-
m ny,nu
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p,m−p 1 m
actual data.
y(k) = . . .
cp,m−p(n1, . . . , nm)
4. LINEAR IDENTIFICATION RESULTS
With focus on the time-domain respresentation, the model identified using state space modeling is represented in the form of matrices as follows:
where
m=0 p=0 n1 ,nm p
× Y y(k − ni)
i=1
m
Y
i=p+1
u(k − ni) (10)
Fig. 6. Simulation of (—) the system output pressure and (- -) the polynomial model with ℓ = 3
Fig. 7. Comparison of fitness percentages using various linear and nonlinear models
the superiority of the NARMAX method over the linear identification methods in the parameters estimation of
ny,nu ny
. ≡ .
ny nu
. . . . .
(11)
polynomial models of the prescribed hydraulic pumping system.
n1 ,nm
n1 =1 n2 =1
nm=1
The fitness level of the ARMAX modeled data was found
and the upper limit is ny if the summation refers to factors in y(k − ni) or nu for factors in u(k − ni). Assuming stability, then in steady–state for constant inputs we may
to be the best as shown in Fig. 7 along with the nonlinear model. One possible reason is the influence of disturbance. Unlike the ARX model, the ARMAX model structure
write y¯
= y(k − 1) = y(k − 3) = . . . = y(k − ny ),
includes disturbance dynamics. ARMAX models are useful
u¯ = u(k − 1) = u(k − 2) = . . . = u(k − nu) and (10)
when you have dominating disturbances that enter early in
is rewritten as
l
m ny,nu
the process, such as at the input. The ARMAX model has
more flexibility in the handling of disturbance modeling than the ARX model. The Box-Jenkins (BJ) structure pro-
y¯ = . . .
cp,m−p(n1, . . . , nm)y¯pu¯m (12)
vides a complete model with disturbance properties mod-
m=0 p=0 n1 ,nm
where constants .ny,nu cp,m−p(n1, . . . , nm) are the coeffi-
cients of the term clusters Ωypum−p , which contains terms of the form yp(k − i)um(k − j) for m + p ≤ l. Such coeffi- cients are called cluster coefficients and are represented as
Σypum .
If max[p] = 1 in the dynamical model (10), such a model is closely related to a Hammerstein type [10] and the steady– state output can be expressed as [11]:
l
eled separately from system dynamics. The Box-Jenkins model is useful when you have disturbances that enter late in the process. For example, measurement noise on the output is a disturbance late in the process.
6. CONCLUSION
In this paper, linear and nonlinear identification meth- ods have been closely examined for the purpose of pa- rameters estimation of polynomial models of a 15 kW hydraulic pumping system. The objective has been to
y¯ =
.0 + .u u¯ + . . m
(13)
determine models with good performance in both transient
1 − .