Documentation

# nlgreyest

Estimate nonlinear grey-box model parameters

## Description

example

sys= nlgreyest(data,init_sys) estimates the parameters of a nonlinear grey-box model, init_sys, using time-domain data, data.

example

sys= nlgreyest(data,init_sys,options) specifies additional model estimation options.

## Examples

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z = iddata(y,u,0.2,'Name','Two tanks');

The data contains 3000 input-output data samples of a two tank system. The input is the voltage applied to a pump, and the output is the liquid level of the lower tank.

Specify file describing the model structure for a two-tank system. The file specifies the state derivatives and model outputs as a function of time, states, inputs, and model parameters.

FileName = 'twotanks_c';

Specify model orders [ny nu nx].

Order = [1 1 2];

Specify initial parameters (Np = 6).

Parameters = {0.5;0.0035;0.019; ...
9.81;0.25;0.016};

Specify initial initial states.

InitialStates = [0;0.1];

Specify as continuous system.

Ts = 0;

Create idnlgrey model object.

nlgr = idnlgrey(FileName,Order,Parameters,InitialStates,Ts, ...
'Name','Two tanks');

Set some parameters as constant.

nlgr.Parameters(1).Fixed = true;
nlgr.Parameters(4).Fixed = true;
nlgr.Parameters(5).Fixed = true;

Estimate the model parameters.

nlgr = nlgreyest(z,nlgr);

Create estimation option set for nlgreyest to view estimation progress, and to set the maximum iteration steps to 50.

opt = nlgreyestOptions;
opt.Display = 'on';
opt.SearchOptions.MaxIterations = 50;

z = iddata(y,u,0.1,'Name','DC-motor');

The data is from a linear DC motor with one input (voltage), and two outputs (angular position and angular velocity). The structure of the model is specified by dcmotor_m.m file.

Create a nonlinear grey-box model.

file_name = 'dcmotor_m';
Order = [2 1 2];
Parameters = [1;0.28];
InitialStates = [0;0];

init_sys = idnlgrey(file_name,Order,Parameters,InitialStates,0, ...
'Name','DC-motor');

Estimate the model parameters using the estimation options.

sys = nlgreyest(z,init_sys,opt);

## Input Arguments

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Time-domain estimation data, specified as an iddata object. data has the same input and output dimensions as init_sys.

If you specify the InterSample property of data as 'bl'(band-limited) and the model is continuous-time, the software treats data as first-order-hold (foh) interpolated for estimation.

Constructed nonlinear grey-box model that configures the initial parameterization of sys, specified as an idnlgrey object. init_sys has the same input and output dimensions as data. Create init_sys using idnlgrey.

Estimation options for nonlinear grey-box model identification, specified as an nlgreyestOptions option set.

## Output Arguments

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Nonlinear grey-box model with the same structure as init_sys, returned as an idnlgrey object. The parameters of sys are estimated such that the response of sys matches the output signal in the estimation data.

Information about the estimation results and options used is stored in the Report property of the model. Report has the following fields:

Report FieldDescription
Status

Summary of the model status, which indicates whether the model was created by construction or obtained by estimation.

Method

Name of the simulation solver and the search method used during estimation.

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:

FieldDescription
FitPercent

Normalized root mean squared error (NRMSE) measure of how well the response of the model fits the estimation data, expressed as a percentage.

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 the model parameters. Structure with the following fields:

FieldDescription
InitialValuesStructure with values of parameters and initial states before estimation.
ParVectorValue of parameters after estimation.
Free

Logical vector specifying the fixed or free status of parameters during estimation

FreeParCovarianceCovariance of the free parameters.
X0Value of initial states after estimation.
X0CovarianceCovariance of the initial states.

OptionsUsed

Option set used for estimation. If no custom options were configured, this is a set of default options. See nlgreyestOptions 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 in the MATLAB® documentation.

DataUsed

Attributes of the data used for estimation — Structure with the following fields:

FieldDescription
Name

Name of the data set.

Type

Data type — For idnlgrey models, this is set to 'Time domain data'.

Length

Number of data samples.

Ts

Sample time. This is equivalent to data.Ts.

InterSample

Input intersample behavior. 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.

The value of Intersample has no effect on estimation results for discrete-time models.

InputOffset

Empty, [], for nonlinear estimation methods.

OutputOffset

Empty, [], for nonlinear estimation methods.

Termination

Termination conditions for the iterative search used for prediction error minimization. Structure with the following fields:

FieldDescription
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.