# oe

Estimate output-error polynomial model using time-domain or frequency-domain data

## Description

Output-error (OE) models are a special configuration of polynomial models, having only two active polynomials—B and F. OE models represent conventional transfer functions that relate measured inputs to outputs while also including white noise as an additive output disturbance. You can estimate OE models using time- and frequency-domain data. The tfest command offers the same functionality as oe. For tfest, you specify the model orders using number of poles and zeros rather than polynomial degrees. For continuous-time estimation, tfest provides faster and more accurate results, and is recommended.

example

sys = oe(data,[nb nf nk]) estimates an OE 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.

oe estimates sys using the measured input-output data data, which can be in the time or the frequency domain. The orders [nb nf nk] define the number of parameters in each component of the estimated polynomial.

example

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.

example

sys = oe(data,___,opt) estimates a polynomial model using the option set opt to specify estimation behavior. You can use this syntax with any of the previous input-argument combinations.

example

[sys,ic] = oe(___) returns the estimated initial conditions as an initialCondition object. Use this syntax if you plan to simulate or predict the model response using the same estimation input data and then compare the response with the same estimation output data. Incorporating the initial conditions yields a better match during the first part of the simulation.

## Examples

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Estimate an OE polynomial from time-domain data using two methods to specify input delay.

Set the orders of the B and F polynomials nb and nf. Set the input delay nk to one sample. Compute the model sys.

nb = 2;
nf = 2;
nk = 1;
sys = oe(z1,[nb nf nk]);

Compare the simulated model response with the measured output.

compare(z1,sys)

The plot shows that the fit percentage between the simulated model and the estimation data is greater than 70%.

Instead of using nk, you can also use the name-value pair argument 'InputDelay' to specify the one-sample delay.

nk = 0;
sys1 = oe(z1,[nb nf nk],'InputDelay',1);
figure
compare(z1,sys1)

The results are identical.

sys.Report
ans =
Status: 'Estimated using OE'
Method: 'OE'
InitialCondition: 'zero'
Fit: [1x1 struct]
Parameters: [1x1 struct]
OptionsUsed: [1x1 idoptions.polyest]
RandState: [1x1 struct]
DataUsed: [1x1 struct]
Termination: [1x1 struct]

sys.Report.Termination
ans = struct with fields:
WhyStop: 'Near (local) minimum, (norm(g) < tol).'
Iterations: 3
FirstOrderOptimality: 0.0708
FcnCount: 7
UpdateNorm: 1.4809e-05
LastImprovement: 5.1744e-06

The report includes information on the number of iterations and the reason the estimation stopped iterating.

The idfrd object data 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;
nf = 3;
sys = oe(data,[nb nf]);

Evaluate the goodness of 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 the 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 regularized model m2 has a reduced variance compared to the unregularized model m1.

Load the estimation data data and sample time Ts.

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

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

View the estimation sample time Ts that you loaded.

Ts
Ts = 1.0000e-03

This value matches the property data.Ts.

data.Ts
ans = 1.0000e-03

You can estimate a continuous model from data by limiting the input and output frequency bands to the Nyquist frequency. To do so, specify the estimation prefilter option 'WeightingFilter' to define a passband from 0 to 0.5*pi/Ts rad/s. The software ignores any response values with frequencies outside of that passband.

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

Set the Ts property to 0 to treat data as continuous-time data.

data.Ts = 0;

Estimate the continuous model.

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

Estimate an OE polynomial model sys and return the initial conditions in ic.

nb = 2;
nf = 2;
nk = 1;
[sys,ic] = oe(z1i,[nb,nf,nk]);
ic
ic =
initialCondition with properties:

A: [2x2 double]
X0: [2x1 double]
C: [0.9428 0.4824]
Ts: 0.1000

ic is an initialCondition object that encapsulates the free response of sys, in state-space form, to the initial state vector in X0. You can incorporate ic when you simulate sys with the z1i input signal and compare the response with the z1i output signal.

## Input Arguments

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Estimation data, specified as an iddata object, an frd object, or an idfrd object.

For time-domain estimation, data must be 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 (Control System Toolbox) or idfrd)

• iddata object with properties specified as follows:

• InputData — Fourier transform of the input signal

• OutputData — Fourier transform of the output signal

• Domain'Frequency'

Time-domain estimation data must be uniformly sampled. By default, the software sets the sample time of the model to the sample time of the estimation data.

For multiexperiment data, the sample times and intersample behavior of all the experiments must match.

You can compute discrete-time models from time-domain data or discrete-time frequency-domain data. Use tfest to compute continuous-time models.

OE model orders, specified as a 1-by-3 vector or a vector of integer matrices.

