Estimate parameters of nonlinear ARX model
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Syntax
sys = nlarx(data,orders)
sys = nlarx(data,regressors)
sys = nlarx(___,output_fcn)
sys = nlarx(data,linmodel)
sys = nlarx(data,linmodel,output_fcn)
sys = nlarx(data,sys0)
sys = nlarx(___,Options)
sys = nlarx(___,Name,Value)
Description
Estimate New Nonlinear ARX Model
example
sys = nlarx(data,orders)
estimates a nonlinear ARX model to fit the given estimation data data
using the specified ARX model orders orders
and the default wavelet network output function.
data
can be in the form of a timetable, a comma-separated pair of numeric matrices, a single numeric matrix, or an iddata object. If data
is a timetable, you can select specific input and output channels to use for estimation by specifying the channel names in the InputName and OutputName name-value arguments.
Use this syntax when you extend an ARX linear model, or when you use only regressors that are linear with consecutive lags.
example
sys = nlarx(data,regressors)
estimates a nonlinear ARX model using the specified regressor set regressors
.
Use this syntax when you have linear regressors that have nonconsecutive lags, or when you also have any combination of polynomial regressors, periodic regressors, and custom regressors.
When you use regressors
to estimate a MIMO system and data
is a timetable, you must also specify the output variables in the OutputName name-value argument.
example
sys = nlarx(___,output_fcn)
specifies the output function that maps the regressors to the model output. You can use this syntax with any of the previous input argument combinations.
Extend Existing Linear Model
example
sys = nlarx(data,linmodel)
uses a linear ARX model linmodel
to specify the model orders and the initial 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.
When you use this syntax, the software initializes the offset value to 0
. In some cases, you can improve the estimation results by overriding this initialization with the command sys.OutputFcn.Offset.Value = NaN
.
example
sys = nlarx(data,linmodel,output_fcn)
specifies the output function to use for model estimation.
Refine Existing Nonlinear ARX Model
example
sys = nlarx(data,sys0)
estimates or refines the parameters of the nonlinear ARX model sys0
.
Use this syntax to:
Estimate the parameters of a model previously created using the
idnlarx
constructor. Prior to estimation, you can configure the model properties using dot notation.Update the parameters of a previously estimated model to improve the fit to the estimation data. In this case, the estimation algorithm uses the parameters of
sys0
as initial guesses.
Specify Additional Options
example
sys = nlarx(___,Options)
specifies additional configuration options for the model estimation.
sys = nlarx(___,Name,Value)
uses additional model options specified by one or more name-value arguments. For example, specify the input and output signal variable names that correspond with the variables to use from MIMO timetable data using sys = nlarx(data,regressors,'InputName',["u1","u3"],'OutputName',["y1","y4"])
.
Examples
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Estimate Nonlinear ARX Model
Open Live Script
Load the estimation data.
load twotankdata;
Create an iddata
object from the estimation data with a sample time of 0.2 seconds.
Ts = 0.2;z = iddata(y,u,Ts);
Estimate the nonlinear ARX model using ARX model orders to specify the regressors.
sysNL = nlarx(z,[4 4 1])
sysNL =Nonlinear ARX model with 1 output and 1 input Inputs: u1 Outputs: y1Regressors: Linear regressors in variables y1, u1Output function: Wavelet network with 11 unitsSample time: 0.2 secondsStatus: Estimated using NLARX on time domain data "z". Fit to estimation data: 96.84% (prediction focus)FPE: 3.482e-05, MSE: 3.431e-05
sys
uses the default idWaveletNetwork
function as the output function.
For comparison, compute a linear ARX model with the same model orders.
sysL = arx(z,[4 4 1]);
Compare the model outputs with the original data.
compare(z,sysNL,sysL)
The nonlinear model has a much better fit to the data than the linear model.
Estimate Nonlinear ARX Model Using Linear Regressor Set
Open Live Script
Specify a linear regressor that is equivalent to an ARX model order matrix of [4 4 1]
.
