Efficient constrained sensor placement for observability of linear systems

TL;DR

This paper investigates the computational complexity of sensor placement for ensuring observability in linear systems, providing NP-hardness results, identifying tractable subclasses, and proposing approximation strategies with practical applications.

Contribution

It establishes NP-completeness for sensor placement problems under certain constraints and offers efficient solutions for specific system structures, along with approximation algorithms.

Findings

Sensor placement for observability is NP-complete in general.

Certain directed tree structures allow linear-time solutions.

A greedy algorithm achieves a (1 - 1/e)-approximate solution for maximizing observable states.

Abstract

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 sensors in order to maximize the number of states of the system that are structurally observable 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 \((1-\frac{1}{e})\)-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.