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 Perturbations are modeling errors of system, environment, noise F=0. OK

4. Basic Notions of Stability Stability An important property of dynamic systems Stability. . . An “insensitivity” to small perturbations Perturbations are modeling errors of system, environment F=0. OK

5. Basic Notions of Stability Stability An important property of dynamic systems Stability. . . An “insensitivity” to small perturbations Perturbations are modeling errors of system, environment F=0. OK

6. 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 F=0. 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. With Isolated equilibrium at The equilibrium point (or motion) is called stable in the sense of Lyapunov if: For all there exists such that whenever Lyapunov Stability Defn1.1: Stability of autonomous ODE, isolated equilibrium Hahn 1967 Slotine, Li

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

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

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

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

16. With Isolated equilibrium at The equilibrium point (or motion) is called attractive if: There exists an such that whenever 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. With Isolated equilibrium at Attractivity Example Defn 1.2: Attractivity of autonomous ODE,isolated equilibrium Denominator always positive Switches on

18. With Isolated equilibrium at 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. The set M is called stable in the sense of Lyapunov if: For all there exists such that whenever 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. The motion is stable if: 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. With Isolated equilibrium at The equilibrium point (or motion) is called stable (Lyapunov) if: For all there exists such that whenever Uniform Stability Defn2.1: Stability of non-autonomous ODE, isolated equilibrium Defn 2.2: Uniform Stability of non-autonomous ODE, isolated equilibrium

22. With Isolated equilibirum at Definitions Defn2.1: Stability of non-autonomous ODE, isolated equilibrium Stable, not uniformly stable system [Dunbar]

23. Definitions-Wrap Up Slide 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: Ifthere 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. Let there be a region be defined by: Let on M is largest invariant set Let there be two more regions E and M: Lasalle Theorem Thm 1.3: Stability of Invariant Set of Autonomous ODE (Lasalle’s 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 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 Barbalet [Slotine,Li]

33. Theorems-Wrap Up Slide • 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 -> 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