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),…,xm(t)]T to a scalar output y(t) using the following relationship:

`$y\left(t\right)={y}_{0}+Χ{\left(t\right)}^{T}PL+S\left(Χ\left(t\right)\right)$`

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

example

````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. ```

example

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

example

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

example

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

Input Arguments

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

This argument sets the `S.NonlinearFcn.NumberOfUnits` property.

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

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

Properties

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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.

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

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`.

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`.

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 NameDescriptionDefault
`InputProjection `

Projection matrix Q, specified as an m-by-q matrix. Q transforms the detrended input vector $\left(X-\overline{X}\right)$ 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

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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 List of all regressors Output function: Sigmoid network with 15 units Sample time: 1 seconds Status: Estimated using NLARX on time domain data "ze". Fit to estimation data: 74.27% (prediction focus) FPE: 6.11, MSE: 0.6547 ```

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

`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 Inputs: {1x0 cell} Outputs: {1x0 cell} 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: absent 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: 55.55% FPE: 309.4, MSE: 218 ```

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Algorithms

`idSigmoidNetwork` uses an iterative search technique for estimating parameters.

Version History

Introduced in R2007a

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Behavior changed in R2022a

Not recommended starting in R2021b

Not recommended starting in R2021a