ECE 688
Nonlinear Systems
Winter 2025
Topics
- List:
- Introduction to nonlinear models and phenomena Examples.
- Mathematical preliminaries Norms, basic topology, continuity and differentiation.
- Dynamical systems and differential equations Dynamical systems, vector fields and local flows, existence and uniqueness.
- Key concepts in dynamics Equilibria and closed orbits, invariant sets, Nagumo’s theorem, limit sets, linearization of nonlinear systems.
- Stability theory Notions of stability, Lyapunov’s direct method, the invariance principle, exponential stability and linearization, converse theorems.
- Introduction to nonlinear stabilization Stabilization using linearization, control Lyapunov functions, Artstein-Sontag theorem, Brockett’s necessary conditions for continuous stabilizability.
- Dissipative systems Dissipative systems and Lyapunov stability, finite-gain stability, interconnected dissipative systems, application to mechanical control systems. 2
Readings
- Nonlinear SystemsOptional suggested textbook.
- Nonlinear Systems Analysis, 2nd edition. M. Vidyasagar (2002).
- Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach. W. Haddad and V. Chellaboina (2008).
- L2-Gain and Passivity Techniques in Nonlinear Control, A. van der Schaft (1996).
- Differential Equations, Dynamical Systems, and Linear Algebra. M. Hirsch and S. Smale (1974).
- Course notes available on the course website.
Winter 2022
Topics
- List:
- Introduction to nonlinear models and phenomena Examples.
- Mathematical preliminaries Norms, basic topology, continuity and differentiation.
- Dynamical systems and differential equations Dynamical systems, vector fields and local flows, existence and uniqueness.
- Key concepts in dynamics Equilibria and closed orbits, invariant sets, limit sets, linearization of nonlinear systems.
- Stability theory Notions of stability, Lyapunov’s direct method, the invariance principle, exponential stability and linearization, converse theorems.
- Introduction to nonlinear stabilization Control Lyapunov functions, Artstein-Sontag theorem, Brockett’s necessary conditions for continuous stabilizability.
- Passive systems Definitions, passivity-based stabilization, application to Hamiltonian control systems, damp- ing control.
Readings
- Nonlinear SystemsOptional suggested textbook.
- Nonlinear Systems Analysis, 2nd edition. M. Vidyasagar (2002).
- V. Chellaboina (2008).
- Differential Equations, Dynamical Systems, and Linear Algebra. M. Hirsch and S. Smale (1974).
- Course notes available on the course website.
Winter 2017
Topics
- Mathematical preliminaries Functions, Norms, topology of ℝⁿ, continuity on ℝⁿ, differentiability on ℝⁿ.
- The vocabulary of dynamical systems Phase and integral curves, phase portraits, state transition function, phase flows, vector fields, existence and uniqueness of solutions, equilibria, closed orbits, invariant sets and limit sets.
- Lyapunov stability Autonomous systems, invariance principle, sign definite functions, domain of attraction, linearization, converse theorems, stability and small perturbations.
- Input-output stability “ℒ spaces” and their extensions, input-output maps, small gain theorem, linear systems with nonlinear feedback.
- Input-to-state stability Cascade connected systems, feedback connected systems, small gain theorem for ISS systems.
- Dissipative systems Definitions, relationship with Lyapunov stability, classes of dissipative systems, control affine systems with quadratic supply rates, linear systems, absolute stability problem.
- Introduction to output regulation Centre-manifold theory, tracking for nonlinear control systems (local theory), single-input single-output control affine systems with relative degree.
Readings
- Nonlinear Systems Analysis, 2nd edition, M. Vidyasagar.
- Nonlinear Control Systems II, A. Isidori.
- L2-Gain and Passivity Techniques in Nonlinear Control, A. van der Schaft.
- Nonlinear Systems: Analysis, stability and control, S. Sastry.
Sources
- Course catalogue ↗2026–2027
- course_outline ↗Winter 2025
- course_outline ↗Winter 2022
- course_outline ↗Winter 2017
- Official public PDF outline or course description. Materials and topic extracts may be partial; consult the linked source for the full list. Display title supplied from the current catalogue; use the source for the historical topic title.
- Official public PDF outline or course description. Materials and topic extracts may be partial; consult the linked source for the full list.
- Public departmental course outline. Source may be archival; no current-term assignment is implied. Bibliographic details are preserved in textbook notes.