idPiecewiseLinear

Create a piecewise-linear nonlinearity estimator 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 $$y=F(x,\theta )$$, 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 (x_k,y_k), k = 1,...,n, such that y_k = F(x_k). 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 x_i and y_i 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 x_k = [-2,1,4] and the corresponding nonlinearity values are y_k = [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.

Creation

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.

Properties

idPiecewiseLinear object properties include:

NumberofUnits — Number of breakpoints

10 (default) | positive integer

Number of breakpoints, specified as an integer.

BreakPoints — Break points and corresponding nonlinearity values at the breakpoints

[ ] (default) | vector | matrix

Break points, x_k, and the corresponding nonlinearity values at the breakpoints, y_k, 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, ...., x_n;y₁, y₂, ..., y_n].

• 1-by-n vector — The specified vector is interpreted as the x values of the break points: x₁,x₂, ...., x_n. 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.

Free — Option to fix or free the values in the mapping object

true (default) | logical scalar

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.

Examples

Create a Default Piecewise-Linear Nonlinearity Estimator

NL = idPiecewiseLinear;

Specify the number of break points.

NL.NumberOfUnits = 5;

Estimate a Hammerstein Model with Piecewise-Linear Nonlinearity

Load estimation data.

load twotankdata;
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,[]);

Version History

Introduced in R2007a

R2021b: Use of previous idnlarx and idnlhw mapping object names is not recommended.

Starting in R2021b, the mapping objects (also known as nonlinearities) used in the nonlinear components of the idnlarx and idnlhw objects have been renamed. The following table lists the name changes.

Pre-R2021b Name	R2021b Name
wavenet	idWaveletNetwork
sigmoidnet	idSigmoidNetwork
treepartition	idTreePartition
customnet	idCustomNetwork
saturation	idSaturation
deadzone	idDeadZone
pwlinear	idPiecewiseLinear
poly1d	idPolynomial1D
unitgain	idUnitGain
linear	idLinear
neuralnet	idFeedforwardNetwork

Scripts with the old names still run normally, although they will produce a warning. Consider using the new names for continuing compatibility with newly developed features and algorithms. There are no plans to exclude the use of these object names at this time.