# tfestOptions

Option set for `tfest`

## Description

Use a `tfestOptions` object to specify options for estimating transfer function models using the `tfest` function. You can specify options such as the estimation objective, the handling of initial conditions, and the numerical search method to be used in estimation.

## Creation

### Syntax

``opt = tfestOptions``
``opt = tfestOptions(Name,Value)``

### Description

example

````opt = tfestOptions` creates the default option set for estimating a transfer function model using `tfest`. To modify the properties of this option set for your specific application, use dot notation.```

example

````opt = tfestOptions(Name,Value)` creates an option set with properties specified using one or more name-value arguments.```

## Properties

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Algorithm used to initialize the values of the numerator and denominator of the output of `tfest`, specified as one of the following values:

• `'iv'` — Instrument Variable approach.

• `'svf'` — State Variable Filters approach.

• `'gpmf'` — Generalized Poisson Moment Functions approach.

• `'n4sid'` — Subspace state-space estimation approach.

• `'all'` — Combination of all of the preceding approaches. The software tries all these methods and selects the method that yields the smallest value of the prediction error norm.

This property is applicable only for estimation of continuous-time transfer functions using time-domain data

Option set for the initialization algorithm used to initialize the values of the numerator and denominator of the output of `tfest`, specified as a structure with the fields in the following table.

Field NameDescriptionDefault
`N4Weight`

Calculates the weighting matrices used in the singular-value decomposition step of the `'n4sid'` algorithm. Applicable when `InitializeMethod` is `'n4sid'`. Options are shown in the following table.

OptionDescription
`'MOESP'`

Uses the MOESP (Multivariable Output Error State Space) algorithm by Verhaegen.

`'CVA'`

Uses the canonical variate algorithm (CVA) by Larimore.

`'SSARX'`

A subspace identification method that uses an ARX-estimation-based algorithm to compute the weighting.

Specifying this option allows unbiased estimates when using data that is collected in closed-loop operation. For more information about the algorithm, see [6].

`'auto'`

The software automatically determines if the MOESP algorithm or the CVA algorithm is used in the singular-value decomposition step.

`'auto'`
`N4Horizon`

Determines the forward and backward prediction horizons used by the `'n4sid'` algorithm. Applicable when `InitializeMethod` is `'n4sid'`.

`N4Horizon` is a row vector with three elements, `[r sy su]`, where:

• `r` is the maximum forward prediction horizon. The algorithm uses up to `r` step-ahead predictors.

• `sy` is the number of past outputs.

• `su` is the number of past inputs that are used for the predictions.

See pages 209 and 210 in [1] for more information. These numbers can have a substantial influence on the quality of the resulting model, and there are no simple rules for choosing them. Making `'N4Horizon'` a `k`-by-3 matrix means that each row of `'N4Horizon'` is tried, and the value that gives the best (prediction) fit to data is selected. `k` is the number of guesses of `[r sy su]` combinations.

If `N4Horizon = 'auto'`, the software uses the Akaike Information Criterion (AIC) for the selection of `sy` and `su`.

`'auto'`
`FilterTimeConstant`

Time constant of the differentiating filter used by the `iv`, `svf`, and `gpmf` initialization methods (see [4] and [5]).

`FilterTimeConstant` specifies the cutoff frequency of the differentiating filter, Fcutoff, as:

`${F}_{cutoff}=\frac{\text{FilterTimeConstant}}{{T}_{s}}$`

Ts is the sample time of the estimation data.

Specify `FilterTimeConstant` as a positive number, typically less than 1. A good value of `FilterTimeConstant` is the ratio of Ts to the dominating time constant of the system.

`0.1`
`MaxIterations`Maximum number of iterations. Applicable when `InitializeMethod` is `'iv'`. `30`
`Tolerance`Convergence tolerance. Applicable when `InitializeMethod` is `'iv'`.`0.01`

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

• `'zero'` — All initial conditions are taken as zero.

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

• `'backcast'` — The necessary initial conditions are estimated by a backcasting (backward filtering) process, described in [2].

• `'auto'` — An automatic choice among the preceding options is made, guided by the data.

