Nested performance bounds and approximate solutions for the sensor placement problem

This paper considers the placement of m sensors at n > m possible locations. Given noisy observations, knowledge of the state correlation matrix, and a mean-square error criterion (equivalently maximizing an efficacy cost criterion), the problem is formulated as an integer programming problem. Computing the solution for large m and n is infeasible, requiring us to look at approximate algorithms and bounding optimal performance. Approximate algorithms include greedy algorithms and variations based on examining the efficacy cost function and projection-based methods that all run in polynomial time of m and n. A sequence of nested bounds are found that upper bound the optimal performance (with analysis based on using matrix pencils and generalized eigenvectors). Finally, we show through simulations that the approximate algorithms perform well and provide tight implementable lower bounds to optimal performance and the nested bounds provide upper bounds to optimal performance with tighter bounds achieved with increasing complexity. The sensor placement problem has many energy applications where we are often confronted with limited resources. Some examples include where to place environmental sensors for an area in which there are many distributed solar photovoltaic generators and where to place grid monitors on an electrical distribution microgrid.