Documentation

nlhw

Estimate a Hammerstein-Wiener model

Syntax

  • sys = nlhw(Data,Orders) example
  • sys = nlhw(Data,Orders,InputNL,OutputNL) example
  • sys = nlhw(Data,LinModel) example
  • sys = nlhw(Data,LinModel,InputNL,OutputNL) example

Description

example

sys = nlhw(Data,Orders) creates and estimates a Hammerstein-Wiener model using the estimation data, model orders and delays, and default piecewise linear functions as input and output nonlinearity estimators.

example

sys = nlhw(Data,Orders,InputNL,OutputNL) specifies InputNL and OutputNL as the input and output nonlinearity estimators, respectively.

example

sys = nlhw(Data,LinModel) uses a linear model to specify the model orders and delays, and default piecewise linear functions for the input and output nonlinearity estimators.

example

sys = nlhw(Data,LinModel,InputNL,OutputNL) specifies InputNL and OutputNL as the input and output nonlinearity estimators, respectively.

example

sys = nlhw(Data,sys0) refines or estimates the parameters of a Hammerstein-Wiener model, sys0, using the estimation data.

Use this syntax to:

  • Update the parameters of a previously estimated model to improve the fit to the estimation data. In this case, the estimation algorithm uses the parameters of sys0 as initial guesses.

  • Estimate the parameters of a model previously created using the idnlhw constructor. Prior to estimation, you can configure the model properties using dot notation.

example

sys = nlhw(___,Options) specifies additional model estimation options. Use Options with any of the previous syntaxes.

Examples

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Estimate a Hammerstein-Wiener Model

load iddata3
m1=nlhw(z3,[4 2 1]);

Estimate a Hammerstein Model with Saturation

Load data.

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

Create a saturation object with lower limit of 0, and upper limit of 5.

InputNL = saturation('LinearInterval',[0 5]);

Estimate model with no output nonlinearity.

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

Estimate a Hammerstein-Wiener Model with a Custom Network Nonlinearity

Define custom unit function and save it as gaussunit.m.

function [f, g, a] = GAUSSUNIT(x)
[f, g, a] = gaussunit(x)
f =  exp(-x.*x);
if nargout>1
  g = - 2*x.*f;
  a = 0.2;
end

Load data.

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

Estimate Hammerstein-Wiener model using the custom Gauss unit function.

H = @gaussunit;
CNetw = customnet(H);
m = nlhw(z1,[5 1 3],CNetw,[]);

Estimate Default Hammerstein-Wiener Model Using an Input-Output Polynomial Model of OE Structure

Estimate linear OE model.

load throttledata.mat
Tr = getTrend(ThrottleData);
Tr.OutputOffset = 15;
DetrendedData = detrend(ThrottleData, Tr);
opt = oeOptions('Focus','simulation');
LinearModel = oe(DetrendedData,[1 2 1],opt);

Estimate Hammerstein-Wiener model using OE model as its linear component and saturation as its output nonlinearity.

sys = nlhw(ThrottleData,LinearModel,[],'saturation');

Estimate a Hammerstein-Wiener Model Using idnlhw to First Define the Model Properties

Load data.

load iddata1;

Construct a Hammerstein-Winer model using idnlhw to define the model properties 'b' and 'f'.

sys0 = idnlhw([2,2,0],[],'wavenet');
sys0.b{1} = [0.8,1];
sys0.f{1} = [1,-1.2,0.5];

Estimate the model.

sys = nlhw(z1,sys0);

Estimate a Hammerstein-Winer model using nlhw to define the model properties 'b' and 'f'/

sys2 = nlhw(z1,[2,2,0],[],'wavenet','b',{[0.8,1]},'f',{[1,-1.2,0.5]});

Compare the two estimated models to see that they are equivalent.

compare(z1, sys,'g',sys2,'r--');

Refine a Hammerstein-Wiener Model Using Successive Calls of nlhw

Estimate a Hammerstein-Wiener Model.

