# Input-Output Polynomial Models

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:

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

A(q), B(q), F(q), C(q) and D(q) are polynomial matrices in terms of the time-shift operator q-1. u(t) is the input, and `nk` is the input delay. y(t) is the output and e(t) is the disturbance signal.

Each polynomial has an independent order, or number of estimable coefficients. For example, if A(q) has an order of 2, then theA polynomial has the form A(q) = 1 + a1q-1 + a2q-2.

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?

## Apps

 System Identification Identify models of dynamic systems from measured data

## Functions

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 `idpoly` Polynomial model with identifiable parameters `arx` Estimate parameters of ARX, ARIX, AR, or ARI model `armax` Estimate parameters of ARMAX, ARIMAX, ARMA, or ARIMA model using time-domain data `bj` Estimate Box-Jenkins polynomial model using time domain data `iv4` ARX model estimation using four-stage instrumental variable method `ivx` ARX model estimation using instrumental variable method with arbitrary instruments `oe` Estimate output-error polynomial model using time-domain or frequency-domain data `polyest` Estimate polynomial model using time- or frequency-domain data `pem` Prediction error minimization for refining linear and nonlinear models
 `arxstruc` Compute loss functions for single-output ARX models `ivstruc` Compute loss functions for sets of ARX model structures using instrumental variable method `selstruc` Select model order for single-output ARX models `struc` Generate model-order combinations for single-output ARX model estimation
 `arxRegul` Determine regularization constants for ARX model estimation `delayest` Estimate time delay (dead time) from data `init` Set or randomize initial parameter values
 `polydata` Access polynomial coefficients and uncertainties of identified model `getpvec` Obtain model parameters and associated uncertainty data `setpvec` Modify values of model parameters `getpar` Obtain attributes such as values and bounds of linear model parameters `setpar` Set attributes such as values and bounds of linear model parameters `setPolyFormat` Specify format for B and F polynomials of multi-input polynomial model
 `armaxOptions` Option set for `armax` `arxOptions` Option set for `arx` `arxRegulOptions` Option set for `arxRegul` `bjOptions` Option set for `bj` `iv4Options` Option set for `iv4` `oeOptions` Option set for `oe` `polyestOptions` Option set for `polyest`

## Topics

### Polynomial Model Basics

What Are Polynomial Models?

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.

### Estimate Polynomial 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.

Estimate Models Using armax

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`.

### Set Polynomial Model Options

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.

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