Nonlinear ARX model

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Description

An idnlarx model represents a nonlinear ARX model, which is an extension of the linear ARX structure and contains linear and nonlinear functions.

A nonlinear ARX model consists of model regressors and an output function. The output function contains one or more mapping objects, one for each model output. Each mapping object can include a linear and a nonlinear function that act on the model regressors to give the model output and a fixed offset for that output. This block diagram represents the structure of a single-output nonlinear ARX model in a simulation scenario.

The software computes the nonlinear ARX model output y intwo stages:

It computes regressor values from the current and past input values and the past output data.
In the simplest case, regressors are delayed inputs and outputs, such as u(t–1) and y(t–3). These kind of regressors are called linear regressors. You specify linear regressors using the linearRegressor object. You can also specify linear regressors by using linear ARX model orders as an input argument. For more information, see Nonlinear ARX Model Orders and Delay. However, this second approach constrains your regressor set to linear regressors with consecutive delays. To create polynomial regressors, use the polynomialRegressor object. To create periodic regressors that contain the sine and cosine functions of delayed input and output variables , use the periodicRegressor object. You can also specify custom regressors, which are nonlinear functions of delayed inputs and outputs. For example, u(t–1)y(t–3) is a custom regressor that multiplies instances of input and output together. Specify custom regressors using the customRegressor object.
You can assign any of the regressors as inputs to the linear function block of the output function, the nonlinear function block, or both.
It maps the regressors to the model output using an output function block. The output function block can include multiple mapping objects, with each mapping object containing linear, nonlinear, and offset blocks in parallel. For example, consider the following equation:
$F (x) = L^{T} (x - r) + g (Q (x - r)) + d$
Here, x is a vector of the regressors, and r is the mean of x. $F (x) = L^{T} (x - r) + y_{0}$ is the output of the linear function block. $g (Q (x - r)) + y_{0}$ represents the output of the nonlinear function block. Q is a projection matrix that makes the calculations well-conditioned. d is a scalar offset that is added to the combined outputs of the linear and nonlinear blocks. The exact form of F(x) depends on your choice of output function. You can select from the available mapping objects, such as tree-partition networks, wavelet networks, and multilayer neural networks. You can also exclude either the linear or the nonlinear function block from the output function.
When estimating a nonlinear ARX model, the software computes the model parameter values, such as L, r, d, Q, and other parameters specifying g.

The resulting nonlinear ARX models are idnlarx objects that store all model data, including model regressors and parameters of the output function. For more information about these objects, see Nonlinear Model Structures.

For more information on the idnlarx model structure, see What are Nonlinear ARX Models?.

For idnlarx object properties, see Properties.

Creation

You can obtain an idnlarx object in one of two ways.

Use the nlarx command to both construct an idnlarx object and estimate the model parameters.
```
sys = nlarx(data,reg)
```
Use the idnlarx constructor to create the nonlinear ARX model and then estimate the model parameters using nlarx or pem.
```
sys = idnlarx(output_name,input_name,reg)
```

Syntax

sys = idnlarx(output_name,input_name,orders)

sys = idnlarx(output_name,input_name,Regressors)

sys = idnlarx(___,OutputFcn)

sys = idnlarx(linmodel)

sys = idnlarx(linmodel,OutputFcn)

sys = idnlarx(___,Name,Value)

Description

Specify Model Directly

example

sys = idnlarx(output_name,input_name,orders) specifies a set of linear regressors using ARX model orders. Use this syntax when you extend an ARX linear model, or when you plan to use only regressors that are linear with consecutive lags.

example

sys = idnlarx(output_name,input_name,Regressors) creates a nonlinear ARX model with the output and input names of output_name and input_name, respectively, and a regressor set in Regressors that contains any combination of linear, polynomial, periodic, and custom regressors. The software constructs sys using the default wavelet network ('idWaveletNetwork') mapping object for the output function.

example

sys = idnlarx(___,OutputFcn) specifies the output function OutputFcn that maps the regressors to the model output. You can use this syntax with any of the previous input argument combinations.

Initialize Model Values Using Linear Model

example

sys = idnlarx(linmodel) uses a linear model linmodel to extract certain properties such as names, units, and sample time, and to initialize the values of the linear coefficients of the model. Use this syntax when you want to create a nonlinear ARX model as an extension of, or an improvement upon, an existing linear model.

example

sys = idnlarx(linmodel,OutputFcn) specifies the output function OutputFcn that maps the regressors to the model output.

Specify Model Properties

sys = idnlarx(___,Name,Value) specifies additional properties of the idnlarx model structure using one or more name-value arguments.

Input Arguments

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`orders` — ARX model orders
`nlarx` orders `[na nb nk]`

ARX model orders, specified as the matrix [na nb nk]. na denotes the number of delayed outputs, nb denotes the number of delayed inputs, and nk denotes the minimum input delay. The minimum output delay is fixed to 1. For more information on how to construct the orders matrix, see arx.

When you specify orders, the software converts the order information into a linear regressor form in the idnlarx Regressors property. For an example, see Create Nonlinear ARX Model Using ARX Model Orders.