An order matrix of [4 4 1]
specifies that both input and output regressor sets contain four regressors with lags ranging from 1 to 4. For example, represents the second input regressor.
Specify the output and input names.
output_name = 'y1';input_name = 'u1';names = {output_name,input_name};
Specify the output and input lags.
output_lag = [1 2 3 4];input_lag = [1 2 3 4];lags = {output_lag,input_lag};
Create the linear regressor object.
lreg = linearRegressor(names,lags)
lreg = Linear regressors in variables y1, u1 Variables: {'y1' 'u1'} Lags: {[1 2 3 4] [1 2 3 4]} UseAbsolute: [0 0] TimeVariable: 't'
Load the estimation data and create an iddata object.
load twotankdataz = iddata(y,u,0.2);
Estimate the nonlinear ARX model.
sys = nlarx(z,lreg)
sys =Nonlinear ARX model with 1 output and 1 input Inputs: u1 Outputs: y1Regressors: Linear regressors in variables y1, u1Output function: Wavelet network with 11 unitsSample time: 0.2 secondsStatus: Estimated using NLARX on time domain data "z". Fit to estimation data: 96.84% (prediction focus)FPE: 3.482e-05, MSE: 3.431e-05
View the regressors.
getreg(sys)
ans = 8x1 cell {'y1(t-1)'} {'y1(t-2)'} {'y1(t-3)'} {'y1(t-4)'} {'u1(t-1)'} {'u1(t-2)'} {'u1(t-3)'} {'u1(t-4)'}
Compare the model output to the estimation data.
compare(z,sys)
Estimate Nonlinear ARX Model from Time Series Data
Open Live Script
Create time and data arrays.
dt = 0.01;t = 0:dt:10;y = 10*sin(2*pi*t)+rand(size(t));
Create an iddata
object with no input signal specified.
z = iddata(y',[],dt);
Estimate the nonlinear ARX model.
sys = nlarx(z,2)
sys =Nonlinear time series model Outputs: y1Regressors: Linear regressors in variables y1Output function: Wavelet network with 8 unitsSample time: 0.01 secondsStatus: Estimated using NLARX on time domain data "z". Fit to estimation data: 92.92% (prediction focus)FPE: 0.2568, MSE: 0.2507
Specify and Customize Output Function
Open Live Script
Estimate a nonlinear ARX model that uses the mapping function idSigmoidNetwork
as its output function.
Load the data and divide it into the estimation and validation data sets ze
and zv
.
load twotankdata.mat u yz = iddata(y,u,'Ts',0.2);ze = z(1:1500);zv = z(1501:end);
Configure the idSigmoidNetwork
mapping function. Fix the offset to 0.2 and the number of units to 15.
s = idSigmoidNetwork;s.Offset.Value = 0.2;s. NonlinearFcn.NumberOfUnits = 15;
Create a linear model regressor specification that contains four output regressors and five input regressors.
reg1 = linearRegressor({'y1','u1'},{1:4,0:4});
Create a polynomial model regressor specification that contains the squares of two output terms and three input terms.
reg2 = polynomialRegressor({'y1','u1'},{1:2,0:2},2);
Set estimation options for the search method and maximum number of iterations.
opt = nlarxOptions('SearchMethod','fmincon')';opt.SearchOptions.MaxIterations = 40;
Estimate the nonlinear ARX model.
sys = nlarx(ze,[reg1;reg2],s,opt);
Validate sys
by comparing the simulated model response to the validation data set.
compare(zv,sys)
Add Output Function to Extend and Improve Linear Model
Open Live Script
Estimate a linear model and improve the model by adding an idTreePartition
output function.
