Optimal state estimation in paper measurement systems

This paper presents a new approach to estimating the cross directional profile (CD), its covariance matrix and the machine direction (MD) scan average based on measurements from a mechanical scanning sensor traversing a moving web. Data sampling in this application is nonuniform. In the uniform sampling case, the exponential matrix is constant and may be computed once at the beginning of execution. However mechanical scanners sends data only while "on-sheet"; i.e., no data is available during head carriage turnaround. This requires computation of the exponential matrix each databox interval requiring in the order of n/sup 2/ calculations where n is the number of databoxes on the sheet. A solution to this computationally intensive problem is provided. It is shown that the exponential matrix can be computed as order n degree of complexity with small error, creating a computationally tractable algorithm for online use. Several new properties of the Kalman formulation as applied to profile estimation are shown. A method to reduce the computational requirement for both the prediction and the estimation portions of the algorithm and an efficient method of competing matrix exponentials for symmetric matrices is shown. Extensive computer analysis of the resulting algorithm shows that the state estimate and the covariance matrix equations can be solved in real-time with current scanner computing systems. Real-time results from a field scanner are also shown.