Page 187 - Dynamics and Control of Nuclear Reactors

184 CHAPTER 13 Boiling water reactors

n
X
ð
xtðÞ ¼ a i xt iΔtÞ + vtðÞ (13.5)
i¼1
where
x(t) is a stationary random signal (measured by neutron detectors).
v(t) is a model prediction error.
{a i , i¼1, 2, …, n} is a set of model parameters.
Δt¼data sampling interval (sec).
n¼model order.
The model parameters {a i ,i¼1, 2, …, n}and the AR model order n are estimated using
the measurements such that the model prediction error is minimized. The least-squares
approach uses the given sampled measurements {x(1), x(2), …,x(N)},where Nis
the total data points. More general forms of the AR model, called an auto-regression
moving average (ARMA) model, are used in some applications. See Ref. [11] for
details.
An example illustrates the evaluation of a time series model. Consider the form
of Eq. (13.5) for a second-order fit. The analysis begins with the third measured
value of x(t).

x 3ðÞ ¼ a 1 x 2ðÞ + a 2 x 1ðÞ + v 3ðÞ
x 4ðÞ ¼ a 1 x 3ðÞ + a 2 x 2ðÞ + v 4ðÞ
x 5ðÞ ¼ a 1 x 4ðÞ + a 2 x 3ðÞ + v 5ðÞ (13.6)
⋮
ð
xNðÞ ¼ a 1 xN 1ð Þ + a 2 xN 2Þ + vNðÞ
Note that a 1 and a 2 are unknown parameters whose values are sought by using all the
available measurements {x(1), x(2), …, x(N)}. An efficient approach for estimating
the parameters is to minimize the error between the left hand side and the right hand
side (also called the model prediction error by minimizing a squared error function
shown below, with respect to (a 1 ,a 2 ):
N
X
2
ð
ð
ð
Min J ¼ ð xkðÞ a 1 xk 1Þ a 2 xk 2ÞÞ a1, a2Þ (13.7)
k¼3
The two parameters are estimated by solving the two equations obtained from
∂J ∂J
¼ 0 and ¼ 0 (13.8)
∂a 1 ∂a 2
The two equations are then simplified by collecting the terms multiplying a 1 and a 2
and solving for the 2-dimensional vector (a 1 ,a 2 ) to give the following solution:
N N N
2 3 1 2 3
X X X
2
ð
ð
ð
xk 1Þ xk 1Þxk 2Þ xkðÞxk 1Þ
ð
6 7 6 7
a 1 6 k¼3 k¼3 7 6 k¼3 7
¼ 6 7 6 7 (13.9)
6 N N 7 6 N 7
a 2 X X X
4 2 5 4 5
ð
ð
xk 1Þxk 2Þ xk 2Þ xkðÞxk 2Þ
ð
ð
k¼3 k¼3 k¼3