nonlinear-systems

UC Berkeley ME C237 (EE C222) - Nonlinear Systems (Spring 2022)

Topics

Planar Dynamical Systems

• Linearization about Equilibria
• Jacobian Linearization
• Hartman-Grobman Theorem
• Stable-Unstable Manifold Theorem
• Closed Orbits
• Bifurcations
• Bendixson's Theorem
• Limit Set & Cycle
• Invariant Regions
• Poincare-Bendixson Theorem

Mathematical Background

• Groups and Fields
• Vector Spaces, Algebras, Norms, etc.
• Contraction Mapping Theorem
• Lipschitz Continuity
• Caratheodory Conditions
• Existence and Uniqueness Theorems for ODEs
• Bellman-Gronwall Lemma
• Solution of Implicit ODEs
• Waveform Relaxation

Lyapunov Stability Theory

• Basic Stability Theorems of Lyapunov
  • Energy-like Functions
  • Stability Theorems in the Sense of Lyapunov
  • Exponential Stability Theorems
• LaSalle's Invariance Principle
• Generalizations of LaSalle's Principle
• Instability Theorems
• Stability of LTV Systems
• Indirect Method of Lyapunov
• Region of Attraction

Lyapunov-based Control

• Sum of Squares
• Control Lyapunov Functions
• Min-norm Control
• Sontag Control

Other Nonlinear Control Techniques

• Adaptive Control
• Back-stepping Control

Linearization by State Feedback

• Lie Derivative, Bracket, etc.
• SISO: Normal Form and Observability
• SISO: I/O Linearization
• SISO: Zero Dynamics
• SISO: Inversion and Exact Tracking
• SISO: Full State Linearization
• SISO: Approximate I/O Linearization for Nonregular Systems
• MIMO: I/O Linearization by Static State Feedback
• MIMO: Full State Linearization
• MIMO: Dynamic Extension
• Sliding Mode Control
• Switching Control

Design Examples

• Ball and Beam
• Nonlinear Flight Control
• Differential Flatness
• Geometric Control of Quadrotos with Suspended Payloads
• Bipedal Robot Running
• Control Barrier Functions (CBF)