5 research outputs found
Verification and Synthesis of Robust Control Barrier Functions: Multilevel Polynomial Optimization and Semidefinite Relaxation
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.Comment: Accepted to IEEE Conference on Decision and Control (CDC) 202
Utopia point method based robust vector polynomial optimization scheme
In this paper, we focus on a class of robust vector polynomial optimization
problems (RVPOP in short) without any convex assumptions. By
combining/improving the utopia point method (a nonlinear scalarization) for
vector optimization and "joint+marginal" relaxation method for polynomial
optimization, we solve the RVPOP successfully. Both theoratical and
computational aspects are considered
A Moment-SOS Hierarchy for Robust Polynomial Matrix Inequality Optimization with SOS-Convexity
We study a class of polynomial optimization problems with a robust polynomial
matrix inequality (PMI) constraint where the uncertainty set itself is defined
also by a PMI. These can be viewed as matrix generalizations of semi-infinite
polynomial programs, since they involve actually infinitely many PMI
constraints in general. Under certain SOS-convexity assumptions, we construct a
hierarchy of increasingly tight moment-SOS relaxations for solving such
problems. Most of the nice features of the moment-SOS hierarchy for the usual
polynomial optimization are extended to this more complicated setting. In
particular, asymptotic convergence of the hierarchy is guaranteed and finite
convergence can be certified if some flat extension condition holds true. To
extract global minimizers, we provide a linear algebra procedure for recovering
a finitely atomic matrix-valued measure from truncated matrix-valued moments.
As an application, we are able to solve the problem of minimizing the smallest
eigenvalue of a polynomial matrix subject to a PMI constraint. If SOS-convexity
is replaced by convexity, we can still approximate the optimal value as closely
as desired by solving a sequence of semidefinite programs, and certify global
optimality in case that certain flat extension conditions hold true. Finally,
an extension to the non-convexity setting is provided under a rank one
condition. To obtain the above-mentioned results, techniques from real
algebraic geometry, matrix-valued measure theory, and convex optimization are
employed.Comment: 35 page
Min-max and robust polynomial optimization
Polynomial optimization, Min-max optimization, Robust optimization, Semidefinite relaxations,