Load the estimation data.
load throttledata ThrottleData
Estimate a linear ARX model linsys
with orders [2 2 1]
.
linsys = arx(ThrottleData,[2 2 1]);
Create an idnlarx
template model that uses linsys
and specifies idTreePartition
as the output function.
sys0 = idnlarx(linsys,idTreePartition);
Fix the linear component of sys0
so that during estimation, the linear portion of sys0
remains identical to linsys
. Set the offset component value to NaN
.
sys0.OutputFcn.LinearFcn.Free = false;sys0.OutputFcn.Offset.Value = NaN;
Estimate the free parameters of sys0
, which are the nonlinear-function parameters and the offset.
sys = nlarx(ThrottleData,sys0);
Compare the fit accuracies for the linear and nonlinear models.
compare(ThrottleData,linsys,sys)
Estimate Nonlinear ARX Model Using Custom Network Mapping Object
Open Script
Generating a custom network mapping object requires the definition of a user-defined unit function.
Define the unit function and save it as gaussunit.m
.
function [f,g,a] = gaussunit(x)% Custom unit function nonlinearity.%% Copyright 2015 The MathWorks, Inc.f = exp(-x.*x);if nargout>1 g = -2*x.*f; a = 0.2;end
Create a custom network mapping object using a handle to the gaussunit
function.
H = @gaussunit;CNet = idCustomNetwork(H);
Load the estimation data.
load iddata1
Estimate a nonlinear ARX model using the custom network.
sys = nlarx(z1,[1 2 1],CNet)
sys =<strong>Nonlinear ARX model with 1 output and 1 input</strong> Inputs: u1 Outputs: y1Regressors: Linear regressors in variables y1, u1Output function: Custom Network with 10 units and "gaussunit" unit functionSample time: 0.1 secondsStatus: Estimated using NLARX on time domain data "z1". Fit to estimation data: 64.35% (prediction focus)FPE: 3.58, MSE: 2.465
Estimate MIMO Nonlinear ARX Model
Open Live Script
Load the estimation data.
load motorizedcamera;
Create an iddata
object.
z = iddata(y,u,0.02,'Name','Motorized Camera','TimeUnit','s');
z
is an iddata
object with six inputs and two outputs.
Specify the model orders.
Orders = [ones(2,2),2*ones(2,6),ones(2,6)];
Specify different mapping functions for each output channel.
NL = [idWaveletNetwork(2),idLinear];
Estimate the nonlinear ARX model.
sys = nlarx(z,Orders,NL)
sys =Nonlinear ARX model with 2 outputs and 6 inputs Inputs: u1, u2, u3, u4, u5, u6 Outputs: y1, y2Regressors: Linear regressors in variables y1, y2, u1, u2, u3, u4, u5, u6Output functions: Output 1: Wavelet network with 2 units Output 2: Linear with offsetSample time: 0.02 secondsStatus: Estimated using NLARX on time domain data "Motorized Camera".Fit to estimation data: [98.82;98.77]% (prediction focus) FPE: 0.4839, MSE: 0.9762
Estimate MIMO Nonlinear ARX Model with Same Mapping Function for All Outputs
Open Live Script
Load the estimation data and create an iddata
object z
. z
contains two output channels and six input channels.
load motorizedcamera;z = iddata(y,u,0.02);
Specify a set of linear regressors that uses the output and input names from z
and contains:
2 output regressors with 1 lag.
6 input regressor pairs with 1 and 2 lags.
names = [z.OutputName; z.InputName];lags = {1,1,[1,2],[1,2],[1,2],[1,2],[1,2],[1,2]};reg = linearRegressor(names,lags);
Estimate a nonlinear ARX model using an idSigmoidNetwork
mapping function with four units for all output channels.
sys = nlarx(z,reg,idSigmoidNetwork(4))
sys =Nonlinear ARX model with 2 outputs and 6 inputs Inputs: u1, u2, u3, u4, u5, u6 Outputs: y1, y2Regressors: Linear regressors in variables y1, y2, u1, u2, u3, u4, u5, u6Output functions: Output 1: Sigmoid network with 4 units Output 2: Sigmoid network with 4 unitsSample time: 0.02 secondsStatus: Estimated using NLARX on time domain data "z". Fit to estimation data: [98.86;98.79]% (prediction focus)FPE: 2.641, MSE: 0.9233
Specify Linear, Polynomial, and Custom Regressors
Open Live Script
Load the estimation data z1
, which has one input and one output, and obtain the output and input names.
load iddata1 z1;names = [z1.OutputName z1.InputName]
names = 1x2 cell {'y1'} {'u1'}
Specify L
as the set of linear regressors that represents , , and .