load iddata3
sys = nlhw(z3,[4 2 1],'sigmoidnet','wavenet');

Refine the model, sys.

sys = nlhw(z3,sys);

Estimate Hammerstein-Wiener Model Using an Estimation Option Set

Create estimation option set for nlhw to view estimation progress and to set the maximum iteration steps to 50.

opt = nlhwOptions;
opt.Display = 'on';
opt.SearchOption.MaxIter = 50;

Load data and estimate the model.

load iddata3
sys = nlhw(z3,[4 2 1],'sigmoidnet','deadzone',opt);

Related Examples

Input Arguments

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Data — Time domain dataiddata object

Time-domain estimation data, specified as an iddata

Orders — Order and delays of the linear subsystem transfer function [nb nf nk] vector of positive integers | [nb nf nk] vector of matrices

Order and delays of the linear subsystem transfer function, specified as a [nb nf nk] vector.

Dimensions of Orders:

  • For a SISO transfer function, Orders is a vector of positive integers.

    nb is the number of zeros plus 1, nf is the number of poles, and nk is the input delay.

  • For a MIMO transfer function with nu inputs and ny outputs, Orders is a vector of matrices.

    nb, nf, and nk are ny-by-nu matrices whose i-jth entry specifies the orders and delay of the transfer function from the jth input to the ith output.

InputNL — Input static nonlinearitystring | nonlinearity estimator object | array of nonlinearity estimator objects or strings

Input static nonlinearity estimator, specified as a nonlinearity estimator object or string representing the nonlinearity estimator type.

'pwlinear' or pwlinear object
(default)
Piecewise linear function
'sigmoidnet' or sigmoidnet objectSigmoid network
'wavenet' or wavenet objectWavelet network
'saturation' or saturation objectSaturation
'deadzone' or deadzone objectDead zone
'poly1d' or poly1d objectOne-
dimensional polynomial
'unitgain' or [] or unitgain objectUnit gain
customnet objectCustom network

Specifying a string creates a nonlinearity estimator object with default settings. Use object representation instead to configure the properties of a nonlinearity estimator.

InputNL = wavenet;
InputNL.NumberOfUnits = 10;

Alternatively, use the associated input nonlinearity estimator function with Name-Value pair arguments.

InputNL = wavenet('NumberOfUnits',10);

For nu input channels, you can specify nonlinear estimators individually for each input channel by setting InputNL to an nu-by-1 array of nonlinearity estimators.

InputNL = [sigmoidnet('NumberofUnits',5); deadzone([-1,2])]
To specify the same nonlinearity for all inputs, specify a single input nonlinearity estimator.

OutputNL — Output static nonlinearitystring | nonlinearity estimator object | array of nonlinearity estimator objects

Output static nonlinearity estimator, specified as a nonlinearity estimator object or string representing the nonlinearity estimator type.

'pwlinear' or pwlinear object
(default)
Piecewise linear function
'sigmoidnet' or sigmoidnet objectSigmoid network
'wavenet' or wavenet objectWavelet network
'saturation' or saturation objectSaturation
'deadzone' or deadzone objectDead zone
'poly1d' or poly1d objectOne-
dimensional polynomial
'unitgain' or [] or unitgain objectUnit gain
customnet objectCustom network

Specifying a string creates a nonlinearity estimator object with default settings. Use object representation instead to configure the properties of a nonlinearity estimator.

OutputNL = sigmoidnet;
OutputNL.NumberOfUnits = 10;

Alternatively, use the associated input nonlinearity estimator function with Name-Value pair arguments.

OutputNL = sigmoidnet('NumberOfUnits',10);

For ny output channels, you can specify nonlinear estimators individually for each output channel by setting OutputNL to an ny-by-1 array of nonlinearity estimators. To specify the same nonlinearity for all outputs, specify a single output nonlinearity estimator.

