Using horizon estimation and nonlinear optimization for grey-box identification

• (Moving) horizon estimation used for state estimation may be used also for grey-box identification.
• If there is process noise horizon estimation will lead to biased estimates even in the linear case.
• In the linear case, bias can be avoided by adding a Riccati equation based term to the objective function.
• If the process noise enters linearly, and not in the nonlinear states, the bias can still be eliminated.
• In the general nonlinear case the bias correction term is approximated in an EKF similar fashion.

An established method for grey-box identification is to use maximum-likelihood estimation for the nonlinear case implemented via extended Kalman filtering. In applications of (nonlinear) model predictive control a more and more common approach for the state estimation is to use moving horizon estimation, which employs (nonlinear) optimization directly on a model for a whole batch of data. This paper shows that, in the linear case, horizon estimation may also be used for joint parameter estimation and state estimation, as long as a bias correction based on the Kalman filter is included. For the nonlinear case two special cases are presented where the bias correction can be determined without approximation. A procedure how to approximate the bias correction for general nonlinear systems is also outlined.