L = linearRegressor(names,{1,[2 5]});
Specify P
as the polynomial regressor .
P = polynomialRegressor(names(1),1,2);
Specify C
as the custom regressor . Use an anonymous function handle to define this function.
C = customRegressor(names,{2 3},@(x,y)x.*y)
C = Custom regressor: y1(t-2).*u1(t-3) VariablesToRegressorFcn: @(x,y)x.*y Variables: {'y1' 'u1'} Lags: {[2] [3]} Vectorized: [1] TimeVariable: 't'
Combine the regressors in the column vector R
.
R = [L;P;C]
R = [3 1] array of linearRegressor, polynomialRegressor, 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 y1 Order: 2 Variables: {'y1'} Lags: {[1]} UseAbsolute: [0] AllowVariableMix: [0] AllowLagMix: [0] TimeVariable: 't'------------------------------------3. Custom regressor: y1(t-2).*u1(t-3) VariablesToRegressorFcn: @(x,y)x.*y Variables: {'y1' 'u1'} Lags: {[2] [3]} Vectorized: [1] TimeVariable: 't'
Estimate a nonlinear ARX model with R
.
sys = nlarx(z1,R)
sys =Nonlinear ARX model with 1 output and 1 input Inputs: u1 Outputs: y1Regressors: 1. Linear regressors in variables y1, u1 2. Order 2 regressors in variables y1 3. Custom regressor: y1(t-2).*u1(t-3)Output function: Wavelet network with 1 unitsSample time: 0.1 secondsStatus: Estimated using NLARX on time domain data "z1". Fit to estimation data: 59.73% (prediction focus)FPE: 3.356, MSE: 3.147
View the full regressor set.
getreg(sys)
ans = 5x1 cell {'y1(t-1)' } {'u1(t-2)' } {'u1(t-5)' } {'y1(t-1)^2' } {'y1(t-2).*u1(t-3)'}
Estimate Nonlinear ARX Model with No Linear Term in Output Function
Open Live Script
Load the estimation data.
load iddata1;
Create a sigmoid network mapping object with 10 units and no linear term.
SN = idSigmoidNetwork(10,false);
Estimate the nonlinear ARX model. Confirm that the model does not use the linear function.
sys = nlarx(z1,[2 2 1],SN);sys.OutputFcn.LinearFcn.Use
ans = logical 0
Specify Nonlinear ARX Orders and Linear Parameters Using Linear ARX Model
Open Live Script
Load the estimation data.
load throttledata;
Detrend the data.
Tr = getTrend(ThrottleData);Tr.OutputOffset = 15;DetrendedData = detrend(ThrottleData,Tr);
Estimate the linear ARX model.
LinearModel = arx(DetrendedData,[2 1 1]);
Estimate the nonlinear ARX model using the linear model. The model orders, delays, and linear parameters of NonlinearModel
are derived from LinearModel
.
NonlinearModel = nlarx(ThrottleData,LinearModel)
NonlinearModel =Nonlinear ARX model with 1 output and 1 input Inputs: Step Command Outputs: Throttle Valve PositionRegressors: Linear regressors in variables Throttle Valve Position, Step CommandOutput function: Wavelet network with 7 unitsSample time: 0.01 secondsStatus: Estimated using NLARX on time domain data "ThrottleData".Fit to estimation data: 99.03% (prediction focus) FPE: 0.1127, MSE: 0.1039
Estimate Nonlinear ARX Model Using Constructed idnlarx
Object
Open Live Script
Load the estimation data.
load iddata1;
Create an idnlarx
model.
sys = idnlarx([2 2 1]);
Configure the model using dot notation to:
Use a sigmoid network mapping object.
