Switched Region Stabilization of Polynomial Dynamical Systems

Abstract: We consider the problem of automatically synthesizing region stabilizing controllers for polynomial dynamical systems. The goal of our synthesis is to produce switching controllers that can switch between finitely many modes to ensure that the resulting closed loop dynamics are guaranteed to reach a given small target region around the equilibrium (and stay there). Our approach uses the notion of a control Lyapunov function, an extension to standard Lyapunov functions for non-autonomous (controlled) systems. We show that a relatively simple modification to the standard control Lyapunov formulation works for the switched system case, and yields controllers with guaranteed minimum dwell times, that can be implemented inside a computer. We show how such an implementation can be performed inside a model-predictive control (MPC) scheme.

Next, we present schemes for automatically discovering control Lyapunov functions by reducing the problem to solving a system of quantified constraints over the reals. We examine the solution of these constraints using a procedure called Counter-Example Guided Inductive Synthesis (CEGIS) driven by nonlinear delta-satisfiability solvers such as dReal, or through a Linear Matrix Inequality (LMI) relaxation of the nonlinear constraints. We conclude by discussing the issue of numerical errors in LMI solving and some alternatives involving Bernstein polynomials.

Joint work with Hadi Ravanbakhsh (University of Colorado, Boulder) and Amin Ben Sassi (University of Colorado, Boulder)

Bio: Sriram Sankaranarayanan is an assistant professor of Computer Science at the University of Colorado, Boulder. His research interests include automatic techniques for reasoning about the behavior of computer and cyber-physical systems. Sriram obtained a PhD in 2005 from Stanford University where he was advised by Zohar Manna and Henny Sipma.

Subsequently he worked as a research staff member at NEC research labs in Princeton, NJ. He has been on the faculty at CU Boulder since 2009. Sriram has been the recipient of awards including the President’s Gold Medal from IIT Kharagpur (2000), Siebel Scholarship (2005), the CAREER award from NSF (2009), Dean’s award for outstanding junior faculty (2012), outstanding teaching (2014), and the Provost’s faculty achievement award (2014).