Documentation

# oe

Estimate Output-Error polynomial model using time or frequency domain data

## Syntax

sys = oe(data,[nb nf nk])
sys = oe(data,[nb nf nk],Name,Value)
sys = oe(data,init_sys)
sys = oe(data,___,opt)

## Description

sys = oe(data,[nb nf nk]) estimates an Output-Error model, sys, represented by:

$y\left(t\right)=\frac{B\left(q\right)}{F\left(q\right)}u\left(t-nk\right)+e\left(t\right)$

y(t) is the output, u(t) is the input, and e(t) is the error.

sys is estimated for the time- or frequency-domain, measured input-output data, data. The orders, [nb nf nk], parameterize the estimated polynomial.

sys = oe(data,[nb nf nk],Name,Value) specifies model structure attributes using additional options specified by one or more Name,Value pair arguments.

sys = oe(data,init_sys) uses the linear system init_sys to configure the initial parameterization of sys.

sys = oe(data,___,opt) estimates a polynomial model using the option set, opt, to specify estimation behavior.

## Input Arguments

 data Estimation data. For time domain estimation, data is an iddata object containing the input and output signal values. For frequency domain estimation, data can be one of the following: Recorded frequency response data (frd or idfrd)iddata object with its properties specified as follows:InputData — Fourier transform of the input signalOutputData — Fourier transform of the output signalDomain — 'Frequency' For multi-experiment data, the sample times and inter-sample behavior of all the experiments must match. [nb nf nk] Output error model orders. For a system represented by: $y\left(t\right)=\frac{B\left(q\right)}{F\left(q\right)}u\left(t-nk\right)+e\left(t\right)$ where y(t) is the output, u(t) is the input and e(t) is the error. nb — Order of the B polynomial + 1. nb is an Ny-by-Nu matrix. Ny is the number of outputs and Nu is the number of inputs.nf — Order of the F polynomial. nf is an Ny-by-Nu matrix. Ny is the number of outputs and Nu is the number of inputs.nk — Input delay, expressed as the number of samples. nk is an Ny-by-Nu matrix. Ny is the number of outputs and Nu is the number of inputs. The delay appears as leading zeros of the B polynomial. For estimation using continuous-time data, only specify [nb nf] and omit nk. init_sys Linear system that configures the initial parameterization of sys. You obtain init_sys by either performing an estimation using measured data or by direct construction. If init_sys is an idpoly model of Output-Error structure, oe uses the parameter values of init_sys as the initial guess for estimating sys. Use the Structure property of init_sys to configure initial guesses and constraints for B(q) and F(q). For example: To specify an initial guess for the F(q) term of init_sys, set init_sys.Structure.F.Value as the initial guess. To specify constraints for the B(q) term of init_sys:Set init_sys.Structure.B.Minimum to the minimum B(q) coefficient valuesSet init_sys.Structure.B.Maximum to the maximum B(q) coefficient valuesSet init_sys.Structure.B.Free to indicate which B(q) coefficients are free for estimation If init_sys is not a polynomial model of Output-Error structure, the software first converts init_sys to an Output-Error structure model. oe uses the parameters of the resulting model as the initial guess for estimating sys. If opt is not specified, and init_sys was created by estimation, then the estimation options from init_sys.Report.OptionsUsed are used. opt Estimation options. opt is an option set, created using oeOptions, that specifies estimation options including: Estimation objectiveHandling of initial conditionsNumerical search method and the associated options

### Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

 'InputDelay' Input delay for each input channel, specified as a scalar value or numeric vector. For continuous-time systems, specify input delays in the time unit stored in the TimeUnit property. For discrete-time systems, specify input delays in integer multiples of the sample time Ts. For example, InputDelay = 3 means a delay of three sample times. For a system with Nu inputs, set InputDelay to an Nu-by-1 vector. Each entry of this vector is a numerical value that represents the input delay for the corresponding input channel. You can also set InputDelay to a scalar value to apply the same delay to all channels. Default: 0 'IODelay' Transport delays. IODelay is a numeric array specifying a separate transport delay for each input/output pair. For continuous-time systems, specify transport delays in the time unit stored in the TimeUnit property. For discrete-time systems, specify transport delays as integers denoting delay of a multiple of the sample time Ts. You can specify IODelay as an alternative to the nk value. Doing so simplifies the model structure by reducing the number of leading zeros the B polynomial. In particular, you can represent max(nk-1,0) leading zeros as input/output delays using IODelay instead. For a MIMO system with Ny outputs and Nu inputs, set IODelay to a Ny-by-Nu array. Each entry of this array is a numerical value that represents the transport delay for the corresponding input/output pair. You can also set IODelay to a scalar value to apply the same delay to all input/output pairs. Default: 0 for all input/output pairs

## Output Arguments

sys

Output-Error polynomial model that fits the estimation data, returned as a idpoly model object. This model is created using the specified model orders, delays, and estimation options.

