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Nonlinear Systems: an Introduction to Lyapunov Stability Theory

Nonlinear Systems: an Introduction to Lyapunov Stability Theory. A. Pascoal , April 2013 (draft under revision). Linear based control laws. Nonlinear control laws. + Good engineering intuition for linear designs (local stability and performance).

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Nonlinear Systems: an Introduction to Lyapunov Stability Theory

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  1. Nonlinear Systems: an Introduction to Lyapunov Stability Theory A. Pascoal , April 2013 (draft under revision)

  2. Linear based control laws Nonlinear control laws +Good engineering intuition for linear designs (local stability and performance) +Powerful robust stability analysis tools +Possible deep physical insight -Lack of global stability and performance results -Need for stronger theoretical background -Poor physical intuition -Limited tools for performance analysis Linear versus Nonlinear Control Nonlinear Plant

  3. AUV speed control Dynamics Nonlinear Plant Objective: generate T(t) so that tracks the reference speed Tracking error Error Dynamics Nonlinear Control: Key Ingredients

  4. Nonlinear Control Law Nonlinear Control: Key Ingredients Error Dynamics

  5. Tracking error tends to zero exponentially fast. Simple and elegant! Catch: the nonlinear dynamics are known EXACTLY. Key idea: i) use “simple” concepts, ii) deal with robustness against parameter uncertainty. Nonlinear Control: Key Ingredients New tools are needed: LYAPUNOV theory

  6. fv v v m/f SIMPLE EXAMPLE Lyapunov theory of stability: a soft Intro (freemass, subjected to a simplemotionresisting force) t v=0 isanequilibriumpoint; dv/dt=0 when v=0! v=0 isattractive (trajectories converge to 0) 0 v

  7. fv v (energyfunction) Lyapunov theory of stability: a soft Intro How can one prove thatthetrajectoriesgo to theequilibriumpoint WITHOUT SOLVING thedifferentialequation? v 0 V positive andboundedbelowby zero; dV/dt negative impliesconvergence of V to 0!

  8. fv v Lyapunov theory of stability: a soft Intro What are the BENEFITS of this seemingly strange approach to investigate convergence of the trajectories to an equilibrium point? f(v) v f a general dissipative force e.g. v|v| Q-I V positive andboundedbelowby zero; dV/dt negative impliesconvergenceof V to 0! 0 v Q-III Very general formofnonlinearequation!

  9. Q-positive definite Lyapunov theory of stability: a soft Intro 2-D case State vector

  10. 1x1 1x2 2x1 V positive andboundedbelowby zero; dV/dt negative impliesconvergenceof V to 0! x tends do 0! Lyapunov theory of stability: a soft Intro 2-D case

  11. y-measuredfromspringatrest Equilibriumpointxeq: dx/dt=0 y mg Examine the ZERO eq. Point! Lyapunov theory of stability: a soft Intro Shifting Istheoriginalwaysthe TRUE origin? Examine if yeq is “attractive”!

  12. Examine the ZERO eq. Point! Lyapunov theory of stability: a soft Intro Shifting Is the origin always the TRUE origin? xref(t) is a solution Examine if xref(t) is “attractive”!

  13. Lyapunov theory of stability: a soft Intro ControlAction u y Nonlinear plant Staticcontrol law Investigateif 0 isattractive!

  14. Lyapunov Theory Stabilityofthe zero solution The zero solutionis STABLE if e d x-space 0

  15. Lyapunov Theory Attractivenessofthe zero solution The zero solutionislocally ATTRACTIVE if a x-space 0

  16. Lyapunov Theory The zero solutionislocally ASYMPTOTICALLY STABLE if itis STABLE and ATTRACTIVE Onemayhaveattractivenessbut NOT Stability! e d (thetwoconditions are required for AsymptoticStability!)

  17. LyapunovTheory (a formal approach) Key Ingredients for Nonlinear Control

  18. Lyapunov Theory (the two conditions are required for Asymptotic Stability!) e d

  19. Lyapunov Theory There are atleastthreewaysofassessingthestability (of anequilibriumpointof a) system: • Solve thedifferentialequation (brute-force) • Linearize thedynamicsand examine thebehaviour • oftheresulting linear system (local results for hyperbolic • eq. pointsonly) • Use Lypaunov´s direct method (elegant and powerful, • may yield global results)

  20. Lyapunov Theory

  21. If thentheoriginisgloballyasymptoticallystable Lyapunov Theory

  22. Whathappenswhen Isthesituationhopeless? (Let M bethelargestinvariantsetcontainedinW. Thenallsolutions converge to M. If M istheorigin, theresultsfollows) Krazovskii-LaSalle Lyapunov Theory No! SupposetheonlytrajectoryofthesystementirelycontainedinW isthenulltrajectory. Then, theoriginisasymptoticallystable

  23. y Krazovskii-La Salle Lyapunov Theory

  24. y Krazovskii-La Salle Lyapunov Theory f(.), k(.) – 1st and 3rd quadrants f(0)=k(0)=0 V(x)>0!

  25. y TrajectoryleavesW unless x1=0! W M istheorigin. Theoriginisasymptoticallystable! Examine dynamics here! Krazovskii-La Salle Lyapunov Theory

  26. Nonlinear Systems: an Introduction to Lyapunov Stability Theory

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