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 values in the following table.

ValueDescription
`[]` 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, specify `[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 the estimation data.

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

`'inv'`

Applicable for estimation using frequency-response data only. Use $1/|G\left(\omega \right)|$ as the weighting filter, where G(ω) is the complex frequency-response data. Use this option for capturing relatively low amplitude dynamics in data, or for fitting data with high modal density. This option also makes it easier to specify channel-dependent weighting filters for MIMO frequency-response data.

`'invsqrt'`

Applicable for estimation using frequency-response data only. Use $1/\sqrt{|G\left(\omega \right)|}$ as the weighting filter. Use this option for capturing relatively low amplitude dynamics in data, or for fitting data with high modal density. This option also makes it easier to specify channel-dependent weighting filters for MIMO frequency-response data.

Option to enforce stability of the estimated model, specified as `true` or `false`.

Use this option when estimating models using frequency-domain data. Models estimated using time-domain data are always stable.

Option to generate parameter covariance data, specified as `true` or `false`.

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

Option 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.

Input-channel intersample behavior for transformations between discrete time and continuous time, specified as `'auto'`, `'zoh'`,`'foh'`, or `'bl'`.

The definitions of the three behavior values are as follows:

• `'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.

`iddata` objects have a similar property, `data.InterSample`, that contains the same behavior value options. When the `InputInterSample` value is `'auto'` and the estimation data is in an `iddata` object `data`, the software uses the `data.InterSample` value. When the estimation data is instead contained in a timetable or a matrix pair, with the `'auto'` option, the software uses `'zoh'`.

The software applies the same option value to all channels and all experiments.

Removal of offset from time-domain input data during estimation, specified as 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 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 `'noise'` and using the identity matrix for `W`.

This option is relevant for only multi-output models.

Options for regularized estimation of model parameters, specified as a structure with the fields in the following table. For more information on regularization, see Regularized Estimates of Model Parameters.

Field NameDescriptionDefault
`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 0 implies no regularization.

`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.

`1`
`Nominal`

The nominal value towards which the free parameters are pulled during estimation.

The default value of 0 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.

0

Numerical search method used for iterative parameter estimation, specified as the one of the values in the following table.

`SearchMethod`Description
`'auto'`

Automatic method selection

A combination of the line search algorithms, `'gn'`, `'lm'`, `'gna'`, and `'grad'`, is tried in sequence at each iteration. The first descent direction leading to a reduction in estimation cost is used.

`'gn'`

Subspace Gauss-Newton least-squares search

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

`'gna'`

Eigenvalues less than `gamma*max(sv)` of the Hessian are ignored, where sv contains the singular values of the Hessian. The Gauss-Newton direction is computed in the remaining subspace. gamma has the initial value `InitialGnaTolerance` (see `Advanced` in `'SearchOptions'` 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 fewer than five bisections. This value is decreased by the factor `2*LMStep` each time a search is successful without any bisections.

`'lm'`

Levenberg-Marquardt least squares search

Each parameter value is `-pinv(H+d*I)*grad` from the previous value. 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.

`'grad'`

Steepest descent least-squares search

`'lsqnonlin'`

Trust-region-reflective algorithm of `lsqnonlin` (Optimization Toolbox)

This algorithm requires Optimization Toolbox™ software.

`'fmincon'`

Constrained nonlinear solvers

You can use the sequential quadratic programming (SQP) and trust-region-reflective algorithms of the `fmincon` (Optimization Toolbox) solver. If you have Optimization Toolbox software, you can also use the interior-point and active-set algorithms of the `fmincon` solver. Specify the algorithm in the `SearchOptions.Algorithm` option. The `fmincon` algorithms might result in improved estimation results in the following scenarios:

• Constrained minimization problems when bounds are imposed on the model parameters.

• Model structures where the loss function is a nonlinear or nonsmooth function of the parameters.

• Multiple-output model estimation. A determinant loss function is minimized by default for multiple-output model estimation. `fmincon` algorithms are able to minimize such loss functions directly. The other search methods such as `'lm'` and `'gn'` minimize the determinant loss function by alternately estimating the noise variance and reducing the loss value for a given noise variance value. Hence, the `fmincon` algorithms can offer better efficiency and accuracy for multiple-output model estimations.

