Safety and stability are essential properties of control systems. Control Barrier Functions (CBFs) and Control Lyapunov Functions (CLFs) have been proposed to ensure safety and stability respectively. However, previous approaches typically verify and synthesize the CBFs and CLFs separately, satisfying their respective constraints, without proving that the CBFs and CLFs are compatible with each other, namely at every state, there exists control actions that satisfy both the CBF and CLF constraints simultaneously. There exists some recent works that synthesized compatible CLF and CBF, but relying on nominal polynomial or rational controllers, which is just a sufficient but not necessary condition for compatibility. In this work, we investigate verification and synthesis of compatible CBF and CLF independent from any nominal controllers. We derive exact necessary and sufficient conditions for compatibility, and further formulate Sum-Of-Squares program for the compatibility verification. Based on our verification framework, we also design an alternating nominal-controller-free synthesis method. We evaluate our method in a linear toy, a non-linear toy, and a power converter example.

翻译：安全性与稳定性是控制系统的基本属性。控制屏障函数与控制李雅普诺夫函数分别被提出以确保安全性与稳定性。然而，以往方法通常分别验证与综合CBF和CLF，仅满足各自约束，并未证明二者相互兼容，即在每个状态均存在同时满足CBF与CLF约束的控制动作。近期虽有研究综合兼容的CLF与CBF，但其依赖标称多项式或有理控制器，这仅为兼容性的充分非必要条件。本研究致力于独立于任何标称控制器的兼容CBF与CLF验证与综合。我们推导出兼容性的精确充要条件，并进一步构建兼容性验证的平方和规划。基于此验证框架，我们还设计了一种无需标称控制器的交替综合方法。我们在线性示例、非线性示例及功率变换器案例中评估了所提方法。