Documentation

# procestOptions

Options set for procest

## Syntax

opt = procestOptions
opt = procestOptions(Name,Value)

## Description

opt = procestOptions creates the default options set for procest.

opt = procestOptions(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 quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

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

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

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

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

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

Handling of additive noise (H) during estimation for the model

$y=G\left(s\right)u+H\left(s\right)e$

e is white noise, u is the input and y is the output.

H(s) is stored in the NoiseTF property of the numerator and denominator of idproc models.

DisturbanceModel is specified as one of the following values:

• 'none'H is fixed to one.

• 'estimate'H is treated as an estimation parameter. The software uses the value of the NoiseTF property as the initial guess.

• 'ARMA1' — The software estimates H as a first-order ARMA model

$\frac{1+cs}{1+ds}$

• 'ARMA2' — The software estimates H as a second-order ARMA model

$\frac{1+{c}_{1}s+{c}_{2}{s}^{2}}{1+{d}_{1}s+{d}_{2}{s}^{2}}$

• 'fixed' — The software fixes the value of the NoiseTF property of the idproc model as the value of H.

### Note

A 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.

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

If EstimateCovariance 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 one of the following values:

• 'estimate' — The software treats the input offsets as an estimation parameter.

• 'auto' — The software chooses the method to handle input offsets based on the estimation data and the model structure. The estimation either assumes zero input offset or estimates the input offset.

For example, the software estimates the input offset for a model that contains an integrator.

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

Use [] to specify no offsets.

In case of multi-experiment 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.

• A parameter object, constructed using param.Continuous, that imposes constraints on how the software estimates the input offset.

For example, create a parameter object for a 2-input model estimation. Specify the first input offset as fixed to zero and the second input offset as an estimation parameter.

opt = procestOptions;
u0 = param.Continuous('u0',[0;NaN]);
u0.Free(1) = false;
opt.Inputoffset = u0;

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 the comma-separated pair consisting of 'SearchMethod' and one of the following:

• 'auto' — A combination of the line search algorithms, 'gn', 'lm', 'gna', and 'grad' methods 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 there is no improvement in this direction, the function tries the gradient direction.

• 'gna' — Adaptive subspace Gauss-Newton search. 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, where 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.

• 'grad' — Steepest descent least squares search.

• 'lsqnonlin' — Trust-region-reflective algorithm of lsqnonlin. Requires Optimization Toolbox™ software.

• 'fmincon' — Constrained nonlinear solvers. You can use the sequential quadratic programming (SQP) and trust-region-reflective algorithms of the fmincon 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 may result in improved estimation results in the following scenarios:

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

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

• Multi-output model estimation. A determinant loss function is minimized by default for multi-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 multi-output model estimations.

Option set for the search algorithm, specified as the comma-separated pair consisting of 'SearchOptions' and 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 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 SearchMethod '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

Advanced search settings, specified as an option set for lsqnonlin.

For more information, see the Optimization Options table in Optimization Options (Optimization Toolbox).

Use optimset('lsqnonlin') to create a default option set.

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

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 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.

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 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 procest, returned as a procestOptions option set.

## Examples

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opt = procestOptions;

Create an option set for procest setting Focus to 'simulation' and turning on the Display.

opt = procestOptions('Focus','simulation','Display','on');

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

opt = procestOptions;
opt.Focus = 'simulation';
opt.Display = 'on';

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## References

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

[2] 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.