In this paper, we study the Robust Minimal Controllability and Observability Problem (rMCOP). The scenario that motivated this question is related to the design of a drone formation to execute some task, where the decision of which nodes to equip with a more expensive communication system represents a critical economic choice. Given a linear time-invariant system for each of the vehicles, this problem consists of identifying a minimal subset of state variables to be actuated and measured, ensuring that the overall formation model is both controllable and observable while tolerating a prescribed level of inputs/outputs that can fail. Based on the tools in the available literature, a naive approach would consist of enumerating separately all possible minimal solutions for the controllability and observability parts. Then, iterating over all combinations to find the maximum intersection of sensors/actuators in the independent solutions, yielding a combinatorial problem. The presented solution couples the design of both controllability and observability parts through a polynomial reformulation as a minimum set multi-covering problem under some mild assumptions. In this format, the algorithm has the following interesting attributes: (i) only requires the solution of a single covering problem; 9ii) using polynomial approximations algorithms, one can obtain close-to-optimal solutions to the rMCOP.

中文翻译：

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鲁棒最小可控性和可观察性问题

在本文中，我们研究了鲁棒最小可控性和可观测性问题 (rMCOP)。引发这个问题的场景与无人机编队的设计有关，以执行某些任务，其中决定为哪些节点配备更昂贵的通信系统代表了一个关键的经济选择。给定每辆车的线性时不变系统，这个问题包括确定要驱动和测量的状态变量的最小子集，确保整个编队模型既可控又可观察，同时容忍规定水平的输入/输出那可能会失败。基于现有文献中的工具，一种简单的方法包括分别枚举可控性和可观察性部分的所有可能的最小解决方案。然后，迭代所有组合以找到独立解决方案中传感器/执行器的最大交集，从而产生组合问题。所提出的解决方案通过多项式重新公式将可控性和可观测性部分的设计结合为一些温和假设下的最小集多重覆盖问题。在这种格式中，该算法具有以下有趣的属性：（i）只需要解决单个覆盖问题；9ii) 使用多项式近似算法，可以获得接近最优的 rMCOP 解。所提出的解决方案通过多项式重新公式将可控性和可观测性部分的设计结合为一些温和假设下的最小集多重覆盖问题。在这种格式中，该算法具有以下有趣的属性：（i）只需要解决单个覆盖问题；9ii) 使用多项式近似算法，可以获得接近最优的 rMCOP 解。所提出的解决方案通过多项式重新公式将可控性和可观测性部分的设计结合为一些温和假设下的最小集多重覆盖问题。在这种格式中，该算法具有以下有趣的属性：（i）只需要解决单个覆盖问题；9ii) 使用多项式近似算法，可以获得接近最优的 rMCOP 解。