1. A limit Kalman filter and smoother for systems with unknown inputs
- Author
-
Grigorios Gakis, Malcolm C. Smith, Gakis, G [0000-0002-0806-0863], Smith, MC [0000-0002-0818-1227], and Apollo - University of Cambridge Repository
- Subjects
road profile estimation ,Control and Systems Engineering ,Kalman filter and smoother ,stability ,unknown input ,Computer Science Applications - Abstract
This paper derives the limit of the Kalman filter and smoother as the inverse of the process noise covariance tends to zero (the zero informational limit) in the case that there is direct feedthrough (of full column rank) of the process noise input to the measurements. The filter in the limit is closely related to the filter proposed by Gillijns and De Moor who used a constrained optimisation problem to estimate an unknown input. We derive a second form of the limit filter which takes the form of a standard Kalman filter without unknown inputs. This form is used to derive necessary and sufficient conditions for convergence and stability of the filter. These consist of a controllability condition and a minimum phase condition. The filter and smoother are applied to an automotive example to estimate an unknown road profile. The example illustrates the usefulness of the stability and convergence conditions to inform the choice of a suitable set of sensors.
- Published
- 2023