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# ssestOptions

Option set for ssest

## Syntax

opt = ssestOptions
opt = ssestOptions(Name,Value)

## Description

opt = ssestOptions creates the default options set for ssest.

opt = ssestOptions(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.

### 'InitialState' —

Specify handling of initial states during estimation.

InitialState requires one of the following values:

• 'zero' — The initial state is set to zero.

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

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

• 'auto'ssest chooses the initial state handling method, based on the estimation data. The possible initial state handling methods are 'zero', 'estimate' and 'backcast'.

• Vector of doubles — Specify a column vector of length Nx, where Nx is the number of states. For multi-experiment 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.

• Parametric initial condition object (x0obj) — Specify initial conditions by using idpar to create a parametric initial condition object. You can specify minimum/maximum bounds and fix the values of specific states using the parametric initial condition object. The free entries of x0obj are estimated together with the idss model parameters.

Use this option only for discrete-time state-space models.

### 'N4Weight' —

Weighting scheme used for singular-value decomposition by the N4SID algorithm.

'N4Weight' requires one of the following values:

• 'MOESP' — Uses the MOESP algorithm by Verhaegen [2].

• 'CVA' — Uses the Canonical Variable Algorithm by Larimore [1].

• '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 estimating function chooses between the MOESP and CVA algorithms.

### 'N4Horizon' —

Forward and backward prediction horizons used by the N4SID algorithm.

'N4Horizon' requires one of the following values:

• 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 [4] 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 you specify N4Horizon as a single column, r = sy = su is used.

• 'auto' — The software uses an Akaike Information Criterion (AIC) for the selection of sy and su.

### 'Focus' —

Defines how the errors e between the measured and the modeled outputs are weighed at specific frequencies during the minimization of the prediction error.

Higher weighting at specific frequencies emphasizes the requirement for a good fit at these frequencies.

Focus requires one of the following values:

• 'simulation' — Estimates a stable model using the frequency weighting of the transfer function given by the input spectrum. Typically, this method favors the frequency range where the input spectrum has the most power.

• 'prediction' — Automatically calculates the weighting function as a product of the input spectrum and the inverse of the noise model. This option minimizes the one-step-ahead prediction, which typically favors fitting small time intervals (higher frequency range). From a statistical-variance point of view, this weighting function is optimal . However, this method neglects the approximation aspects (bias) of the fit, and might not result in a stable model. Use 'stability' when you want to ensure a stable model.

• 'stability' — Same as 'prediction', except that this method enforces model stability.

• Passbands — Row vector or matrix containing frequency values that define desired passbands. For example:

```[wl,wh]
[w1l,w1h;w2l,w2h;w3l,w3h;...]```

where wl and wh represent upper and lower limits of a passband. For a matrix with several rows defining frequency passbands, the algorithm uses union of frequency ranges to define the estimation passband.

• SISO filter — Enter any SISO linear filter in any of the following ways:

• A single-input-single-output (SISO) linear system.

• The {A,B,C,D} format, which specifies the state-space matrices of the filter.

• The {numerator, denominator} format, which specifies the numerator and denominator of the filter transfer function

This option calculates the weighting function as a product of the filter and the input spectrum to estimate the transfer function. To obtain a good model fit for a specific frequency range, you must choose the filter with a passband in this range. The estimation result is the same if you first prefilter the data using idfilt.

• Weighting vector — For frequency-domain data only, enter a column vector of weights for 'Focus'. This vector must have the same size as length of 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.

### 'EstCovar' — Control whether to generate parameter covariance datatrue (default) | false

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.

### 'Display' — Specify whether to display the estimation progress'off' (default) | 'on'

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

Display requires one of the following strings:

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

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

### 'InputOffset' — Remove offset from time-domain input data during estimation[] (default) | vector of positive integers

Removes offset from time-domain input data during estimation, specified as a vector of positive integers.

Specify as a column vector of length Nu, where Nu is the number of inputs.

Use [] to indicate no offset.

For multiexperiment data, specify InputOffset as a 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.

### 'OutputOffset' — Remove offset from time-domain output data during estimation[] (default) | vector

Removes offset from time domain output data during estimation, specified as a vector of positive integers or [].

Specify as a column vector of length Ny, where Ny is the number of outputs.

Use [] to indicate no offset.

For multiexperiment 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.

### 'OutputWeight' —

Specifies criterion used during minimization.

OutputWeight can have the following values:

• 'noise' — Minimize $\mathrm{det}\left(E\text{'}*E\right)$, where E represents the prediction error. 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). 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-input, multiple-output models, or the reliability of corresponding data.

This option is relevant for only multi-input, multi-output models.

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

### 'Regularization' —

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

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

### 'SearchMethod' —

Search method used for iterative parameter estimation.

SearchMethod requires 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. 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 the Optimization Toolbox™ software. This search method can only handle the Trace criterion.

• '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.

### 'SearchOption' —

Advanced is a structure with the following fields:

• 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 criteria. The standard deviation is estimated robustly as the median of the absolute deviations from the median and divided by 0.7. For more information on robust norm choices, see section 15.2 of [4].

ErrorThreshold = 0 disables robustification and leads to a purely quadratic criterion. 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 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 InitialState is 'auto'.

Default: 1.05

• DDC — Specifies if the Data Driven Coordinates algorithm [5] is used to estimate freely parameterized state-space models.

Specify DDC as one of the following values:

• 'on' — The free parameters are projected to a reduced space of identifiable parameters using the Data Driven Coordinates algorithm.

• 'off' — All the entries of A, B, and C updated directly using the chosen SearchMethod.

Default: 'on'

## Output Arguments

 opt Option set containing the specified options for ssest.

## Examples

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### Create Default Options Set for State Space Estimation

`opt = ssestOptions;`

### Specify Options for State Space Estimation

Create an options set for ssest using the 'backcast' algorithm to initialize the state and set the Display to 'on'.

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

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

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

## References

[1] Larimore, W.E. "Canonical variate analysis in identification, filtering and adaptive control." Proceedings of the 29th IEEE Conference on Decision and Control, pp. 596–604, 1990.

[2] Verhaegen, M. "Identification of the deterministic part of MIMO state space models." Automatica, Vol. 30, No. 1, 1994, pp. 61–74.

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

[5] McKelvey, T., A. Helmersson,, and T. Ribarits. "Data driven local coordinates for multivariable linear systems and their application to system identification." Automatica, Volume 40, No. 9, 2004, pp. 1629–1635.

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