Some Fundamentals of Stability Theory

Aaron Greenfield

Outline

I. Introduction + Motivation

II. Definitions

III. Theorems

IV. Techniques for Lyapunov Function Construction

Basic Notion of Stability

StabilityAn important property of dynamic systems

Stability. . .An “insensitivity” to small perturbationsPerturbations are modeling errors of system, environment, noise

F=0. OK

Basic Notions of Stability

StabilityAn important property of dynamic systems

Stability. . .An “insensitivity” to small perturbationsPerturbations are modeling errors of system, environment

F=0. OK

Basic Notions of Stability

StabilityAn important property of dynamic systems

Stability. . .An “insensitivity” to small perturbationsPerturbations are modeling errors of system, environment

F=0. OK

StabilityAn important property of dynamic systems

Stability. . .An “insensitivity” to small perturbationsPerturbations are modeling errors of system, environment, unmodeled noise

F=0. OK

Basic Notions of Stability

StabilityWhy might someone in robotics study stability?

Basic Notions of Stability

(1) To ensure acceptable performance of the robot under perturbation

gqCqM

Configuration space trajectorywith constraints

Some Notation

||||

),;( 00

x

txtp

xeAn isolated equilibrium of an ODE

A solution curve to first-order ODE systemwith initial conditions listed

Standard Euclidean Vector Norm

)(xfx

MANY definitions for related stability concepts

),( txfx

Definitions

),,( tuxfx Reduces to above under action of a control

Restrict attention to following classes of differential equations

Autonomous ODE

Non-Autonomous ODE

Stabilizability Question

Definitions Summary Slide

Attractivity

Lyapunov Stability

)(xfx 0exWith Isolated equilibrium at

Defn1.1: Stability of autonomous ODE, isolated equilibrium

Hahn 1967Slotine, Li

The equilibrium point (or motion) is called stable in the senseof Lyapunov if:

For all there exists such that0 0

000 ||),;(|| tttxtp

whenever )( |||| 0 x

),;( 00 txtp

)(xfx 0exWith Isolated equilibrium at

Defn1.1: Stability of autonomous ODE, isolated equilibrium

The equilibrium point (or motion) is called stable in the senseof Lyapunov if:

For all there exists such that0 0

000 ||),;(|| tttxtp

whenever )( |||| 0 xNotes

(1) If 11 (2) There can be a but no minmax

Lyapunov Stability

(Local Concept)

(Unbounded Solutions)

Lagrange Stability

)(xfx 0exWith Isolated equilibrium at

Defn1.1: Stability of autonomous ODE, isolated equilibrium

The equilibrium point (or motion) is called stable in the senseof Lyapunov if:

For all there exists such that0 0

000 )( ||),;(|| tttxtp

whenever |||| 0 x

Lagrange Stability

)(xfx 0exWith Isolated equilibrium at

Defn1.1: Stability of autonomous ODE, isolated equilibrium

The equilibrium point (or motion) is called stable in the senseof Lyapunov if:

For all there exists such that0 0

000 )( ||),;(|| tttxtp

whenever |||| 0 xLagrange Stable

Lagrange Stability

)(xfx 0exWith Isolated equilibrium at

Defn1.1: Stability of autonomous ODE, isolated equilibrium

The system is Lagrange stable if:

For all there exists such that00000 )( ||),;(|| tttxtp

whenever |||| 0 xNotes

(1) Bounded Solutions

(2) Independent Concepta) Lyapunov, Lagrangeb) Not Lyapunov, Lagrangec) Lyapunov, Not Lagranged) Not Lyapunov, Not Lagrange

(2) Stability completely separate concepta) Stable, Attractiveb) Unstable, Unattractivec) Stable, Unattractived) Unstable, Attractive

Attractive)(xfx 0exWith Isolated equilibrium at

Defn 1.2: Attractivity of autonomous ODE,isolated equilibrium

The equilibrium point (or motion) is called attractive if:

There exists an such that0

et

xtxtp

),;(lim 00 whenever ||:|| 00 xx

(3) Unstable yet attractive, Vinograd

Notes

(1) Asymptotic concept, no transient notion

Attractivity Example)(xfx 0exWith Isolated equilibrium at

Defn 1.2: Attractivity of autonomous ODE,isolated equilibrium

0;0 if 0x ))(1)((

)(22222

52

yxyxyx

yxyxx

0;0 if 0y ))(1)((

)2(22222

2

yxyxyx

xyyy

Denominator always positive

y Switches on xy 2

Asymptotic Stability

)(xfx 0exWith Isolated equilibrium at

Defn 1.3: Asymptotic stability of autonomous ODE, isolated equilibrium

Asymptotically stable equals both stable and attractive

Defn 1.4: Global Asymptotic stability of autonomous ODE, isolated equilibrium

Global Asymptotic Stability is both stable and attractive for

[Hahn]