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

Estimation command used.

InitialCondition

Handling of initial conditions during model estimation, returned as one of the following values:

• 'zero' — The initial conditions were set to zero.

• 'estimate' — The initial conditions were treated as independent estimation parameters.

• 'backcast' — The initial conditions were estimated using the best least squares fit.

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

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 model parameters.

OptionsUsed

Option set used for estimation. If no custom options were configured, this is a set of default options. See oeOptions 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, returned as a structure with the following fields:

FieldDescription
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. 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.

## Examples

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Obtain the estimation data.

filename = fullfile(matlabroot,'examples','ident','oe_data1.mat');

data, an idfrd object, contains the continuous-time frequency response for the following model:

$G\left(s\right)=\frac{s+3}{{s}^{3}+2{s}^{2}+s+1}$

Estimate the model.

nb = 2;
nk = 3;
sys = oe(data,[nb nk]);

Evaluate the goodness of the fit.

compare(data,sys);

Estimate a high-order OE model from data collected by simulating a high-order system. Determine the regularization constants by trial and error and use the values for model estimation.

Estimate an unregularized OE model of order 30.

m1 = oe(m0simdata,[30 30 1]);

Obtain a regularized OE model by determining Lambda value using trial and error.

opt = oeOptions;
opt.Regularization.Lambda = 1;
m2 = oe(m0simdata,[30 30 1],opt);

Compare the model outputs with the estimation data.

opt = compareOptions('InitialCondition','z');
compare(m0simdata,m1,m2,opt);

The regularized model, m2, produces a better fit than the unregularized model, m1.

Compare the variance in the model responses.

h = bodeplot(m1,m2);
opt = getoptions(h);
opt.PhaseMatching = 'on';
opt.ConfidenceRegionNumberSD = 3;
opt.PhaseMatching = 'on';
setoptions(h,opt);
showConfidence(h);

The variance of the regularized model m2 is reduced compared to the unregularized model m1.

Obtain the estimation data.

filename = fullfile(matlabroot,'examples','ident','oe_data2.mat');

data, an iddata object, contains the discrete-time frequency response for the following model:

$G\left(s\right)=\frac{1000}{s+500}$

The sample time for data, Ts, is 0.001 seconds.

Treat data as continuous-time data. When you plot data, the input/output signals are band-limited, which allows you to treat data as continuous-time data. You can now obtain a continuous-time model.

data.Ts = 0;

Specify the estimation options.

opt = oeOptions('WeightingFilter',[0 0.5*pi/Ts]);

This prefilter choice directs the software to ignore the response values for frequencies higher than 0.5*pi/Ts rad/s.

Estimate the model.

nb = 1;
nf = 3;
sys = oe(data,[nb nf],opt);

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### Output-Error (OE) Model

The general Output-Error model structure is:

$y\left(t\right)=\frac{B\left(q\right)}{F\left(q\right)}u\left(t-{n}_{k}\right)+e\left(t\right)$

The orders of the Output-Error model are:

### Continuous-Time, Output-Error Model

If data is continuous-time frequency-domain data, oe estimates a continuous-time model with transfer function:

$G\left(s\right)=\frac{B\left(s\right)}{F\left(s\right)}=\frac{{b}_{nb}{s}^{\left(nb-1\right)}+{b}_{nb-1}{s}^{\left(nb-2\right)}+...+{b}_{1}}{{s}^{nf}+{f}_{nf}{s}^{\left(nf-1\right)}+...+{f}_{1}}$

The orders of the numerator and denominator are nb and nf, similar to the discrete-time case. However, the delay nk has no meaning and you should omit it when specifying model orders for estimation. Use model = oe(data,[nb nf]). Use the IODelay model property to specify any input-output delays. For example, use model = oe(data,[nb nf], 'IODelay',iod) instead.

## Tips

• To estimate a continuous-time model when data represents continuous-time frequency response data, omit nk.

For example, use sys = oe(data,[nb nf]).

## Algorithms

The estimation algorithm minimizes prediction errors.

## Alternatives

Output-Error models are a special configuration of polynomial models, having only two active polynomials - B and F. For such models, it may be more convenient to use a transfer function (idtf) model and its estimation command, tfest.

Also, tfest is the recommended command for estimating continuous-time models.