Properties

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`Regressors` — Regressor specification
`linearRegressor` object | `polynomialRegressor` object | `periodicRegressor` object | `customRegressor` object | column array of regressor specification objects

Regressor specification, specified as a column vector containing one or more regressor specification objects, which are the linearRegressor objects, polynomialRegressor objects, periodicRegressor objects, and customRegressor objects. Each object specifies a formula for generating regressors from lagged variables. For example:

L = linearRegressor({'y1','u1'},{1,[2 5]}) generates the regressors y₁(t–1), u₁(t–2), and u₂(t–5).
P = polynomialRegressor('y2',4:7,2) generates the regressors y₂(t–4)², y₂(t–5)²,y₂(t–6)², and y₂(t–7)².
SC = periodicRegressor({'y1','u1'},{1,2}) generates the regressors y₁(t-1)), cos(y₁(t-1)), sin(u₁(t-2)), and cos(u₁(t-2)).
C = customRegressor({'y1','u1','u2'},{1 2 2},@(x,y,z)sin(x.*y+z)) generates the single regressor sin(y₁(t–1)u₁(t–2)+u₂(t–2)
.

For an example that implements these regressors, see Create and Combine Regressor Types.

To add regressors to an existing model, create a vector of specification objects and use dot notation to set Regressors to this vector. For example, the following code first creates the idnlarx model sys and then adds the regressor objects L, P, SC, and C to the regressors of sys.

sys = idnlarx({'y1','y2'},{'u1','u2'});R = [L;P;SC;C];sys.Regressors = R;

For an example of creating and using a linear regressor set, see Create Nonlinear ARX Model Using Linear Regressors.

`OutputFcn` — Output function
`'idWaveletNetwork'` (default) | `'idLinear'` | `[]` | `''` | `'idSigmoidNetwork'` | `'idTreePartition'` | `'idGaussianProcess'` | `'idTreeEnsemble'` | `'idSupportVectorMachine'` | mapping object | array of mapping objects

Output function that maps the regressors of the idnlarx model into the model output, specified as a column array containing zero or more of the following strings or mapping objects:

`'idWaveletNetwork'` or idWaveletNetwork object	Wavelet network
`'idLinear'` or `''` or `[]` or idLinear object	Linear function
`'idSigmoidNetwork'` or idSigmoidNetwork object	Sigmoid network
`'idTreePartition'` or idTreePartition object	Binary tree partition regression model
`'idGaussianProcess'` or idGaussianProcess object	Gaussian process regression model (requires Statistics and Machine Learning Toolbox™)
`'idTreeEnsemble'` or idTreeEnsemble	Regression tree ensemble model (requires Statistics and Machine Learning Toolbox)
`'idSupportVectorMachine'` or idSupportVectorMachine	Kernel-based Support Vector Machine (SVM) regression model with constraints (requires Statistics and Machine Learning Toolbox)
`'idNeuralNetwork'` or idNeuralNetwork object	Multilayer neural network (requires Statistics and Machine Learning Toolbox or Deep Learning Toolbox™)
idCustomNetwork object	Custom network — Similar to `idSigmoidNetwork`, but with a user-defined replacement for the sigmoid function

The idWaveletNetwork, idSigmoidNetwork, idTreePartition, and idCustomNetwork objects contain both linear and nonlinear components. You can remove (not use) the linear components of idWaveletNetwork, idSigmoidNetwork, and idCustomNetwork by setting the LinearFcn.Use value to false.

The idTreeEnsemble and idSupportVectorMachine objects contain only a nonlinear component. The idLinear function, as the name implies, has only a linear component.

Specifying a character vector, for example 'idSigmoidNetwork', creates a mapping object with default settings. Alternatively, you can specify mapping object properties in two other ways:

Create the mapping object using arguments to modify default properties.
```
MO = idSigmoidNetwork(15)
```
Create a default mapping object first and then use dot notation to modify properties.
```
MO = idSigmoidNetwork;MO.NumberOfUnits = 15
```

For n_y output channels, you can specify mapping objects individually for each channel by setting OutputFcn to an array of n_y mapping objects. For example, the following code specifies OutputFcn using dot notation for a system with two input channels and two output channels.

sys = idnlarx({'y1','y2'},{'u1','u2'});sys.OutputFcn = [idWaveletNetwork; idSigmoidNetwork]

To specify the same mapping for all outputs, specify OutputFcn as a character vector or a single mapping object.

OutputFcn represents a static mapping function that transforms the regressors of the nonlinear ARX model into the model output. OutputFcn is static because it does not depend on the time. For example, if $y (t) = y_{0} + a_{1} y (t - 1) + a_{2} y (t - 2) + \dots + b_{1} u (t - 1) + b_{2} u (t - 2) + \dots$ , then OutputFcn is a linear function represented by the idLinear object.

For an example of specifying the output function, see Specify Output Function for Nonlinear ARX Model.

`RegressorUsage` — Regressor assignments
table with logical entries

Regressor assignments to the linear and nonlinear components of the nonlinear ARX model, specified as an n_r-by-n_c table with logical entries that specify which regressors to use for which component. Here, n_r is the number of regressors. n_c is the total number of linear and nonlinear components in OutputFcn. The rows of the table correspond to individual regressors. The row names are set to regressor names. If the table value for row i and component index j is true, then the ith regressor is an input to the linear or nonlinear component j.

For multi-output systems, OutputFcn contains one mapping object for each output. Each mapping object can use both linear and nonlinear components or only one of the two components.

