Complexity of constrained sensor placement problems for optimal observability

This article studies two problems related to observability and efficient constrained sensor placement in linear time-invariant discrete-time systems with partial state observations: (i) We impose the condition that both the set of outputs and the state that each output can measure are pre-specified. We establish that for any fixed k>2, the problem of placing the minimum number of sensors/outputs required to ensure that the structural observability index is at most k, is NP-complete. Conversely, we identify a subclass of systems whose structures are directed trees with self-loops at every state vertex, for which the problem can be solved in linear time. (ii) Assuming that the set of states that each given output can measure is given, we prove that the problem of selecting a pre-assigned number of outputs in order to maximize the number of states of the system that are structurally observable (i.e., to maximize the size of the observable subgraph) is also NP-hard. As an application, we identify suitable conditions on the system structure under which there exists an efficient greedy strategy, which we provide, to obtain a (1−1e)-approximate solution. An illustration of the techniques developed for this problem is given on the benchmark IEEE 118-bus power network containing roughly 400 states in its linearized model.