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Option set for tfest
opt = tfestOptions
opt = tfestOptions(Name,Value)
creates
the default option set for opt
= tfestOptionstfest
.
creates
an option set with the options specified by one or more opt
= tfestOptions(Name,Value
)Name,Value
pair
arguments.
Specify optional
commaseparated 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
.
'InitializeMethod'
— Algorithm used to initialize the numerator and denominator'iv'
(default)  'svf'
 'gpmf'
 'n4sid'
 'all'
Algorithm used to initialize the values of the numerator and denominator of the
output of tfest
, applicable only for estimation of
continuoustime transfer functions using timedomain 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 statespace 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.
'InitializeOptions'
— Option set for the initialization algorithmOption 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 singularvalue decomposition step of the 'n4sid'
algorithm.
Applicable when InitializeMethod
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 closedloop 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
singularvalue decomposition step.
Default:
'auto'
N4Horizon
— Determines the forward and backward
prediction horizons used by the 'n4sid'
algorithm. Applicable
when InitializeMethod
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
stepahead 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
by3 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, F_{cutoff},
as:
$${F}_{cutoff}=\frac{\text{FilterTimeConstant}}{{T}_{s}}$$
T_{s} 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
T_{s} to the dominating time constant
of the system.
Default:
0.1
MaxIterations
— Maximum number of iterations.
Applicable when InitializeMethod
is 'iv'
.
Default:
30
Tolerance
— Convergence tolerance. Applicable when
InitializeMethod
is 'iv'
.
Default:
0.01
'InitialCondition'
— Handling of initial conditions'auto'
(default)  'zero'
 'estimate'
 'backcast'
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.
'WeightingFilter'
— Weighting prefilter[]
(default)  vector  matrix  cell array  linear system  'inv'
 'invsqrt'
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
timedomain data and in FrequencyUnit
for frequencydomain
data, where TimeUnit
and FrequencyUnit
are
the time and frequency units of the estimation data.
SISO filter — Specify a singleinputsingleoutput (SISO) linear filter in one of the following ways:
A SISO LTI model
{A,B,C,D}
format, which specifies
the statespace 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 frequencydomain
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 frequencyresponse data
only. Use $$1/G(\omega )$$ as the weighting filter, where
G(ω) is the complex frequencyresponse
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 channeldependent weighting filters for MIMO frequencyresponse
data.
'invsqrt'
— Applicable for estimation using
frequencyresponse data only. Use $$1/\sqrt{G(\omega )}$$ 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 channeldependent weighting
filters for MIMO frequencyresponse data.
'EnforceStability'
— Control whether to enforce stability of modelfalse
(default)  true
Control whether to enforce stability of estimated model, specified
as the commaseparated pair consisting of 'EnforceStability'
and
either true
or false
.
Use this option when estimating models using frequencydomain data. Models estimated using timedomain data are always stable.
Data Types: logical
'EstimateCovariance'
— Control whether to generate parameter covariance datatrue
(default)  false
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.
'Display'
— Specify whether to display the estimation progress'off'
(default)  'on'
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 progressviewer
window.
'off'
— No progress or results
information is displayed.
'InputOffset'
— Removal of offset from timedomain input data during estimation[]
(default)  vector of positive integers  matrixRemoval of offset from timedomain input data during estimation,
specified as the commaseparated 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.
NubyNe matrix
— For multiexperiment data, specify InputOffset
as
an NubyNe 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'
— Removal of offset from timedomain output data during estimation[]
(default)  vector  matrixRemoval of offset from timedomain output data during estimation,
specified as the commaseparated 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.
NybyNe matrix
— For multiexperiment data, specify OutputOffset
as
a NybyNe 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'
— Weighting of prediction errors in multioutput estimations[]
(default)  'noise'
 positive semidefinite symmetric matrixWeighting of prediction errors in multioutput estimations, specified as one of the following values:
'noise'
— Minimize $$\mathrm{det}(E\text{'}*E/N)$$, 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.
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 multipleoutput 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 multioutput models.
'Regularization'
— Options for regularized estimation of model parametersOptions 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 semidefinite matrix. The length must be equal to the number of free parameters of the model.
For blackbox models, using the default value is recommended.
For structured and greybox 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'
— Numerical search method used for iterative parameter estimation'auto'
(default)  'gn'
 'gna'
 'lm'
 'grad'
 'lsqnonlin'
 'fmincon'
Numerical search method used for iterative parameter estimation,
specified as the commaseparated 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 GaussNewton 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
J^{T}J. If there is no
improvement in this direction, the function tries the gradient direction.
'gna'
— Adaptive subspace GaussNewton search.
Eigenvalues less than gamma*max(sv)
of the Hessian are
ignored, where sv contains the singular values of the
Hessian. The GaussNewton 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'
— LevenbergMarquardt
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'
— Trustregionreflective
algorithm of lsqnonlin
. Requires Optimization
Toolbox™ software.
'fmincon'
— Constrained nonlinear solvers. You can
use the sequential quadratic programming (SQP) and trustregionreflective
algorithms of the fmincon
solver. If you have
Optimization
Toolbox software, you can also use the interiorpoint and activeset
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.
Multioutput model estimation. A determinant loss function
is minimized by default for multioutput 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 multioutput model estimations.
'SearchOptions'
— Option set for the search algorithmOption set for the search algorithm, specified as the commaseparated 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 Name  Description  Default  