Set Stability

Defn 1.5: Stability of an invariant set M,autonomous ODE

Notes

MyyxMx ||,||inf),(

Now consider stability of objects other than isolated equilibrium point

The set M is called stable in the sense of Lyapunov if:

For all there exists such that0 0

0 tt ε M)ρ(p(t),

whenever δ M),ρ(x0 (1) Attractivity, Asymptotic Stability are comparably redefined

Invariant-Not entered or exited

(2) Use on limit cycles, for example

[Hahn]

Motion Stability

Defn 1.6: Stability of a motion (trajectory), autonomous ODE

The motion is stable if:),;( 0ttp 0x

Now consider stability of objects other than isolated equilibrium point

For all there exists such that0 0000 ||),;(),;(|| ttttpttp 10 xx

whenever |||| 01 x-x

Notes

(1) Just redefined distance again

(2) Error Coordinate Transform

[Hahn]

Uniform Stability),( txfx 0exWith Isolated equilibrium at

Defn2.1: Stability of non-autonomous ODE,isolated equilibrium

The equilibrium point (or motion) is called stable (Lyapunov) if:

For all there exists such that0 0

000 ||),;(|| tttxtp

whenever ),( |||| 00 tx

Defn 2.2: Uniform Stability of non-autonomous ODE, isolated equilibrium

),( |||| 00 tx )( |||| 0 x

Definitions),( txfx 0exWith Isolated equilibirum at

Defn2.1: Stability of non-autonomous ODE,isolated equilibrium

Stable, not uniformly stable system

xtttx )2cossin(

)cos2(exp*)cos2(exp)( 000 ttttxtx

[Dunbar]

Definitions-Wrap Up Slide

Exponential StabilityInput-Output Stability BIBO-BIBSStochastic Stability NotionsStabilizability, Instability, Total

Autonomous ODE Non-Autonomous ODE

Stability of Equilibrium

Lagrange Stability

Attractivity

Asymptotic Stability

Stability of Set

Stability of Motion

Same

Uniform Stability

Not Covered:

Theorems

Restrict attention again to autonomous, non-autonomous ODE

How do we show a specific system has a stability property?

These theorems typically relate existence of a particular function(Lyapunov) function to a particular stability property

Theorem: If there exists a Lyapunov function,

then some stability property

MANY theorems exist which can be used to prove some stability property

Lyapunov Functions

Lyapunov Functions

0)(

00)(

xV

xxV

0)(

fx

VxV

Defn 3.1 Lyapunov function for an autonomous system )(xfx

)(xV Positive Definite around origin

For some neighborhood of origin

Defn 3.2 Lyapunov function for an non-autonomous system ),( txfx txVtxV )(),( 2

[Slotine, Li][Hahn]

0),(

t

Vf

x

VtxV

Dominates Positive Definite Fn

For some neighborhood of origin

Assume V is continuous in x,t is also

Note

V

Stability Theorem

)(xfx Thm 1.1: Stability of Isolated Equilibrium of Autonomous ODE

An isolated equilibrium of is stable if there exists a Lyapunov Function for this system

)(xfx

Proof Sketch 1.1

For all there exists

0 0000 ||),;(|| tttxtp

whenever |||| 0 x

Ifexists 0)(),( xVxV

then

(1) Pick Arbitrary Epsilon, Construct Delta

(2) Consider min of V(x) on Vbound Extreme Value Theorem

|||| x

||:||)(max)( xxVf(3) Define function

(4) If continuous, then by IVT

boundvf )(:),0( )(f

(5) Since

||)(|||||| 0 txx

0)( xV

Stability proof example

)(xfx Thm 1.1: Stability of Isolated Equilibrium of Autonomous ODE

An isolated equilibrium of is stable if there exists a Lyapunov Function for this system

)(xfx

Example- Undamped pendulum

0sin

sin

2

2

1cos1),( V

0*sin),( V

(1) Propose

(2) Derivative

(Kinetic + Potential)

Asymptotic stability theorem

Thm 1.2: Asymptotic Stability of Isolated Equilibrium of Autonomous ODE

An isolated equilibrium of is asymptotically stable if there exists a Lyapunov Function for this system with strictly negative time derivative.

