Abstract

The use of Linear Matrix Inequality and Semidefinite Programming techniques is very common in modern control systems analysis and design. At the same time, positive polynomials can help formulate a large number of problems in robust control, non-linear control and non-convex optimization – consider, for example, the use of Lyapunov functions for stability analysis of equilibria of nonlinear dynamical systems. The fact that polynomial positivity conditions can be formulated efficiently in terms of Linear Matrix Inequalities opens up new directions in nonlinear systems analysis and design.

In this talk I will first present how ideas from dynamical systems, positive polynomials and convex optimization can be used to analyze the stability, robust stability, performance and robust performance of systems described by nonlinear ODEs. I will also discuss briefly how hybrid/switched systems and time-delay systems can be analyzed before describing how other, more interesting analysis questions can be answered using these tools.

This approach for systems analysis, although entirely algorithmic, is currently not scalable to large system instances. To address this, I will first consider the analysis of large-scale networked systems and discuss how the system structure (both the dynamics at the nodes and the topology of the underlying network) can help generate robust functionality conditions that scale with the system size. I will finally talk about some of the most recent work on how to analyze “medium-sized” dynamical systems, combining ideas from graph partitioning and the theory of interconnected systems.