A =  0.1567 0.7600 ,

and an m-dimensional input). Moreover θ is a vector of

parameters that correspond to the unknown values of physical coefficients, material constants, and the like.

. −0.0136 .

B = −0.1191

, CT

. 199.8214 .

= −7.0361 (8)

Let η(t) be the measurements that would be obtained with ideal, noise-free sensors:

η(t) = Hx(t)

Using p as the differential operator, the above state repre- sentation can be written   as

[pI − F (θ)]x(t) = tt(θ)u(t)

Which means that the transfer function from u to η is

η(t) = ttc(p, θ)u(t)

−1

As have been discussed earlier, there are a variety of parametric model structures available to assist in modeling a system. The choice of model structure is based upon an understanding of the system identification method and insight and understanding into the system undergoing identification. The characteristics of both system and disturbance dynamics play a role is the proper model selection. These system identification methods can handle a wide range of system dynamics without knowledge of the actual system physics, thereby reducing the engineering effort required to develop models. With respect to the complexity of the model, the fitness levels and the residual analysis it is concluded that the ARMAX model suits the

ttc(p, θ) = H[pI − F (θ)]

tt(θ)

given system best for the data history provided.

Let the measurements be sampled at the time  instants t = kT , k = 1, 2, . . . and the disturbance effects at those time instants be vT (kT ). Hence the measured output is

y(kT ) = ttc(p, θ)u(t) + vT (kT )

There are several ways of transporting ttc(p, θ) to a representation that is explicitly discrete time. Suppose that the input is constant over the sampling interval  T

Next, we consider a particular class of nonlinear identifi- cation.

5.  NONLINEAR ARMAX MODEL

The nonlinear ARMAX model [10] can be represented   by

y(k) = Fl[y(k − 1), . . . , y(k − ny ), u(k − τd), . . .

u(k − τd − nu), e(k), e(k − 1), . . . , e(k − ne)] (9)

u(t) = uk = u(kT ),  kT ≤ t < (k + 1)T

where n , n

and n

are the maximum lags considered  for

y u e

Then Eq. (7) can be solved from t = kT to t = kT + T ,

yielding

x(kT + T ) = AT (θ)x(kT ) + BT (θ)u(kT )

where

A  (θ) = eF (θ)T

T

¸ (θ)τ

output y(k), input u(k), and noise e(k), respectively, τd is the delay and F is a nonlinear function with nonlinearity degree l ∈ Z+.

The deterministic part of a polynomial NARMAX model, (that is a NARX model) can be expanded as the sum- mation of terms with degrees of nonlinearity in the range 0 ≤ m ≤ ℓ. Each mth order term can contain a pth  order

factor in y(k − ni) and a (m− p)th order factor in u(k − ni)

BT (θ) =

eF tt(θ)dτ

and  is  multiplied  by  a  coefficient c

(n , . . . , n

) as

τ =0

Fig. 5 displays the state-space modeled data versus    the

follows:

ℓ