For an example of viewing and modifying the RegressorUsage property, see Modify Regressor Assignments to Output Function Components.

`Normalization` — Regressor and output data centering and scaling
structure (default)

Regressor and output centering and scaling, specified as a structure. As the following table shows, each field in the structure contains a row vector with a length that is equal to the number of either regressors (n_r) or model outputs (n_y).

Field	Description	Default Element Value
`RegressorCenter`	Row vector of length n_r	`NaN`
`RegressorScale`	Row vector of length n_r	`NaN`
`OutputCenter`	Row vector of length n_y	`NaN`
`OutputScale`	Row vector of length n_y	`NaN`

For a matrix X, with centering vector C and scaling vector S, the software computes the normalized form of X using Xnorm = (X-C)./S.

The following figure illustrates the normalization flow for a nonlinear ARX model.

In this figure:

The algorithm converts the inputs u(t) and y(t) into the regressor set R(t).
The algorithm uses the regressor centering and scaling parameters to normalize R(t) as R_N(t).
R_N(t) provides the input to the mapping function, which then produces the normalized output y_N
The algorithm uses the output scaling and centering parameters to restore the original range, producing y(t).

Typically, the software normalizes the data automatically during model estimation, in accordance with the option settings in nlarxOptions for Normalize and NormalizationOptions. You can also directly assign centering and scaling values by specifying the values in vectors, as described in the previous table. The values that you assign must be real and finite. This approach can be useful, for example, when you are simulating your model using inputs that represent a different operating point from the operating point for the original estimation data. You can assign the values for any field independently. The software will estimate the values of any fields that remain unassigned (NaN).

`Report` — Summary report
report field values

This property is read-only.

Summary report that contains information about the estimation options and results for a nonlinear ARX model obtained using the nlarx command. Use Report to find estimation information for the identified model, including:

Estimation method
Estimation options
Search termination conditions
Estimation data fit

The contents of Report are irrelevant if the model was constructed using idnlarx.

sys = idnlarx('y1','u1',reg);sys.Report.OptionsUsed

`TimeVariable` — Independent variable
`'t'` (default) | character vector

Independent variable for the inputs, outputs, and—when available—internal states, specified as a character vector.

`NoiseVariance` — Noise variance
matrix

Noise variance (covariance matrix) of the model innovations e. The estimation algorithm typically sets this property. However, you can also assign the covariance values by specifying an ny-by-ny matrix.

`Ts` — Sample time
`1` (default) | positive scalar

Sample time, specified as a positive scalar representing the sampling period. This value is expressed in the unit specified by the TimeUnit property of the model.

`TimeUnit` — Units for time variable
`'seconds'` (default) | `'nanoseconds'` | `'microseconds'` | `'milliseconds'` | `'minutes'` | `'hours'` | `'days'` | `'weeks'` | `'months'` | `'years'`

Units for the time variable, the sample time Ts, and any time delays in the model, specified as one of the following values:

'nanoseconds'
'microseconds'
'milliseconds'
'seconds'
'minutes'
'hours'
'days'
'weeks'
'months'
'years'

Changing this property has no effect on other properties, and therefore changes the overall system behavior. Use chgTimeUnit (Control System Toolbox) to convert between time units without modifying system behavior.

`InputName` — Input channel names
`''` for all input channels (default) | character vector | cell array of character vectors

Input channel names, specified as one of the following:

Character vector — For single-input models, for example, 'controls'.
Cell array of character vectors — For multi-input models.

Input names in Nonlinear ARX models must be valid MATLAB^® variable names after you remove any spaces.

Alternatively, use automatic vector expansion to assign input names for multi-input models. For example, if sys is a two-input model, enter:

sys.InputName = 'controls';

The input names automatically expand to {'controls(1)';'controls(2)'}.

When you estimate a model using an iddata object, data, the software automatically sets InputName to data.InputName.

You can use the shorthand notation u to refer to the InputName property. For example, sys.u is equivalent to sys.InputName.

Input channel names have several uses, including:

Identifying channels on model display and plots
Extracting subsystems of MIMO systems
Specifying connection points when interconnecting models

`InputUnit` — Input channel units
`''` for all input channels (default) | character vector | cell array of character vectors

Input channel units, specified as one of the following:

Character vector — For single-input models, for example, 'seconds'.
Cell array of character vectors — For multi-input models.

Use InputUnit to keep track of input signal units. InputUnit has no effect on system behavior.

`InputGroup` — Input channel groups
structure with no fields (default) | structure

Input channel groups. The InputGroup property lets you assign the input channels of MIMO systems into groups and refer to each group by name. Specify input groups as a structure. In this structure, field names are the group names, and field values are the input channels belonging to each group. For example:

sys.InputGroup.controls = [1 2];sys.InputGroup.noise = [3 5];

creates input groups named controls and noise that include input channels 1, 2 and 3, 5, respectively. You can then extract the subsystem from the controls inputs to all outputs using:

sys(:,'controls')

`OutputName` — Output channel names
`''` for all output channels (default) | character vector | cell array of character vectors

Output channel names, specified as one of the following:

Character vector — For single-output models. For example, 'measurements'.
Cell array of character vectors — For multi-output models.

Output names in Nonlinear ARX models must be valid MATLAB variable names after you remove any spaces.