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  0.01  
MaxIterations  Maximum number of iterations during lossfunction minimization, specified as a positive
integer. The iterations stop when Setting
Use
 20  
Advanced  Advanced search settings, specified as a structure with the following fields:

SearchOptions
Structure When SearchMethod
is Specified
as 'lsqnonlin'
Field Name  Description  Default 

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  1e5 
StepTolerance  Termination tolerance on the estimated parameter values, specified as a positive scalar. The value of  1e6 
MaxIterations  Maximum number of iterations during lossfunction minimization, specified as a positive
integer. The iterations stop when The value of
 20 
Advanced  Advanced search settings, specified as an option set
for 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 Name  Description  Default 

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.  1e6 
StepTolerance  Termination tolerance on the estimated parameter values, specified as a positive scalar.  1e6 
MaxIterations  Maximum number of iterations during loss function minimization, specified as a positive
integer. The iterations stop when  100 
'Advanced'
— Additional advanced optionsAdditional 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, 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 frequencydomain data, the software sets ErrorThreshold
to
zero. For timedomain data that contains outliers, try setting ErrorThreshold
to 1.6
.
Default: 0
MaxSize
— Specifies the
maximum number of elements in a segment when inputoutput 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 rightmost pole to test the stability of continuoustime models.
A model is considered stable when its rightmost pole is to the left
of s
.
Default: 0
z
— Specifies the maximum
distance of all poles from the origin to test stability of discretetime
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
$$\frac{\Vert {y}_{p,z}{y}_{meas}\Vert}{\Vert {y}_{p,e}{y}_{meas}\Vert}>\text{AutoInitThreshold}$$
y_{meas} is the measured output.
y_{p,z} is the predicted output of a model estimated using zero initial states.
y_{p,e} is the predicted output of a model estimated using estimated initial states.
Applicable when InitialCondition
is 'auto'
.
Default: 1.05
opt
— Option set for tfest
tfestOptions
option setOption set for tfest
,
returned as an tfestOptions
option set.
opt = tfestOptions;
Create an options set for tfest
using the 'n4sid'
initialization algorithm and set the Display
to 'on'
.
opt = tfestOptions('InitializeMethod','n4sid','Display','on');
Alternatively, use dot notation to set the values of opt
.
opt = tfestOptions; opt.InitializeMethod = 'n4sid'; opt.Display = 'on';
[1] Ljung, L. System Identification: Theory for the User. Upper Saddle River, NJ: PrenticeHall 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 GradientBased 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. “Continuoustime 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 ContinuousTime 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.
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