Applied Optimisation

1. The Basic Optimisation Problem
   1. General Formulation;
   2. Special Cases: Linear Programs; Convex Optimisation Problems;
   3. Motivating Examples
2. Linear Programming
   1. General form;
   2. Simplex Method;
   3. Applications in Network Flow;
3. Convex Optimisation
   1. Algorithms for Convex Optimisation;
   2. Applications of Convex Optimisation;
   3. Convex Optimisation in Communications and Signal Processing;
   4. Convex Optimisation in Wireless Communications
4. General Optimisation
   1. Mixed Integer Programming;
   2. Algorithms and Heuristics;
   3. Applications to VLSI and other relevant applications

Background Material

• Basic Real Analysis
• Basic Linear Algebra

Lecture Notes

To Appear

Further Reading

• Linear and Nonlinear Programming. D. Luenberger and Y. Le. Springer, 2008 (3rd Ed)
• Convex Optimization. S. Boyd and L. Vandenberghe. Cambridge University Press, 2004
• Integer and Combinatorial Optimization. L.A. Wolsey and G.L. Nemhauser. Wiley, 1999

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Background Material

• Basic Real Analysis
• Basic Linear Algebra
• Introduction to Dynamic Systems

Lecture Notes

To Appear

Further Reading

• Stability theory for swithched and hybrid systems. R. Shorten, F. Wirth, O. Mason, K. Wulff and C. King. SIAM Review 49(4): 545-592, 2007
• Switching in systems and control. D. Liberzon. Birkhauser, 2003.

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;