Sensitivity models for nonlinear filters with application to recursive parameter estimation for nonlinear state-space models

If a recursive prediction error method is applied for the parameter estimation of state-space models, a filter must be used as the predictor, resulting in an adaptive filtering problem. Whereas for linear systems the optimal linear filter, the Kalman filter, can be used, approximate nonlinear filters have to be employed for nonlinear systems. For the parameter estimation, the gradient of the prediction error, and maybe the gradient of its covariance matrix as well, are required. These can be obtained from a sensitivity model; however, the derivation of such a sensitivity model is tedious and has only been done for the extended Kalman filter. Sensitivity models for four common continuous-discrete nonlinear filters are derived: the extended Kalman filter, the first-order bias-corrected filter, the truncated second-order filter, and the modified Gaussian second-order filter. The derivation of these sensitivity models relies heavily on the use of matrix differential calculus, and utilises a reformulation of the approximate nonlinear filters derived earlier. Elementwise derivatives and bookkeeping over indices are thus entirely avoided. An adaptive filter is applied for real-time parameter identification and tracking for a laboratory three-tank process