(484d) Online Estimation for Nonlinear Periodic Systems

There are many issues related to nonlinear systems that hinder the ability to implement a robust, online estimation procedure. The most important factor of these is noise in the parameters and measurements. If the noise in a measurement is accounted for, then an estimation method such as moving horizon estimation can calculate more accurately the model's parameters. With these known parameters, which may change over time, it is then possible to apply a self-sustained model predictive control scheme online.

An unscented Kalman filter is capable of performing an online estimation of the true mean values for the measurements and accounts for all process noise [1]. It is relatively easy to implement, although it assumes Gaussian noise profiles and provides calculated values for the mean and covariance. In non-linear systems, even if the noise distribution for the parameters in the non-linear model is Gaussian, the end effect of this noise will not necessarily be Gaussian. Therefore, it is then advantageous to compare this unscented Kalman filter technique with a more known estimation method, such as Monte Carlo simulations [2].

Various sets of Monte Carlo simulations were carried out on the Lotka-Volterra predator-prey model, and an unscented Kalman filter algorithm was also applied to the same data. Both processes provided a close and accurate representation of the noise-less data. As expected, however, the Monte Carlo simulations, while very time-extensive, were able to more accurately determine the error profiles caused by the introduced Gaussian process noise. The unscented Kalman filter, on the other hand, was able to obtain a very close representation of the noise-less data, and also at a much shorter calculation time. Because of this, the unscented Kalman filter is most likely suited for an online estimation component.

Reference:

[1] E. A. Wan and R. Van der Merwe, ?The unscented Kalman filter for nonlinear estimation,? in Proc. Symp. Adaptive Syst. Signal Process., Commun. Contr., Lake Louise, AB, Canada, Oct. 2000.

[2] Doucet, J. F. G. de Freitas, and N. J. Gordon, ?An introduction to sequential Monte Carlo methods,? in Sequential Monte Carlo Methods in Practice, A. Doucet, J. F. G. de Freitas, and N. J. Gordon, Eds. New York: Springer-Verlag, 2001.