Class representing neural network nonlinearity estimator for nonlinear ARX models


net_estimator = neuralnet(Network)


neuralnet is the class that encapsulates the neural network nonlinearity estimator. A neuralnet object lets you use networks, created using Neural Network Toolbox™ software, in nonlinear ARX models.

The neural network nonlinearity estimator defines a nonlinear function y=F(x), where F is a multilayer feed-forward (static) neural network, as defined in the Neural Network Toolbox software. y is a scalar and x is an m-dimensional row vector.

You create multi-layer feed-forward neural networks using Neural Network Toolbox commands such as feedforwardnet, cascadeforwardnet and linearlayer. When you create the network:

  • Designate the input and output sizes to be unknown by leaving them at the default value of zero (recommended method). When estimating a nonlinear ARX model using the nlarx command, the software automatically determines the input-output sizes of the network.

  • Initialize the sizes manually by setting input and output ranges to m-by-2 and 1-by-2 matrices, respectively, where m is the number of nonlinear ARX model regressors and the range values are minimum and maximum values of regressors and output data, respectively.

See Examples for more information.

Use evaluate(net_estimator,x) to compute the value of the function defined by the neuralnet object net_estimator at input value x. When used for nonlinear ARX model estimation, x represents the model regressors for the output for which the neuralnet object is assigned as the nonlinearity estimator.

You cannot use neuralnet when the Focus option in nlarxOptions is 'simulation' because this nonlinearity estimator is considered to be nondifferentiable for estimation. Minimization of simulation error requires differentiable nonlinear functions.


net_estimator = neuralnet(Network) creates a neural network nonlinearity estimator based on the feed-forward (static) network object Network created using Neural Network Toolbox commands feedforwardnet, cascadeforwardnet, and linearlayer. Network must represent a static mapping between the inputs and output without I/O delays or feedback. The number of outputs of the network, if assigned, must be one. For a multiple-output nonlinear ARX models, create a separate neuralnet object for each output—that is, each estimator must represent a single-output network object.


NetworkNeural network object, typically created using the Neural Network Toolbox commands feedforwardnet, cascadeforwardnet, and linearlayer.

After creating the object, you can use get or dot notation to access the object property values. For example:

% List Network property value

You can also use the set function to set the value of particular properties. For example:

set(d, 'Network', net_obj)
The first argument to set must be the name of a MATLAB® variable.


Create a neural network nonlinearity estimator using a feed-forward neural network with three hidden layers, transfer functions of types logsig, radbas, and purelinand unknown input and output sizes:

% Create a neural network.
net = feedforwardnet([4 6 1]);
net.layers{1}.transferFcn = 'logsig';
net.layers{2}.transferFcn = 'radbas';
net.layers{3}.transferFcn = 'purelin';
% View the network diagram.
% Create a neuralnet estimator.
net_estimator = neuralnet(net);

Create a single-layer, cascade-forward network with unknown input and output sizes and use this network for nonlinear ARX model estimation:

  1. Create a cascade-forward neural network with 20 neurons and unknown input-output sizes.

    net =  cascadeforwardnet(20);
  2. Create a neural network nonlinearity estimator.

    net_estimator = neuralnet(net);
  3. Estimate nonlinear ARX model.

    % Create estimation data.
    load twotankdata
    Data = iddata(y, u, 0.2);
    % Estimate model.
    Model = nlarx(Data, [2 2 1], net_estimator);  
    % Compare model response to measured output signal.
    compare(Data, Model) 

Initialize the input-output sizes of a two-layer feed-forward neural network based on estimation data and use this network for nonlinear ARX estimation:

  1. Create estimation data.

    % Load estimation data.
    load iddata7 z7
    % Use only first 200 samples for estimation.
    z7 = z7(1:200); 
  2. Create a template Nonlinear ARX model with no nonlinearity.

    model = idnlarx([4 4 4 1 1], []); 

    This model has six regressors and is simply used to define the regressors. The range of regressor values for input-output data in z7 is then used to set the input ranges in the neural network object, as shown in the next steps.

  3. Obtain the model regressor values.

    R = getreg(model, 'all', z7); 

    R is a matrix of regressor values for z7.

  4. Create a two-layer, feed-forward neural network and initialize the network input and output dimensions to 2 and 1, respectively.

    % Use 5 neurons for first layer and 7 for second layer.
    net = feedforwardnet([5 7]); 
    % Determine input range.
    InputRange = [min(R); max(R)].';
    % Initialize input dimensions of estimator.
    net.inputs{1}.range = InputRange;
    % Determine output range.
    OutputRange = [min(z7.OutputData), max(z7.OutputData)];
    % Initialize output dimensions of estimator.
    net.outputs{net.outputConnect}.range = OutputRange;
    % Create neuralnet estimator.
    net_estimator = neuralnet(net);
  5. Specify the nonlinearity estimator in the model.

    model.Nonlinearity = net_estimator;
  6. Estimate the parameters of the network to minimize the prediction error between data and model.

    % Estimate model.
    model = nlarx(z7, model);
    % Compare model's predicted response to measured output signal.
    compare(z7(1:100), model,1)

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The nlarx command uses the train method of the network object, defined in the Neural Network Toolbox software, to compute the network parameter values.

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