Nested performance bounds and approximate solutions for the sensor placement problem

Muhammad Sharif Uddin, Anthony Kuh, Aleksandar Kavcic, Toshihisa Tanaka
2014 APSIPA Transactions on Signal and Information Processing  
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
more » ... d 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. This paperdevelops a number of tools and algorithms to analyze the sensor placement problem. The problem is formulated as an optimization problem where a limited number of sensors is placed to maximize an efficacy cost criterion. The solution involves solving an integer programming problem that becomes infeasible when the number of sensors and locations becomes large. The paper makes two key contributions; development of set of approximation algorithms (greedy algorithms and variations using the efficacy cost and also a projection-based method) that run in polynomial time in the number of parameters and derives a set of analytical nested performance upper bounds to the optimal solution based on the structure of the data correlation matrix. Simulations are conducted on a number of experiments from random correlation matrices, to a simulated sensor network, to an IEEE 57-bus test system [1] that show how tight the nested performance upper bounds are
doi:10.1017/atsip.2014.3 fatcat:qhomcc23a5aj7h7e63focby7b4