Minimum jointly structural input and output selection

Large-scale dynamical systems are becoming pervasive today and the costs associated with maintenance and processing should be kept to the minimum. In multi-agent systems, whereas onboard capabilities often allow local communication, some agents need to be equipped with more expensive communication and computation devices that enable proper regulation and monitoring of the system. As such, there is an implicit combinatorial problem associated with deciding the smallest number of agents (or, more generically, state variables) that require simultaneous actuation and sensing. Due to the potential freedom or uncertainty of the weights in such a multi-agent system, we rely on structural systems theory and, in this paper, we address the problem of determining the minimum number of state variables that need to be simultaneously actuated and measured to ensure structural controllability and observability, respectively. We notice that a possible ‘separation principle’ to first find the solution to either problem and then putting them together does not necessarily hold an optimal solution. Remarkably, we show that despite the combinatorial nature of the problem, it is possible to design a solution with polynomial computational complexity, which contrasts with the non-structural version of the problem (i.e., a minimum number of a simultaneous actuator and sensor placement to guarantee controllability and observability, respectively) that it is NP-hard.