LPV system identification with unknown scheduling variable

A Linear Parameter Varying (LPV) system is a system whose parameters depend on an exogenous variable, the so-called scheduling variable. LPV systems are generally controlled using LPV controllers i.e., controller whose parameters also depend on the scheduling variable. In order to be able to use an LPV controller for an LPV system, it is generally assumed that the scheduling variable of the LPV system is either fully known or measurable. Here, we consider the case where the scheduling variable has to be identified based on data collected on the system. We do that in the case of LPV systems that are successively operated at different constant values of the scheduling variable. We show how to identify the maximum likelihood estimate of the unknown constant scheduling variable and we derive the statistical properties of this estimate. Moreover, we also develop an optimal experimental design framework in order to optimally design the experimental conditions of the identification experiment yielding the estimate of the scheduling variable.