idSigmoidNetwork

Sigmoid network function for nonlinear ARX and Hammerstein-Wiener models

Description

An idSigmoidNetwork object implements a sigmoid network function, and is a nonlinear mapping function for estimating nonlinear ARX and Nonlinear Hammerstein-Wiener models. The mapping function, which is also referred to as a nonlinearity, uses a combination of linear weights, an offset and a nonlinear function to compute its output. The nonlinear function contains sigmoid unit functions that operate on a ridge combination (weighted linear sum) of inputs.

Mathematically, idSigmoidNetwork is a function that maps m inputs X(t) = [x(t₁),x₂(t),…,x_m(t)]^T to a scalar output y(t) using the following relationship:

$$y(t)=y_0+{\rm X}{(t)}^{T}PL+S({\rm X}(t))$$

Here:

• X(t) is an m-by-1 vector of inputs, or regressors.

• y₀ is the output offset, a scalar.

• P is an m-by-p projection matrix, where m is the number of regressors and is p is the number of linear weights. m must be greater than or equal to p.

• L is a p-by-1 vector of weights.

• S(X) is a sum of dilated and translated sigmoid functions. The total number of sigmoid functions is referred to as the number of units n of the network.

For the definition of the sigmoid function term S(X) , see More About.

Use idSigmoidNetwork as the value of the OutputFcn property of an idnlarx model or the InputNonlinearity and OutputLinearity properties of an idnlhw object. For example, specify idSigmoidNetwork when you estimate an idnlarx model with the following command.

sys = nlarx(data,regressors,idSigmoidNetwork)

When nlarx estimates the model, it essentially estimates the parameters of the idSigmoidNetwork function.

You can configure the idSigmoidNetwork function to disable components and fix parameters. To omit the linear component, set LinearFcn.Use to false. To omit the offset, set Offset.Use to false. To specify known values for the linear function and the offset, set their Value attributes directly and set the corresponding Free attributes to False. Use evaluate to compute the output of the function for a given vector of inputs.

Creation

Syntax

S = idSigmoidNetwork

S = idSigmoidNetwork(numUnits)

S = idSigmoidNetwork(numUnits,UseLinearFcn)

S = idSigmoidNetwork(numUnits,UseLinearFcn,UseOffset)

Description

S = idSigmoidNetwork creates a idSigmoidNetwork object S that uses 10 units. The number of inputs is determined during model estimation and the number of outputs is 1.

S = idSigmoidNetwork(numUnits) specifies the number of sigmoid functions numUnits.

S = idSigmoidNetwork(numUnits,UseLinearFcn) specifies whether the function uses a linear function as a subcomponent.

S = idSigmoidNetwork(numUnits,UseLinearFcn,UseOffset) specifies whether the function uses an offset term y₀ parameter.

Input Arguments

numUnits — Number of units

10 (default) | positive integer

Number of units, specified as a positive integer. numUnits determines the number of sigmoid functions.

This argument sets the S.NonlinearFcn.NumberOfUnits property.

UseLinearFcn — Option to use linear function

true (default) | false

Option to use the linear function subcomponent, specified as true or false. This argument sets the value of the S.LinearFcn.Use property.

UseOffset — Option to use offset term

true (default) | false

Option to use an offset term, specified as true or false. This argument sets the value of the S.Offset.Use property.

Properties

Inputs — Input signal names

cell array

Input signal names for the inputs to the mapping object, specified as a 1-by-m cell array, where m is the number of input signals. This property is determined during estimation.

Outputs — Output signal name

cell array

Output signal name for the output of the mapping object, specified as a 1-by-1 cell array. This property is determined during estimation.

LinearFcn — Parameters of linear function

linear function property values (default)

Parameters of the linear function, specified as follows:

• Use — Option to use the linear function in the sigmoid network, specified as a scalar logical. The default value is true.

• Value — Linear weights that compose L', specified as a 1-by-p vector.

• InputProjection — Input projection matrix P, specified as an m-by-p matrix, that transforms the detrended input vector of length m into a vector of length p. For Hammerstein-Wiener models, InputProjection is equal to 1.

• Free — Option to update entries of Value during estimation, specified as a 1-by-p logical vector. The software honors the Free specification only if the starting value of Value is finite. The default value is true.

Offset — Parameters of offset term

offset property values

Parameters of the offset term, specified as follows:

• Use — Option to use the offset in the sigmoid network, specified as a scalar logical. The default value is true.

• Value — Offset value, specified as a scalar.

• Free — Option to update Value during estimation, specified as a scalar logical. The software honors the Free specification of false only if the value of Value is finite. The default value is true.

NonlinearFcn — Parameters of nonlinear function

nonlinear function property values

Parameters of the nonlinear function, specified as follows:

• NumberOfUnits — Number of units, specified as a positive integer. NumberOfUnits determines the number of sigmoid functions.

• Parameters — Parameters of idSigmoidNetwork, specified as in the following table:

  Field Name	Description	Default
  InputProjection	Projection matrix Q, specified as an m-by-q matrix. Q transforms the detrended input vector $$(X-\overline{X})$$ of length m into a vector of length q. Typically, Q has the same dimensions as the linear projection matrix P. In this case, q is equal to p, which is the number of linear weights. For Hammerstein-Wiener models, InputProjection is equal to 1.	[]
  OutputCoefficient	Sigmoid function output coefficients si, specified as an n-by-1 vector.	[]
  Translation	Translation matrix, specified as an n-by-q matrix of translation row vectors ci.	[]
  Dilation	Dilation coefficients bi, specified as an n-by-1 vector.	[]

• Free — Option to estimate parameters, specified as a logical scalar. If all the parameters have finite values, such as when the idSigmoidNetwork object corresponds to a previously estimated model, then setting Free to false causes the parameters of the nonlinear function S(X) to remain unchanged during estimation. The default value is true.

