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Option set for ssest
opt = ssestOptions
opt = ssestOptions(Name,Value)
creates
the default option set for opt
= ssestOptionsssest
.
creates an option set with the options specified by one or more
opt
= ssestOptions(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 statespace parameters'auto'
(default)  'n4sid'
 'lsrf'
Algorithm used to initialize the statespace parameter values for
ssest
, specified as one of the following
values:
'auto'
— ssest
selects automatically:
lsrf
, if the system is
nonMIMO, the data is frequencydomain, and the
statespace parameters are realvalued.
n4sid
otherwise (timedomain,
MIMO, or with complexvalued statespace
parameters).
'n4sid'
— Subspace statespace
estimation approach — can be used with all systems (see n4sid
).
'lsrf'
— Leastsquares rational function
estimationbased approach [7] (see ContinuousTime Transfer Function Estimation Using ContinuousTime FrequencyDomain Data) — can provide
higheraccuracy results for nonMIMO frequencydomain systems
with realvalued statespace parameters, but cannot be used for
any other systems (timedomain, MIMO, or with complexvalued
statespace parameters).
'InitialState'
— Handling of initial states'auto'
(default)  'zero'
 'estimate'
 'backcast'
 vector  parametric initial condition object
(x0obj
)Handling of initial states during estimation, specified as 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 multiexperiment 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 discretetime statespace models.
'N4Weight'
— Weighting scheme used for singularvalue decomposition by the N4SID algorithm'auto'
(default)  'MOESP'
 'CVA'
 'SSARX'
Weighting scheme used for singularvalue decomposition by the N4SID algorithm, specified as 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 closedloop operation. For more information about the algorithm, see [6].
'auto'
— The estimating function
chooses between the MOESP and CVA algorithms.
'N4Horizon'
— Forward and backwardprediction horizons used by the N4SID
algorithm'auto'
(default)  vector [r sy su]
 k
by3 matrixForward and backward prediction horizons used by the N4SID algorithm, specified as 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
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 [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
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
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'
— Error to be minimized'prediction'
(default)  'simulation'
Error to be minimized in the loss function during estimation,
specified as the commaseparated pair consisting of 'Focus'
and
one of the following values:
'prediction'
— The onestep
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.
'WeightingFilter'
— Weighting prefilter[]
(default)  vector  matrix  cell array  linear systemWeighting 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 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.
'invsqrt'
— Applicable for frequencydomain
data only, with InitializeMethod
set to
'lsrf'
only. Uses $$1/\sqrt{G(\omega )}$$ as the weighting filter, where
G(ω) is the complex
frequencyresponse data. Use this option for capturing
relatively low amplitude dynamics in data.
'inv'
— Applicable for frequencydomain
data only, with InitializeMethod
set to
'lsrf'
only. Uses $$1/G(\omega )$$ as the weighting filter. Similarly to
'invsqrt'
, this option captures
relatively lowamplitude dynamics in data. Use it when
'invsqrt'
weighting produces an estimate
that is missing dynamics in the lowamplitude regions.
'inv'
is more sensitive to noise than
'invsqrt'
.
'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
.
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 [4].
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.
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 InitialState
is
'auto'
.
Default:
1.05
DDC
— Specifies if the Data Driven
Coordinates algorithm [5] is used to estimate freely parameterized statespace
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'
opt
— Option set for ssest
ssestOptions
option setOption set for ssest
,
returned as an ssestOptions
option set.
opt = ssestOptions;
Create an option 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';
[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 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] Ljung, L. System Identification: Theory for the User. Upper Saddle River, NJ: PrenticeHall 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.
[7] Ozdemir, A. A., and S. Gumossoy. "Transfer Function Estimation in System identification Toolbox via Vector Fitting." Proceedings of the 20th World Congress of the International Federation of Automatic Control. Toulouse, France, July 2017.
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