Lyapunov Stability Theory

M. S. FadaliProfessor of EE

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Outline

Equilibrium of nonlinear system .

Stability definitions.

Lyapunov’s (second) direct method.

Class functions.

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Equilibrium Point

Consider autonomous systems.

= open connected subset of locally Lipschitz

Equilibrium point

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Lyapunov Stability

The equilibrium is stable if0 such that

Otherwise, is unstable.

Can stay arbitrarily close to the equilibrium.

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x1 

x2

Convergence

The equilibrium is convergent if

s.t.

_|Å→

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Convergence and Stability

Independent properties. Convergence does not imply stability. Example: Trajectories always go to a circle

of radius r before converging. Unstable.

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x1 

x2

r 

Asymptotic Stability

The equilibrium is asymptotically stable if it is both

1. Stable (i.s. Lyapunov)

2. Convergent.

Definition does not include rate of convergence.

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Equilibrium at the Origin W.l.og., assume of

If equilibrium is at , translate the axes:

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Example

Equilibrium:

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Exponential Stability

The equilibrium is exponentially stable if s.t.

The equilibrium is globally exponentially stable if the condition holds exponentially stable

asymptotically stable stable

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Lyapunov’s 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.

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Directional Derivative

Consider a continuous function f with continuous partial derivatives.

Directional derivative of at in the direction

_|Å→ Some authors assume

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Lie Derivative

The Lie derivative of along

Example

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Definite Functions

= open connected subset of

Positive Definite:

Negative Definite:

Positive Semidefinite:

Negative Semidefinite: 14

Example: Quadratic Forms

Positive Definite

Negative Definite

Positive Semidefinite

Negative Semidefinite

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Lyapunov Function

Positive definite:

Decreasing (or non-increasing) along the trajectories of the system.

Derivative negative definite (or semidefinite)

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Lyapunov Stability Theorem

Given a C1 positive definite function in

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.17

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2

2

2

2

2

4

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6

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-3 -2 -1 0 1-1.5

-1

-0.5

0

0.5

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Lyapunov Surface

Boundary of a region in state space where

Surface encloses all surfaces if

Shape of surface depends on : may or may not be the

sphere

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Lyapunov Stability

: a trajectory starting inside never leaves it.

can choose s. t. the largest ball inside is

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x1 

x2

 

 

Asymptotic Stability If has a negative derivative along the

trajectories, the Lyapunov surface shrinks. decreases inside the Lyapunov surface. The surface continues to shrink along the

trajectories until it converges to its minimum zero value.

The minimum value of the function is (at the equilibrium point).

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Instability : Lyapunov surfaces grow along

any trajectory and grows continuously. diverges along any trajectory. Note: condition is very severe because an

unstable system may have some stable trajectories (e.g. saddle point).

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Remarks

The theorem provides sufficient conditions for stability and sufficient conditions 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 can

provide a necessary and sufficient condition.

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Laypunov Stability Analysis1. Select a positive definite function for

use in stability analysis: Lyapunov function candidate.

2. Evaluate the derivative along the system trajectories.

3. Use the stability theorem: success means that = Lyapunov function

If the test fails, no conclusion (sufficient) Can try another function (it may be

difficult to find one that works)23

Scalar System (Slotine & Li)

Show that the system is asymptotically stable.

Lyapunov function

,

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Example: Frictionless Pendulum

Equilibrium: Physically motivated choice: total energy

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Stability Analysis

Stable equilibrium.

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Example: Pendulum + Friction

Equilibrium: Physically motivated choice: total energy

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Stability Analysis

Stable equilibrium (we know it is asymptotically stable).

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Using a Different

locally

Asymptotically stable equilibrium29

Global Stability (in the Large)

Requires radially unbounded. Radially unbounded

continuously differentiable and as uniformly in .

Example:

if either or or both.

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Example:

closed for For , trajectories can diverge while moving towards a Lyapunov contour associated with a lower value.

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0.4

0.4

0.4

0.4

0.8

0.80.8

0.80.8

0.8

1.2

1.2 1.2

1.2

1.2 1.2

1.6

1.6

1.6

1.6

1.6

1.6

2

2

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

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Why Radially Unbounded? Thm. Global Asymptotic Stability Given a C1 positive definite, radially

unbounded function

If the derivative of along the trajectories of the system is negative definite then is globally asymptotically stable.

Proof: similar to earlier proofs and radially unbounded prevents pathological cases.

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Example

Equilibrium

pos. def. and radially unbounded.

Equilibrium is globally asymptotically stable.

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Chetaev’s TheoremThe equilibrium of ,

continuously differentiable inis unstable if a function satisfyingI.II. arbitrarily close to s.t.

