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tfestOptions

Option set for `tfest`

Syntax

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

Description

`opt = tfestOptions` creates the default option set for `tfest`.

`opt = tfestOptions(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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Algorithm used to initialize the values of the numerator and denominator of the output of `tfest`, applicable only for estimation of continuous-time transfer functions using time-domain data, 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 prediction error norm.

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 following fields:

• `N4Weight` — Calculates the weighting matrices used in the singular-value decomposition step of the `'n4sid'` algorithm. Applicable when `InitMethod` is `'n4sid'`.

`N4Weight` is specified as one of the following values:

• `'MOESP'` — Uses the MOESP algorithm by Verhaegen.

• `'CVA'` — Uses the canonical variable 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 should be used in the singular-value decomposition step.

Default: `‘auto'`

• `N4Horizon` — Determines the forward and backward prediction horizons used by the `'n4sid'` algorithm. Applicable when `InitMethod` 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, and `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 an Akaike Information Criterion (AIC) for the selection of `sy` and `su`.

Default: `'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.

Default: `0.1`

• `MaxIter` — Maximum number of iterations. Applicable when `InitMethod` is `'iv'`.

Default: `30`

• `Tolerance` — Convergence tolerance. Applicable when `InitMethod` is `'iv'`.

Default: `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 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 the 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.

• `'inv'` — Applicable for estimation using frequency-response data only. Use $1/|\left(G\left(jw\right)\right)|$ as the weighting filter. Where G(jw) 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 $1/\sqrt{|G\left(jw\right)|}$ 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.

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

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

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

Numerical 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 [3]. 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). gamma is increased by the factor `LMStep` each time the search fails to find a lower value of the criterion in less than 5 bisections. gamma 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, where 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. This search method can only handle the Trace criterion.

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

• `'auto'` — A choice among the preceding options is made in the algorithm. The descent direction is calculated using `'gn'`, `'gna'`, `'lm'`, and `'grad'` successively, at each iteration 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 [1].

`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 value.

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 conditions.

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 states.

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

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

Default: `1.05`

Output Arguments

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Option set for `tfest`, returned as an `tfestOptions` option set.

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('InitMethod','n4sid','Display','on'); ```

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

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

References

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

[2] Knudsen, T. "New method for estimating ARMAX models," In Proceedings of 10th IFAC Symposium on System Identification, SYSID'94, Copenhagen, Denmark, July 1994, Vol. 2, pp. 611–617.

[3] 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.

[4] Garnier, H., M. Mensler, and A. Richard. "Continuous-time Model Identification From Sampled Data: Implementation Issues and Performance Evaluation" International Journal of Control, 2003, Vol. 76, Issue 13, pp 1337–1357.

[5] Ljung, L. "Experiments With Identification of Continuous-Time Models." Proceedings of the 15th IFAC Symposium on System Identification. 2009.

[6] Jansson, M. "Subspace identification and ARX modeling." 13th IFAC Symposium on System Identification , Rotterdam, The Netherlands, 2003.