idnlarx
Nonlinear ARX model
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 in two 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 thepolynomialRegressor
object. To create periodic regressors that contain the sine and cosine functions of delayed input and output variables , use theperiodicRegressor
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 thecustomRegressor
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:
Here, x is a vector of the regressors, and r is the mean of x. is the output of the linear function block. 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.
Syntax
Description
Specify Model Directly
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.sys
= idnlarx(output_name
,input_name
,orders
)
creates a nonlinear ARX model with the output and input names of
sys
= idnlarx(output_name
,input_name
,Regressors)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.
Initialize Model Values Using Linear Model
uses a linear model sys
= idnlarx(linmodel
)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.
Specify Model Properties
specifies additional properties of
the sys
= idnlarx(___,Name,Value
)idnlarx
model structure using one or more name-value arguments.
Input Arguments
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.
linmodel
— Discrete-time linear model
idpoly
object | idss
object | idtf
object | idproc
object
Discrete-time identified input/output linear model, specified as any linear model
created using estimators, that is, an idpoly
object, an idss
object, an idtf
object, or an idproc
object with Ts
> 0. Create this model using the
constructor function for the object or estimate the model using the associated
estimation command. For example, to create an ARX model, use arx
, and specify the resulting idpoly
object as
linmodel
.
Properties
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 ny output channels, you can specify
mapping objects individually for each channel by setting OutputFcn
to an array of ny 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]
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 , 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
nr-by-nc
table with logical entries that specify which regressors to use for which component.
Here, nr is the number of regressors.
nc 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 (nr) or model outputs (ny).
Field | Description | Default Element Value |
---|---|---|
RegressorCenter | Row vector of length nr | NaN |
RegressorScale | Row vector of length nr | NaN |
OutputCenter | Row vector of length ny | NaN |
OutputScale | Row vector of length ny | 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 RN(t).
RN(t) provides the input to the mapping function, which then produces the normalized output yN
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
ans = []
If you use nlarx
to estimate the model, the fields of
Report
contain information on the estimation data, options, and
results.
load iddata1;
sys = nlarx(z1,reg);
m.Report.OptionsUsed
Option set for the nlarx command: IterativeWavenet: 'auto' Focus: 'prediction' Display: 'off' Regularization: [1x1 struct] SearchMethod: 'auto' SearchOptions: [1x1 idoptions.search.identsolver] OutputWeight: 'noise' Advanced: [1x1 struct]
For more information on this property and how to use it, see Output Arguments in the nlarx
reference page and Estimation Report.
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.Notes sys2.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
Create Nonlinear ARX Model Using ARX Model Orders
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 L
ags
array fory1
,[1 2]
, is equivalent to thena
value of 2. Both forms specify two consecutive output regressors,y1(t-1)
andy1(t-2)
.The
Lags
array foru1
,[5 6 7]
, incorporates the three delays specified by thenb
value of 3, and shifts them by thenk
value of 5. The input regressors are thereforeu1(t-5)
,u1(t-6)
, andu1(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
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
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
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
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: y1 Regressors: Linear regressors in variables y1, u1 Output function: Linear with offset Sample time: 1 seconds Status: Created by direct construction or transformation. Not estimated.
Create and Combine Regressor Types
Create a regressor set that includes linear, polynomial, periodic, and custom regressors.
Specify L
as the set of linear regressors , , and .
L = linearRegressor({'y1','u1'},{1, [2 5]});
Specify P
as the set of polynomial regressors , ,, and .
P = polynomialRegressor('y2',4:7,2);
Specify SC as the set of periodic regressors , , , and .
SC = periodicRegressor({'y1','u1'},{1,2});
Specify C
as the custom regressor , 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, y2 Regressors: 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 automatically Sample time: 1 seconds Status: Created by direct construction or transformation. Not estimated.
Create Nonlinear ARX Model Using Linear Model
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: y1 Regressors: Linear regressors in variables y1, u1 Output function: Wavelet network with number of units chosen automatically Sample time: 0.1 seconds Status: 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 z1 ssModel = 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: y1 Regressors: Linear regressors in variables y1, u1 Output function: Wavelet network with number of units chosen automatically Sample time: 0.1 seconds Status: Created by direct construction or transformation. Not estimated.
Modify Regressor Assignments to Output Function Components
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, y2 Regressors: Linear regressors in variables y1, y2, u1, u2 Output functions: Output 1: Wavelet network with number of units chosen automatically Output 2: Sigmoid network with 10 units Sample time: 1 seconds Status: 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
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 R2007aR2023b: 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. |
You can no longer change
order values in an existing |
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. |
See Also
nlarx
| linearRegressor
| polynomialRegressor
| periodicRegressor
| customRegressor
| idnlarx/findop
| getreg
| idnlarx/linearize
| pem
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