Sensor Network Design via Observability Analysis and Principal Component Analysis

This paper extends a recently developed technique for sensor network design such that interactions between individual sensors are taken into account via principal component analysis. In past work, the trace of the empirical observability gramian was determined to be the most promising measure for determining the location of a single sensor. The extension to placing multiple sensors was performed by interpreting the trace of the gramian as the sum of the diagonal elements and defining interactions between sensors by comparing the magnitude of diagonal entries of empirical observability gramians computed for different sensors. However, the diagonal entries of the gramian only represent the variance of the output measurements for perturbations in a state. The covariances, which are given by the entries of the gramian not on the diagonal, are neglected using such an approach. The presented work remedies this situation, as all the information contained in the empirical observability gramian is considered. Principal component analysis is used to extract the contribution of a sensor placed at a specific location on the overall sensor network. Two approaches are presented for designing sensor networks. The first technique places sensors sequentially, such that each new sensor maximizes the amount of new information that can be gained from the system. This technique is straightforward to implement with the newly developed principal component analysis-based technique for evaluating the system's empirical observability gramian. The second methodology designs the entire sensor network by using genetic algorithms to solve an optimization problem that maximizes observability of the system. The first technique has the advantage that it is easier to implement, while the second method will generally result in a larger amount of information that can be gained about the system. Both techniques are illustrated with a case study representing a distillation column.