Distributed parameter system identification using spatial filtering and Karhunen-Loeve modes

This paper presents a procedure for empirically generating a reduced order model of a distributed parameter system. Two problems arise in generating this model: state estimation and system identification. These problems are coupled by the fact that the optimal state estimator, the Kalman filter, requires a model of the system being estimated. In general, state estimation for a distributed parameter system can be performed by filtering in time, "space," or both. Spatial filtering is a suboptimal state estimation technique with the advantage that it decouples the state estimation and system identification problems. When the number of sensors is large, the procedure yields good estimates. The equations for the optimal spatial filter are presented along with the least squares solution of the system identification problem. Together, they generate a reduced order model of the distributed parameter system. This reduced order model is defined by the reduced order state which in turn depends on the choice of a finite dimensional basis. Choosing this basis to consist of terms in the Karhunen-Loeve expansion results in simplifications to the equations which are presented. The statistics required to generate the spatial filter are also defined in terms of the Karhunen-Loeve expansion. A recursive formulation of the entire process is presented.