Neural Identification of Controlled Dynamical Systems and Recurrent Networks - Neural Networks: Methodology and Applications - page 263

The residual of the regression of Y ( k +1) onto Y ( k )is

HA X ( k )

Y ( k +1) − − HBu ( k )

X ( k )] + HV ( k +1)+ W ( k +1) .

= HA [ X ( k ) −

It is exactly the innovation term, which was defined in the previous section
in the variational formulation of the filtering problem. From now on, the
innovation will be written

HA X ( k ) ϑ ( k +1)= Y ( k +1) − − HBu ( k ) .

Note that the innovation at time k + 1 in independent from Y ( k ).
The optimal estimate of the state at time k + 1 can be split into the sum
of two terms:

• A prediction term, which depends on the previous available information,

equal to A X ( k )+ Bu ( k );

• A correction term, which depends linearly on the innovation ϑ ( k +1) at

time k + 1 and is equal to

K k +1 ϑ ( k +1)= K k +1 [ Y ( k +1) − HAX ( k ) − H B u ( k )] ,

where K k +1 is called the Kalman gain of the filter at time k +1.
Thus, the filter is recursive and defined by the following formula
X ( k +1)= A X ( k )+ Bu ( k )+ K k +1 ϑ ( k +1) .

That computation shows that the Bayesian estimate is an innovation fil-
ter. The Kalman gain is the matrix coe cient of the linear regression of the
state X ( k +1)attime k + 1 onto the innovation ϑ ( k + 1). That coe cient
is computed from the covariance matrix according to the following formula
(linear regression has been recalled in Chap. 2):

K k +1 =Cov[ X ( k +1) , ϑ ( k + 1)]Var[ ϑ ( k +1)] − 1 .

To compute the Kalman gain, the covariance matrix of the errors must
be computed. The details are given in the appendix of this chapter. We just
state here the results.
If P k stands for the covariance matrix of the estimation error X ( k )

X ( k )
and P k +1 for the covariance matrix of the prediction error X ( k +1) − A X ( k ) −
Bu ( k ), then the Kalman gain is given by the following formula:

−

K k +1 = P k +1 H T [ HP k +1 H T + R ] − 1 ,
where the dynamics of matrices P k and P k +1 are determined by the following
updating equations, the so-called covariance propagation equations:
P k +1 = AP k A T + Q
P k +1 =( I K k +1 H )( AP k A T + Q )( I K k +1 H ) T + K k +1 RK T

− −

k +1 .