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：我们研究了具有有界加性不确定性及控制凸多项式约束的控制仿射多项式系统中鲁棒控制屏障函数（CBF）的验证与综合问题。首先将鲁棒CBF的验证与综合表述为多层级多项式优化问题（POP）：其中验证问题在三个层级上分别优化不确定性、控制与状态，而综合问题则额外优化所选参数化CBF候选函数中的参数。通过引入不确定性与控制内层优化的KKT条件，我们证明验证问题可简化为单层级POP，综合问题则转化为最小-最大POP，这一简化催生了多层级半定松弛方法。针对验证问题，应用Lasserre矩松弛层级；针对综合问题，我们建立其与现有鲁棒最小-最大POP松弛技术的联系——该技术首先利用平方和规划求取验证POP未知值函数的严格递增多项式下界，再通过Lasserre层级最大化该下界。两种半定松弛均能保证渐近全局收敛至最优解。最后以受控范德波尔振荡器（含/不含加性不确定性）为例，深入研究了所提框架的性能。