Alternatively, use automatic vector expansion to assign output names for multi-output models. For example, if sys is a two-output model, enter:

sys.OutputName = 'measurements';

The output names automatically expand to {'measurements(1)';'measurements(2)'}.

When you estimate a model using an iddata object, data, the software automatically sets OutputName to data.OutputName.

You can use the shorthand notation y to refer to the OutputName property. For example, sys.y is equivalent to sys.OutputName.

Output channel names have several uses, including:

Identifying channels on model display and plots
Extracting subsystems of MIMO systems
Specifying connection points when interconnecting models

`OutputUnit` — Output channel units
`''` for all output channels (default) | character vector | cell array of character vectors

Output channel units, specified as one of the following:

Character vector — For single-output models. For example, 'seconds'.
Cell array of character vectors — For multi-output models.

Use OutputUnit to keep track of output signal units. OutputUnit has no effect on system behavior.

`OutputGroup` — Output channel groups
structure with no fields (default) | structure

Output channel groups. The OutputGroup property lets you assign the output channels of MIMO systems into groups and refer to each group by name. Specify output groups as a structure. In this structure, field names are the group names, and field values are the output channels belonging to each group. For example:

sys.OutputGroup.temperature = [1];sys.OutputGroup.measurement = [3 5];

creates output groups named temperature and measurement that include output channels 1, and 3, 5, respectively. You can then extract the subsystem from all inputs to the measurement outputs using:

sys('measurement',:)

`Name` — System Name
`''` (default) | character vector

System name, specified as a character vector. For example, 'system 1'.

`Notes` — Notes on system
`0`-by-`1` string (default) | string | character vector

Any text that you want to associate with the system, specified as a string or a cell array of character vectors. The property stores whichever data type you provide. For instance, if sys1 and sys2 are dynamic system models, you can set their Notes properties as follows.

sys1.Notes = "sys1 has a string.";sys2.Notes = 'sys2 has a character vector.';sys1.Notessys2.Notes

ans = "sys1 has a string."ans = 'sys2 has a character vector.'

`UserData` — Data to associate with system
`[]` (default) | any MATLAB data type

Any data you want to associate with the system, specified as any MATLAB data type.

Object Functions

For information about object functions for idnlarx, see Nonlinear ARX Models.

Examples

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Create Nonlinear ARX Model Using ARX Model Orders

Open Live Script

Create an idnlarx model by specifying an ARX model order vector.

Create an order vector of the form [na nb nk], where na and nb are the orders of the A and B ARX model polynomials and nk is the number of input/output delays.

na = 2;nb = 3;nk = 5;orders = [na nb nk];

Construct a nonlinear ARX model sys.

output_name = 'y1';input_name = 'u1';sys = idnlarx(output_name,input_name,[2 3 5]);

View the output function.

disp(sys.OutputFcn)

Wavelet Network Nonlinear Function: Wavelet network with number of units chosen automatically Linear Function: uninitialized Output Offset: uninitialized NonlinearFcn: '<Wavelet and scaling function units and their parameters>' LinearFcn: '<Linear function parameters>' Offset: '<Offset parameters>' EstimationOptions: '<Estimation options>'

By default, the model uses a wavelet network, represented by a idWaveletNetwork object, for the output function. The idWaveletNetwork object includes linear and nonlinear components.

View the Regressors property.

disp(sys.Regressors)

Linear regressors in variables y1, u1 Variables: {'y1' 'u1'} Lags: {[1 2] [5 6 7]} UseAbsolute: [0 0] TimeVariable: 't'

The idnlarx constructor transforms the model orders into the Regressors form.

The Lags array for y1, [1 2], is equivalent to the na value of 2. Both forms specify two consecutive output regressors, y1(t-1) and y1(t-2).
The Lags array for u1, [5 6 7], incorporates the three delays specified by the nb value of 3, and shifts them by the nk value of 5. The input regressors are therefore u1(t-5), u1(t-6), and u1(t-7).

View the regressors.

getreg(sys)

ans = 5x1 cell {'y1(t-1)'} {'y1(t-2)'} {'u1(t-5)'} {'u1(t-6)'} {'u1(t-7)'}

You can use the orders syntax to specify simple linear regressors. However, to create more complex regressors, use the regressor commands linearRegressor, polynomialRegressor, and customRegressor to create a combined regressor for the Regressors syntax.

Create Nonlinear ARX Model Using Linear Regressors

Open Live Script

Construct an idnlarx model by specifying linear regressors.

Create a linear regressor that contains two output lags and two input lags.

output_name = 'y1';input_name = 'u1';var_names = {output_name,input_name};output_lag = [1 2];input_lag = [1 2];lags = {output_lag,input_lag};reg = linearRegressor(var_names,lags)

reg = Linear regressors in variables y1, u1 Variables: {'y1' 'u1'} Lags: {[1 2] [1 2]} UseAbsolute: [0 0] TimeVariable: 't'

The model contains the regressors y(t-1), y(t-2), u(t-1), and u(t-2).

