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 − .