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# armaxOptions

Option set for `armax`

## Syntax

`opt = armaxOptionsopt = armaxOptions(Name,Value)`

## Description

`opt = armaxOptions` creates the default options set for `armax`.

`opt = armaxOptions(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 conditions during estimation, specified as one of the following values:

• `'zero'` — The initial conditions are set to zero.

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

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

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

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

This option is not available for multi-output models with a non-diagonal A polynomial array.

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.

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. Singular values of the Jacobian matrix less than `GnPinvConst*eps*max(size(J))*norm(J)` are discarded when computing the search direction. J is the Jacobian matrix. The Hessian matrix is approximated by JTJ. If there is no improvement in this direction, the function tries the gradient direction.

• `'gna'` — An adaptive version of subspace Gauss-Newton approach, suggested by Wills and Ninness [1]. Eigenvalues less than `gamma*max(sv)` of the Hessian are ignored, where sv are the singular values of the Hessian. The Gauss-Newton direction is computed in the remaining subspace. gamma has the initial value `InitGnaTol` (see `Advanced` for more information). This value is increased by the factor `LMStep` each time the search fails to find a lower value of the criterion in less than 5 bisections. This value is decreased by the factor `2*LMStep` each time a search is successful without any bisections.

• `'lm'` — Uses the Levenberg-Marquardt method so that the next parameter value is `-pinv(H+d*I)*grad` from the previous one. H is the Hessian, I is the identity matrix, and grad is the gradient. d is a number that is increased until a lower value of the criterion is found.

• `'lsqnonlin'` — Uses `lsqnonlin` optimizer from Optimization Toolbox™ software. You must have Optimization Toolbox installed to use this option. This search method can handle only the Trace criterion.

• `'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 and 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 condition.

The initial condition 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 conditions.

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

Applicable when `InitialCondition` is `'auto'`.

Default: `1.05`

## Output Arguments

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Option set for `armax`, returned as an `armaxOptions` option set.

## Examples

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```opt = armaxOptions; ```

Create an option set for `armax` to use the 'simulation' Focus and to set the Display to 'on'.

```opt = armaxOptions('Focus','simulation','Display','on'); ```

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

```opt = armaxOptions; opt.Focus = 'simulation'; 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.