Construct the idnlarx model and view the regressors.

sys = idnlarx(output_name,input_name,reg);getreg(sys)

ans = 4x1 cell {'y1(t-1)'} {'y1(t-2)'} {'u1(t-1)'} {'u1(t-2)'}

View the output function.

disp(sys.OutputFcn)

Wavelet Network Nonlinear Function: Wavelet network with number of units chosen automatically Linear Function: uninitialized Output Offset: uninitialized NonlinearFcn: '<Wavelet and scaling function units and their parameters>' LinearFcn: '<Linear function parameters>' Offset: '<Offset parameters>' EstimationOptions: '<Estimation options>'

View the regressor usage table.

disp(sys.RegressorUsage)

 y1:LinearFcn y1:NonlinearFcn ____________ _______________ y1(t-1) true true y1(t-2) true true u1(t-1) true true u1(t-2) true true

All the regressors are inputs to both the linear and nonlinear components of the idWaveletNetwork object.

Create and Configure Nonlinear ARX Model

Open Live Script

Create a nonlinear ARX model with a linear regressor set.

Create a linear regressor that contains three output lags and two input lags.

output_name = 'y1';input_name = 'u1';var_names = {output_name,input_name};output_lag = [1 2 3];input_lag = [1 2];lags = {output_lag,input_lag};reg = linearRegressor(var_names,lags)

reg = Linear regressors in variables y1, u1 Variables: {'y1' 'u1'} Lags: {[1 2 3] [1 2]} UseAbsolute: [0 0] TimeVariable: 't'

Construct the nonlinear ARX model.

sys = idnlarx(output_name,input_name,reg);

View the Regressors property.

disp(sys.Regressors)

Linear regressors in variables y1, u1 Variables: {'y1' 'u1'} Lags: {[1 2 3] [1 2]} UseAbsolute: [0 0] TimeVariable: 't'

sys uses idWavenetNetwork as the default output function. Reconfigure the output function to idSigmoidNetwork.

sys.OutputFcn = 'idSigmoidNetwork';disp(sys.OutputFcn)

Sigmoid Network Nonlinear Function: Sigmoid network with 10 units Linear Function: uninitialized Output Offset: uninitialized NonlinearFcn: '<Sigmoid units and their parameters>' LinearFcn: '<Linear function parameters>' Offset: '<Offset parameters>'

Specify Output Function for Nonlinear ARX Model

Open Live Script

Specify the sigmoid network output function when you construct a nonlinear ARX model.

Assign variable names and specify a regressor set.

output_name = 'y1';input_name = 'u1';r = linearRegressor({output_name,input_name},{1 1});

Construct a nonlinear ARX model that specifies the idSigmoidNetwork output function. Set the number of terms in the sigmoid expansion to 15.

sys = idnlarx(output_name,input_name,r,idSigmoidNetwork(15));

View the output function specification.

disp(sys.OutputFcn)

Sigmoid Network Nonlinear Function: Sigmoid network with 15 units Linear Function: uninitialized Output Offset: uninitialized NonlinearFcn: '<Sigmoid units and their parameters>' LinearFcn: '<Linear function parameters>' Offset: '<Offset parameters>'

Create Nonlinear ARX Model Without Nonlinear Mapping Function

Open Live Script

Construct an idnlarx model that uses only linear mapping in the output function. An argument value of [] is equivalent to an argument value of idLinear.

sys = idnlarx([2 2 1],[])

sys =Nonlinear ARX model with 1 output and 1 input Inputs: u1 Outputs: y1Regressors: Linear regressors in variables y1, u1Output function: Linear with offsetSample time: 1 secondsStatus: Created by direct construction or transformation. Not estimated.

Create and Combine Regressor Types

Open Live Script

Create a regressor set that includes linear, polynomial, periodic, and custom regressors.

Specify L as the set of linear regressors $y_{1} (t - 1)$ , $u_{1} (t - 2)$ , and $u_{1} (t - 5)$ .

L = linearRegressor({'y1','u1'},{1, [2 5]});

Specify P as the set of polynomial regressors $y_{2} {(t - 4)}^{2}$ , $y_{2} {(t - 5)}^{2}$ , $y_{2} {(t - 6)}^{2}$ , and $y_{2} {(t - 7)}^{2}$ .

P = polynomialRegressor('y2',4:7,2);

Specify SC as the set of periodic regressors $\sin (y_{1} (t - 1))$ , $\cos (y_{1} (t - 1))$ , $\sin (u_{1} (t - 2))$ , and $\cos (u_{1} (t - 2))$ .

SC = periodicRegressor({'y1','u1'},{1,2});

Specify C as the custom regressor $\sin (y_{1} (t - 1) u_{1} (t - 2) + u_{2} (t - 2))$ , using the @ symbol to create an anonymous function handle.

C = customRegressor({'y1','u1','u2'},{1 2 2},@(x,y,z)sin(x.*y+z));

Combine the regressors into one regressor set R.

