Approach based on the concept of model error

Abstract

There are many real life situations where accurate identification of nonlinear terms (parameters) in the model of a dynamic system is required. In principle as well as in practice, the parameter estimation methods can be applied to nonlinear problems. We recall here that the estimation before modelling approach uses two steps in the estimation procedure and the extended Kalman filter can be used for joint state/parameter estimation. As such, the Kalman filter cannot determine the deficiency or discrepancy in the model of the system used in the filter, since it pre-supposes availability of an accurate state-space model. Assume a situation where we are given the measurements from a nonlinear dynamic system and we want to determine the state estimates. In this case, we use the extended Kalman filter and we need to have the knowledge of the nonlinear function f and h. Any discrepancy in the model will cause model errors that will tend to create a mismatch of the estimated states with the true state of the system. In the Kalman filter, this is usually handled or circumvented by including the process noise term Q. This artifice would normally work well, but it still could have some problems: i) deviation from the Gaussian assumption might degrade the performance of the algorithm; and ii) the filtering algorithm is dependent on the covariance matrix P of the state estimation error, since this is used for computation of Kalman gain K. Since the process noise is added to this directly, as GQGT term, one would have some doubt on the accuracy of this approach. In fact, the inclusion of the 'process noise' term in the filter does not improve the model, since the model could be deficient, although the trick can get a good match of the states. Estimates would be more dependent on the current measurements. This approach will work if the measurements are dense in time, i.e., high frequency of measurements, and are accurate.