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EEC251 – Nonlinear Systems

3 units – Spring Quarter; alternate years

Lecture: 3 hours

Prerequisite: EEC 250

Grading: Letter; homework.

Catalog Description:

Nonlinear differential equations, second-order systems, approximation methods, Lyapunov stability, absolute stability, Popov criterion, circle criterion, feedback linearization techniques.

Expanded Course Description:

Basic analysis techniques for nonlinear systems.

  1. Normed Linear Vector Spaces; Inner Product Spaces
    1. Convergence
    2. Cauchy sequences, Banach spaces, Hilbert spaces
    3. Continuity
  2. Existence and Uniqueness of Solutions
    1. Contraction mapping theorem
    2. Lipschitz condition
    3. Bellman-Gronwall Lemma
    4. Small gain theorem
  3. Nonlinear Differential Equations
    1. Autonomy
    2. Equilibrium points
  4. Second-Order Systems
    1. Phase plane portrait
    2. Limit cycles
    3. Bendixson’s theorem
    4. Poincaré-Bendixson theorem
    5. Index theory
  5. Approximation Methods
    1. Krylov-Baguliubov method
    2. Describing functions technique, optimal quasilinear
  6. Lyapunov Stability
    1. Stable, uniformly stable, attractive, asymptotically stable, globally asymptotically stable equilibrium point
    2. Class K functions, positive definite functions, decrescent functions, radially unbounded functions
    3. Derivative along trajectories
    4. Lyapunov’s direct method
    5. Invariant set theorems; La Salle’s theorem
    6. Instability theorems
    7. Lyapunov stability of linear, time-invariant systems
    8. Lyapunov’s linearization method
    9. Krasovskii’s method
  7. The Luré Problem (Absolute Stability)
    1. Positive real transfer functions
    2. Kalman-Yacubovich Lemma
    3. Aizerman’s conjecture; Kalman’s conjecture
    4. Circle criterion
    5. Popov criterion
  8. Feedback linearization
    1. Vector fields, forms
    2. Diffeomorphisms
    3. Inverse function theorem; Implicit function theorem
    4. Lie derivative; Lie bracket
    5. Complete integrability; involutivity
    6. Frobenius theorem
    7. Reachability
    8. Single-input feedback linearization
    9. Kronecker indices
    10. Brunovsky canonical form


  1. M. Vidyasagar, Nonlinear Systems Analysis, Prentice-Hall, 1992.
  2. J. J. Slotine and W. Li, Applied Nonlinear Control, Prentice-Hall, 1991 (recommended).
  3. Course notes.

Instructor: Gündes


Last revised: December 1991