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。近年来，针对离散时间动力系统的多步最优控制问题被提出，以便在模型预测控制（MPC）框架中，通过离散时间控制障碍函数在满足输入约束和安全关键要求的同时，强制系统稳定性。现有工作通常侧重于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阶为一的控制障碍函数上。此外，当MPC问题涉及较大预测时域时，无论对于相对阶为一还是高阶控制障碍函数，其实时计算都具有挑战性。本文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阶的控制障碍函数。在所提公式中，非线性系统动力学以及建模为离散热门高阶控制障碍函数（DHOCBF）的安全约束在每个时间步被线性化。我们的公式对于任意相对阶的控制障碍函数普遍有效。快速计算性能的优势以及安全性保证通过数值结果进行了分析与验证。