Preferential Estimation via Tuning of the Kalman Filter

Abstract

Estimation problems have been traditionally formulated so as to minimize the estimation error of the full state vector. However, in applications that involve the tracking of only a few unmeasured variables, it is sufficient to limit the attention along certain directions in state space. This way, it is hoped that better accuracy can be obtained along the desired directions, possibly at the cost of poorer estimates along the other directions. This problem, termed preferential estimation, is formally formulated in this paper as a least-squares minimization problem. Using calibration measurements of the preferred variables, the above mentioned problem is solved numerically via tuning of the Kalman filter. The approach is illustrated in simulation on the optimization of a penicillin fermentation process, where preferential estimation is used successfully to reduce the error in tracking a single unmeasured variable, the substrate concentration.

Preferential Estimation for Uncertain Linear Systems at Steady State: Application to Filamentous Fungal Fermentation

“…The concept of preferential estimation, as introduced in [9], consists of estimating certain linear combinations of the states more accurately than others. The concepts therein are summarized next.…”

“…Hence, the same also holds for L. Preferential estimation is formulated as the minimization of the mean-squared estimation error J of the preferred variables z [9]:…”

“…Note also that, if the projection matrix L = I n×n is chosen, Problem (3) corresponds to minimizing the error inx, i.e. the standard full-state estimation (SE) defined in [9] as the minimization of the mean-squared error inx.…”

“…However, in the presence of model uncertainty, the estimated variables are typically biased, and the Kalman filter is no longer optimal. It was shown that, in this case, it is possible to reduce the error in selected variables [9]. The bias caused by model-plant mismatch opens the way to compromises between bias and variance on the one hand, and bias in the preferred variables and bias in the other variables on the other.…”

“…The bias-variance and bias-bias tradeoffs are studied qualitatively in [9]. The objective of the present paper is to provide a more comprehensive analysis of these tradeoffs for the case of linear systems operating at steady state.…”