In real-world problems, uncertainties (e.g., errors in the measurement, precision errors) often lead to poor performance of numerical algorithms when not explicitly taken into account. This is also the case for control problems, where optimal solutions can degrade in quality or even become infeasible. Thus, there is the need to design methods that can handle uncertainty. In this work, we consider nonlinear multi-objective optimal control problems with uncertainty on the initial conditions, and in particular their incorporation into a feedback loop via model predictive control (MPC). In multi-objective optimal control, an optimal compromise between multiple conflicting criteria has to be found. For such problems, not much has been reported in terms of uncertainties. To address this problem class, we design an offline/online framework to compute an approximation of efficient control strategies. This approach is closely related to explicit MPC for nonlinear systems, where the potentially expensive optimization problem is solved in an offline phase in order to enable fast solutions in the online phase. In order to reduce the numerical cost of the offline phase, we exploit symmetries in the control problems. Furthermore, in order to ensure optimality of the solutions, we include an additional online optimization step, which is considerably cheaper than the original multi-objective optimization problem. We test our framework on a car maneuvering problem where safety and speed are the objectives. The multi-objective framework allows for online adaptations of the desired objective. Alternatively, an automatic scalarizing procedure yields very efficient feedback controls. Our results show that the method is capable of designing driving strategies that deal better with uncertainties in the initial conditions, which translates into potentially safer and faster driving strategies.

中文翻译：

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不确定性下非线性系统的显式多目标模型预测控制

在实际问题中，不确定性（例如，测量误差、精度误差）如果没有明确考虑，通常会导致数值算法的性能不佳。控制问题也是如此，其中最佳解决方案可能会降低质量甚至变得不可行。因此，需要设计可以处理不确定性的方法。在这项工作中，我们考虑了在初始条件下具有不确定性的非线性多目标最优控制问题，特别是它们通过模型预测控制 (MPC) 并入反馈回路。在多目标最优控制中，必须找到多个相互冲突的准则之间的最优折衷。对于此类问题，关于不确定性的报道并不多。为了解决这个问题类，我们设计了一个离线/在线框架来计算有效控制策略的近似值。这种方法与非线性系统的显式 MPC 密切相关，其中潜在昂贵的优化问题在离线阶段解决，以便在在线阶段实现快速解决方案。为了降低离线阶段的数值成本，我们利用控制问题中的对称性。此外，为了确保解决方案的最优性，我们包括一个额外的在线优化步骤，这比原始多目标优化问题便宜得多。我们在以安全和速度为目标的汽车操纵问题上测试我们的框架。多目标框架允许对所需目标进行在线调整。或者，自动标定程序产生非常有效的反馈控制。我们的结果表明，该方法能够设计出更好地处理初始条件下不确定性的驾驶策略，从而转化为潜在的更安全和更快的驾驶策略。