Assign a name.
sys.Nonlinearity = 'idSigmoidNetwork';sys.Name = 'Model 1';
Estimate a nonlinear ARX model with the structure and properties specified in the idnlarx
object.
sys = nlarx(z1,sys)
sys =Nonlinear ARX model with 1 output and 1 input Inputs: u1 Outputs: y1Regressors: Linear regressors in variables y1, u1Output function: Sigmoid network with 10 unitsName: Model 1Sample time: 0.1 secondsStatus: Estimated using NLARX on time domain data "z1". Fit to estimation data: 69.03% (prediction focus)FPE: 2.918, MSE: 1.86
Estimate Nonlinear ARX Model and Avoid Local Minima
Open Live Script
If an estimation stops at a local minimum, you can perturb the model using init
and re-estimate the model.
Load the estimation data.
load iddata1;
Estimate the initial nonlinear model.
sys1 = nlarx(z1,[4 2 1],'idSigmoidNetwork');
Randomly perturb the model parameters to avoid local minima.
sys2 = init(sys1);
Estimate the new nonlinear model with the perturbed values.
sys2 = nlarx(z1,sys1);
Estimate Nonlinear ARX Model Using Specific Options
Open Live Script
Load the estimation data.
load twotankdata;
Create an iddata
object from the estimation data.
z = iddata(y,u,0.2);
Create an nlarxOptions
option set specifying a simulation error minimization objective and a maximum of 10 estimation iterations.
opt = nlarxOptions;opt.Focus = 'simulation';opt.SearchOptions.MaxIterations = 10;
Estimate the nonlinear ARX model.
sys = nlarx(z,[4 4 1],idSigmoidNetwork(3),opt)
sys =Nonlinear ARX model with 1 output and 1 input Inputs: u1 Outputs: y1Regressors: Linear regressors in variables y1, u1Output function: Sigmoid network with 3 unitsSample time: 0.2 secondsStatus: Estimated using NLARX on time domain data "z". Fit to estimation data: 85.86% (simulation focus)FPE: 3.791e-05, MSE: 0.0006853
Estimate Regularized Nonlinear ARX Model with Large Number of Units
Open Live Script
Load the regularization example data.
load regularizationExampleData.mat nldata;
Create an idSigmoidnetwork
mapping object with 30 units and specify the model orders.
MO = idSigmoidNetwork(30);Orders = [1 2 1];
Create an estimation option set and set the estimation search method to lm
.
opt = nlarxOptions('SearchMethod','lm');
Estimate an unregularized model.
sys = nlarx(nldata,Orders,MO,opt);
Configure the regularization Lambda
parameter.
opt.Regularization.Lambda = 1e-8;
Estimate a regularized model.
sysR = nlarx(nldata,Orders,MO,opt);
Compare the two models.
compare(nldata,sys,sysR)
The large negative fit result for the unregularized model indicates a poor fit to the data. Estimating a regularized model produces a significantly better result.
Estimate Nonlinear ARX Model Using Optimal Subset of Regressors
Open Live Script
Load the estimation data.
load twotankdata y u
Create an iddata
object from the data. Use the first 1000 samples for estimation and the remaining samples for validation.