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, the elements of [nb nf nk] are as follows:

• nb — Order of the B(q) polynomial + 1, which is equivalent to the length of the B(q) polynomial. 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.

• nk — Input delay, expressed as the number of samples. nk is an Ny-by-Nu matrix. The delay appears as leading zeros of the B polynomial.

For estimation using continuous-time frequency-domain data, specify only [nb nf] and omit nk. For an example, see Estimate Continuous-Time OE Model Using Frequency Response.

Linear system that configures the initial parameterization of sys, specified as an idpoly model, another linear model, or a structure. You obtain init_sys either by performing an estimation using measured data or by direct construction.

If init_sys is an idpoly model of the OE structure, oe uses the parameter values of init_sys as the initial guess for estimating sys. The sample time of init_sys must match the sample time of the data.

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 values.

• Set init_sys.Structure.B.Maximum to the maximum B(q) coefficient values.

• Set init_sys.Structure.B.Free to indicate which B(q) coefficients are free for estimation.

If init_sys is not a polynomial model of the OE structure, the software first converts init_sys to an OE structure model. oe uses the parameters of the resulting model as the initial guess for estimating sys.

If you do not specify opt and init_sys was obtained by estimation, then the software uses estimation options from init_sys.Report.OptionsUsed.

Estimation options, specified as an oeOptions option set. Options specified by opt include:

• Estimation objective

• Handling of initial conditions

• Numerical search method and the associated options

For examples of specifying estimation options, see Estimate Continuous Model Using Band-Limited Discrete-Time Frequency-Domain Data.

### Name-Value 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.

Example: 'InputDelay',1

Input delays for each input channel, specified as the comma-separated pair consisting of 'InputDelay' and a numeric vector.

• For continuous-time models, specify 'InputDelay' in the time units stored in the TimeUnit property.

• For discrete-time models, specify 'InputDelay' in integer multiples of the sample time Ts. For example, setting 'InputDelay' to 3 specifies a delay of three sampling periods.

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.

To apply the same delay to all channels, specify 'InputDelay' as a scalar.

For an example, see Estimate OE Polynomial Model.

Transport delays for each input-output pair, specified as the comma-separated pair consisting of 'IODelay' and a numeric array.

• For continuous-time models, specify 'IODelay' in the time units stored in the TimeUnit property.

• For discrete-time models, specify 'IODelay' in integer multiples of the sample time Ts. For example, setting 'IODelay' to 4 specifies a transport delay of four sampling periods.

For a system with Nu inputs and Ny outputs, set 'IODelay' to an Ny-by-Nu matrix. Each entry is an integer value representing the transport delay for the corresponding input-output pair.

To apply the same delay to all channels, specify 'IODelay' as a scalar.

You can specify 'IODelay' as an alternative to the nk value. Doing so simplifies the model structure by reducing the number of leading zeros in the B polynomial. In particular, you can represent max(nk-1,0) leading zeros as input-output delays using 'IODelay' instead.

## Output Arguments

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OE polynomial model that fits the estimation data, returned as an idpoly model object. This model is created using the specified model orders, delays, and estimation options. The sample time of sys matches the sample time of the estimation data. Therefore, sys is always a discrete-time model when estimated from time-domain data. For continuous-time model identification using time-domain data, use tfest.

The Report property of the model stores information about the estimation results and options used. 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 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 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.

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, returned as a 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.

Estimated initial conditions, returned as an initialCondition object or an object array of initialCondition values.

• For a single-experiment data set, ic represents, in state-space form, the free response of the transfer function model (A and C matrices) to the estimated initial states (x0).

• For a multiple-experiment data set with Ne experiments, ic is an object array of length Ne that contains one set of initialCondition values for each experiment.

If oe returns ic values of 0 and the you know that you have non-zero initial conditions, set the 'InitialCondition' option in oeOptions to 'estimate' and pass the updated option set to oe. For example:

opt = oeOptions('InitialCondition','estimate')
[sys,ic] = oe(data,np,nz,opt)
The default 'auto' setting of 'InitialCondition' uses the 'zero' method when the initial conditions have a negligible effect on the overall estimation-error minimization process. Specifying 'estimate' ensures that the software estimates values for ic.

For more information, see initialCondition. For an example of using this argument, see Obtain Initial Conditions.

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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 the following 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 sample delay nk does not exist in the continuous case, and you should not specify nk when you command the estimation. Instead, express any system delay using the name-value pair argument 'IODelay' along with the system delay in the time units that are stored in the property TimeUnit. For example, suppose that your continuous system has a delay of iod seconds. Use model = oe(data,[nb nf],'IODelay',iod).