Verification and Synthesis of Robust Control Barrier Functions: Multilevel Polynomial Optimization and Semidefinite Relaxation

TL;DR

This paper develops a multilevel polynomial optimization framework using semidefinite relaxations for verifying and synthesizing robust control barrier functions in uncertain polynomial systems, ensuring global optimality.

Contribution

It introduces a novel reduction of verification and synthesis problems to single-level and min-max polynomial optimization problems, enabling global solutions via semidefinite relaxations.

Findings

Effective verification and synthesis of robust CBFs demonstrated on Van der Pol Oscillator.

Semidefinite relaxations guarantee asymptotic convergence to global optima.

Framework handles bounded uncertainties and convex polynomial control constraints.

Abstract

We study the problem of verification and synthesis of robust control barrier functions (CBF) for control-affine polynomial systems with bounded additive uncertainty and convex polynomial constraints on the control. We first formulate robust CBF verification and synthesis as multilevel polynomial optimization problems (POP), where verification optimizes -- in three levels -- the uncertainty, control, and state, while synthesis additionally optimizes the parameter of a chosen parametric CBF candidate. We then show that, by invoking the KKT conditions of the inner optimizations over uncertainty and control, the verification problem can be simplified as a single-level POP and the synthesis problem reduces to a min-max POP. This reduction leads to multilevel semidefinite relaxations. For the verification problem, we apply Lasserre's hierarchy of moment relaxations. For the synthesis problem, we draw connections to existing relaxation techniques for robust min-max POP, which first use sum-of-squares programming to find increasingly tight polynomial lower bounds to the unknown value function of the verification POP, and then call Lasserre's hierarchy again to maximize the lower bounds. Both semidefinite relaxations guarantee asymptotic global convergence to optimality. We provide an in-depth study of our framework on the controlled Van der Pol Oscillator, both with and without additive uncertainty.