nlarx

Estimate parameters of nonlinear ARX model

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)

Description

Specify Regressors

sys = nlarx(data,orders) estimates a nonlinear ARX model to fit the given estimation data using the specified ARX model 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 a data object. Use this syntax when you extend an ARX linear model, or when you use only regressors that are linear with consecutive lags.

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 non-consecutive lags, or when you also have any combination of polynomial regressors, periodic regressors, and custom regressors.

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.

Specify Linear Model

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.

sys = nlarx(data,linmodel,output_fcn) specifies the output function to use for model estimation.

Refine Existing Model

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 Options

sys = nlarx(___,Options) specifies additional configuration options for the model estimation.

Examples

Estimate Nonlinear ARX Model

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: y1

Regressors:
  Linear regressors in variables y1, u1

Output function: Wavelet network with 11 units
Sample time: 0.2 seconds

Status:                                          
Estimated using NLARX on time domain data "z".   
Fit to estimation data: 96.84% (prediction focus)
FPE: 3.482e-05, MSE: 3.431e-05                   
More information in model's "Report" property.

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

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, ${\mathit{u}}_1\left(\mathit{t}-2\right)$ 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'

  Regressors described by this set

Load the estimation data and create an iddata object.

load twotankdata
z = 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: y1

Regressors:
  Linear regressors in variables y1, u1

Output function: Wavelet network with 11 units
Sample time: 0.2 seconds

Status:                                          
Estimated using NLARX on time domain data "z".   
Fit to estimation data: 96.84% (prediction focus)
FPE: 3.482e-05, MSE: 3.431e-05                   
More information in model's "Report" property.

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

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: y1

Regressors:
  Linear regressors in variables y1

Output function: Wavelet network with 8 units
Sample time: 0.01 seconds

Status:                                          
Estimated using NLARX on time domain data "z".   
Fit to estimation data: 92.92% (prediction focus)
FPE: 0.2568, MSE: 0.2507                         
More information in model's "Report" property.

Specify and Customize Output Function

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 y
z = 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

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

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: y1

Regressors:
  Linear regressors in variables y1, u1

Output function: Custom Network with 10 units and "gaussunit" unit function
Sample time: 0.1 seconds

Status:                                                      
Termination condition: Maximum number of iterations reached..
Number of iterations: 20, Number of function evaluations: 613
                                                             
Estimated using NLARX on time domain data "z1".              
Fit to estimation data: 64.35% (prediction focus)            
FPE: 3.58, MSE: 2.465                                        
More information in model's "Report" property.

Estimate MIMO Nonlinear ARX Model

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, y2

Regressors:
  Linear regressors in variables y1, y2, u1, u2, u3, u4, u5, u6

Output functions:
  Output 1:  Wavelet network with 2 units
  Output 2:  Linear with offset

Sample time: 0.02 seconds

Status:                                                      
Termination condition: Maximum number of iterations reached..
Number of iterations: 20, Number of function evaluations: 710
                                                             
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                                     
More information in model's "Report" property.

Estimate MIMO Nonlinear ARX Model with Same Mapping Function for All Outputs

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, y2

Regressors:
  Linear regressors in variables y1, y2, u1, u2, u3, u4, u5, u6

Output functions:
  Output 1:  Sigmoid network with 4 units
  Output 2:  Sigmoid network with 4 units

Sample time: 0.02 seconds

Status:                                                      
Termination condition: Maximum number of iterations reached..
Number of iterations: 20, Number of function evaluations: 177
                                                             
Estimated using NLARX on time domain data "z".               
Fit to estimation data: [98.86;98.79]% (prediction focus)    
FPE: 2.641, MSE: 0.9233                                      
More information in model's "Report" property.

Specify Linear, Polynomial, and Custom Regressors

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 ${\mathit{y}}_1\left(\mathit{t}-1\right)$, ${\mathit{u}}_1\left(\mathit{t}-2\right)$, and ${\mathit{u}}_1\left(\mathit{t}-5\right)$.

L = linearRegressor(names,{1,[2 5]});

Specify P as the polynomial regressor ${{\mathit{y}}_1\left(\mathit{t}-1\right)}^2$.

P = polynomialRegressor(names(1),1,2);

Specify C as the custom regressor ${\mathit{y}}_1\left(\mathit{t}-2\right)$${\mathit{u}}_1\left(\mathit{t}-3\right)$. 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'

  Regressors described by this set

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'

 
Regressors described by this set

Estimate a nonlinear ARX model with R.

sys = nlarx(z1,R)

sys =

Nonlinear ARX model with 1 output and 1 input
  Inputs: u1
  Outputs: y1

Regressors:
  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 units
Sample time: 0.1 seconds

Status:                                          
Estimated using NLARX on time domain data "z1".  
Fit to estimation data: 59.73% (prediction focus)
FPE: 3.356, MSE: 3.147                           
More information in model's "Report" property.

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

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

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 Position

Regressors:
  Linear regressors in variables Throttle Valve Position, Step Command

Output function: Wavelet network with 7 units
Sample time: 0.01 seconds

Status:                                                  
Estimated using NLARX on time domain data "ThrottleData".
Fit to estimation data: 99.03% (prediction focus)        
FPE: 0.1127, MSE: 0.1039                                 
More information in model's "Report" property.

