# 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`