This course is about convex optimization, a class of mathematical optimization with numerous applications in science and engineering problems. The main advantage of formulating a problem as a convex optimization is that it can be solved numerically efficient and tractable. This course has been designed for graduate students of electrical engineering. After introducing different forms of convex optimization such as linear programming and least-squares, the focus will be on Linear Matrix Inequalities (LMI) and its applications in electrical engineering.
Introduction
Mathematical optimization
Least-squares and linear programming
Convex optimization
Nonlinear optimization
Convex sets
Affine and convex sets
Operations that preserve convexity
Convex functions
Basic properties and examples
Operations that preserve convexity
The conjugate function
Quasiconvex functions
Convex optimization problems
Linear optimization problems
Quadratic optimization problems
Geometric programming
Vector optimization
Linear Matrix Inequalities (LMI)
What are LMI's and what are they good for?
Stability: linear time-invariant, time-varying or non-linear systems
Performance
Dissipativity
Quadratic performance and specializations (H_{1}, passivity)
H_{2} performance and generalizations
Synthesis
State-feedback and estimation problems
Output feedback synthesis
Multi-objective Control
Youla parametrization and genuine multi-objective controller synthesis
Robust controller design
Parameter Robust Stability
Robust stability against time-invariant and time-varying uncertainties
Parameter dependent Lyapunov functions
Semi-infinite LMI problems and relaxations
Robust Optimization and Lagrange Duality
Introduction to robust optimization and robust LMI problems
Lagrange duality
How to construct tractable relaxations
Dynamic Robustness
Linear fractional representations
Robust stability tests with multipliers
Relations to the structured singular value
LPV synthesis
Linear parametrically-varying controller synthesis
Direct approach
Multiplier approach
Polynomial optimization
Sum of Square (SOS) optimization
S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, Cambridge, 2004.
C. Scherer and S. Weiland, Linear Matrix Inequalities in Control, Lecture Notes, 2005.
S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory (Studies in Applied Mathematics), SIAM, Philadelphia, 1994.
Project 30%
Midterm 30%
Final 40%