Examples

Estimate Nonlinear ARX Model with idSigmoidNetwork as Output Function

Load the data z7 and create a subset to use as estimation data.

load iddata7 z7
ze = z7(1:300);

Create and configure an idSigmoidNetwork mapping object. Fix the offset to 0.2 and the number of units to 15.

S = idSigmoidNetwork;
S.Offset.Value = 0.2;
S.Offset.Free = false;
S.NonlinearFcn.NumberOfUnits = 15;

Create linear and polynomial model regressors. Use the input and output variable names from z7 as the variable names for the regressors.

var_names = [z7.OutputName;z7.InputName]

var_names = 3x1 cell
    {'y1'}
    {'u1'}
    {'u2'}

Reg1 = linearRegressor(var_names,{1:4,0:4,1});
Reg2 = polynomialRegressor(var_names,{1:2,0:2,0},2);

Set the estimation options.

opt = nlarxOptions('SearchMethod','fmincon');
opt.SearchOptions.MaxIterations = 40;

Estimate the nonlinear ARX model.

sys = nlarx(ze,[Reg1;Reg2],S,opt)

sys =

Nonlinear ARX model with 1 output and 2 inputs
  Inputs: u1, u2
  Outputs: y1

Regressors:
  1. Linear regressors in variables y1, u1, u2
  2. Order 2 regressors in variables y1, u1, u2

Output function: Sigmoid network with 15 units
Sample time: 1 seconds

Status:                                         
Estimated using NLARX on time domain data "ze". 
Fit to estimation data: 71.5% (prediction focus)
FPE: 7.497, MSE: 0.8034

Estimate Hammerstein-Wiener Model That Uses idSigmoidNetwork

Estimate a Hammerstein-Wiener model that uses idSigmoidNetwork as the output nonlinearity.

Load the data

load throttledata

Create an idSigmoidNetwork mapping object that has 15 units and that has no input nonlinearity or offset.

S = idSigmoidNetwork(15,false,false)

S = 
Sigmoid Network

 Nonlinear Function: Sigmoid network with 15 units
 Linear Function: not in use
 Output Offset: not in use

    NonlinearFcn: '<Sigmoid units and their parameters>'
       LinearFcn: '<Linear function parameters>'
          Offset: '<Offset parameters>'

Estimate a Hammerstein-Wiener model.

sys = nlhw(ThrottleData,[4 4 0],[],S)

sys =

Hammerstein-Wiener model with 1 output and 1 input

Linear transfer function corresponding to the orders nb = 4, nf = 4, nk = 0

Input nonlinearity: None
Output nonlinearity: Sigmoid network with 15 units
Sample time: 0.01 seconds

Status:                                                 
Estimated using NLHW on time domain data "ThrottleData".
Fit to estimation data: 59.67%                          
FPE: 254.7, MSE: 179.5

More About

Sigmoid Nonlinear Function S(X)

The sigmoid nonlinear function is a sum of the dilated and translated sigmoid functions, and is described by the following equation:

$$S(X)={\displaystyle \sum _{i=1}^n{s}_i}f(X^{T}Q{b}_i+c_i)$$

Here:

• Q is an m-by-q projection matrix, where m ≥ q.

• s₁, s₂, …, s_n are scalar weights called output coefficients.

• b₁, b₂, …, b_n are q-by-1 vectors called dilation coefficients.

• c₁, c₂, …, c_n are scalars called translations.

• $$f(z)=\frac{1}{e^{-z}+1}.$$ is the sigmoid function, also called a unit function of the sigmoid network. Here, z is a scalar of the form b_iX^TQ + c_i.

Algorithms

idSigmoidNetwork uses an iterative search technique for estimating parameters.

Version History

Introduced in R2007a

R2022a: Previous idnlarx data normalization information moved from mapping object properties to idnlarx Normalization property

Information related to data normalization was moved from the idSigmoidNetwork mapping object level to the model level. The Normalization property of the idnlarx model contains the data centering and scaling information that the estimation process computes. In addition, the regressor-selection process for the mapping objects has also moved to the model level. The model now passes the actual regressor names rather than the selection indices to the mapping object, eliminating the need for an index property at the mapping object level.

The following table summarizes the mapping object subproperties that were eliminated. For more information, see the Normalization property of idnlarx.

Main Properties / Subproperties	Input	Output	LinearMdl	Offset	NonlinearMdl
Mean	X	X
Range	X	X
Minimum			X	X	X
Maximum			X	X	X
SelectedInputIndex			X		X

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.

R2021a: Use of previous nonlinearity estimator properties is not recommended

Starting in R2021a, the properties of the mapping objects, previously known as nonlinearity estimators, have been reorganized. These objects, which were renamed in R2021b, are idWaveletNetwork (W), idSigmoidNetwork (S), idTreePartition (T), idCustomNetwork (C), and idLinear (L). The property changes do not apply to idFeedforwardNetwork. The use of the pre-R2021a properties in the following table is discouraged. However, the software still accepts commands that set these properties. There are no plans to exclude such commands at this time.

Pre-R2021a Property	R2021a Property	Applicable Mapping Objects
NumberOfUnits	NonlinearFcn.NumberOfUnits	W,S,T,C
LinearTerm	LinearFcn.Use, Offset.Use	W,S,C
Parameters	Split into three pieces: LinearFcn.Value Offset.Value NonlinearFcn.Parameters	W,S,T,C,L linear (L) excludes NonlinearFcn.Parameters.
Options	NonlinearFcn.Structure	W,T