Hybrid Systems

1. Fundamentals
   1. Motivating examples: Automotive transmission control; Internet congestion control; Automated highway systems.
   2. Definitions and models of switched and hybrid systems: Finite automata; Ordinary differential equations; Differential inclusions; Deterministic and stochastic hybrid systems.
   3. Existence and uniqueness of solutions for switched and hybrid systems.
2. General Stability Theory
   1. Practical importance of stability
   2. Definitions and core concepts: Stability, attractivity, asymptotic stability and exponential stability
   3. Lyapunov functions
   4. Definition of key stability problems for switched systems
   5. Basic results on periodic systems – Floquet theory
3. Stability under Arbitrary Switching
   1. Common Lyapunov functions and converse theorems
   2. Common quadratic Lyapunov functions
   3. Non-quadratic Lyapunov functions
   4. Triangular and Lie-algebraic results
   5. Instability theorems
4. Further Stability Questions
   1. Dwell-time and slow-switching regimes
   2. Multiple Lyapunov Functions
   3. State-dependent switching
   4. Piecewise quadratic and piecewise linear Lyapunov functions
   5. Stabilising switching laws
5. Applications
   1. ABS control – Supervisory/Adaptive control; Multiple-models
      switching & tuning; ABS design.

   2. TCP Congestion control & Synchronised communication networks;

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