Ts = 0.2;z = iddata(y,u,Ts);ze = z(1:1000);zv = z(1001:end);
Create an nlarxOptions
option set. Specify a simulation error minimization objective, 'lm'
least squares search, and a maximum of 10 estimation iterations. Display progress during estimation.
opt = nlarxOptions('Focus','simulation','SearchMethod','lm','Display','on');opt.SearchOptions.MaxIterations = 10;
Estimate a nonlinear ARX model, using ARX model orders to specify the regressors and an idSigmoidNetwork
mapping function. The model uses all candidate regressors. To view regressor usage information, at the MATLAB® command prompt, enter sys.RegressorUsage
.
orders = [8 8 1];outputFcn = idSigmoidNetwork;sys = nlarx(ze,orders,outputFcn,opt);allRegressors = getreg(sys);
Sparsify the model (reduce the regressors in use) by using the "log-sum"
measure.
opt.SearchOptions.MaxIterations = 20;opt.SparsifyRegressors = true;opt.SparsificationOptions.MaxIterations = 10;opt.SparsificationOptions.Lambda = 2;sysr1 = nlarx(ze,sys,opt);T = sysr1.RegressorUsage;inUse = any(T{:,:},2);fprintf('Regressors in use: %s\n', strjoin(allRegressors(inUse),', '))
Regressors in use: y1(t-1), y1(t-5), u1(t-2), u1(t-3), u1(t-4), u1(t-7), u1(t-8)
Sparsify the model again using the "l1"
measure.
opt.SparsificationOptions.SparsityMeasure = 'l1';opt.SparsificationOptions.Lambda = 2.2;sysr2 = nlarx(ze,sys,opt);T = sysr2.RegressorUsage;inUse = any(T{:,:},2);fprintf('Regressors in use: %s\n', strjoin(allRegressors(inUse),', '))
Regressors in use: y1(t-1), y1(t-5), u1(t-2), u1(t-3), u1(t-4), u1(t-6), u1(t-7), u1(t-8)
Sparsify the model again using the "l0"
measure.
opt.SparsificationOptions.SparsityMeasure = 'l0';opt.SparsificationOptions.Lambda = 2.2;sysr3 = nlarx(ze,sys,opt);T = sysr3.RegressorUsage;InUse = any(T{:,:},2);fprintf('Regressors in use: %s\n', strjoin(allRegressors(inUse),', '))
Regressors in use: y1(t-1), y1(t-5), u1(t-2), u1(t-3), u1(t-4), u1(t-6), u1(t-7), u1(t-8)
Compare the full regressor model and three sparse regressor models against the validation data.
compare(zv,sys,sysr1,sysr2,sysr3)
Input Arguments
collapse all
data
— Estimation data
timetable | numeric matrix pair | numeric matrix | iddata
object | cell array of timetables | cell array of matrices
Uniformly sampled estimation data, specified as a timetable, a comma-separated matrix pair, as single matrix, or a data object, as the following sections describe.
Timetable
Specify data
as a timetable that uses a regularly spaced time vector. tt
contains variables representing input and output channels.
When you use timetables for estimation, you can use all the variables or specify a subset of channels to use.
For multiexperiment data, specify data as an Ne-by-1 cell array of timetables, where Ne is the number of experiments. The sample times of all the experiments must match.
To select individual input and output channels to use for estimation, use the InputName and OutputName name-value arguments. If your model has more than one output and you use the regressors argument, then you must specify the output channel names.
For example, use the following command to select input channels u1
and u3
and output channels y2
and y4
from a timetable that contains three input variables and four output variables.
sys = nlarx(tt,orders,'InputName',["u1" "u3"],'OutputName',["y2" "y4"])
Comma-Separated Matrix Pair
Specify data
as a comma-separated pair of real-valued matrices that contain input and output time-domain signal values. When you specify matrix-based data, the software assumes a sample time of 1 second. You can change the sample time after estimation by setting the property sys.Ts
.
For SISO systems, specify
data
as a pair of Ns-element numeric column vectors that contain uniformly sampled input and output time-domain signal values. Here, Ns is the number of samples.For MIMO systems, specify
u
,y
as an input/output matrix pair with the following dimensions:u
— Ns-by-Nu, where Nu is the number of inputs.y
— Ns-by-Ny, where Ny is the number of outputs.
For multiexperiment data, specify data as a pair of Ne-by-1 cell arrays of matrices, where Ne is the number of experiments. The sample times of all the experiments must match.
