Log-concave Observers
Toivo Henningsson, Karl Johan Åström · Lund University Publications (Lund University) · 2006
The Kalman filter is the optimal state observer in the case of linear dynamics and Gaussian noise. In this paper, the observer problem is studied when process noise and measurements are generalized from Gaussian to log-concave. This generalization is of interest for example in the case where observations only give information that the signal is in a given range. It turns out that the optimal observer preserves log-concavity. The concept of strong log-concavity is introduced and two new theorems are derived to compute upper bounds on optimal observer covariance in the log-concave case. The theory is applied to a system with threshold based measurements, which are log-concave but far from Gaussian.