Convex optimization methods based on Linear Matrix Inequalities (LMIs) have dramatically expanded our ability to analyze and design complex multivariable control systems. This course explores material from the broad areas of robust and optimal control, with an emphasis on formulating systems analysis and controller design problems using LMIs. Topics covered will include: Historical context of robust and optimal control; Fundamentals of optimization, linear matrix inequalities and semidefinite programming; Lyapunov equations and inequalities; H-Infinity and H2 performance criteria for dynamic systems; Dissipative dynamical systems; The generalized plant framework for optimal control; LMI solutions of H2 and H-Infinity state and output feedback control problems; Uncertain systems: linear and nonlinear uncertainty modelling, linear fractional representations; Robust stability and performance analysis of uncertain systems; Introduction to integral quadratic constraints.