R = [L;P;SC;C]

R = [4 1] array of linearRegressor, polynomialRegressor, periodicRegressor, customRegressor objects.------------------------------------1. Linear regressors in variables y1, u1 Variables: {'y1' 'u1'} Lags: {[1] [2 5]} UseAbsolute: [0 0] TimeVariable: 't'------------------------------------2. Order 2 regressors in variables y2 Order: 2 Variables: {'y2'} Lags: {[4 5 6 7]} UseAbsolute: [0] AllowVariableMix: [0] AllowLagMix: [0] TimeVariable: 't'------------------------------------3. Periodic regressors in variables y1, u1 with 1 Fourier terms Variables: {'y1' 'u1'} Lags: {[1] [2]} W: [1] NumTerms: [1] UseSin: [1] UseCos: [1] TimeVariable: 't' UseAbsolute: [0 0]------------------------------------4. Custom regressor: sin(y1(t-1).*u1(t-2)+u2(t-2)) VariablesToRegressorFcn: @(x,y,z)sin(x.*y+z) Variables: {'y1' 'u1' 'u2'} Lags: {[1] [2] [2]} Vectorized: [1] TimeVariable: 't'

Create a nonlinear ARX model.

sys = idnlarx({'y1','y2'},{'u1','u2'},R)

sys =Nonlinear ARX model with 2 outputs and 2 inputs Inputs: u1, u2 Outputs: y1, y2Regressors: 1. Linear regressors in variables y1, u1 2. Order 2 regressors in variables y2 3. Periodic regressors in variables y1, u1 with W = 1, and 1 Fourier terms 4. Custom regressor: sin(y1(t-1).*u1(t-2)+u2(t-2))Output functions: Output 1: Wavelet network with number of units chosen automatically Output 2: Wavelet network with number of units chosen automaticallySample time: 1 secondsStatus: Created by direct construction or transformation. Not estimated.

Create Nonlinear ARX Model Using Linear Model

Open Live Script

Use a linear ARX model instead of a regressor set to construct a nonlinear ARX model.

Construct a linear ARX model using idpoly.

A = [1 -1.2 0.5];B = [0.8 1];LinearModel = idpoly(A, B, 'Ts', 0.1);

Specify input and output names for the model using dot notation.

LinearModel.OutputName = 'y1';LinearModel.InputName = 'u1';

Construct a nonlinear ARX model using the linear ARX model.

m1 = idnlarx(LinearModel)

m1 =Nonlinear ARX model with 1 output and 1 input Inputs: u1 Outputs: y1Regressors: Linear regressors in variables y1, u1Output function: Wavelet network with number of units chosen automaticallySample time: 0.1 secondsStatus: Created by direct construction or transformation. Not estimated.

You can create a linear ARX model from any identified discrete-time linear model.

Estimate a second-order state-space model from estimation data z1.

load iddata1 z1ssModel = ssest(z1,2,'Ts',0.1);

Construct a nonlinear ARX model from ssModel. The software uses the input and output names that ssModel extracts from z1.

m2 = idnlarx(ssModel)

m2 =Nonlinear ARX model with 1 output and 1 input Inputs: u1 Outputs: y1Regressors: Linear regressors in variables y1, u1Output function: Wavelet network with number of units chosen automaticallySample time: 0.1 secondsStatus: Created by direct construction or transformation. Not estimated.

Modify Regressor Assignments to Output Function Components

Open Live Script

Modify regressor assignments by modifying the RegressorUsage table.

Construct a nonlinear ARX model that has two inputs and two outputs.

Create the variable names and the regressors.

varnames = {'y1','y2','u1','u2'};lags = {[1 2 3],[1 2],[1 2],[1 3]};reg = linearRegressor(varnames,lags);

Create an output function specification fcn that uses idWaveletNetwork for mapping regressors to output y1 and idSigmoidNetwork for mapping regressors to output y2. Both mapping objects contain linear and nonlinear components.

fcn = [idWaveletNetwork;idSigmoidNetwork];

Construct the nonlinear ARX model.

output_name = {'y1' 'y2'};input_name = {'u1' 'u2'};sys = idnlarx(output_name,input_name,reg,fcn)

sys =Nonlinear ARX model with 2 outputs and 2 inputs Inputs: u1, u2 Outputs: y1, y2Regressors: Linear regressors in variables y1, y2, u1, u2Output functions: Output 1: Wavelet network with number of units chosen automatically Output 2: Sigmoid network with 10 unitsSample time: 1 secondsStatus: Created by direct construction or transformation. Not estimated.

Display the RegressorUsage table.

disp(sys.RegressorUsage)

 y1:LinearFcn y1:NonlinearFcn y2:LinearFcn y2:NonlinearFcn ____________ _______________ ____________ _______________ y1(t-1) true true true true y1(t-2) true true true true y1(t-3) true true true true y2(t-1) true true true true y2(t-2) true true true true u1(t-1) true true true true u1(t-2) true true true true u2(t-1) true true true true u2(t-3) true true true true

The rows of the table represent the regressors. The first two columns of the table represent the linear and nonlinear components of the mapping to output y1 (idWaveletNetwork). The last two columns represent the two components of the mapping to output y2 (idSigmoidNetwork).

In this table, all the input and output regressors are inputs to all components.

Remove the y2(t-2) regressor from the y2 nonlinear component.

sys.RegressorUsage{4,4} = false;disp(sys.RegressorUsage)

 y1:LinearFcn y1:NonlinearFcn y2:LinearFcn y2:NonlinearFcn ____________ _______________ ____________ _______________ y1(t-1) true true true true y1(t-2) true true true true y1(t-3) true true true true y2(t-1) true true true false y2(t-2) true true true true u1(t-1) true true true true u1(t-2) true true true true u2(t-1) true true true true u2(t-3) true true true true

The table displays false for this regressor-component pair.

Store the regressor names in Names.

