Traditional Problem Definition - Computational Intelligence in Time Series Forecasting - page 70

is introduced in which the polynomials P and Q are uniquely determined given the
polynomial A and number of prediction steps d . Now, multiplying the above
CARIMA model (2.22) by

d P ' and using the Diophantine equation, the

d

predicted value will be

ytd PButd ( ' ) ( ) Qyt P td ( ) . [ ( ) (2.23)

d d d

Next, to determine the predictive control law, the future set-points w ( t+d ), d =
1, 2, … should be given, or it is supposed that they have a constant value w . The
control objectives would then be to find the control law that will drive the system
output y ( t+d ) as close as possible to the set points w ( t+d ). This value is obtained
by minimizing the cost function

-

n n

½

2 2

Jn n ( , ) ( [ yt d ( ) wt d ( )]

2

O ( d )[ ' ut d ( )]

2

,

¦ ¦

® ¾ ¿

12

¯

dn d 1

1

which is the expectation value, in which Ȝ ( d ) is a weighting factor of control
sequences, and 12 nn are the minimum and maximum cost horizons. But still, the

solution found in this way is the open-loop feedback-optimal control . To find the
corresponding closed-loop control we will proceed as follows.
The CARIMA Equation (2.23), after ignoring the future noise component
ȟ( t+d ), is written as

yt ( ' d ) G ut ( d 1) Q yt ( )

d d

where,

GPB

d d

with

1 2

Gg gz gz ...

d d 0 d 1 d 2

Writing now the above equation for d = 1, 2, …, n , the set of generated prediction
equations will be

yt ( ' ) G ut ( ) Qyt ( )

1 1

yt ( ' ) G ut ( ) Q yt ( )

2 2

… … ...

yt ( ' n ) G ut ( n 1 ) Q yt ( )

n n