Single Matrix
Specify data
as a single real-valued matrix with Ny+Nu columns that contain the output signal values followed by the input signal values. Note that this order is the opposite of the order used for the comma-separated matrix pair form of data
. When you specify matrix-based data, the software assumes a sample time of 1 second. You can change the sample time after estimation by setting the property sys.Ts
.
Data Object
An estimation data object, specified as a time-domain iddata
object that contains uniformly sampled input and output values. The data object can have one or more output channels and zero or more input channels. By default, the software sets the sample time of the model to the sample time of the estimation data.
For more information about working with estimation data types, see Data Domains and Data Types in System Identification Toolbox.
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 linear regressor form in the idnlarx
Regressors
property. For an example, see Create Nonlinear ARX Model Using ARX Model Orders.
regressors
— Regressor specification
linearRegressor
object | polynomialRegressor
object | periodicRegressor
| 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).
When you create a regressor set to support estimation with an iddata
object, you can use the input and output names of the object rather than create the names for the regressor function. For instance, suppose you create a linear regressor for a model, plan to use the iddata
object z
to estimate the model. You can use the following command to create the linear regressor.
L = linearRegressor([z.outputName;z.inputName],{1,[2 5]})
For an example of creating and using a SISO linear regressor set, see Estimate Nonlinear ARX Model Using Linear Regressor Set. For an example of creating a MIMO linear regressor set that obtains variable names from the estimation data set, see Estimate MIMO Nonlinear ARX Model with Same Mapping Function for All Outputs.
output_fcn
— Output function
'idWaveletNetwork'
(default) | 'idLinear'
| []
| 'idSigmoidNetwork'
| 'idTreePartition'
| 'idTreePartition'
| 'idGaussianProcess'
| 'idTreeEnsemble'
| 'idSupportVectorMachine'
| 'idNeuralNetwork'
| 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 objects:
'idWaveletNetwork' or idWaveletNetwork object | Wavelet network |
'linear' 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. |
Use a string, such as 'idSigmoidNetwork'
, to use the default properties of the mapping function object. Use the object itself, such as idSigmoidNetwork
, when you want to configure the properties of the mapping object.
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
object, as the name implies, has only a linear component.
output_fcn
is static in that it depends only upon the data values at a specific time, but not directly on time itself. For example, if the output function y(t) is equal to y0 + a1 y(t–1) + a2 y(t–2) + … b1 u(t–1) + b2 u(t–2) + …, then output_fcn
is a linear function that the linear
mapping object represents.
Specifying a character vector, for example 'idSigmoidNetwork'
, creates a mapping object with default settings. Alternatively, you can specify mapping object properties in two 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 output_fcn
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];
To specify the same mapping for all outputs, specify OutputFcn
as a character vector or a single mapping object.
output_fcn
represents a static mapping function that transforms the regressors of the nonlinear ARX model into the model output. output_fcn
is static because it does not depend on time. For example, if , then output_fcn
is a linear function represented by the idLinear object.
For an example of specifying the output function, see Specify and Customize Output Function.
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 an estimator such as arx, armax, tfest, or ssest. For example, to create a state-space idss
model, estimate the model using ssest.
sys0
— Nonlinear ARX model
idnlarx
model
Nonlinear ARX model, specified as an idnlarx
model. sys0
can be:
A model previously estimated using
nlarx
. The estimation algorithm uses the parameters ofsys0
as initial guesses. In this case, use init to slightly perturb the model properties to avoid trapping the model in local minima.sys = init(sys);sys = nlarx(data,sys);
A model previously created using the
idnlarx
constructor and with properties set using dot notation. For example, use the following to create an idnlarx object, set its properties, and estimate the model.sys1 = idnlarx('y1','u1',Regressors);sys1.OutputFcn = 'idTreePartition';sys1.Ts = 0.02;sys1.TimeUnit = 'Minutes';sys1.InputName = 'My Data';sys2 = nlarx(data,sys1);
The preceding code is equivalent to the following nlarx command.
sys2 = nlarx(data,Regressors,'idTreePartition','Ts',0.02,'TimeUnit','Minutes', ...'InputName','My Data');
Options
— Estimation options
nlarxOptions
option set
Estimation options for nonlinear ARX model identification, specified as an nlarxOptions option set. Available options include:
Minimization objective
Normalization options
Regularization options
Name-Value Arguments
Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN
, where Name
is the argument name and Value
is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose Name
in quotes.
