Virtual sensors for linear dynamic systems: structure and identification

Consider a linear system with input u and outputs y and z. Assume that u(t) and y(t) are measured for all times t and that z(t) is measured only for t<=tmax, but it is of interest to know z(t) for t>tmax. Such a situation may arise when the sensor measuring z fails and it is important to recover this variable, e.g. for feedback control. Another case arises when the sensor measuring z is too complex and costly to be used, except for an initial set of experiments. Assuming that z is observable from the couple (u,y), the standard approach consist of a two-step procedure: identify a model first, then design an observer/Kalman filter based on the identified model. Noticing that an estimator of z(t) is a system with (u(t),y(t)) as input that gives an estimate of z(t) as output, the problem of directly identifying an estimator model from the available noisy data in the time interval (0,tmax) is investigated. When stochastic noise is considered, the direct procedure can be carried out using standard techniques and performs better or like the two-step approach. In the case of Set Membership noise, a procedure for the identification of direct virtual sensors is presented. An example related to the vertical dynamics of vehicles with controlled suspension shows the effectiveness of the presented approaches.