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Input-output polynomial models, including ARX, ARMAX, output-error, and Box-Jenkins
model structures

A polynomial model uses a generalized notion of transfer functions to express the
relationship between the input, *u*(*t*), the output
*y*(*t*), and the noise
*e*(*t*) using an equation of the form:

$$A(q)y(t)=\frac{B(q)}{F(q)}u(t-nk)+\frac{C(q)}{D(q)}e(t)$$

*A*(*q*), *B*(*q*),
*F*(*q*), *C*(*q*) and
*D*(*q*) are polynomial matrices in terms of the
time-shift operator *q ^{-1}*.

`nk`

is the
input delay. Each polynomial has an independent *order*, or number of estimable
coefficients. For example, if *A*(*q*) has an order of 2,
then the*A* polynomial has the form
*A*(*q*) = 1 +
*a _{1}*

In practice, not all the polynomials are simultaneously active. Simpler polynomial forms, such as ARX, ARMAX, Output-Error, and Box-Jenkins provide model structures suitable for specific objectives such as handling nonstationary disturbances or providing completely independent parameterization for dynamics and noise. For more information about these model types, see What Are Polynomial Models?

System Identification | Identify models of dynamic systems from measured data |

Polynomial model structures including ARX, ARMAX, output-error, and Box-Jenkins.

**Data Supported by Polynomial Models**

Use time-domain and frequency-domain data to estimate discrete-time and continuous-time models.

**Preliminary Step – Estimating Model Orders and Input Delays**

To estimate polynomial models, you must provide input delays and model orders.

**Estimate Polynomial Models in the App**

Import data into the app, specify model orders, delays and estimation options.

**Estimate Polynomial Models at the Command Line**

Specify model orders, delays, and estimation options.

**Polynomial Sizes and Orders of Multi-Output Polynomial Models**

Size of *A*, *B*, *C*, *D*,
and *F* polynomials for multi-output models.

This example shows how to estimate a linear, polynomial model
with an ARMAX structure for a three-input and single-output (MISO)
system using the iterative estimation method `armax`

.

**Specifying Initial States for Iterative Estimation Algorithms**

When you use the `pem`

or `polyest`

functions to estimate ARMAX, Box-Jenkins (BJ), Output-Error (OE), you
must specify how the algorithm treats initial conditions.

**Polynomial Model Estimation Algorithms**

Choose between the ARX and IV algorithms for ARX and AR model estimation.