# 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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