Some Fundamentals of Stability Theory - PowerPoint PPT Presentation

About This Presentation
Title:

Some Fundamentals of Stability Theory

Description:

An important property of dynamic systems. Stability. ... Positive Definite and Decrescent. Decrescent [Slotine,Li] Barbalet's Lemma ... – PowerPoint PPT presentation

Number of Views:648
Avg rating:3.0/5.0
Slides: 40
Provided by: csC76
Learn more at: http://www.cs.cmu.edu
Category:

less

Transcript and Presenter's Notes

Title: Some Fundamentals of Stability Theory


1
Some Fundamentals of Stability Theory
  • Aaron Greenfield

2
Outline
  • Introduction Motivation
  • Definitions
  • Theorems
  • Techniques for Lyapunov Function Construction

3
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
4
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
7
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
8
Some Notation

An isolated equilibrium of an ODE
A solution curve to first-order ODE system with
initial conditions listed
Standard Euclidean Vector Norm
9
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
10
Definitions Summary Slide
Attractivity
11
Lyapunov Stability
Defn1.1 Stability of autonomous ODE, isolated
equilibrium
Hahn 1967 Slotine, Li
12
Lyapunov Stability
Defn1.1 Stability of autonomous ODE, isolated
equilibrium
Notes
(Local Concept)
(1) If
(Unbounded Solutions)
13
Lagrange Stability
Defn1.1 Stability of autonomous ODE, isolated
equilibrium
14
Lagrange Stability
Defn1.1 Stability of autonomous ODE, isolated
equilibrium
Lagrange Stable
15
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
16
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
17
Attractivity Example
Defn 1.2 Attractivity of autonomous ODE,isolated
equilibrium
Denominator always positive
Switches on
18
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
19
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
20
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
21
Uniform Stability
Defn2.1 Stability of non-autonomous
ODE, isolated equilibrium
Defn 2.2 Uniform Stability of non-autonomous
ODE, isolated equilibrium
22
Definitions
Defn2.1 Stability of non-autonomous
ODE, isolated equilibrium
Stable, not uniformly stable system
Dunbar
23
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
24
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
25
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
26
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
27
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
28
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
29
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
30
Lasalle Theorem example
Lasalles Theorem Example
Example- Damped pendulum
(1) Propose
(2) Derivative
Asymptotic Stability of Origin
31
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
32
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
33
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
34
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
35
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
36
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
37
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
38
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
39
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
Write a Comment
User Comments (0)
About PowerShow.com