Summary

Lyapunov and dissipation inequalities play a central role in the numerical solution of many design and analysis problems in control theory. Quadratic constraints greatly enhance this theory by accommodating parametric and non-parametric uncertainty, including nonlinearities and unmodeled dynamics. This course focuses on the application of these methods to solve a variety of dynamical systems problems. First, we address systems with known dynamics and review basic Lyapunov and dissipativity theory to obtain stability and performance conditions, as well as reachable set characterizations for safety. Second, we introduce the quadratic constraint framework to describe uncertainties, and combine this framework with Lyapunov/dissipativity theory for robust stability, performance, and safety. Third, we introduce computational techniques for these analyses, such as semidefinite programming and sum-of-squares methods. Finally, we showcase the power of the methodology with a variety of case studies, including: (i) analysis of optimization algorithms, (ii) design and analysis of feedback systems with neural network controllers, and (iii) robustness analysis in flight control, power systems, and other applications. Numerical examples and code will be provided so that students can quickly integrate the methods into their own research.

Topics

• Dissipation inequalities, including Lyapunov inequalities and bounded real lemma.

• Quadratic constraints to describe model uncertainty.

• Combination of dissipation inequalities and quadratic constraints for robust stability, performance, and reachability analysis.

• Emerging applications in convergence analysis of optimization algorithms, stability and performance analysis of neural network controllers, flight control, power systems, etc.