# oe

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

## Syntax

## 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.

### Estimate OE Model

estimates an OE model `sys`

= oe(`tt`

,```
[nb
nf nk]
```

)`sys`

using the data contained in the variables
of timetable `tt`

. The software uses the first *Nu*
variables as inputs and the next *Ny* variables as outputs, where
*Nu* and *Ny* are determined from the dimensions of
the specified polynomial orders.

`sys`

is represented by the equation

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

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

The orders `[nb nf nk]`

define the number of parameters in each
component of the estimated polynomial.

To select specific input and output channels from `tt`

, use
name-value syntax to set `'InputName'`

and
`'OutputName'`

to the corresponding timetable variable names.

uses the time-domain input and output signals in the comma-separated matrices
`sys`

= oe(`u`

,`y`

,`[nb nf nk]`

)`u`

,`y`

. The software assumes that the data sample
time is 1 second. To change the sample time, set `Ts`

using name-value
syntax.

uses the time-domain or frequency-domain data in the data object
`sys`

= oe(`data`

,```
[nb
nf nk]
```

)`data`

.

specifies model structure attributes using additional options specified by one or more
name-value pair arguments. You can use this syntax with any of the previous input-argument
combinations.`sys`

= oe(___,`Name,Value`

)

### Configure Initial Parameters

### Specify Additional Estimation Options

### Return Estimated Initial Conditions

`[`

returns the estimated initial conditions as an `sys`

,`ic`

] = oe(___)`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

### Estimate OE Polynomial Model

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

Load the estimation data.

load sdata1 tt1

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(tt1,[nb nf nk]);

Compare the simulated model response with the measured output.

compare(tt1,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(tt1,[nb nf nk],'InputDelay',1);
figure
compare(tt1,sys1)
```

The results are identical.

You can view more information about the estimation by exploring the `idpoly`

property `sys.Report`

.

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]

For example, find out more information about the termination conditions.

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.

### Estimate Continuous-Time OE Model Using Frequency Response

Load the estimation data.

load oe_data1 data;

`The idfrd`

object `data`

contains the continuous-time frequency response for the following model:

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

Estimate the model.

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

Evaluate the goodness of fit.

compare(data,sys);

### Estimate OE Model Using Regularization

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.

Load the data.

load regularizationExampleData.mat m0simdata

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`

.

### Estimate Continuous Model Using Band-Limited Discrete-Time Frequency-Domain Data

Load the estimation data `data`

and sample time `Ts`

.

load oe_data2.mat data Ts

An `iddata`

object `data`

contains the discrete-time frequency response for the following model:

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

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);

### Obtain Initial Conditions

Load the data, which consists of input and output data in matrix form and the sample time.

load sdata1i umat1i ymat1i Ts1i

Estimate an OE polynomial model `sys`

and return the initial conditions in `ic`

.

```
nb = 2;
nf = 2;
nk = 1;
[sys,ic] = oe(umat1i,ymat1i,[nb,nf,nk],'Ts',Ts1i);
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 `umat1i`

input signal and compare the response with the `ymat1i`

output signal.

## Input Arguments

`tt`

— Timetable-based estimation data

timetable | cell array of timetables.

Estimation data, specified 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

The software determines the number of input and output channels to use for estimation from the
dimensions of the specified polynomial orders. The input/output channel selection
depends on whether the `'InputName'`

and
`'OutputName'`

name-value arguments are specified.

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.

For more information about working with estimation data types, see Data Domains and Data Types in System Identification Toolbox.

`u`

, `y`

— Matrix-based estimation data

matrices | cell array of matrices

Estimation data, specified for SISO systems as a comma-separated pair of
*N _{s}*-by-1 real-valued matrices that
contain uniformly sampled input and output time-domain signal values. Here,

*N*is the number of samples.

_{s}For MIMO systems, specify `u`

,`y`

as an
input/output matrix pair with the following dimensions:

`u`

—*N*-by-_{s}*N*, where_{u}*N*is the number of inputs._{u}`y`

—*N*-by-_{s}*N*, where_{y}*N*is the number of outputs._{y}

For multiexperiment data, specify `u`

,`y`

as a
pair of 1-by-*N _{e}* cell arrays, where

*N*is the number of experiments. The sample times of all the experiments must match.

_{e}For time series data, which contains only outputs and no inputs, specify
`[]`

,`y`

.

#### Limitations

For more information about working with estimation data types, see Data Domains and Data Types in System Identification Toolbox.

`data`

— Estimation data

`iddata`

object | `frd`

object | `idfrd`

object

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:

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.

