Some Fundamentals of Stability Theory

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Some Fundamentals of Stability Theory

• Aaron Greenfield

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Outline

• Introduction Motivation
• Definitions
• Theorems
• Techniques for Lyapunov Function Construction

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Basic Notion of Stability
Stability
An important property of dynamic systems
Stability. . .

An insensitivity to small perturbations Pertur
bations are modeling errors of system,
environment, noise
F0. OK
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Basic Notions of Stability
Stability
An important property of dynamic systems
Stability. . .

An insensitivity to small perturbations Pertur
bations are modeling errors of system, environment
F0. OK
5
Basic Notions of Stability
Stability
An important property of dynamic systems
Stability. . .

An insensitivity to small perturbations Pertur
bations are modeling errors of system, environment
F0. OK
6
Basic Notions of Stability
Stability
An important property of dynamic systems
Stability. . .

An insensitivity to small perturbations Pertur
bations are modeling errors of system,
environment, unmodeled noise
F0. OK
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Basic Notions of Stability
Stability
Why might someone in robotics study stability?
(1) To ensure acceptable performance of the robot
under perturbation

Configuration space trajectory with constraints
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Some Notation

An isolated equilibrium of an ODE
A solution curve to first-order ODE system with
initial conditions listed
Standard Euclidean Vector Norm
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Definitions
MANY definitions for related stability concepts
Restrict attention to following classes of
differential equations
Autonomous ODE
Non-Autonomous ODE
Reduces to above under action of a control
Stabilizability Question
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Definitions Summary Slide
Attractivity
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Lyapunov Stability
Defn1.1 Stability of autonomous ODE, isolated
equilibrium
Hahn 1967 Slotine, Li
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Lyapunov Stability
Defn1.1 Stability of autonomous ODE, isolated
equilibrium
Notes
(Local Concept)
(1) If
(Unbounded Solutions)
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Lagrange Stability
Defn1.1 Stability of autonomous ODE, isolated
equilibrium
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Lagrange Stability
Defn1.1 Stability of autonomous ODE, isolated
equilibrium
Lagrange Stable
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Lagrange Stability
Defn1.1 Stability of autonomous ODE, isolated
equilibrium
Notes
(1) Bounded Solutions
(2) Independent Concept
a) Lyapunov, Lagrange b) Not Lyapunov,
Lagrange c) Lyapunov, Not Lagrange d) Not
Lyapunov, Not Lagrange
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Attractive
Defn 1.2 Attractivity of autonomous ODE,isolated
equilibrium
Notes
(1) Asymptotic concept, no transient notion
(2) Stability completely separate concept a)
Stable, Attractive b) Unstable, Unattractive c)
Stable, Unattractive d) Unstable, Attractive
(3) Unstable yet attractive, Vinograd
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Attractivity Example
Defn 1.2 Attractivity of autonomous ODE,isolated
equilibrium
Denominator always positive
Switches on
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Asymptotic Stability
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
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Set Stability
Now consider stability of objects other than
isolated equilibrium point
Defn 1.5 Stability of an invariant set
M, autonomous ODE
Invariant-Not entered or exited
Notes
(1) Attractivity, Asymptotic Stability are
comparably redefined
(2) Use on limit cycles, for example
Hahn
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Motion Stability
Now consider stability of objects other than
isolated equilibrium point
Defn 1.6 Stability of a motion (trajectory),
autonomous ODE
For all there exists such
that
whenever
Notes
(1) Just redefined distance again
(2) Error Coordinate Transform
Hahn
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Uniform Stability
Defn2.1 Stability of non-autonomous
ODE, isolated equilibrium
Defn 2.2 Uniform Stability of non-autonomous
ODE, isolated equilibrium
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Definitions
Defn2.1 Stability of non-autonomous
ODE, isolated equilibrium
Stable, not uniformly stable system
Dunbar
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Definitions-Wrap Up Slide
Autonomous ODE Non-Autonomous ODE
Stability of Equilibrium Lagrange Stability Attractivity Asymptotic Stability Stability of Set Stability of Motion Same Uniform Stability
Exponential Stability Input-Output Stability
BIBO-BIBS Stochastic Stability Notions Stabilizabi
lity, Instability, Total
Not Covered
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Theorems
How do we show a specific system has a stability
property?
MANY theorems exist which can be used to prove
some stability property
Restrict attention again to autonomous,
non-autonomous ODE
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
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Lyapunov Functions
Lyapunov Functions
Defn 3.1 Lyapunov function for an autonomous
system
Positive Definite around origin
For some neighborhood of origin
Defn 3.2 Lyapunov function for an non-autonomous
system
Dominates Positive Definite Fn
For some neighborhood of origin
Note
Assume V is continuous in x,t is also
Slotine, Li Hahn
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Stability Theorem
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
Proof Sketch 1.1
If
(1) Pick Arbitrary Epsilon, Construct Delta
(2) Consider min of V(x) on Vbound
Extreme Value Theorem
then
(3) Define function
For all there exists
(4) If continuous, then by IVT
whenever
(5) Since
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Stability proof example
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
Example- Undamped pendulum
(1) Propose
(Kinetic Potential)
(2) Derivative
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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.
Small Proof Sketch 1.2
(1) Stability from prev, Need Attractivity
(2) EVT with Ball not entered
(3) Construct a sequence of Epsilon balls
Notes
Local
Global
Radial Unbounded, Barbashin Extension
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Lasalle Theorem
Thm 1.3 Stability of Invariant Set of Autonomous
ODE (Lasalles Theorem)
Use and Limit Cycle Stability
Then M is attractive, that is
Small Proof Sketch 1.3
(1) Define Positive Limit Set
Properties Invariant, Non-Empty, ATTRACTIVE!!
(2) Show
Lasalle 1975
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Lasalle Theorem example
Lasalles Theorem Example
Example- Damped pendulum
(1) Propose
(2) Derivative
Asymptotic Stability of Origin
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Uniform Stability Theorem
Theorems for Non-Autonomous ODE
Stability and Asymptotic Stability remain the same
Stability
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 exists A Lyapunov
function with
and there exists a function such that
Decrescent
Small Proof Sketch 1.4
Positive Definite and Decrescent
Slotine,Li
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Barbalets Lemma
Thm 1.5 Barbalets 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
Then
Barbalet
Slotine,Li
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Theorems-Wrap Up Slide
Autonomous ODE Non-Autonomous ODE
Lyapunov implies stability Lyapunov implies a.s Lasalles Theorem for sets Same Uniform Stability Barbalets Lemma

