The ARX Model

The ARX model is a linear difference equation that relates the input u(t) to the output y(t) as follows:

y(t) + a_1 y(t-1) + ... + a_na y(t-na) = b_1 u(t-nk) + ... + b_nb u(t-nk-nb+1)

The structure is thus entirely defined by the three integers na, nb, and nk. na is the number of poles, nb+1 is the number of zeros, and nk is the pure time delay (the dead time) in the system. For a sampled data control system, typically nk=1 if there is no dead time.

For multi-input systems nb and nk are row vectors, where the i'th element gives the order/delay associated with the i'th input.

To estimate an ARX model, open Parametric Models from the pop-up menu Estimate in the ident window. The orders na, nb, and nk can either be directly entered into the edit box in the Parametric Models dialog, or selected using the pop-up menus in the Order Editor.

There are two methods to estimate the coefficients a and b: Least Squares (ARX) and Instrumental Variabels (IV),

Note that there is a quite useful possibility to estimate many ARX models simultaneously. Push the Order selection button, and then the Estimate button.

Help topics.

Estimating Many ARX Models Simultaneously When ARX has been selected, there is a possibility to compute many models simultaneously. Just enter each order as a vector using the : operator, and models for all combinations of orders will be computed.

Pushing the button Order selection fills out the Orders field with a default choice of orders.

In the multi-input case, there is an option to specify one order vector for all inputs, like [1:4 1:5 0:3]. Then the input orders (nb) will be the same for all inputs.

When the multi-model option has been chosen, a special dialog window will open and the fit to validation data for the different models will be displayed. You can then select which models to insert into the model board by clicking in this plot.

Help topics.

Least Squares and Instrumental Variable Methods There are two methods to estimate the coefficients a and b: Least Squares (LS) and Instrumental Variabels (IV).

Least Squares:
  Minimizes the sum of squares of the RHS minus the LHS of the expression above, with respect to a and b. This is obtained by selecting the ARX radio box.
Instrumental Variables:
  Determines a and b so that the error (RHS-LHS) becomes uncorrelated with certain linear combinations of the inputs. This is obtained by selecting the IV radio box.

The methods are described in more detail in the manual under ARX and IV4.

Help topics.

(file iduiarx.htm)