D-optimal spatio-temporal sampling design for identification of distributed parameter systems

Abstract

The aim of this paper is to develop a method for optimal node activation in large-scale sensor networks whose measurements are to be used to estimate unknown parameters of a distributed parameter system. Given a partition of the observation horizon into a finite number of intervals, the problem consists in selecting gaged sites on each interval so that the determinant of the Fisher information matrix associated with the estimated parameters be maximal. A major novelty here is that the numbers of active sensors may differ from interval to interval in accordance with the amount of information about the parameters expected in these intervals. An additional requirement is that, given a spatially-varying cost of taking measurements, the resulting total cost of the experiment must not exceed a fixed budget. The combinatorial nature of the sensor selection problem is circumvented by operating on the spatial density of sensors, rather than on the sensor locations. The original problem then reduces to maximizing the determinant of the sum of finite convex combinations of some nonnegative definite matrices subject to a linear inequality constraint reflecting the limited experimental budget and additional box constraints on the weights of this combination. Optimality conditions are indicated and simplicial decomposition is applied to obtain numerical solutions. As a result, a simple computational scheme is obtained which can be implemented without resorting to sophisticated numerical software.