Discrete Stochastic Processes and Optimal Filtering

7.3. Kalman filtering

Vector or multivariate approach given:

– X_K : state multivector (n×1)

– x_K : state vector of results

– Y_K : multivector of observations (m×1)

– y_K : vector of observations and results

7.3.1. State equation

with A(K)= state matrix (n×n), deterministic matrix

and N_K = process noise vector (l×1)

that we will choose centered, white and of correlation matrix (covariance matrix in the general case).

:correlation matrix of the process noise vector N_K

: deterministic matrix

7.3.2. Observation equation

with

H(K): matrix of measurements or of observations (m×n), deterministic matrix;

W_K : measurement noise vector of observations vector (p×1) that we choose, like N_K, centered, white and of correlation matrix(covariance matrix in the general case);

correlation matrix of the measurement noise vector W_K

G(K) : (m × p) : deterministic matrix

The noises N_K and W_K are independent, and, as they are centered:

∀K and j

We will suppose, in what follows, that W_K ⊥ X₀.

By iteration ...