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greyest

Linear grey-box model estimation

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Syntax

sys= greyest(data,init_sys)
sys= greyest(data,init_sys,opt)
[sys,x0] = greyest(___)

Description

sys= greyest(data,init_sys)estimates a linear grey-box model,sys, using time or frequency domain data,data. The dimensions of the inputs and outputs ofdataandinit_sys, anidgreymodel, must match.sysis an identifiedidgreymodel that has the same structure asinit_sys.

sys= greyest(data,init_sys,opt)estimates a linear grey-box model using the option set,opt, to configure the estimation options.

[sys,x0] = greyest(___)returns the value of the initial states computed during estimation. You can use this syntax with any of the previous input-argument combinations.

Input Arguments

data

Estimation data.

输入和输出的尺寸dataandinit_sysmust match.

For time-domain estimation,datais aniddataobject containing the input and output signal values.

For frequency domain estimation,datacan be one of the following:

  • Recorded frequency response data (frd(Control System Toolbox)oridfrd)

  • iddataobject with itsDomainproperty set to'Frequency'

init_sys

Identified linear grey-box model that configures the initial parameterization ofsys.

init_sys, anidgreymodel, must have the same input and output dimensions asdata.

opt

Estimation options.

optis an option set, created usinggreyestOptions, which specifies options including:

  • Estimation objective

  • Initialization choice

  • Disturbance model handling

  • Numerical search method to be used in estimation

输出参数

sys

Estimated grey-box model, returned as anidgreymodel. This model is created using the specified initial system, and estimation options.

Information about the estimation results and options used is stored in theReportproperty of the model.Reporthas 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.
InitialState

Handling of initial states during estimation, returned as one of the following:

  • 'model'— The initial state is parameterized by the ODE file used by theidgreymodel.

  • 'zero'— The initial state is set to zero.

  • 'estimate'— The initial state is treated as an independent estimation parameter.

  • 'backcast'— The initial state is estimated using the best least squares fit.

  • Vector of doubles of lengthNx, whereNx是number of states. For multiexperiment data, a matrix withNecolumns, whereNe是number of experiments.

This field is especially useful to view how the initial states were handled when theInitialStateoption in the estimation option set is'auto'.
DisturbanceModel

Handling of the disturbance component (K) during estimation, returned as one of the following values:

  • 'model'—Kvalues are parameterized by the ODE file used by theidgreymodel.

  • 'fixed'— The value of theKproperty of theidgreymodel is fixed to its original value.

  • 'none'—Kis fixed to zero.

  • 'estimate'—Kis treated as an independent estimation parameter.

This field is especially useful to view the how the disturbance component was handled when theDisturbanceModeloption in the estimation option set is'auto'.
Fit

Quantitative assessment of the estimation, returned as a structure. SeeLoss Function and Model Quality Metricsfor 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 percentagefitpercent= 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. SeegreyestOptionsfor 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, seerng.
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

∞ -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, theTerminationfield is omitted.

For more information on usingReport, seeEstimation Report.

x0

Initial states computed during the estimation, returned as a matrix containing a column vector corresponding to each experiment.

This array is also stored in theParametersfield of the modelReportproperty.

Examples

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Open Live Script

Estimate the parameters of a DC motor using the linear grey-box framework.

Load the measured data.

load(fullfile(matlabroot,'toolbox','ident','iddemos','data','dcmotordata')); data = iddata(y, u, 0.1,'Name','DC-motor'); data.InputName ='Voltage'; data.InputUnit ='V'; data.OutputName = {'Angular position','Angular velocity'}; data.OutputUnit = {'rad','rad/s'}; data.Tstart = 0; data.TimeUnit ='s';

datais aniddataobject containing the measured data for the outputs, the angular position, the angular velocity. It also contains the input, the driving voltage.

Create a grey-box model representing the system dynamics.

For the DC motor, choose the angular position (rad) and the angular velocity (rad/s) as the outputs and the driving voltage (V) as the input. Set up a linear state-space structure of the following form:

x ˙ ( t ) = [ 0 1 0 - 1 τ ] x ( t ) + [ 0 G τ ] u ( t )

y ( t ) = [ 1 0 0 1 ] x ( t ) .

τ 是time constant of the motor in seconds, and G 静态增益从输入到v角elocity in rad/(V*s) .

G = 0.25; tau = 1; init_sys = idgrey('motorDynamics',tau,'cd',G,0);

The governing equations in state-space form are represented in the MATLAB® filemotorDynamics.m. To view the contents of this file, enteredit motorDynamics.mat the MATLAB command prompt.

G is a known quantity that is provided tomotorDynamics.mas an optional argument.

τ is a free estimation parameter.

init_sysis anidgreymodel associated withmotor.m.

Estimate τ .

sys =老龄化最严重的(数据,init_sys);

sysis anidgreymodel containing the estimated value of τ .

To obtain the estimated parameter values associated withsys, usegetpvec(sys).

Analyze the result.

opt = compareOptions('InitialCondition','zero'); compare(data,sys,Inf,opt)

Figure contains 2 axes objects. Axes object 1 contains 2 objects of type line. These objects represent DC-motor (Angular position), sys: 98.35%. Axes object 2 contains 2 objects of type line. These objects represent DC-motor (Angular velocity), sys: 84.42%.

sysprovides a 98.35% fit for the angular position and an 84.42% fit for the angular velocity.

Open Live Script

Estimate the parameters of a DC motor by incorporating prior information about the parameters when using regularization constants.

The model is parameterized by static gainGand time constant τ . From prior knowledge, it is known thatGis about 4 and τ is about 1. Also, you have more confidence in the value of τ thanGand would like to guide the estimation to remain close to the initial guess.

Load estimation data.

loadregularizationExampleData.matmotorData

The data contains measurements of motor's angular position and velocity at given input voltages.

Create anidgreymodel for DC motor dynamics. Use the functionDCMotorODEthat represents the structure of the grey-box model.

mi = idgrey(@DCMotorODE,{'G', 4;'Tau', 1},'cd',{}, 0); mi = setpar(mi,'label','default');

If you want to view theDCMotorODEfunction, type:

typeDCMotorODE.m
function [A,B,C,D] = DCMotorODE(G,Tau,Ts) %DCMOTORODE ODE file representing the dynamics of a DC motor parameterized %by gain G and time constant Tau. % % [A,B,C,D,K,X0] = DCMOTORODE(G,Tau,Ts) returns the state space matrices % of the DC-motor with time-constant Tau and static gain G. The sample % time is Ts. % % This file returns continuous-time representation if input argument Ts % is zero. If Ts>0, a discrete-time representation is returned. % % See also IDGREY, GREYEST. % Copyright 2013 The MathWorks, Inc. A = [0 1;0 -1/Tau]; B = [0; G/Tau]; C = eye(2); D = [0;0]; if Ts>0 % Sample the model with sample time Ts s = expm([[A B]*Ts; zeros(1,3)]); A = s(1:2,1:2); B = s(1:2,3); end

Specify regularization options Lambda.

opt = greyestOptions; opt.Regularization.Lambda = 100;

Specify regularization options R.

opt.Regularization.R = [1, 1000];

You specify more weighting on the second parameter because you have more confidence in the value of τ thanG.

Specify the initial values of the parameters as regularization option θ *.

opt.Regularization.Nominal ='model';

Estimate the regularized grey-box model.

sys= greyest(motorData, mi, opt);

Extended Capabilities

Version History

Introduced in R2012a

See Also

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