The robust minimal controllability and observability problem

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.