Names = sys.RegressorUsage.Properties.RowNames;

Determine the indices of the rows that contain y1 or y2 and set the corresponding values of y1:NonlinearFcn to False.

idx = contains(Names,'y1')|contains(Names,'y2');sys.RegressorUsage{idx,2} = false;disp(sys.RegressorUsage)

 y1:LinearFcn y1:NonlinearFcn y2:LinearFcn y2:NonlinearFcn ____________ _______________ ____________ _______________ y1(t-1) true false true true y1(t-2) true false true true y1(t-3) true false true true y2(t-1) true false true false y2(t-2) true false true true u1(t-1) true true true true u1(t-2) true true true true u2(t-1) true true true true u2(t-3) true true true true

The table values reflect the new assignments.

The RegressorUsage table provides complete flexibility for individually controlling regressor assignments.

More About

expand all

Definition of idnlarx States

The states of an idnlarx object are an ordered list of delayed input and output variables that define the structure of the model. The toolbox uses this definition of states for creating the initial state vector that sim, predict, and compare use for simulation and prediction. idnlarx/linearize also uses this definition for linearization of nonlinear ARX models.

This toolbox provides several options to facilitate how you specify the initial states. For example, you can use findstates and data2state to search for state values in simulation and prediction applications. For linearization, use findop. You can also specify the states manually.

The states of an idnlarx model depend on the maximum delay in each input and output variable used by the regressors. If a variable p has a maximum delay of D samples, then it contributes D elements to the state vector at time t: p(t–1), p(t–2), ..., p(t–D).

For example, if you have a single-input, single-output idnlarx model.

m = idnlarx([2 3 0],'idWaveletNetwork','CustomRegressors',{'y1(t-10)*u1(t-1)'});

This model has these regressors.

getreg(m)

ans = 6x1 cell {'y1(t-1)' } {'y1(t-2)' } {'u1(t)' } {'u1(t-1)' } {'u1(t-2)' } {'y1(t-10)*u1(t-1)'}

The regressors show that the maximum delay in the output variable y1 is 10 samples and the maximum delay in the input u1 is two samples. Thus, this model has a total of 12 states:

X(t) = [y1(t-1),y2(t-2),…,y1(t-10),u1(t-1),u1(t-2)] (1)

Note

The state vector includes the output variables first, followed by input variables.

As another example, consider the 2-output and 3-input model.

m = idnlarx([2 0 2 2 1 1 0 0; 1 0 1 5 0 1 1 0],[idWaveletNetwork; idLinear]);

This model has these regressors.

getreg(m)

ans = 11x1 cell {'y1(t-1)'} {'y1(t-2)'} {'u1(t-1)'} {'u1(t-2)'} {'u2(t)' } {'u2(t-1)'} {'u2(t-2)'} {'u2(t-3)'} {'u2(t-4)'} {'u2(t-5)'} {'u3(t)' }

The maximum delay in output variable y1 is two samples. This delay occurs in the regressor set for output 1. The maximum delays in the three input variables are 2, 5, and 0, respectively. Thus, the state vector is:

X(t) = [y1(t-1), y1(t-2), u1(t-1), u1(t-2), u2(t-1), u2(t-2), u2(t-3), u2(t-4), u2(t-5)]

Variables y2 and u3 do not contribute to the state vector because the maximum delay in these variables is zero.

A simpler way to determine states by inspecting regressors is to use getDelayInfo, which returns the maximum delays in all I/O variables across all model outputs. For the multi-input multi-output model m, getDelayInfo returns:

maxDel = getDelayInfo(m)

maxDel = 1×5 2 0 2 5 0

maxDel contains the maximum delays for all input and output variables in the order (y1, y2, u1, u2, u3). The total number of model states is sum(maxDel) = 9.

The set of states for an idnlarx model is not required to be minimal.

Version History

Introduced in R2007a

expand all

R2023b: New neural network mapping object creates neural networks from both Statistics and Machine Learning Toolbox and Deep Learning Toolbox

The idNeuralNetwork mapping object creates neural networks using both the regression networks of Statistics and Machine Learning Toolbox and the shallow or deep networks of Deep Learning Toolbox. This mapping object replaces and enhances the functionality of idFeedforwardNetwork, which is limited to the shallow networks of Deep Learning Toolbox. For more information, see idNeuralNetwork.

R2022a: Normalization and regressor selection moved from mapping object properties to `idnlarx` object

Information related to data normalization was moved from the mapping object properties to the idnlarx Normalization property. In addition, the regressor-selection process for the mapping objects was moved to idnlarx. The model now passes the actual regressor names rather than the selection indices to the mapping object.

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 `idnlarx` properties is not recommended.

Starting in R2021a, several properties of idnlarx have been modified or replaced.

These changes affect the syntaxes in both idnlarx and nlarx. The use of the pre-R2021a properties in the following table is discouraged. However, the software still accepts calling syntaxes that include these properties. There are no plans to exclude these syntaxes at this time. The command syntax that uses ARX model orders continues be a recommended syntax.

Pre-R2021a Property	R2021a Property	Usage
ARX model orders `na,nb,nk`	`Regressors`, which can include linearRegressor, polynomialRegressor, and customRegressor objects.	`na,nb,nk` remains a valid `idnlarx` and `nlarx` input argument that the software converts to a `linearRegressor` object. You can no longer change order values in an existing `idnlarx` model by dot assignment or by using the `set` function. Create a new model object instead.
`customRegressors`	`Regressors`	Use polynomialRegressor or customRegressor to create regressor objects and add the objects to the `Regressors` array.
`NonlinearRegressors`	`RegressorUsage`	`RegressorUsage` is a table that contains regressor assignments to linear and nonlinear output components. Change assignments by modifying the corresponding `RegressorUsage`table entries.
`Nonlinearity`	`OutputFcn`	Change is in name only. Property remains an object or an array or objects that map regressor inputs to an output.

