Lypanov's Ppt

7/28/2019 Lypanov's Ppt

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Lyapunov Stability Theory

M. S. Fadali

Professor of EE

1

Outline

Stability of an equilibrium of a nonlinear

system .

Lyapunovs (first, indirect) linearization

method.

Lyapunovs (second) direct method.

Linear time-invariant case.

2

Lyapunovs Linearization Method Linearize nonlinear system in vicinity

of equilibrium :

. Find the eigenvalues of the linearized system.

The equilibrium of the nonlinear system is: asymptotically stable ifall the eigenvalues are in the

open LHP.

unstable if one or more of its eigenvalues is in theopen RHP.

Inconclusive for LHP eigenvalues and one ormore eigenvalues on the imaginary axis.

3

Example Determine the stability of the equilibrium

of the mechanical system at the origin

Equilibrium with

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Nonlinear State Equations Physical state variables

State Equations

5

Linearization and Stability

Equilibrium state

Linearized model with

Characteristic polynomial and stability

,

Stable ,

6

Lyapunovs Direct Method

Examine stability of nonlinear system

directly.

Generalize concept of energy function.

Gives sufficient stability or instability

conditions (in general).

Possible difficulty, choice of suitable

Lyapunov (generalized energy) function.

7

Lyapunov Stability

there exists a such that

Defined for an equilibrium point (origin)

Start near the equilibrium and stay near the

equilibrium.

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x1

x2


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Asymptotic Stability

For an equilibrium point

1. Lyapunov stabilty

2. Convergence to the origin, i.e. there

exists a such that

_|

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Directional Derivative

Consider a continuous function fwith

continuous partial derivatives. Directional derivative of at in the

direction

_|

Some authors assume

;

10

Lyapunov Function

Positive definite:

Decreasing (or non-increasing) along the

trajectories of the system.

Derivative negative (or semidefinite)

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Quadratic form

Replacing terms in the first summation by

gives

Assume a symmetric matrix with no lossof generality

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Positive Definite Matrix P>0

for any nonzero

All its eigenvalues are positive

/ /

for any nonzero

Common choice of Lyapunov Function

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Negative Definite Matrix All its eigenvalues are negative

If is positive definite, then is negativedefinite / /

for any nonzero

Negative semi-definite Matrix: eigenvalues

are negative or zero .

Look for negative definite

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Lyapunov Stability Theorem

Given a positive definite function

If the derivative of along the

trajectories of the system isa) negative semi-definite then the equilibrium is

stable in the sense of Lyapunov.

b) negative definite then the equilibrium is

asymptotically stable.

c) positive definite then the equilibrium is

unstable.

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Justification

Lypunov function gets smaller with the

length of .

If the function has a negative derivativealong the trajectories, it is getting smaller.

The function continues to get smaller

along the trajectories until it reaches a

minimum: convergence.

The minimum value of the function is zero

(at the equilibrium point).

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La Salles Principle

If the derivative of along the

trajectories of the system isnegative semi-definite but is only zero for

trivial trajectories (ones that imply )

then the equilibrium is asymptotically

stable.

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Remarks

The theorem provides sufficient

conditions for stability and sufficientconditions for instability.

If the test fails, there is no conclusion.

It is often difficult to find a suitable

Lyapunov function for nonlinear systems.

For linear systems, the theorem canprovide a necessary and sufficient

condition.

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Scalar System (Slotine & Li)

Show that the system is asymptoticallystable.

Lyapunov function

,

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Nonlinear Spring-Mass-Damper

Show that the system is asymptotically

stable.

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Solution Use energy as the Lyapunov function

(stable i.s. Lyap.)

(asymptot. stable)

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Linear Time-invariant Case

The LTI system

is asymptotically stable if and only if for

any positive definite matrix there exists

a positive definite symmetric solution to

the Lyapunov equation

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Proof

Use a quadratic Lyapunov function

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Remarks

Recall that the original Lyapunov theorem

only gives a sufficient condition.

If we start with (i.e. with Lyapunovfunction) and solve for , the condition

the test may or may not work.

If we start with (i.e. with the derivative

and we find a the condition is necessary

and sufficient.

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Example

Determine the stability of the system with

state matrix

using the Lyapunov equation with .

Note: The system is clearly stable byinspection since is in companion form.

Choosing P=I will not work! No conclusion.

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Solution

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0 6

1 5

0 1

6 5

1 0

0 1

Multiply

12 6 56 5 2 10

1 0

0 1

Equate to obtain three equations in three unknowns.

Equivalent Linear System

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12 6 56 5 2 10

1 0

0 1

0 12 0

1 5 60 2 10

1

01

1/12

1 2 /10 7/60

6 5 7/10 5/12 67/60

67/60 1/121/12 7/60 1.1167 0.083330.08333 0.1167

MAPLE

Compute:

with(LinearAlgebra):

Transpose(A).P+P.A

Solve the equivalent linear system:

LinearSolve(M,B)

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MATLAB

Solve a different equation.

Identical to our equation with

replaced by .

Eigenvalues are the same!

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MATLAB Example

>> A=[0,1;-6,-5];

>> Q=eye(2)>> P=lyap(A,eye(2))

P =

0.5333 -0.5000

-0.5000 0.7000

>> eig(P)

ans =

0.1098

1.1236

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To Get Earlier Answer

>> P=lyap(A',eye(2))

P =

1.1167 0.0833

0.0833 0.1167

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1167.008333.0

08333.01167.1P

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