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findstatesOptions

Option set for findstates

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Syntax

opt = findstatesOptions
opt = findstatesOptions(Name,Value)

Description

opt = findstatesOptions creates the default option set for findstates. Use dot notation to customize the option set, if needed.

example

opt = findstatesOptions(Name,Value) creates an option set with options specified by one or more Name,Value pair arguments. The options that you do not specify retain their default value.

example

Examples

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Open Live Script

Create an option set for findstates by configuring a specification object for the initial states.

Identify a fourth-order state-space model from data.

load iddata8 z8;
sys = ssest(z8,4);

z8 is an iddata object containing time-domain system response data. sys is a fourth-order idss model that is identified from the data.

Configure a specification object for the initial states of the model.

x0obj = idpar([1;nan(3,1)]);
x0obj.Free(1) = false;
x0obj.Minimum(2) = 0;
x0obj.Maximum(2) = 1;

x0obj specifies estimation constraints on the initial conditions. The value of the first state is specified as 1 when x0obj is created. x0obj.Free(1) = false specifies the first initial state as a fixed estimation parameter. The second state is unknown. But, x0obj.Minimum(2) = 0 and x0obj.Maximum(2) = 1 specify the lower and upper bounds of the second state as 0 and 1, respectively.

Create an option set for findstates to identify the initial states of the model.

opt = findstatesOptions;
opt.InitialState = x0obj;

Identify the initial states of the model.

x0_estimated = findstates(sys,z8,Inf,opt);
Open Live Script

Create an option set for findstates where:

  • Initial states are estimated such that the norm of prediction error is minimized. The initial values of the states corresponding to nonzero delays are also estimated.

  • Adaptive subspace Gauss-Newton search is used for estimation. 

opt = findstatesOptions('InitialState','d','SearchMethod','gna');

Input Arguments

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Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: findstatesOptions('InitialState','d')

Estimation of initial states, specified as the comma-separated pair consisting of 'InitialState' and one of the following:

  • 'e' — The initial states are estimated such that the norm of prediction error is minimized.

    For nonlinear grey-box models, only those initial states i that are designated as free in the model (sys.InitialStates(i).Fixed = false) are estimated. To estimate all the states of the model, first specify all the Nx states of the idnlgrey model sys as free.

    for i = 1:Nx
sys.InitialStates(i).Fixed = false;
end

    Similarly, to fix all the initial states to values specified in sys.InitialStates, first specify all the states as fixed in the sys.InitialStates property of the nonlinear grey-box model.

  • 'd' — Similar to 'e', but absorbs nonzero delays into the model coefficients. The delays are first converted to explicit model states, and the initial values of those states are also estimated and returned.

    Use this option for discrete-time linear models only.

  • Vector or Matrix — Initial guess for state values, when using nonlinear models. Specify a column vector of length equal to the number of states. For multi-experiment data, use a matrix with Ne columns, where Ne is the number of experiments.

    Use this option for nonlinear models only.

  • x0obj — Specification object created using idpar. Use x0obj to impose constraints on the initial states by fixing their value or specifying minimum or maximum bounds.

    Use x0obj only for nonlinear grey-box models and linear state-space models (idss or idgrey). This option is applicable only for prediction horizon equal to 1 or Inf. See findstates for more details about the prediction horizon.

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 when using multi-output data, specified as the comma-separated pair consisting of 'OutputWeight' and one of the following:

  • [] — No weighting is used. Specifying as [] is the same as eye(Ny), where Ny is the number of outputs.

  • 'noise' — Inverse of the noise variance stored with the model is used for weighting during estimation of initial states.

  • Positive semidefinite matrix, W, of size Ny-by-Ny — This weighting minimizes trace(E'*E*W) for estimation of initial states, where E is the matrix of prediction errors.

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

SearchMethodDescription
'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'

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

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 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.

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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).

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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.

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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

To specify field values in SearchOptions, create a default findstatesOptions set and modify the fields using dot notation. Any fields that you do not modify retain their default values.

opt = findstatesOptions;
opt.SearchOptions.Tolerance = 0.02;
opt.SearchOptions.Advanced.MaxBisections = 30;

Output Arguments

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Option set for findstates, returned as an findstatesOptions option set.

Version History

Introduced in R2012a

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See Also

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