EE 587 – Lecture Notes

Mihailo Jovanovic, University of Southern California, Spring 2018

These notes are courtesy of Daniel Hagen. They should give you a fairly good idea of what I write on board during lectures. Needless to say, Dan's handwriting beats mine by orders of magnitude - great job, Dan!

  1. Course mechanics; Topics; Introduction to nonlinear systems

  2. Examples of nonlinear systems; Equilibrium points; Linearization

  3. Range of nonlinear phenomena (multiple isolated equilibrium points, finite escape time, limit cycles, chaos); Fold bifurcations

  4. Transcritical and pitchfork bifurcations; Phase portraits of 2nd order linear systems

  5. Phase portraits of nonlinear systems near hyperbolic equilibria (Hartman-Grobman); Bendixson's theorem

  6. Bendixson's theorem (examples); Positively invariant sets; Poincare-Bendixson theorem

  7. Poincare-Bendixson theorem; Hopf bifurcations

  8. Center manifold theory; Invariant manifolds

  9. Existence and uniqueness of solutions; Lipschitz continuity

  10. Continuous dependence on initial conditions and parameters; Sensitivity equations

  11. Stability of equilibrium points

  12. Lyapunov functions; Positive definiteness; Radial unboundedness

  13. Lyapunov functions (examples); LaSalle's invariance principle

  14. Lyapunov functions for LTI systems; Algebraic Lyapunov equation

  15. LaSalle's invariance principle for LTI systems; Stability via linearization

  16. Discussion of HW3; Region of attraction; Comparison functions

  17. Comparison functions (continued); Uniform stability; Exponential stability

  18. Lyapunov based stability analysis of time-varying systems; Stability using comparison functions

  19. Exponential stability; Differential Lyapunov equation for linear time-varying system

  20. “Generalization” of LaSalle's invariance principle to linear time-varying systems (much weaker result that for time-invariant systems); Uniform observability; Estimation of constant unknown parameters

  21. Convergence of gradient update law for estimation of constant unknown parameters

  22. Convergence of gradient update law for estimation of constant unknown parameters (examples)

  23. Example of model reference adaptive control (MRAC)

  24. Introduction to integrator backstepping

  25. Integrator backstepping

  26. Control Lyapunov functions; Relative degree and zero dynamics

  27. Feedback linearization; Input-output linearization; Zero dynamics

  28. Normal form

  29. Input-to-State Stability

  30. Notes on “Proximal Augmented Lagrangian Method for Non-Smooth Composite Optimization” slides