This paper addresses synthesizing receding-horizon controllers for nonlinear, control-affine dynamical systems under multiple incompatible hard and soft constraints. Handling incompatibility of constraints has mostly been addressed in literature by relaxing the soft constraints via slack variables. However, this may lead to trajectories that are far from the optimal solution and may compromise satisfaction of the hard constraints over time. In that regard, permanently dropping incompatible soft constraints may be beneficial for the satisfaction over time of the hard constraints (under the assumption that hard constraints are compatible with each other at initial time). To this end, motivated by approximate methods on the maximal feasible subset (maxFS) selection problem, we propose heuristics that depend on the Lagrange multipliers of the constraints. The main observation for using heuristics based on the Lagrange multipliers instead of slack variables (which is the standard approach in the related literature of finding maxFS) is that when the optimization is feasible, the Lagrange multiplier of a given constraint is non-zero, in contrast to the slack variable which is zero. This observation is particularly useful in the case of a dynamical nonlinear system where its control input is computed recursively as the optimization of a cost functional subject to the system dynamics and constraints, in the sense that the Lagrange multipliers of the constraints over a prediction horizon can indicate the constraints to be dropped so that the resulting constraints are compatible. The method is evaluated empirically in a case study with a robot navigating under multiple time and state constraints, and compared to a greedy method based on the Lagrange multiplier.

翻译：本文研究非线性、控制仿射动力系统在多类不相容硬约束与软约束条件下的滚动时域控制器综合问题。现有文献主要通过引入松弛变量放松软约束来处理约束不相容性，但这种方法可能使轨迹偏离最优解，并随时间推移损害硬约束的满足性。基于此，永久性剔除不相容软约束可能更有利于长期维持硬约束的满足性（假设初始时刻硬约束之间相互相容）。受最大可行子集（maxFS）选择问题近似方法的启发，本文提出依赖约束拉格朗日乘子的启发式算法。选用拉格朗日乘子而非松弛变量（相关maxFS求解文献中的标准方法）的核心依据在于：当优化问题可行时，约束的拉格朗日乘子非零，而松弛变量为零。这一特性对非线性动力系统尤为关键——其控制输入通过递归求解受系统动力学与约束约束的代价泛函优化问题获得，预测时域内约束的拉格朗日乘子可指示需要剔除的约束，使剩余约束达到相容状态。通过多时间约束与状态约束下的机器人导航案例进行实证评估，并与基于拉格朗日乘子的贪婪方法进行对比。