LinModel — Discrete time linear modelidpoly | idss with K = 0 | idtf

Discrete-time linear model used to specify the linear subsytem, specified as one of the following:

  • Input-output polynomial model of Output-Error (OE) structure (idpoly)

  • State-space model with no disturbance component (idss with K = 0)

  • Transfer function model (idtf)

Typically, you estimate the model using oe, n4sid, or tfest.

sys0 — Hammerstein-Wiener modelidnlhw object

Hammerstein-Wiener model, specified as an idnlhw object. sys0 can be:

  • A model previously created using idnlhw to specify model properties.

  • A model previously estimated using nlhw, that you want to update using a new estimation data set.

    You can also refine sys0 using the original estimation data set. If the previous estimation stopped when the numerical search was stuck at a local minima of the cost function, use init to first randomize the parameters of sys0. See sys0.Report.Termination for search stopping conditions. Using init does not guarantee a better solution on further refinement.

Options — Estimation optionsnlhwOptions option set

Estimation options for Hammerstein-Wiener model identification, specified as an nlhwOptions option set.

Output Arguments

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sys — Estimated Hammerstein-Wiener modelidnlhw object

Estimated Hammerstein-Wiener model, , returned as an idnlhw object. The model is estimated using the specified model orders and delays, input and output nonlinearity estimators, and estimation options.

Information about the estimation results and options used is stored in the model's Report property. The contents of Report contain the following fields:

Report FieldDescription
Status

Summary of the model status, which indicates whether the model was created by construction or obtained by estimation.

Method

Estimation command used.

Fit

Quantitative assessment of the estimation — Structure with the following fields:

FieldDescription
FitPercent

Normalized root mean squared error (NRMSE) measure of how well the response of the model fits the estimation data, expressed as a percentage. See goodnessOfFit for more information.

FPE

Final prediction error for the model. See fpe for more information.

LossFcn

Value of the loss function, equal to det(E'*E/N), where E is the residual error matrix (one column for each output) and N is the total number of samples.

MSE

Mean squared error (MSE) measure of how well the response of the model fits the estimation data. See goodnessOfFit for more information.

Parameters

Estimated values of the model parameters.

OptionsUsed

Option set used for estimation. If no custom options were configured, this is a set of default options. See nlhwOptions for more information.

RandState

State of the random number stream at the start of estimation. Empty, [], if randomization was not used during estimation.

DataUsed

Attributes of the data used for estimation — Structure with the following fields:

FieldDescription
Name

Name of the data set.

Type

Data type — For idnlhw models, this is set to 'Time domain data'.

Length

Number of data samples.

Ts

Sample time. This is equivalent to Data.Ts.

InterSample

Input intersample behavior. One of the following values:

  • 'zoh' — Zero-order hold maintains a piecewise-constant input signal between samples.

  • 'foh' — First-order hold maintains a piecewise-linear input signal between samples.

  • 'bl' — Band-limited behavior specifies that the continuous-time input signal has zero power above the Nyquist frequency.

The value of Intersample has no effect on estimation results for discrete-time models.

InputOffset

Empty, [], for nonlinear estimation methods.

OutputOffset

Empty, [], for nonlinear estimation methods.

Termination

Termination conditions for the iterative search used for prediction error minimization — Structure with the following fields:

FieldDescription
WhyStop

Reason for terminating the numerical search.

Iterations

Number of search iterations performed by the estimation algorithm.

FirstOrderOptimality

-norm of the gradient search vector when the search algorithm terminates.

FcnCount

Number of times the objective function was called.

UpdateNorm

Norm of the gradient search vector in the last iteration. The field is omitted when 'lsqnonlin' is the search method.

LastImprovement

Criterion improvement in the last iteration, expressed as a percentage. Omitted when 'lsqnonlin' is the search method.

Algorithm

Algorithm used for 'lsqnonlin' search method. Omitted when other search methods are used.

For estimation methods that do not require numerical search optimization, the Termination field is omitted.

For more information, see Estimation Report.

Introduced in R2007a

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