Page 80 - Neural Network Modeling and Identification of Dynamical Systems

68 2. DYNAMIC NEURAL NETWORKS: STRUCTURES AND TRAINING METHODS

The EKF algorithm is initialized as follows. In order to use the Kalman filter, it is required
For k = 0,set to linearize the observation equation. It is possi-
ble to use statistical linearization, i.e., lineariza-
ˆ z(t 0 ) = E[z(t 0 )], tion with respect to the mathematical expecta-
T tion. This gives
P(t 0 ) = E[(z(t 0 ) − E[z(t 0 )])(z(t 0 ) − E[z(t 0 )]) ].
Then, for k = 1,2,..., the following values are w(t k+1 ) = w(t k ) + ζ(t k ),
calculated: ˆ y(t k ) = H(t k )w(t k ) + η(t k ),
• state estimation where the observation matrix has the form

−
ˆ z (t k ) = f(t k , ˆz(t k−1 )); ∂ ˆy ∂e(t k )
H(t k ) = =− =−J(t k ).
∂w T w=w(t k ) ∂w(t k ) T
• estimation of the error covariance z=z(t k )
− − − T Here e(t k ) is the observation error vector at the
P (t k ) = F (t k )P(t k−1 )F (t k ) + Q(t k−1 );
kth estimation step, i.e.,
• the gain matrix
e(t k ) =˜y(t k ) −ˆy(t k ) =˜y(t k ) − f(z(t k ),w(t k )).
−
G(t k ) = P (t k )H(t k ) T
The extended Kalman filter equations for es-
−1
T
−
× H(t k )P (t k )H(t k ) + R(t k ) ; timating w(t k+1 ) in the next step have the form
T
S(t k ) = H(t k )P(t k )H(t k ) + R(t k ),
• correction of the state estimation
T
K(t k ) = P(t k )H(t k ) S(t k ) −1 ,

− −
ˆ z(t k ) =ˆz (t k ) + G(t k ) ˜y(t k ) − h(t k , ˆz (t k )) ; β
P(t k+1 ) = (P(t k ) − K(t k )H(t k )P(t k ))e + Q(t k ),
• correction of the error covariance estimation w(t k+1 ) = w(t k ) + K(t k )e(t k )
−
P(t k ) = (I − G(t k )H(t k ))P (t k ). Here β is the forgetting factor, which affects the
significance of the previous steps.
We will assume that for an ideal ANN model Here K(t k ) is the Kalman gain, S(t k ) is the co-
the observed process is stationary, that is, variance matrix for state prediction errors e(t k ),
w(t k+1 ) = w(t k ), but its states (weight w(t k ))are
and P(t k ) is the covariance matrix for weight es-
“corrupted” by the noise ζ(t k ).
timation errors ( ˆ w(t k ) − w(t k )).
The Kalman filter (KF) in its standard ver-
There are alternative variants of the EKF al-
sion is applicable only for systems whose obser- gorithm, which may prove to be more effective
vations are linear in the estimated parameters, in solving the problems under consideration, in
while the neural network observation equation particular,
is nonlinear, i.e.,
−
P (t k ) = P(t k ) + Q(t k ),
w(t k+1 ) = w(t k ) + ζ(t k ), T
−
S(t k ) = H(t k )P (t k )H(t k ) + R(t k ),
ˆ y(t k ) = f(u(t k ),w(t k )) + η(t k ),
T
−
K(t k ) = P (t k )H(t k ) S(t k ) −1 ,
where u(t k ) are control actions, ζ is the pro-
−
P(t k+1 ) = (I − K(t k )H(t k ))P (t k )
cess noise, and η is the observation noise; these T T
noises are Gaussian random sequences with × (I − K(t k )H(t k )) + K(t k )K(t k ) ,
zero mean and covariance matrices Q and R. w(t k+1 ) = w(t k ) + K(t k )e(t k ).