pwlinear

Create a piecewise-linear nonlinearity estimator object

Syntax

NL = pwlinear
NL = pwlinear(Name,Value)


Description

NL = pwlinear 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 = pwlinear(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

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

Use pwlinear 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 pwlinear object. When using evaluate, the break points have to be initialized manually.

For pwlinear object properties, see Properties.

Examples

Create a Default Piecewise-Linear Nonlinearity Estimator

NL = pwlinear;

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 a pwlinear object, and specify the breakpoints.

InputNL = pwlinear('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 Pair 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 pwlinear nonlinearity. For example, NL= pwlinear('NumberofUnits',5) creates a piecewise-linear nonlinearity estimator object with 5 breakpoints.

Properties

pwlinear 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: []

Output Arguments

NL — Piecewise-linear nonlinearity estimator object

pwlinear object

Piecewise-linear nonlinearity estimator object, returned as a pwlinear object.