Option set for the search algorithm, specified as a search option set with fields that depend on the value of `SearchMethod`.

`SearchOptions` Structure When `SearchMethod` is Specified as `'gn'`, `'gna'`, `'lm'`, `'grad'`, or `'auto'`

Field NameDescriptionDefault
`Tolerance`

Minimum percentage difference between the current value of the loss function and its expected improvement after the next iteration, specified as a positive scalar. When the percentage of expected improvement is less than `Tolerance`, the iterations stop. The estimate of the expected loss-function improvement at the next iteration is based on the Gauss-Newton vector computed for the current parameter value.

`0.01`
`MaxIterations`

Maximum number of iterations during loss-function minimization, specified as a positive integer. The iterations stop when `MaxIterations` is reached or another stopping criterion is satisfied, such as `Tolerance`.

Setting `MaxIterations = 0` returns the result of the start-up procedure.

Use `sys.Report.Termination.Iterations` to get the actual number of iterations during an estimation, where sys is an `idtf` model.

`20`
`Advanced`

Advanced search settings, specified as a structure with the following fields.

Field NameDescriptionDefault
`GnPinvConstant`

Jacobian matrix singular value threshold, specified as a positive scalar. Singular values of the Jacobian matrix that are smaller than `GnPinvConstant*max(size(J)*norm(J)*eps)` are discarded when computing the search direction. Applicable when `SearchMethod` is `'gn'`.

`10000`
`InitialGnaTolerance`

Initial value of gamma, specified as a positive scalar. Applicable when `SearchMethod` is `'gna'`.

`0.0001`
`LMStartValue`

Starting value of search-direction length d in the Levenberg-Marquardt method, specified as a positive scalar. Applicable when `SearchMethod` is `'lm'`.

`0.001`
`LMStep`

Size of the Levenberg-Marquardt step, specified as a positive integer. The next value of the search-direction length d in the Levenberg-Marquardt method is `LMStep` times the previous one. Applicable when `SearchMethod` is `'lm'`.

`2`
`MaxBisections`

Maximum number of bisections used for line search along the search direction, specified as a positive integer.

`25`
`MaxFunctionEvaluations`

Maximum number of calls to the model file, specified as a positive integer. Iterations stop if the number of calls to the model file exceeds this value.

`Inf`
`MinParameterChange `

Smallest parameter update allowed per iteration, specified as a nonnegative scalar.

`0`
`RelativeImprovement`

Relative improvement threshold, specified as a nonnegative scalar. Iterations stop if the relative improvement of the criterion function is less than this value.

`0`
`StepReduction`

Step reduction factor, specified as a positive scalar that is greater than 1. The suggested parameter update is reduced by the factor `StepReduction` after each try. This reduction continues until `MaxBisections` tries are completed or a lower value of the criterion function is obtained.

`StepReduction` is not applicable for a `SearchMethod` of `'lm'` (Levenberg-Marquardt method).

`2`

`SearchOptions` Structure When `SearchMethod` is Specified as `'lsqnonlin'`

Field NameDescriptionDefault
`FunctionTolerance`

Termination tolerance on the loss function that the software minimizes to determine the estimated parameter values, specified as a positive scalar.

The value of `FunctionTolerance` is the same as that of `opt.SearchOptions.Advanced.TolFun`.

`1e-5`
`StepTolerance`

Termination tolerance on the estimated parameter values, specified as a positive scalar.

The value of `StepTolerance` is the same as that of `opt.SearchOptions.Advanced.TolX`.

`1e-6`
`MaxIterations`

Maximum number of iterations during loss-function minimization, specified as a positive integer. The iterations stop when `MaxIterations` is reached or another stopping criterion is satisfied, such as `FunctionTolerance`.

The value of `MaxIterations` is the same as that of `opt.SearchOptions.Advanced.MaxIter`.

