**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} ⊥ *X*_{0}.

By iteration ...