`[nb nf nk]`

— OE model orders

integer row vector | row vector of integer matrices

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

For a system represented by

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

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*N*-by-_{y}*N*matrix._{u}*N*is the number of outputs and_{y}*N*is the number of inputs._{u}`nf`

— Order of the*F*polynomial.`nf`

is an*N*-by-_{y}*N*matrix._{u}`nk`

— Input delay, expressed as the number of samples.`nk`

is an*N*-by-_{y}*N*matrix. The delay appears as leading zeros of the_{u}*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.

`init_sys`

— Linear system

`idpoly`

model | linear model | structure

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`

.

`opt`

— Estimation options

`oeOptions`

option set

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 pairs of arguments as
`Name1=Value1,...,NameN=ValueN`

, where `Name`

is
the argument name and `Value`

is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.

*
Before R2021a, use commas to separate each name and value, and enclose*
`Name`

*in quotes.*

**Example: **`'InputDelay',1`

`InputName`

— Input channel names

string | character vector | string array | cell array of character vectors

Input channel names, specified as a string, character vector, string array, or cell array of character vectors.

If you are using a timetable for the data source, the names in
`InputName`

must be a subset of the timetable variables.

**Example: **`sys = oe(tt,__,'InputName',["u1" "u2"])`

selects
the variables `u1`

and `u2`

as the input channels from
the timetable `tt`

to use for the estimation.

`OutputName`

— Output channel names

string | character vector | string array | cell array of character vectors

Output channel names, specified as a string, character vector, string array, or cell array of character vectors.

If you are using a timetable for the data source, the names in
`OutputName`

must be a subset of the timetable variables.

**Example: **`sys = oe(tt,__,'OutputName',["y1" "y3"])`

selects
the variables `y1`

and `y3`

as the output channels
from the timetable `tt`

to use for the estimation.

`Ts`

— Sample time

`1`

(default) | positive scalar

Sample time, specified as the comma-separated pair consisting of `'Ts'`

and
the sample time in the units specified by `TimeUnit`

. When you use
matrix-based data (`u`

,`y`

), you must specify
`Ts`

if you require a sample time other than the assumed sample
time of 1 second.

To obtain the data sample time for a timetable `tt`

, use the timetable property `tt.Properties.Timestep`

.

**Example: **`oe(umat1,ymat1,___,'Ts',0.08)`

computes a model with sample
time of 0.08 seconds.

`InputDelay`

— Input delays

0 (default) | positive integer vector | integer scalar

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 *N _{u}* inputs, set

`InputDelay`

to an
*N*-by-1 vector. Each entry of this vector is a numerical value that represents the input delay for the corresponding input channel.

_{u}To apply the same delay to all channels, specify `'InputDelay'`

as a scalar.

For an example, see Estimate OE Polynomial Model.

`IODelay`

— Transport delays

0 (default) | scalar | numeric array

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 *N _{u}* inputs and

*N*outputs, set

_{y}`'IODelay'`

to an
*N*-by-

_{y}*N*matrix. Each entry is an integer value representing the transport delay for the corresponding input-output pair.

_{u}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

`sys`

— OE polynomial model

`idpoly`

object

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 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. | ||||||||||||||||||

`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 | ||||||||||||||||||

`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:
| ||||||||||||||||||

`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 | ||||||||||||||||||

`RandState` | State of the random number stream at the start of estimation. Empty,
| ||||||||||||||||||

`DataUsed` | Attributes of the data used for estimation, returned as a structure with the following fields.
| ||||||||||||||||||

`Termination` | Termination conditions for the iterative search used for prediction error minimization, returned as a structure with the following fields:
For estimation methods that do not require numerical search optimization,
the |

For more information on using `Report`

, see Estimation Report.

`ic`

— Initial conditions

`initialCondition`

object | object array of `initialCondition`

values

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 (*x*)._{0}For a multiple-experiment data set with

*N*experiments,_{e}`ic`

is an object array of length*N*that contains one set of_{e}`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)

`'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.

## More About

### Output-Error (OE) Model

The general output-error model structure is:

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

The orders of the output-error model are:

$$\begin{array}{l}nb\text{:}B(q)={b}_{1}+{b}_{2}{q}^{-1}+\mathrm{...}+{b}_{nb}{q}^{-nb+1}\\ nf\text{:}F(q)=1+{f}_{1}{q}^{-1}+\mathrm{...}+{f}_{nf}{q}^{-nf}\end{array}$$

### Continuous-Time Output-Error Model

If `data`

is continuous-time frequency-domain data,
`oe`

estimates a continuous-time model with the following transfer
function:

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

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)`

.

## Version History

**Introduced before R2006a**

### 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.

### 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™.

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