Efficient constrained sensor placement for observability of linear systems

This article studies two problems related to observability and efficient sensor placement in a linear time-invariant discrete-time systems with partial state observations. Firstly, we impose the condition that the available set of outputs and the state that each output can measure are pre-specified. Under this assumption, we establish that the problem of determination of the minimum number of sensors/outputs required to ensure a desired bound, that is any fixed integer greater than or equal to three, on the structural observability index is NP-complete. In the converse direction, we identify a subclass of systems for which the problem can be solved in linear time. Secondly, we assume that the set of states that each given output can measure is known. Under this assumption, we prove that the problem of selecting a fixed number of sensors from the given set of sensors in order to maximize the number of states of the system that are structurally observable by them is also NP-hard. We identify suitable conditions on the system structure under which there exists an efficient greedy strategy, which we provide, to obtain an approximate solution. An illustration of the techniques developed for the second problem is given on the benchmark IEEE 118-bus power network containing roughly $400$ states in its linearized model