Page 727 - The Mechatronics Handbook

0066_Frame_C23 Page 35 Wednesday, January 9, 2002 1:55 PM

Stochastic models including the A polynomial, according to Eqs. (23.69) and (23.70), are known as
autoregressive (AR) models and models including the C polynomial are known as moving-average (MA)
models, whereas the B polynomial determines the effects of the external input (X). Notice that the term
moving average is here somewhat misleading, as there is no restriction that the coefficients should add
to 1 or that the coefficients are nonnegative. An alternative description is finite impulse response or all-
zero filter.
Thus, the full model of Eq. (23.69) is an autoregressive moving average model with external input
−1
−1
(ARMAX) and its pulse transfer function H(z) = B(z )/A(z ) is stable if and only if the poles—that is,
−1
the complex numbers z 1 ,…, z n solving the equation A(z ) = 0—are strictly inside the unit circle, that is,
|z i | < 1. The zeros of the system—that is, the complex numbers z 1 ,…, z n solving the equation B(z ) =
−1
0—may take on any value without any instability arising, although it is preferable to obtain zeros located
strictly inside the unit circle, that is, |z i | < 1 (minimum-phase zeros). By linearity {y k } can be separated
into one purely deterministic process {x k } and one purely stochastic process {v k }:

 Az )Xz() = Bz )Uz()  y k = x k +
1
–
(
1
–
(
  v k
 and  (23.71)
 – 1 – 1 
(
(
 Az )Vz() = Cz )Wz()  Yz() = Xz() + Vz()
The type of decomposition (Eq. (23.71)) that separates the deterministic and stochastic processes is
known as the Wold decomposition.
Prediction and Reconstruction
Consider the problem of predicting the output d steps ahead when the output {y k } is generated by the
ARMA model,

1
–
Az ( – 1 )Yz() = Cz )Wz() (23.72)
(
2
which is driven by a zero-mean white noise {w k } with covariance {w i w j } = s w d ij . In other words,
assuming that observations {y k } are available up to the present time, how should the output d steps ahead
−1
−1
be predicted optimally? Assume that the polynomials A(z ) and C(z ) are mutually prime with no zeros
for |z| ≥ 1. Let the C polynomial be expanded according to the Diophantine equation,
(
(
Cz ( – 1 ) = Az )Fz ) + z Gz ) (23.73)
(
–
1
–
1
1
d
–
–
which is solved by the two polynomials
Fz ( – 1 ) = 1 + f 1 z + … + f nF z – nF , n F = d 1
1
–
–
(23.74)
(
Gz ( – 1 ) = g 0 + g 1 z + … + g nG z – nG , n G = max n A – 1, n C – d)
–
1
−1
Interpretation of z as a backward shift operator and application of Eqs. (23.72) and (23.73) permit
the formulation
1
(
–
(
y k+d = Fz )w k+d + Gz ) (23.75)
–
1
--------------- y k
(
Cz )
–
1
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