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.

翻译：本文研究具有有界加性不确定性和控制变量凸多项式约束的控制仿射多项式系统的鲁棒控制屏障函数（CBF）的验证与综合问题。我们首先将鲁棒CBF验证与综合问题建模为多层多项式优化问题（POP），其中验证问题在三个层级上优化——不确定性、控制变量和状态变量，而综合问题则额外优化所选参数化CBF候选函数的参数。随后我们证明，通过引入不确定性优化和控制变量优化的KKT条件，验证问题可简化为单层POP，综合问题可简化为极小极大POP。这种简化导出了多层半定松弛：对于验证问题，我们采用Lasserre矩松弛层级；对于综合问题，我们将其与现有鲁棒极小极大POP松弛技术建立联系——该技术首先利用平方和规划求解验证POP未知值函数的渐进紧多项式下界，再通过Lasserre层级最大化该下界。两类半定松弛均保证渐近全局收敛到最优解。我们以受控范德波尔振荡器为对象，分别在有无加性不确定性条件下深入研究了所提框架的性能。