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armaxOptions

Option set for armax

Syntax

opt = armaxOptions
opt = 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

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 GnPinvConst*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 InitGnaTol (see Advanced in 'SearchOption' 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 SearchOption.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 'SearchOption' and a search option set with fields that depend on the value of SearchMethod.

SearchOption 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
MaxIter

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

Setting MaxIter = 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.

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Advanced

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

Field NameDescriptionDefault
GnPinvConst

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

10000
InitGnaTol

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
MaxFunEvals

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
MinParChange

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

0
RelImprovement

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).

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SearchOption Structure When SearchMethod is Specified as 'lsqnonlin'

Field NameDescriptionDefault
TolFun

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

The value of TolFun is the same as that of opt.SearchOption.Advanced.TolFun.

1e-5
TolX

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

The value of TolX is the same as that of opt.SearchOption.Advanced.TolX.

1e-6
MaxIter

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

The value of MaxIter is the same as that of opt.SearchOption.Advanced.MaxIter.

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Advanced

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.

SearchOption 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'
TolFun

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

1e-6
TolX

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

1e-6
MaxIter

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

100

Additional advanced options, specified as 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 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

    yp,zymeasyp,eymeas>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.

Introduced in R2012a

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