Convex Optimization in Control

ABOUT THE COURSE:Applications of convex optimization techniques, particularly semidefinite programming and linear matrix inequalities (LMIs), have significantly enriched the theory of optimal and robust control in the past few decades. In this course, first the theory of duality in convex programming and semidefinite programming will be covered. Then, problems pertaining to stability analysis, optimal state feedback controller synthesis, robust stability analysis and robust controller synthesis of linear dynamical systems will be studied and reformulated in terms of LMIs. The underlying theory of dissipativity and integral quadratic constraints will also be covered. Finally, applications of the above tools in the analysis and synthesis of convex optimization algorithms will be discussed. The mathematical treatment will be complemented with an extensive programming and implementation component.INTENDED AUDIENCE: M.Tech and PhD Students in Control Systems specialization as well faculty members from different colleges. The course will also be accessible to final year UG students.PREREQUISITES: The learners should have attended a course on linear systems/control theory from a state-space point of view. Attending an introductory course on optimization would also be helpful.INDUSTRY SUPPORT: Mathworks, General Electric, ABB, DRDO, ISRO, Tata Motors, Adani Defence and Aerospace

Syllabus

Week 1: Introduction to Convex Optimization: Convex Sets, Convex Functions, Separating Hyperplane Theorems
Week 2:Duality: Farka’s Lemma, Strong Duality in Linear Programming, Lagrangian duality, Derivation of Strong Duality Theorem, Optimality Conditions
Week 3:Linear Matrix Inequalities (LMIs), Semidefinite Programming (SDP), SDP duality, Schur Complement Lemma, Examples
Week 4:Linear Dynamical Systems in State-Space Form, Lyapunov Stability, LMI Conditions for Stability of Continuous and Discrete-time Systems
Week 5:LMI Conditions for Controllability and State Feedback, Observability and Observer Design
Week 6:Signal Spaces, Operators, Norms for Systems and Transfer Functions
Week 7:Dissipativity, Bounded Real Lemma and Positive Real Lemma
Week 8:Stabilizing Controllers and LMI Characterizations, H_2 Optimal Control and State Feedback Synthesis via LMIs
Week 9:LMIs for H_Infinity State Feedback Synthesis
Week 10:Uncertain Systems, Quadratic Stability with Affine, Polytopic and Interval Uncertainty
Week 11:Integral Quadratic Constraints (IQC), Robust Stability Analysis and Controller Design via IQC
Week 12:Analysis of Optimization Algorithms as Robust Control Problems: Dissipativity and IQC based approaches