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# greyestOptions

Option set for `greyest`

## Syntax

`opt = greyestOptionsopt = greyestOptions(Name,Value)`

## Description

`opt = greyestOptions` creates the default options set for `greyest`.

`opt = greyestOptions(Name,Value)` creates an option set with the options specified by one or more `Name,Value` pair arguments.

## Input Arguments

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### 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 single quotes (`' '`). You can specify several name and value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.

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

• `'model'` — The initial state is parameterized by the ODE file used by the `idgrey` model. The ODE file must return 6 or more output arguments.

• `'zero'` — The initial state is set to zero. Any values returned by the ODE file are ignored.

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

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

• `'auto'` — The software chooses the method to handle initial states based on the estimation data.

• Vector of doubles — Specify a column vector of length Nx, where Nx is the number of states. For multiexperiment data, specify a matrix with Ne columns, where Ne is the number of experiments. The specified values are treated as fixed values during the estimation process.

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

• `'model'`K values are parameterized by the ODE file used by the `idgrey` model. The ODE file must return 5 or more output arguments.

• `'fixed'` — The value of the `K` property of the `idgrey` model is fixed to its original value.

• `'none'`K is fixed to zero. Any values returned by the ODE file are ignored.

• `'estimate'`K is treated as an independent estimation parameter.

• `'auto'` — The software chooses the method to handle how the disturbance component is handled during estimation. The software uses the `'model'` method if the ODE file returns 5 or more output arguments with a finite value for K. Else, the software uses the `'fixed'` method.

 Note:   Noise model cannot be estimated using frequency domain data.

Error to be minimized in the loss function during estimation, specified as the comma-separated pair consisting of `'Focus'` and one of the following values:

• `'prediction'` — The one-step ahead prediction error between measured and predicted outputs is minimized during estimation. As a result, the estimation focuses on producing a good predictor model.

• `'simulation'` — The simulation error between measured and simulated outputs is minimized during estimation. As a result, the estimation focuses on making a good fit for simulation of model response with the current inputs.

The `Focus` option can be interpreted as a weighting filter in the loss function. For more information, see Loss Function and Model Quality Metrics.

Weighting prefilter applied to the loss function to be minimized during estimation. To understand the effect of `WeightingFilter` on the loss function, see Loss Function and Model Quality Metrics.

Specify `WeightingFilter` as one of the following values:

• `[]` — No weighting prefilter is used.

• Passbands — Specify a row vector or matrix containing frequency values that define desired passbands. You select a frequency band where the fit between estimated model and estimation data is optimized. For example, `[wl,wh]` where `wl` and `wh` represent lower and upper limits of a passband. For a matrix with several rows defining frequency passbands, `[w1l,w1h;w2l,w2h;w3l,w3h;...]`, the estimation algorithm uses the union of the frequency ranges to define the estimation passband.

Passbands are expressed in `rad/TimeUnit` for time-domain data and in `FrequencyUnit` for frequency-domain data, where `TimeUnit` and `FrequencyUnit` are the time and frequency units of the estimation data.

• SISO filter — Specify a single-input-single-output (SISO) linear filter in one of the following ways:

• A SISO LTI model

• `{A,B,C,D}` format, which specifies the state-space matrices of a filter with the same sample time as estimation data.

• `{numerator,denominator}` format, which specifies the numerator and denominator of the filter as a transfer function with same sample time as estimation data.

This option calculates the weighting function as a product of the filter and the input spectrum to estimate the transfer function.

• Weighting vector — Applicable for frequency-domain data only. Specify a column vector of weights. This vector must have the same length as the frequency vector of the data set, `Data.Frequency`. Each input and output response in the data is multiplied by the corresponding weight at that frequency.

Control whether to enforce stability of estimated model, specified as the comma-separated pair consisting of `'EnforceStability'` and either `true` or `false`.

Data Types: `logical`

Controls whether parameter covariance data is generated, specified as `true` or `false`.

If `EstCovar` is `true`, then use `getcov` to fetch the covariance matrix from the estimated model.

Specify whether to display the estimation progress, specified as one of the following values:

• `'on'` — Information on model structure and estimation results are displayed in a progress-viewer window.

• `'off'` — No progress or results information is displayed.

Removal of offset from time-domain input data during estimation, specified as the comma-separated pair consisting of `'InputOffset'` and one of the following:

• A column vector of positive integers of length Nu, where Nu is the number of inputs.

• `[]` — Indicates no offset.

• Nu-by-Ne matrix — For multi-experiment data, specify `InputOffset` as an Nu-by-Ne matrix. Nu is the number of inputs, and Ne is the number of experiments.

Each entry specified by `InputOffset` is subtracted from the corresponding input data.

Removal of offset from time-domain output data during estimation, specified as the comma-separated pair consisting of `'OutputOffset'` and one of the following:

• A column vector of length Ny, where Ny is the number of outputs.

• `[]` — Indicates no offset.

• Ny-by-Ne matrix — For multi-experiment data, specify `OutputOffset` as a Ny-by-Ne matrix. Ny is the number of outputs, and Ne is the number of experiments.

Each entry specified by `OutputOffset` is subtracted from the corresponding output data.

