Optimal fast-rate soft-sensor design for multi-rate processes

Measuring accurate parameters and states at fast rates in some systems may have a significant cost associated with it or it may even be unfeasible. Soft-sensors are a good substitution in these cases. This paper studies the problem of optimal soft-sensor design for multi-rate processes. The main idea is to extend the Kalman filter to the multi-rate case to design a Kalman filter based soft-sensor. The state lifting method is introduced that can be easily used to generalize the minimum variance Kalman filtering method to the multi-rate case for fast-rate estimation. The optimal Kalman gains and covariance matrices are found at fast rate, based on multi-rate input-output data and fast-rate system models. Some examples, especially the one taken from a real mechanical system for air-fuel ratio control, validate the applicability of the proposed method to soft-sensor design in dual-rate and multi-rate processes represented in the state-space form