Estimate Nonlinear ARX Model Using Constructed idnlarx Object

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: y1

Regressors:
  Linear regressors in variables y1, u1

Output function: Sigmoid network with 10 units
Name: Model 1
Sample time: 0.1 seconds

Status:                                                      
Termination condition: Maximum number of iterations reached..
Number of iterations: 20, Number of function evaluations: 353
                                                             
Estimated using NLARX on time domain data "z1".              
Fit to estimation data: 69.03% (prediction focus)            
FPE: 2.918, MSE: 1.86                                        
More information in model's "Report" property.

Estimate Nonlinear ARX Model and Avoid Local Minima

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

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: y1

Regressors:
  Linear regressors in variables y1, u1

Output function: Sigmoid network with 3 units
Sample time: 0.2 seconds

Status:                                                      
Termination condition: Maximum number of iterations reached..
Number of iterations: 10, Number of function evaluations: 114
                                                             
Estimated using NLARX on time domain data "z".               
Fit to estimation data: 85.86% (simulation focus)            
FPE: 3.791e-05, MSE: 0.0006853                               
More information in model's "Report" property.

Estimate Regularized Nonlinear ARX Model with Large Number of Units

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

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

data — Estimation data

timetable | numeric matrix pair | numeric matrix | iddata object

Uniformly sampled estimation data, specified as described in the following sections.

Timetable

Specify data as a timetable that uses a regularly spaced time vector. tt contains variables representing input and output channels. For multiexperiment data, tt is a cell array of timetables of length Ne, where Ne is the number of experiments.

When you use a syntax that specifies orders or linmodel, the software determines the number of input and output channels to use for estimation.

• If 'InputName' and 'OutputName' are not specified, then the software uses the first Nu variables of tt as inputs and the next Ny variables of tt as outputs.

• If 'InputName' and 'OutputName' are specified, then the software uses the specified variables. The number of specified input and output names must be consistent with Nu and Ny.

• For functions that can estimate a time series model, where there are no inputs, 'InputName' does not need to be specified.

When you use a syntax that specifies regressors, by default, the software interprets the last variable in the timetable as the sole output channel and the other variables as input channels. To change the input/output channel selection in this case, specify 'InputName' and 'OutputName'.

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 N_s-by-1 real-valued matrices that contain uniformly sampled input and output time-domain signal values. Here, N_s is the number of samples.

• For MIMO systems, specify u,y as an input/output matrix pair with the following dimensions:

  • u — N_s-by-N_u, where N_u is the number of inputs.

  • y — N_s-by-N_y, where N_y is the number of outputs.

Single Matrix

Specify data as a single real-valued matrix with N_y+N_u 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 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' | 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)
idFeedforwardNetwork object	Neural network — Feedforward network of 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 idFeedforwardNetwork function has only a nonlinear component, which is the network (Deep Learning Toolbox) object of Deep Learning Toolbox. 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 n_y output channels, you can specify mapping objects individually for each channel by setting output_fcn 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.

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 $$y(t)=y_0+a_{1}y(t-1)+a_{2}y(t-2)+\dots +b_{1}u(t-1)+b_{2}u(t-2)+\dots$$, then 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 of sys0 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

Output Arguments

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 the following fields: Field Description FitPercent Normalized root mean squared error (NRMSE) measure of how well the response of the model fits the estimation data, expressed as the percentage fitpercent = 100(1-NRMSE). LossFcn Value of the loss function when the estimation completes. MSE Mean squared error (MSE) measure of how well the response of the model fits the estimation data. FPE Final prediction error for the model. AIC Raw Akaike Information Criteria (AIC) measure of model quality. AICc Small-sample-size corrected AIC. nAIC Normalized AIC. BIC Bayesian Information Criteria (BIC).
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, [], if randomization was not used during estimation. For more information, see rng.
DataUsed	Attributes of the data used for estimation, returned as a structure with the following fields. Field Description Name Name of the data set. Type Data type. Length Number of data samples. Ts Sample time. InterSample Input intersample behavior, returned as one of the following values: 'zoh' — Zero-order hold maintains a piecewise-constant input signal between samples. 'foh' — First-order hold maintains a piecewise-linear input signal between samples. 'bl' — Band-limited behavior specifies that the continuous-time input signal has zero power above the Nyquist frequency. InputOffset Offset removed from time-domain input data during estimation. For nonlinear models, it is []. OutputOffset Offset removed from time-domain output data during estimation. For nonlinear models, it is [].
Termination	Termination conditions for the iterative search used for prediction error minimization, returned as a structure with the following fields: Field Description WhyStop Reason for terminating the numerical search. Iterations Number of search iterations performed by the estimation algorithm. FirstOrderOptimality $$\infty$$-norm of the gradient search vector when the search algorithm terminates. FcnCount Number of times the objective function was called. UpdateNorm Norm of the gradient search vector in the last iteration. Omitted when the search method is 'lsqnonlin' or 'fmincon'. LastImprovement Criterion improvement in the last iteration, expressed as a percentage. Omitted when the search method is 'lsqnonlin' or 'fmincon'. Algorithm Algorithm used by 'lsqnonlin' or 'fmincon' search method. Omitted when other search methods are used. For estimation methods that do not require numerical search optimization, the Termination field is omitted.

For more information on using Report, see Estimation Report.

Algorithms

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 in two stages:

1. 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.

2. It maps the regressors to the model output using an output function block. The output function block can include multiple mapping objectslinear, nonlinear, and offset blocks in parallel. For example, consider the following equation:

   $$F(x)=L^T(x-r)+g\left(Q(x-r)\right)+d$$

   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. $$g\left(Q(x-r)\right)+y_0$$ 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.

Version History

Introduced in R2007a

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™.