• Instability Theorems
• Converse Theorems
• Stabilizability
• Kalman-Yacobovich, other Frequency theorems

Not Covered
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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 Cases Linear Time Invariant
Systems Mechanical Systems
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Construction for Linear System
Construction for a Linear System
P is symmetric P is positive definite
(1) Propose
(2) Time Derivative
If we choose and solve
algebraically for P
As long as A is stable, a solution is known to
exist.
Also an explicit representation of the solution
exists
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Construction for a Mechanical System
Construction for a Mechanical System

• Propose
• (or similar)

Potential Energy
Kinetic Energy
(2) Time Derivative
If we use PD-controller with gravity compensation
then
Asymptotically stable with Lasalle
Sciavicco,Siciliano
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General Construction Techniques
Construction methods for an Arbitrary System
Krasofskii
A quadratic form (ellipsoid) of system velocity
Solve
Variable Gradient
Assume a form for the gradient, i.e
Solve for negative semi-definite gradient
Slotine, Li Hahn
Integrate and hope for positive definite V
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Construction Wrap-Up Slide
(1) Linear System -gt Explicity Solve Lyapunov
Equation
(2) Mechanical System -gt Try a variant of
mechanical energy
(3) Krasovskiis Method Variable Gradient
Problem specific trial and error
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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