Safety is one of the fundamental challenges in control theory. Recently, multi-step optimal control problems for discrete-time dynamical systems were formulated to enforce stability, while subject to input constraints as well as safety-critical requirements using discrete-time control barrier functions within a model predictive control (MPC) framework. Existing work usually focus on the feasibility or the safety for the optimization problem, and the majority of the existing work restrict the discussions to relative-degree one control barrier functions. Additionally, the real-time computation is challenging when a large horizon is considered in the MPC problem for relative-degree one or high-order control barrier functions. In this paper, we propose a framework that solves the safety-critical MPC problem in an iterative optimization, which is applicable for any relative-degree control barrier functions. In the proposed formulation, the nonlinear system dynamics as well as the safety constraints modeled as discrete-time high-order control barrier functions (DHOCBF) are linearized at each time step. Our formulation is generally valid for any control barrier function with an arbitrary relative-degree. The advantages of fast computational performance with safety guarantee are analyzed and validated with numerical results.

翻译：安全性是控制理论中的基本挑战之一。最近，为了在模型预测控制（MPC）框架内使用离散时间控制屏障函数来强制执行稳定性，同时满足输入约束和安全关键要求，提出了针对离散时间动力系统的多步最优控制问题。现有工作通常关注优化问题的可行性或安全性，且大多数现有工作将讨论限制在相对度为一的控制屏障函数。此外，当在MPC问题中考虑较大时间步长时，对于相对度为一或高阶控制屏障函数的实时计算具有挑战性。在本文中，我们提出了一种通过迭代优化求解安全关键MPC问题的框架，该框架适用于任意相对度的控制屏障函数。在所提出的公式中，非线性系统动力学以及建模为离散时间高阶控制屏障函数（DHOCBF）的安全约束在每个时间步进行线性化。我们的公式对于任意相对度的控制屏障函数普遍有效。快速计算性能及安全性保证的优势通过数值结果进行了分析和验证。