Weighting of prediction errors in multi-output estimations, specified as one of the following values:

• `'noise'` — Minimize $\mathrm{det}\left(E\text{'}*E/N\right)$, where E represents the prediction error and `N` is the number of data samples. This choice is optimal in a statistical sense and leads to maximum likelihood estimates if nothing is known about the variance of the noise. It uses the inverse of the estimated noise variance as the weighting function.

 Note:   `OutputWeight` must not be `'noise'` if `SearchMethod` is `'lsqnonlin'`.
• Positive semidefinite symmetric matrix (`W`) — Minimize the trace of the weighted prediction error matrix `trace(E'*E*W/N)` where:

• E is the matrix of prediction errors, with one column for each output, and W is the positive semidefinite symmetric matrix of size equal to the number of outputs. Use W to specify the relative importance of outputs in multiple-output models, or the reliability of corresponding data.

• `N` is the number of data samples.

• `[]` — The software chooses between the `'noise'` or using the identity matrix for `W`.

This option is relevant for only multi-output models.

Options for regularized estimation of model parameters. For more information on regularization, see Regularized Estimates of Model Parameters.

`Regularization` is a structure with the following fields:

• `Lambda` — Constant that determines the bias versus variance tradeoff.

Specify a positive scalar to add the regularization term to the estimation cost.

The default value of zero implies no regularization.

Default: 0

• `R` — Weighting matrix.

Specify a vector of nonnegative numbers or a square positive semi-definite matrix. The length must be equal to the number of free parameters of the model.

For black-box models, using the default value is recommended. For structured and grey-box models, you can also specify a vector of `np` positive numbers such that each entry denotes the confidence in the value of the associated parameter.

The default value of 1 implies a value of `eye(npfree)`, where `npfree` is the number of free parameters.

Default: 1

• `Nominal` — The nominal value towards which the free parameters are pulled during estimation.

The default value of zero implies that the parameter values are pulled towards zero. If you are refining a model, you can set the value to `'model'` to pull the parameters towards the parameter values of the initial model. The initial parameter values must be finite for this setting to work.

Default: 0

Search method used for iterative parameter estimation, specified as one of the following values:

• `gn` — The subspace Gauss-Newton direction.

• `gna` — An adaptive version of subspace Gauss-Newton approach, suggested by Wills and Ninness [1].

• `lm` — Uses the Levenberg-Marquardt method.

• `lsqnonlin` — Uses the trust region reflective algorithm. Requires Optimization Toolbox™ software.

• `grad` — The steepest descent gradient search method.

• `auto` — The algorithm chooses one of the preceding options. The descent direction is calculated using `gn`, `gna`, `lm`, and `grad` successively at each iteration. The iterations continue until a sufficient reduction in error is achieved.

Option set for the search algorithm with fields that depend on the value of `SearchMethod`.

• `ErrorThreshold` — Specifies when to adjust the weight of large errors from quadratic to linear.

Errors larger than `ErrorThreshold` times the estimated standard deviation have a linear weight in the loss function. The standard deviation is estimated robustly as the median of the absolute deviations from the median of the prediction errors, divided by `0.7`. For more information on robust norm choices, see section 15.2 of [2].

`ErrorThreshold = 0` disables robustification and leads to a purely quadratic loss function. When estimating with frequency-domain data, the software sets `ErrorThreshold` to zero. For time-domain data that contains outliers, try setting `ErrorThreshold` to `1.6`.

Default: 0

• `MaxSize` — Specifies the maximum number of elements in a segment when input-output data is split into segments.

`MaxSize` must be a positive integer.

Default: 250000

• `StabilityThreshold` — Specifies thresholds for stability tests.

`StabilityThreshold` is a structure with the following fields:

• `s` — Specifies the location of the right-most pole to test the stability of continuous-time models. A model is considered stable when its right-most pole is to the left of `s`.

Default: 0

• `z` — Specifies the maximum distance of all poles from the origin to test stability of discrete-time models. A model is considered stable if all poles are within the distance `z` from the origin.

Default: `1+sqrt(eps)`

• `AutoInitThreshold` — Specifies when to automatically estimate the initial state.

The initial state is estimated when

`$\frac{‖{y}_{p,z}-{y}_{meas}‖}{‖{y}_{p,e}-{y}_{meas}‖}>\text{AutoInitThreshold}$`
• ymeas is the measured output.

• yp,z is the predicted output of a model estimated using zero initial states.

• yp,e is the predicted output of a model estimated using estimated initial states.

Applicable when `InitialState` is `'auto'`.

Default: `1.05`

## Output Arguments

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Option set for `greyest`, returned as an `greyestOptions` option set.

## Examples

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```opt = greyestOptions; ```

Create an options set for `greyest` using the `'backcast'` algorithm to initialize the state. Specify `Display` as `'on'`.

```opt = greyestOptions('InitialState','backcast','Display','on'); ```

Alternatively, use dot notation to set the values of `opt`.

```opt = greyestOptions; opt.InitialState = 'backcast'; opt.Display = 'on'; ```

## References

[1] Wills, Adrian, B. Ninness, and S. Gibson. "On Gradient-Based Search for Multivariable System Estimates". Proceedings of the 16th IFAC World Congress, Prague, Czech Republic, July 3–8, 2005. Oxford, UK: Elsevier Ltd., 2005.

[2] Ljung, L. System Identification: Theory for the User. Upper Saddle River, NJ: Prentice-Hall PTR, 1999.