III.34

Proof: Main idea Show that a trajectory starting at

arbitrarily close to in the set will cross the sphere , arbitrary.

Unstable: cannot find s.t.

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x1 

x2

>0 

>0 

0 

 

Theorem (Apostol, p.83) If is continuous on a compact subset , metric space, then

s. t. ∈∈ Needed in the proof.

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Proof is nonempty

since arbitrarily close to s.t.

bounded

For closed &

bounded (compact) set have bounds in ∈37

Proof Cont.

Starting at arbitrarily close to , the trajectory leaves as and crosses

Unstable: cannot find s.t.

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Example (Khalil p.126)

locally Lipschitz, satisfy in (

Equilibrium

is unstable39

Functions of Class Class : continuous function withI.

II. strictly increasing.Class : continuous function withI.

II. strictly increasing.III. as

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Lemma 3.1

is positive definite if and only if class functions and

s. t.

If and is radially unbounded, then and can be chosen in class

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Example

Choose

is positive definite and radially unbounded.

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Lemma 3.2

The equilibrium of is stable if and only if a class function and a constant s.t.

Class function : Upper bound for stable trajectory.

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Class

Continuous function s.t.I. For fixed , is in class w.r.t. .II. For fixed , is strictly decreasing

w.r.t. .III. as

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Exponential Stability Given a C1 positive definite function in

. IfI. is negative definite.II. s.t.

Then is exponentially stable. If the conditions hold globally, is

globally exponentially stable.

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Proof

/ ///

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Example

Show that the system is globally exponentially stable (obvious).

Lyapunov function

,

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Invariance Principle

In some cases, we can prove asymptotic stability even if .

Exploit the relationship between the state variables of the system.

(Positively) Invariant Set : is an invariant set w.r.t. a dynamical system

if Starting in , the system never leaves .

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Examples: Invariant Sets

Equilibrium point Limit cycle. Any system trajectory.

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Limit Set

= trajectory of

is a (positive) limit set of if a time sequence

s.t. as , i.e.

_|Å→

= set of points to which tends as

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Example

Asymptotically stable equilibrium point:limit set of any trajectory in its domain of attraction. Stable limit cycle limit set of any sufficiently close trajectory.

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Lemmas

If = bounded solution of

is the (positive) limit set of

Theni is bounded, closed and nonempty.ii approaches as iii is an invariant set w.r.t.

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Theorem 3.6 The equilibrium point of is

asymptotically stable if a function satisfying1. is positive definite 2. is negative semidefinite in a bounded region .3. is only zero for the trivial trajectory

(nonzero on all nontrivial trajectories).

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Theorem 3.7

The equilibrium point of is globally asymptotically stable if a function

satisfying1. is positive definite 2. is negative semidefinite .3. is only zero for the trivial trajectory (nonzero on all nontrivial trajectories)4. is radially unbounded.

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Example: Pendulum with Friction

Equilibrium:

asymptotically stable.55

Theorem 3.7The equilibrium of of is globally asymptotically stable if a function

satisfying1. is positive definite 2. is negative semidefinite in a bounded region .3. is only zero for the trivial trajectory

(nonzero on all nontrivial trajectories)4. is radially unbounded.56

Example

½ , radially unbounded

is globally asymptotically stable.57

La Salle’s Theorem continuously differentiable

compact set invariant w.r.t. the solutions of

= largest invariant set in

Then every solution starting in approaches as

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Remarks Unlike Lyapunov theory, La Salle’s Thm.1. Requires to be continuously

differentiable but not necessarily positive definite.

2. Is applicable to any attractor and not just an equilibrium point i.e. it can be used to determine the stability of a limit cycle.

3. Largest invariant set =union of all invariant sets.

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Corollaries continuously differentiable, positive

definite in .

is negative semidefinite in

no solution remains in other than the trivial trajectory.

Then is an asymptotically stable equilibrium of

If and is radially unbounded then is globally asymptotically stable

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Example

Equilibrium: Invariant Set:

For 1st Quadrant

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Stability of Limit Cycle

pos. semidef. In

iff (a) , or (b) on circle of radius

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Apply La Salle’s Theorem Given define

compact (closed & bounded) by construction and any trajectory starting in

remains in , i.e. is an invariant set

and invariant sets = largest invariant set in , La Salle’s Thm.: every motion in converges to

or (limit cycle)

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Refine Answer measure of distance to the limit cycle

Choose

excludes but includes the limit cycle Limit cycle is stable (attractive) while is

unstable because arbitrarily close points to converge to the limit cycle.

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