# idPiecewiseLinear

Create a piecewise-linear nonlinearity estimator object

## Syntax

NL = idPiecewiseLinear
NL = idPiecewiseLinear(Name,Value)

## Description

NL = idPiecewiseLinear creates a default piecewise-linear nonlinearity estimator object with 10 break points for estimating Hammerstein-Wiener models. The value of the nonlinearity at the break points are set to []. The initial value of the nonlinearity is determined from the estimation data range during estimation using nlhw. Use dot notation to customize the object properties, if needed.

NL = idPiecewiseLinear(Name,Value) creates a piecewise-linear nonlinearity estimator object with properties specified by one or more Name,Value pair arguments. The properties that you do not specify retain their default value.

## Object Description

idPiecewiseLinear is an object that stores the piecewise-linear nonlinearity estimator for estimating Hammerstein-Wiener models.

Use idPiecewiseLinear to define a nonlinear function $y=F\left(x,\theta \right)$, where y and x are scalars, and θ represents the parameters specifying the number of break points and the value of nonlinearity at the break points.

The nonlinearity function, F, is a piecewise-linear (affine) function of x. There are n breakpoints (xk,yk), k = 1,...,n, such that yk = F(xk). F is linearly interpolated between the breakpoints.

F is also linear to the left and right of the extreme breakpoints. The slope of these extensions is a function of xi and yi breakpoints. The breakpoints are ordered by ascending x-values, which is important when you set a specific breakpoint to a different value.

There are minor difference between the breakpoint values you set and the values stored in the object because the toolbox has a different internal representation of breakpoints.

For example, in the following plot, the breakpoints are xk = [-2,1,4] and the corresponding nonlinearity values are yk = [4,3,5].

The value F(x) is computed by evaluate(NL,x), where NL is the idPiecewiseLinear object. When using evaluate, the break points have to be initialized manually.

For idPiecewiseLinear object properties, see Properties.

## Examples

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NL = idPiecewiseLinear;

Specify the number of break points.

NL.NumberOfUnits = 5;

z = iddata(y,u,0.2,'Name','Two tank system');
z1 = z(1:1000);

Create an idPiecewiseLinear object, and specify the breakpoints.

InputNL = idPiecewiseLinear('BreakPoints',[-2,1,4]);

Since BreakPoints is specified as a vector, the specified vector is interpreted as the x-values of the break points. The y-values of the break points are set to 0, and are determined during model estimation.

Estimate model with no output nonlinearity.

sys = nlhw(z1,[2 3 0],InputNL,[]);

## Input Arguments

### Name-Value Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Use Name,Value arguments to specify additional properties of idPiecewiseLinear nonlinearity. For example, NL= idPiecewiseLinear('NumberofUnits',5) creates a piecewise-linear nonlinearity estimator object with 5 breakpoints.

## Properties

idPiecewiseLinear object properties include:

 NumberofUnits Number of breakpoints, specified as an integer. Default: 10 BreakPoints Break points, xk, and the corresponding nonlinearity values at the breakpoints, yk, specified as one of the following: 2-by-n matrix — The x and y values for each of the n break points are specified as [x1,x2, ...., xn;y1, y2, ..., yn].1-by-n vector — The specified vector is interpreted as the x values of the break points: x1,x2, ...., xn. All the y values of the break points are set to 0. When the nonlinearity object is created, the breakpoints are ordered by ascending x-values. This is important to consider if you set a specific breakpoint to a different value after creating the object. Default: [] Free Option to fix or free the values in the mapping object, specified as a logical scalar. When you set an element of Free to false, the object does not update during estimation. Default: true

## Output Arguments

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Piecewise-linear nonlinearity estimator object, returned as an idPiecewiseLinear object.

## Compatibility Considerations

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Not recommended starting in R2021b