)(xfx

Small Proof Sketch 1.2

Notes

||||,)( xxV

Radial Unbounded, Barbashin Extension

locally 0)( xVLocal Global

globally 0)( xV

(1) Stability from prev, Need Attractivity

(2) EVT with

“Ball” not entered)(

)(

xV

xV txtp ||);(:|| 0

(3) Construct a sequence of Epsilon balls

lxVx )(:lG

Let there be a region be defined by:

Let onLet there be two more regions E and M:

lGEM

VE

);0(: 1 M is largest invariantset

Lasalle Theorem

)(xfx Thm 1.3: Stability of Invariant Set of Autonomous ODE(Lasalle’s Theorem)

Small Proof Sketch 1.3

lG0)( xV

Then M is attractive, that is 0)),((lim

Mtpt

} as );( and }{:{)( 0 nonn tpxtptpx

(1) Define Positive Limit Set

Properties:Invariant, Non-Empty, ATTRACTIVE!!

(2) Show EVx )0()( 10

Use: and Limit Cycle Stability0)( xV

[Lasalle 1975]

Lasalle Theorem example

Lasalle’s Theorem Example

Example- Damped pendulum

0sin b

bsin

2

2

1cos1),( V(1) Propose

(2) Derivative *sin),(V

02 b

)}0,{()0(1 VE

)0,0(: M Asymptotic Stability of Origin

Uniform Stability Theorem

),( txfx Theorems for Non-Autonomous ODE

Small Proof Sketch 1.4

Stability and Asymptotic Stability remain the same

0),( txV Stability

0),( txV Asymptotic Stability

Thm 1.4: Uniform (Stability) Asymptotic Stability of Non-Autonomous ODE, Isolated Equilibrium point

The equilibrium is uniformly (Stable) asymptotically stable if there existsA Lyapunov function with and there existsa function such that:

)0),(( txV 0),( txV

)(),( 1 xVtxV

||)(||),(||)(|| xtxVx Positive Definite and Decrescent

||))((||),()()( txtxV

Decrescent

[Slotine,Li]||))((||)( tx

Barbalet’s Lemma

),( txfx Thm 1.5: Barbalet’s Lemma as used in Stability (Used for Non-Autonomous ODE)

If there exists a scalar function such that:(1)(2) (3) is uniformly continuous in time

),( txV

0V),( txV

Then 0),(lim

txVt

*sin),(V

02 b

Barbalet 0)(lim

tt

boundlower a has V

[Slotine,Li]

Theorems-Wrap Up Slide

Instability TheoremsConverse Theorems

StabilizabilityKalman-Yacobovich, other Frequency theorems

Autonomous ODE Non-Autonomous ODE

Lyapunov implies stability

Lyapunov implies a.s

Lasalle’s Theorem for sets

Same

Uniform Stability

Barbalet’s Lemma

Not Covered:

Techniques for Lyapunov Construction

Theorems relate function existence with stability

How then to show a Lyapunov function exists? Construct it

In general, Lyapunov function construction is an art.

Special CasesLinear Time Invariant SystemsMechanical Systems

Construction for Linear System

Construction for a Linear System Axx

PxxxV T)(

xPAPAx

PAxxAxVxVTT

T

)(

2)(

(1) Propose

(2) Time Derivative

0

dtQeeP AttAT

P is symmetricP is positive definite

QPAPA T If we choose and solve algebraically for P: 0Q

As long as A is stable, a solution is known to exist.

Also an explicit representation of the solution exists:

Construction for a Mechanical System

Construction for a Mechanical System )()( qgqCqqM

qKqqqMqqqV pTT

2

1)(

2

1),(

Kinetic Energy Potential Energy

(1) Propose(or similar)

(2) Time DerivativeqKqqqMqqqMqqqV pTTT )(

2

1)(),(

[Sciavicco,Siciliano]

gqKqK dp If we use PD-controller with gravity compensation

then

qKqqqV DT ),( Asymptotically stable with Lasalle

General Construction Techniques

PffxV T)( A quadratic form (ellipsoid) of system velocityKrasofskii

Variable Gradient

[Slotine, Li][Hahn]

fx

fPP

x

fffPfPffxV

TTTT )**()(

Solve 0**

Qx

fPP

x

f T

x

VdxxV0

)(

Assume a form for the gradient, i.e

n

jjiji xaV

1Solve for negative semi-definite gradient

Integrate and hope for positive definite V

Construction methods for an Arbitrary System )(xfx

Construction Wrap-Up Slide

(1) Linear System -> Explicity Solve Lyapunov Equation

(2) Mechanical System -> Try a variant of mechanical energy

(3) Krasovskii’s Method Variable Gradient

Problem specific trial and error

Conclusion

Motivated why stability is an important concept

Looked at a variety of definitions of various forms of stability

Looked at theorems relating Lyapunov functions to these notions of stability

Looked at some methods to construct Lyapunov functions for particular problems