Neural Identification of Controlled Dynamical Systems and Recurrent Networks - Neural Networks: Methodology and Applications - page 239

4.1.7 Linear Gaussian Model as an Example of a Continuous-State
Dynamical System with Noise
Engineers are commonly dealing with state noise in continuous state dynami-
cal systems. In that case, probability calculus is more complex and cannot be
solved analytically except for the case of linear models with additive gaussian
noise. We will describe that model because it is frequently used for Kalman
filtering.
Let us consider the linear controlled dynamical system with state equation
x ( k +1)= Ax ( k )+ Bu ( k )+ Cv ( k +1) ,
where ( v ( k )) is a centered reduced white gaussian noise, i.e., a sequence of
random independent identically distributed random vectors which follow a
gaussian distribution with 0 mean and identity covariance matrix.
If x ( k ) is a gaussian vector with expectation equal to m ( k ) and covariance
matrix equal to P ( k ), then, from the stability of gaussian law under linear
transform, x ( k + 1) is a gaussian vector with mean equal to
m ( k +1)= Am ( k )+ Bu ( k )
and with covariance matrix equal to
P ( k +1)= AP ( k ) A T + CC T ,
where A T and C T are transposed matrices A and C .
We recall that if P is the covariance matrix of a random vector x ,that
takes its values in a finite-dimensional vector space E ,andif A is a linear
mapping defined on E , then the covariance matrix of the random vector Ax
is AP A T . (We merge here the notations for linear mapping A and its ma-
trix representation in the reference basis). That result will be crucial for the
computation of the Kalman filter.
The above equation is termed the propagation equation of covariance.
Then we can determine the asymptotic behavior of gaussian stochastic process
( x ( k )) for long times. If matrix A is stable (i.e., if the magnitude of all its
eigenvalues is smaller than 1), the gaussian process converges when times goes
to infinity towards a gaussian law with 0 mean and with covariance matrix
P ∞ which is the solution of the following equation:
P ∞ = AP ∞ A T + CC T .
Conversely, if A is not stable (i.e., if there exists an eigenvalue whose
modulus is larger than or equal to 1) then there does not exist a stationary
regime and the process diverges for long time. The linear model is said to be
unstable.