`20`

`SearchOptions` Structure When `SearchMethod` is Specified as `'fmincon'`

Field NameDescriptionDefault
`Algorithm`

`fmincon` optimization algorithm, specified as one of the following:

• `'sqp'` — Sequential quadratic programming algorithm. The algorithm satisfies bounds at all iterations, and it can recover from `NaN` or `Inf` results. It is not a large-scale algorithm. For more information, see Large-Scale vs. Medium-Scale Algorithms (Optimization Toolbox).

• `'trust-region-reflective'` — Subspace trust-region method based on the interior-reflective Newton method. It is a large-scale algorithm.

• `'interior-point'` — Large-scale algorithm that requires Optimization Toolbox software. The algorithm satisfies bounds at all iterations, and it can recover from `NaN` or `Inf` results.

• `'active-set'` — Requires Optimization Toolbox software. The algorithm can take large steps, which adds speed. It is not a large-scale algorithm.

For more information about the algorithms, see Constrained Nonlinear Optimization Algorithms (Optimization Toolbox) and Choosing the Algorithm (Optimization Toolbox).

`'sqp'`
`FunctionTolerance`

Termination tolerance on the loss function that the software minimizes to determine the estimated parameter values, specified as a positive scalar.

`1e-6`
`StepTolerance`

Termination tolerance on the estimated parameter values, specified as a positive scalar.

`1e-6`
`MaxIterations`

Maximum number of iterations during loss function minimization, specified as a positive integer. The iterations stop when `MaxIterations` is reached or another stopping criterion is satisfied, such as `FunctionTolerance`.

`100`

Additional advanced options, specified as a structure with the fields in the following table.

Field NameDescriptionDefault
`ErrorThreshold`

Error threshold at which 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 [1].

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

`0`
`MaxSize`

Maximum number of elements in a segment when input-output data is split into segments.

`MaxSize` must be a positive integer value.

`250000`
`StabilityThreshold`

Threshold for stability tests.

Field NameDescriptionDefault
`s`

Location of the right-most pole.

The software uses `s` to test the stability of continuous-time models. A model is considered stable when its right-most pole is to the left of `s`.

`0`
`z`

Maximum distance of all poles from the origin.

The software uses `z` to test the stability of discrete-time models. A model is considered stable if all poles are within the distance `z` from the origin.

`1+sqrt(eps)`

`AutoInitThreshold`

Threshold at which to automatically estimate initial conditions.

The software estimates the initial conditions when:

`$\frac{‖{y}_{p,z}-{y}_{meas}‖}{‖{y}_{p,e}-{y}_{meas}‖}>\text{AutoInitThreshold}$`
`1.05`

## Examples

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`opt = tfestOptions;`

Create an options set for `tfest` using the `'n4sid'` initialization algorithm and set the `Display` to `'on'`.

`opt = tfestOptions('InitializeMethod','n4sid','Display','on');`

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

```opt = tfestOptions; opt.InitializeMethod = 'n4sid'; opt.Display = 'on';```

## References

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

[2] Knudsen, T. "A New method for estimating ARMAX models," IFAC Proceedings Volumes 27, no. 8 (July 1994): 895–901. https://doi.org/10.1016/S1474-6670(17)47823-2.

[3] Wills, Adrian, B. Ninness, and S. Gibson. “On Gradient-Based Search for Multivariable System Estimates.” IFAC Proceedings Volumes 38, No 1 (2005): 832–837. https://doi.org/10.3182/20050703-6-CZ-1902.00140.

[4] Garnier, H., M. Mensler, and A. Richard. “Continuous-time Model Identification From Sampled Data: Implementation Issues and Performance Evaluation” International Journal of Control 76, no 13 (January 2003): 1337–57. https://doi.org/10.1080/0020717031000149636.

[5] Ljung, Lennart. “Experiments With Identification of Continuous-Time Models.” IFAC Proceedings Volumes 42, no. 10 (2009):1175–80. https://doi.org/10.3182/20090706-3-FR-2004.00195.

[6] Jansson, Magnus. “Subspace identification and ARX modeling.” IFAC Proceedings Volumes 36 no. 16 (September 2003): 1585–90. https://doi.org/10.1016/S1474-6670(17)34986-8

## Version History

Introduced in R2012b

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