AI 中文总结
研究具有仿射安全约束的线性系统，设计满足高阶CBF约束且使原点全局指数稳定的线性反馈控制器，刻画满足约束的线性增益矩阵类别及稳定条件，还表明能在该类别约束下解决LQR和鲁棒控制问题并通过仿真验证。
AI 中文摘要
控制障碍函数（CBF）已成为受安全约束的自主系统的重要控制器设计工具。尽管很受欢迎，但最近的研究表明基于CBF的控制器可能会使系统的内部动态不稳定。本文考虑具有仿射安全约束的线性系统，并设计满足高阶CBF（HOCBF）约束的线性反馈控制器，同时使原点全局指数稳定。我们首先刻画了全局满足HOCBF约束的所有线性增益矩阵的确切类别，包括该类别非空的充要条件。然后，通过利用最近引入的CBF输出动态和CBF内部动态的概念，我们给出了该类别中存在稳定增益矩阵的充要条件。最后，我们表明通过代数 Riccati 方程（ARE）和线性矩阵不等式（LMI）等标准线性控制技术，可以在这类安全稳定增益矩阵的约束下解决线性二次调节器（LQR）和鲁棒控制问题。我们在一个仿真示例中说明了我们的结果。
英文摘要
Control barrier functions (CBFs) have become an important controller design tool for autonomous systems subject to safety constraints. Despite their popularity, recent works have shown that CBF-based controllers can destabilize the internal dynamics of the system. In this paper, we consider linear systems with affine safety constraints and design linear feedback controllers that satisfy high-order CBF (HOCBF) constraints while rendering the origin globally exponentially stable. We first characterize the exact class of all linear gain matrices that globally satisfy the HOCBF constraints, including necessary and sufficient conditions for when this class is nonempty. Then, by leveraging the recently introduced notion of CBF output dynamics and CBF internal dynamics, we provide the necessary and sufficient conditions for the existence of stabilizing gain matrices within that class. Finally, we show that Linear Quadratic Regulator (LQR) and robust control problems can be solved while being constrained within this class of safe and stabilizing gain matrices, through standard linear control techniques such as algebraic Riccati equations (AREs) and Linear Matrix Inequalities (LMIs). We illustrate our results in a simulation example.