Example: sys = nlarx(data,regressors,'InputName',"u2")
InputName
— Input channel names
string | character vector | string array | cell array of character vectors
Input channel names, specified as a string, character vector, string array, or cell array of character vectors.
If you are using a timetable for the data source, the names in InputName
must be a subset of the timetable variables.
Example: sys = nlarx(tt,__,'InputName',["u1" "u2"])
selects the variables u1
and u2
as the input channels from the timetable tt
to use for the estimation.
OutputName
— Output channel names
string | character vector | string array | cell array of character vectors
Output channel names, specified as a string, character vector, string array, or cell array of character vectors.
If you are using a timetable for the data source, the names in OutputName
must be a subset of the timetable variables.
Example: sys = nlarx(tt,__,'OutputName',["y1" "y3"])
selects the variables y1
and y3
as the output channels from the timetable tt
to use for the estimation.
Output Arguments
collapse all
sys
— Nonlinear ARX model
idnlarx
object
Nonlinear ARX model that fits the given estimation data, returned as an idnlarx object. This model is created using the specified model orders, nonlinearity estimator, and estimation options.
Information about the estimation results and options used is stored in the Report
property of the model. The contents of Report
depend upon the choice of nonlinearity and estimation focus you specified for nlarx
. Report
has the following fields:
Report Field | Description |
---|---|
Status | Summary of the model status, which indicates whether the model was created by construction or obtained by estimation |
Method | Estimation command used |
Fit | Quantitative assessment of the estimation, returned as a structure. See Loss Function and Model Quality Metrics for more information on these quality metrics. The structure has these fields.
|
Parameters | Estimated values of model parameters |
OptionsUsed | Option set used for estimation. If no custom options were configured, this is a set of default options. See nlarxOptions for more information. |
RandState | State of the random number stream at the start of estimation. Empty, |
DataUsed | Attributes of the data used for estimation, returned as a structure with the following fields.
|
Termination | Termination conditions for the iterative search used for prediction error minimization, returned as a structure with these fields.
For estimation methods that do not require numerical search optimization, the |
For more information on using Report
, see Estimation Report.
Algorithms
collapse all
Nonlinear ARX Model Structure
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:
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.
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.
R2022b: Time-domain estimation data is accepted in the form of timetables and matrices
Most estimation, validation, analysis, and utility functions now accept time-domain input/output data in the form of a single timetable that contains both input and output data or a pair of matrices that contain the input and output data separately. These functions continue to accept iddata objects as a data source as well, for both time-domain and frequency-domain data.
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.
R2018a: Advanced Options are deprecated for SearchOptions
when SearchMethod
is 'lsqnonlin'
Specification of lsqnonlin
- related advanced options are deprecated, including the option to invoke parallel processing when estimating using the lsqnonlin
search method, or solver, in Optimization Toolbox™.
See Also
idnlarx | nlarxOptions | isnlarx | goodnessOfFit | aic | fpe | polynomialRegressor | periodicRegressor | linearRegressor | customRegressor
Topics
- Estimate Nonlinear ARX Models at the Command Line
- Estimate Nonlinear ARX Models Initialized Using Linear ARX Models
- Identifying Nonlinear ARX Models
- Validate Nonlinear ARX Models
- Using Nonlinear ARX Models
- Loss Function and Model Quality Metrics
- Regularized Estimates of Model Parameters
- Estimation Report
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