System Identification Toolbox

Polynomial Representation of Transfer Functions

Rather than specifying the functions G and H in (3-10) in terms of functions of the frequency variable , you can describe them as rational functions of and specify the numerator and denominator coefficients in some way.

A commonly used parametric model is the ARX model that corresponds to

(3-12)

where B and A are polynomials in the delay operator :

(3-13)

Here, the numbers na and nb are the orders of the respective polynomials. The number nk is the number of delays from input to output. The model is usually written

(3-14)

or explicitly

(3-15)

Note that (3-14) - (3-15) apply also to the multivariable case, with ny output channels and nu input channels. Then and the coefficients become ny-by-ny matrices, and the coefficients become ny-by-nu matrices.

Another very common, and more general, model structure is the ARMAX structure

(3-16)

Here, and are as in (3-13), while

An Output-Error (OE) structure is obtained as

(3-17)

with

The so-called Box-Jenkins (BJ) model structure is given by

(3-18)

with

All these models are special cases of the general parametric model structure.

(3-19)

The variance of the white noise is assumed to be .

Within the structure of (3-19), virtually all of the usual linear black-box model structures are obtained as special cases. The ARX structure is obviously obtained for . The ARMAX structure corresponds to . The ARARX structure (or the "generalized least squares model") is obtained for , while the ARARMAX structure (or "extended matrix model") corresponds to . The Output-Error model is obtained with , while the Box-Jenkins model corresponds to . (See Section 4.2 in Ljung (1999) for a detailed discussion.)

The same type of models can be defined for systems with an arbitrary number of inputs. They have the form

(3-20)

The System Identification Problem

State-Space Representation of Transfer Functions