This crate implements a standard linear Kalman filter and smoothing for
vectors of arbitrary dimension. Implementation method relies on rulinalg
library for linear algebra computations. Most inputs and outputs rely
therefore on (derived) constructs from
rulinalg
library, in particular
Vector<f64>
and
Matrix<f64>
structs.
Currently, implementation method assumes that Kalman filter is time-invariant and is based on the equations detailed below. Notations in the below equations correspond to annotations in the source code.
Measurement and state equations:
z_{t} = H_{t} x_{t} + v_{t}
wherev_{t} ~ N(0, R_{t})
x_{t} = F_{t} x_{t-1} + B_{t} u_{t} + w_{t}
wherew_{t} ~ N(0, Q_{t})
Kalman filter equations:
P_{t|t-1} = F_{t} P_{t-1|t-1} F'_{t} + Q_{t}
x_{t|t-1} = F_{t} x_{t-1|t-1} + B_{t} u_{t}
K_{t} = P_{t|t-1} H'_{t} * (H_{t} P_{t|t-1} H'_{t} + R_{t})^{-1}
P_{t|t} = (Id - K_{t} H_{t}) * P_{t|t-1}
x_{t|t} = x_{t|t-1} + K_{t} * (z_{t} - H_{t} x_{t|t-1})
Kalman smoothing equations:
J_{t} = P_{t|t} F'_{t} P_{t+1|t}^{-1}
x_{t|T} = x_{t|t} + J_{t} * (x_{t+1|T} - x_{t+1|t})
P_{t|T} = P_{t|t} - J_{t} * (P_{t+1|T} - P_{t+1|t}) * J'_{t}
Nomenclature:
(x_{t+1|t}, P_{t+1|t})
will be referred to as predicted state variables.(x_{t|t}, P_{t|t})
will be referred to as filtered state variables.(x_{t|T}, P_{t|T})
will be referred to as smoothed state variables.
For now, it is assumed here that B_{t}
matrix is null and that Q_{t}, R_{t}, H_{t}
and F_{t}
matrices are constant over time.