1. Some Fundamentals of Stability Theory
Aaron Greenfield

2. Outline Introduction + Motivation Definitions Theorems
Techniques for Lyapunov Function Construction

3. Basic Notion of Stability
An important property of dynamic systems
Stability. . .
An “insensitivity” to small perturbations
Perturbations are modeling errors of system, environment, noise
Have a dynamic system and consider its evolution in time. Now if we perturb the system in some way, either by initial condition, starting time, or some paramater of evolution we would expect the system to still exhibit similar evolution in time if it is a stable system
F=0. OK

4. Basic Notions of Stability
An important property of dynamic systems
Stability. . .
An “insensitivity” to small perturbations
Perturbations are modeling errors of system, environment
Have a dynamic system and consider its evolution in time. Now if we perturb the system in some way, either by initial condition, starting time, or some paramater of evolution we would expect the system to still exhibit similar evolution in time if it is a stable system
F=0. OK

5. Basic Notions of Stability
An important property of dynamic systems
Stability. . .
An “insensitivity” to small perturbations
Perturbations are modeling errors of system, environment
Have a dynamic system and consider its evolution in time. Now if we perturb the system in some way, either by initial condition, starting time, or some paramater of evolution we would expect the system to still exhibit similar evolution in time if it is a stable system
F=0. OK

6. An important property of dynamic systems
Basic Notions of Stability
Stability
An important property of dynamic systems
Stability. . .
An “insensitivity” to small perturbations
Perturbations are modeling errors of system, environment, unmodeled noise
Potentially add a 4th Slide with a spring
F=0. OK

7. Why might someone in robotics study stability?
Basic Notions of Stability
Stability
Why might someone in robotics study stability?
(1) To ensure acceptable performance of the robot under perturbation
Like to see if we are controlling our robot in a specific way, and we have some inevitable modeling errors, can we still expect good performance
Stabilty+mot-Try to use stability to aid in motion planning tasks for dynamic systems. Especially used for systems with (config space) constraints
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
With Isolated equilibrium at
Defn1.1: Stability of autonomous ODE, isolated equilibrium
The equilibrium point (or motion) is called stable in the sense
of Lyapunov if:
For all there exists such that
whenever
Hahn 1967
Slotine, Li

12. Lyapunov Stability
With Isolated equilibrium at
Defn1.1: Stability of autonomous ODE, isolated equilibrium
The equilibrium point (or motion) is called stable in the sense
of Lyapunov if:
For all there exists such that
whenever
Notes
(Local Concept)
(1) If
(2) There can be a but no
(Unbounded Solutions)

13. Lagrange Stability
With Isolated equilibrium at
Defn1.1: Stability of autonomous ODE, isolated equilibrium
The equilibrium point (or motion) is called stable in the sense
of Lyapunov if:
For all there exists such that
whenever

14. Lagrange Stability
With Isolated equilibrium at
Defn1.1: Stability of autonomous ODE, isolated equilibrium
The equilibrium point (or motion) is called stable in the sense
of Lyapunov if:
For all there exists such that
whenever
Lagrange Stable

15. Lagrange Stability
With Isolated equilibrium at
Defn1.1: Stability of autonomous ODE, isolated equilibrium
The system is Lagrange stable if:
For all there exists such that
whenever
Notes
(1) Bounded Solutions
a) Lyapunov, Lagrange
b) Not Lyapunov, Lagrange
c) Lyapunov, Not Lagrange
d) Not Lyapunov, Not Lagrange
(2) Independent Concept

16. Attractive
With 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 that
whenever
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
With Isolated equilibrium at
Defn 1.2: Attractivity of autonomous ODE,isolated equilibrium
Denominator always positive
Switches on
Perturb slightly, very far away from origin, unstable. Comes back eventually so its attractive

18. Asymptotic Stability Defn 1.3: Asymptotic stability of autonomous ODE,
With 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]

19. Set Stability Defn 1.5: Stability of an invariant set M,
Now consider stability of objects other than isolated equilibrium point
Defn 1.5: Stability of an invariant set M,
autonomous ODE
The set M is called stable in the sense of Lyapunov if:
For all there exists such that
whenever
Invariant-Not entered or exited
Notes
Up to this point, considered stability of equilbria. Can extend to set by defining a new set distance function, otherwise exactly the same
(1) Attractivity, Asymptotic Stability
are comparably redefined
(2) Use on limit cycles, for example
[Hahn]

20. Motion Stability Defn 1.6: Stability of a motion (trajectory),
Now consider stability of objects other than isolated equilibrium point
Defn 1.6: Stability of a motion (trajectory),
autonomous ODE
The motion is stable if:
For all there exists such that
whenever
We can ascribe stability to this object just by redefining our distance metric again
Notes
(1) Just redefined distance again
(2) Error Coordinate Transform
[Hahn]

21. Uniform Stability Defn2.1: Stability of non-autonomous ODE,
With 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 that
whenever
New concept is uniform stability. Delta can be a function of time, but is not a function of time. An example illustrates this
Defn 2.2: Uniform Stability of non-autonomous ODE,
isolated equilibrium

22. Definitions Defn2.1: Stability of non-autonomous ODE,
With Isolated equilibirum at
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
Stabilizability, 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
Lyapunov Function is an “energy” function
For some neighborhood of origin
Note
Assume
V is continuous in x,t
is also
[Slotine, Li]
[Hahn]

26. Stability Theorem If then
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
Assume existence of Lyapunov function
(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
Assume existence of Lyapunov function
(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
(Lasalle’s Theorem)
Use: and Limit Cycle Stability
Let there be a region be defined by:
Let on
Let there be two more regions E and M:
M is largest invariant
set
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
Lasalle’s 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. Barbalet’s Lemma 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
Then
Some replacement for Lasalle for non-autonomous systems
Barbalet
[Slotine,Li]

33. Theorems-Wrap Up Slide
Autonomous ODE
Non-Autonomous ODE
Lyapunov implies stability
Lyapunov implies a.s
Lasalle’s Theorem for sets
Same
Uniform Stability
Barbalet’s 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
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 -> Explicity Solve Lyapunov Equation
(2) Mechanical System -> Try a variant of mechanical energy
(3) Krasovskii’s 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