may not be able to solve exactly?

• Notice that the notion of stability is only concerned with what hap-pens in a small neighborhood of the equilibrium point, and aswe zoom in closer and closer, the vector field starting looking likethat of a linear system, so we do the obvious thing:Linearization: We approximate the nonlinear system by a linearsystem.Eigenvalue Analysis: We evaluate the eigenvalues of the lin-earization to obtain information about the stability of the nonlinearsystem.

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Visualizing Dynamical Systems

¥ Hamiltonian (Energy) Methods

• The pendulum example we considered is special in that it is con-servative, and hence, by looking at level sets of the energy, wecan also get a sense of how the system behaves.

−6 −4 −2 0 2 4 6

−4

−2

0

2

4

−2

0

2

4

6

8

10

x

0.5 y y−cos(x)

y

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¥ Phase Portraits

• Instead of plotting position or velocity against time, in a time-

series plot, we can often gain insight by a Phase portrait,where we plot velocity against position as a parametric plot.

• Returning to the pendulum example, we have the following phaseportrait,

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¥ Phase Portraits

• Periodic solutions show up as closed orbits.

•We can see from the nearby trajectories whether a equilibriumpoint is stable or unstable.

• Phase portraits allow us to get a sense of the different types ofbehavior which may occur in a dynamical system.

• In the pendulum example, we clearly see the distinction betweenoscillating modes, and whirling modes.

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¥ Phase Portraits

• It might seem to you that the whirlingmotion of a pendulum is a periodic or-bit, but how do we see that from thephase portrait?

• If we recall that we need to make theidentification θ = π = −π, we canwrap the phase plane into a cylinder,and the whirling modes become closedcurves as expected of periodic orbits.

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¥ More Phase Portraits

• Consider the more complicated example of a damped pendulum.The phase portrait is more complicated, and is shown below,

θ ’ = ω ω ’ = − sin(θ) − D ω

D = 0.1

−10 −8 −6 −4 −2 0 2 4 6 8 10

−4

−3

−2

−1

0

1

2

3

4

θ

ω

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¥ Extended Phase Portraits

• The time evolution of adamped pendulum is moreinteresting.

•We can combine time-seriesplots and phase portraits, bylooking at the Extended

Phase Portrait, which isa parametric plot of posi-tion, velocity and time.

• The time-series and phaseportrait are projections ofthe extended phase portrait.

θ ’ = ω ω ’ = − sin(θ) − D ω

D = 0.1

−2−1

01

23

−2

−1

0

1

2

0

10

20

30

40

50

60

70

80

90

θω

t

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Computational Tools

¥ MATLAB and PPLANE6

• A good program for phase plane analysis is PPLANE6, which iswritten for MATLAB. The homepage is,

http : //math.rice.edu/~dfield/

23

Computational Tools

¥ MATLAB and PPLANE6

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Computational Tools

¥ Numerical Integration

• How does a computer compute the solution of a nonlinear differ-ential equation?

• Given the equation,x = f (x),

we could think of computing the solution at a discrete set of timeintervals, spaced at ∆t = 0.1.

•We could then make the approximation,

x =∆x

∆t,

from which we have,

xn+1 − xn = ∆x = ∆tf (xn).

• This method is known as the Forward Euler method.

25

Computational Tools

¥ Numerical Integration

• A more accurate and stable numerical integration algorithm is theRunge-Kutta method, which is very popular. It is given by,

k1 = f (xn)∆t

k2 = f (xn + k1/2)∆t

k3 = f (xn + k2/2)∆t

k4 = f (xn + k3)∆t

xn+1 = xn +1

6(k1 + 2k2 + 2k3 + k4)

• Numerical integration algorithms in software like MATLAB aremore sophisticated, but are based on algorithms like the Runge-Kutta method above.

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Resources

¥ Related Courses at Caltech

•CDS 140 Introduction to Dynamics

•CDS 201 Applied Operator Theory

•ACM 110 Introduction to Numerical Analysis

¥ Webpages

• Control and Dynamical Systems Homepage

http : //www.cds.caltech.edu/

•MATLAB Homepage

http : //www.mathworks.com

• PPLANE6 Homepage

http : //math.rice.edu/~dfield/

Control and Dynamical Systems

cds 101 precourse phase plane analysis and stabilitymurray/courses/cds101/fa02/p...cds 101 precourse...

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