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Nonlinear ARX model - MATLAB (2024)

Description

Creation

Syntax

Description

Specify Model Directly

Initialize Model Values Using Linear Model

Specify Model Properties

Input Arguments

orders — ARX model ordersnlarx orders [na nb nk]

Properties

Regressors — Regressor specification linearRegressor object | polynomialRegressor object | periodicRegressor object | customRegressor object | column array of regressor specification objects

OutputFcn — Output function 'idWaveletNetwork' (default) | 'idLinear' | [] | '' | 'idSigmoidNetwork' | 'idTreePartition' | 'idGaussianProcess' | 'idTreeEnsemble' | 'idSupportVectorMachine' | mapping object | array of mapping objects

RegressorUsage — Regressor assignments table with logical entries

Normalization — Regressor and output data centering and scaling structure (default)

Report — Summary report report field values

TimeVariable — Independent variable 't' (default) | character vector

NoiseVariance — Noise variance matrix

Ts — Sample time 1 (default) | positive scalar

TimeUnit — Units for time variable 'seconds' (default) | 'nanoseconds' | 'microseconds' | 'milliseconds' | 'minutes' | 'hours' | 'days' | 'weeks' | 'months' | 'years'

InputName — Input channel names '' for all input channels (default) | character vector | cell array of character vectors

InputUnit — Input channel units '' for all input channels (default) | character vector | cell array of character vectors

InputGroup — Input channel groups structure with no fields (default) | structure

OutputName — Output channel names '' for all output channels (default) | character vector | cell array of character vectors

OutputUnit — Output channel units '' for all output channels (default) | character vector | cell array of character vectors

OutputGroup — Output channel groups structure with no fields (default) | structure

Name — System Name '' (default) | character vector

Notes — Notes on system 0-by-1 string (default) | string | character vector

UserData — Data to associate with system [] (default) | any MATLAB data type

Object Functions

Examples

Create Nonlinear ARX Model Using ARX Model Orders

Create Nonlinear ARX Model Using Linear Regressors

Create and Configure Nonlinear ARX Model

Specify Output Function for Nonlinear ARX Model

Create Nonlinear ARX Model Without Nonlinear Mapping Function

Create and Combine Regressor Types

Create Nonlinear ARX Model Using Linear Model

Modify Regressor Assignments to Output Function Components

More About

Definition of idnlarx States

Version History

R2023b: New neural network mapping object creates neural networks from both Statistics and Machine Learning Toolbox and Deep Learning Toolbox

R2022a: Normalization and regressor selection moved from mapping object properties to idnlarx object

R2021b: Use of previous idnlarx and idnlhw mapping object names is not recommended.

R2021a: Use of previous idnlarx properties is not recommended.

See Also

Topics

MATLAB Command

Americas

Europe

Asia Pacific

References

`orders` — ARX model orders
`nlarx` orders `[na nb nk]`

`Regressors` — Regressor specification
`linearRegressor` object | `polynomialRegressor` object | `periodicRegressor` object | `customRegressor` object | column array of regressor specification objects

`OutputFcn` — Output function
`'idWaveletNetwork'` (default) | `'idLinear'` | `[]` | `''` | `'idSigmoidNetwork'` | `'idTreePartition'` | `'idGaussianProcess'` | `'idTreeEnsemble'` | `'idSupportVectorMachine'` | mapping object | array of mapping objects

`RegressorUsage` — Regressor assignments
table with logical entries

`Normalization` — Regressor and output data centering and scaling
structure (default)

`Report` — Summary report
report field values

`TimeVariable` — Independent variable
`'t'` (default) | character vector

`NoiseVariance` — Noise variance
matrix

`Ts` — Sample time
`1` (default) | positive scalar

`TimeUnit` — Units for time variable
`'seconds'` (default) | `'nanoseconds'` | `'microseconds'` | `'milliseconds'` | `'minutes'` | `'hours'` | `'days'` | `'weeks'` | `'months'` | `'years'`

`InputName` — Input channel names
`''` for all input channels (default) | character vector | cell array of character vectors

`InputUnit` — Input channel units
`''` for all input channels (default) | character vector | cell array of character vectors

`InputGroup` — Input channel groups
structure with no fields (default) | structure

`OutputName` — Output channel names
`''` for all output channels (default) | character vector | cell array of character vectors

`OutputUnit` — Output channel units
`''` for all output channels (default) | character vector | cell array of character vectors

`OutputGroup` — Output channel groups
structure with no fields (default) | structure

`Name` — System Name
`''` (default) | character vector

`Notes` — Notes on system
`0`-by-`1` string (default) | string | character vector

`UserData` — Data to associate with system
`[]` (default) | any MATLAB data type

R2022a: Normalization and regressor selection moved from mapping object properties to `idnlarx` object

R2021b: Use of previous `idnlarx` and `idnlhw` mapping object names is not recommended.

R2021a: Use of previous `idnlarx` properties is not recommended.