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# linear

Class representing linear nonlinearity estimator for nonlinear ARX models

## Syntax

lin=linear
lin=linear('Parameters',Par)

## Description

linear is an object that stores the linear nonlinearity estimator for estimating nonlinear ARX models.

lin=linear instantiates the linear object.

lin=linear('Parameters',Par) instantiates the linear object and specifies optional values in the Par structure. For more information about this structure, see linear Properties.

## linear Properties

You can include property-value pairs in the constructor to specify the object.

After creating the object, you can use get or dot notation to access the object property values. For example:

```% List Parameters values
get(lin)
% Get value of Parameters property
lin.Parameters```
Property NameDescription
Parameters

Structure containing the following fields:

• LinearCoef: m-by-1 vector L.

• OutputOffset: Scalar d.

## Examples

Estimate a nonlinear ARX model using the linear estimator with custom regressors for the following system:

y(t) = a1y(t–1) + a2y(t–2) + a3u(t–1) + a4y(t–1)u(t–2) + a5|u(t)|u(t–3) + a6,

where u is the input and y is the output.

```% Create regressors y(t-1), y(t-2) and u(t-1).
orders = [2 1 1];
% Create an idnlarx model using linear estimator with custom regressors.
model = idnlarx(orders, linear, 'InputName', 'u', 'OutputName', 'y',...
'CustomRegressors', {'y(t-1)*u(t-2)','abs(u(t))*u(t-3)'})
% Estimate the model parameters a1, a2, ... a6.
EstimatedModel = nlarx(data, model)
```
 Note:   The nonlinearity in the model is described by custom regressors only.

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### Tips

• linear is a linear (affine) function $y=F\left(x\right)$, defined as follows:

$F\left(x\right)=xL+d$

y is scalar, and x is a 1-by-m vector.

• Use evaluate(lin,x) to compute the value of the function defined by the linear object lin at x.

• When creating a nonlinear ARX model using the constructor (idnlarx) or estimator (nlarx), you can specify a linear nonlinearity estimator using [], instead of entering linear explicitly. For example:

`m=idnlarx(orders,[]);`

### Algorithms

When the idnlarx property Focus is 'Prediction', linear uses a fast, noniterative initialization and iterative search technique for estimating parameters. In most cases, iterative search requires only a few iterations.

When the idnlarx property Focus='Simulation', linear uses an iterative technique for estimating parameters.