ssestOptions

Option set for ssest

Description

Use an ssestOptions object to specify options for estimating state-space models using the ssest function. You can specify options such as the handling of initial states, stability enforcement, and the numerical search method to be used in estimation.

Creation

Syntax

opt = ssestOptions

opt = ssestOptions(Name,Value)

Description

opt = ssestOptions creates the default option set for ssest. To modify the properties of this option set for your specific application, use dot notation.

opt = ssestOptions(Name,Value) creates an option set with the properties specified using one or more name-value arguments.

Properties

InitializeMethod — Algorithm used to initialize the state-space parameters

'auto' (default) | 'n4sid' | 'lsrf'

Algorithm used to initialize the state-space parameter values for ssest, specified as one of the following values:

• 'auto' — ssest selects automatically:

  • lsrf, if the system is non-MIMO, the data is frequency-domain, and the state-space parameters are real-valued.

  • n4sid otherwise (time-domain, MIMO, or with complex-valued state-space parameters).

• 'n4sid' — Subspace state-space estimation approach — can be used with all systems (see n4sid).

• 'lsrf' — Least-squares rational function estimation-based approach [7] (see Continuous-Time Transfer Function Estimation Using Continuous-Time Frequency-Domain Data) — can provide higher-accuracy results for non-MIMO frequency-domain systems with real-valued state-space parameters, but cannot be used for any other systems (time-domain, MIMO, or with complex-valued state-space 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 multiple-experiment 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 discrete-time state-space models.

N4Weight — Weighting scheme used for singular-value decomposition by the N4SID algorithm

'auto' (default) | 'MOESP' | 'CVA' | 'SSARX'

Weighting scheme used for singular-value decomposition by the N4SID algorithm, specified as one of the following values:

• 'MOESP' — Uses the MOESP algorithm by Verhaegen [2].

• 'CVA' — Uses the Canonical Variate 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 closed-loop operation. For more information about the algorithm, see [6].

• 'auto' — The estimating function chooses between the MOESP and CVA algorithms.

N4Horizon — Forward- and backward-prediction horizons used by the N4SID algorithm

'auto' (default) | vector [r sy su] | k-by-3 matrix

Forward 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 step-ahead 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-by-3 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 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.

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 values in the following table.

Value	Description
[]	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, specify [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 the estimation data. {numerator,denominator} format, which specifies the numerator and denominator of the filter as a transfer function with the same sample time as the 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.
'inv'	Applicable for estimation using frequency-response data only. Use $$1/|G(\omega )|$$ as the weighting filter, where G(ω) is the complex frequency-response 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 channel-dependent weighting filters for MIMO frequency-response data.
'invsqrt'	Applicable for estimation using frequency-response 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 channel-dependent weighting filters for MIMO frequency-response data.

EnforceStability — Control whether to enforce stability of model

false (default) | true

Control whether to enforce stability of estimated model, specified as the comma-separated pair consisting of 'EnforceStability' and either true or false.

EstimateCovariance — Option to generate parameter covariance data

true (default) | false

Option to generate parameter covariance data, specified as true or false.

If EstimateCovariance is true, then use getcov to fetch the covariance matrix from the estimated model.

Display — Option to display estimation progress

'off' (default) | 'on'

Option 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.

InputInterSample — Input-channel intersample behavior

'auto' | 'zoh' | 'foh' | 'bl'

Input-channel intersample behavior for transformations between discrete time and continuous time, specified as 'auto', 'zoh','foh', or 'bl'.

The definitions of the three behavior values are as follows:

• 'zoh' — Zero-order hold maintains a piecewise-constant input signal between samples.

• 'foh' — First-order hold maintains a piecewise-linear input signal between samples.

• 'bl' — Band-limited behavior specifies that the continuous-time input signal has zero power above the Nyquist frequency.

iddata objects have a similar property, data.InterSample, that contains the same behavior value options. When the InputInterSample value is 'auto' and the estimation data is in an iddata object data, the software uses the data.InterSample value. When the estimation data is instead contained in a timetable or a matrix pair, with the 'auto' option, the software uses 'zoh'.

The software applies the same option value to all channels and all experiments.

InputOffset — Removal of offset from time-domain input data during estimation

[] (default) | vector of positive integers | matrix

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.

OutputOffset — Removal of offset from time-domain output data during estimation

[] (default) | vector | matrix

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.

OutputWeight — Weighting of prediction errors in multi-output estimations

[] (default) | 'noise' | positive semidefinite symmetric matrix

Weighting of prediction errors in multi-output estimations, specified as one of the following values:

• 'noise' — Minimize $$\det (E\text{'}*E/N)$$, 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.

  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 'noise' and using the identity matrix for W.

This option is relevant for only multi-output models.

Regularization — Options for regularized estimation of model parameters

structure

Options for regularized estimation of model parameters, specified as a structure with the fields in the following table. For more information on regularization, see Regularized Estimates of Model Parameters.

Field Name	Description	Default
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 0 implies no regularization.	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.	1
Nominal	The nominal value towards which the free parameters are pulled during estimation. The default value of 0 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.	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 one of the values in the following table.

SearchMethod	Description
'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.

SearchOptions — Option set for search algorithm

search option set

Option set for the search algorithm, specified as 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 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	Advanced search settings, specified as a structure with the following fields. Field Name Description Default 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). 2

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 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

SearchOptions Structure When SearchMethod Is Specified as 'fmincon'

Field Name	Description	Default
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 — Additional advanced options

structure

Additional advanced options, specified as a structure with the fields in the following table.

Field Name	Description	Default
ErrorThreshold	Error threshold at which 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]. An ErrorThreshold value of 0 disables robustification and leads to a purely quadratic loss function. When estimating with frequency-domain data, the software sets ErrorThreshold to 0. For time-domain data that contains outliers, try setting ErrorThreshold to 1.6.	0
MaxSize	Maximum number of elements in a segment when input-output data is split into segments. MaxSize must be a positive integer value.	250000
StabilityThreshold	Threshold for stability tests. Field Name Description Default s Location of the right-most pole. The software uses s to test the stability of continuous-time models. A model is considered stable when its right-most pole is to the left of s. 0 z Maximum distance of all poles from the origin. The software uses z to test the stability of discrete-time models. A model is considered stable if all poles are within the distance z from the origin. 1+sqrt(eps)
AutoInitThreshold	Threshold at which to automatically estimate initial conditions. The software estimates the initial conditions when: $$\frac{\Vert y_{p,z}-y_{meas}\Vert }{\Vert y_{p,e}-y_{meas}\Vert }>\text{AutoInitThreshold}$$	1.05
DDC	Specifies if the Data Driven Coordinates algorithm [5] is used to estimate freely parameterized state-space 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 are updated directly using the specified SearchMethod.

Examples

Create Default Option Set for State Space Estimation

opt = ssestOptions

Option set for the ssest command:

      InitializeMethod: 'auto'
          InitialState: 'auto'
              N4Weight: 'auto'
             N4Horizon: 'auto'
               Display: 'off'
      InputInterSample: 'auto'
           InputOffset: []
          OutputOffset: []
    EstimateCovariance: 1
          OutputWeight: []
                 Focus: 'prediction'
       WeightingFilter: []
      EnforceStability: 0
          SearchMethod: 'auto'
         SearchOptions: '<Optimization options set>'
        Regularization: [1x1 struct]
              Advanced: [1x1 struct]

Specify Options for State Space Estimation

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")

Option set for the ssest command:

      InitializeMethod: 'auto'
          InitialState: 'backcast'
              N4Weight: 'auto'
             N4Horizon: 'auto'
               Display: 'on'
      InputInterSample: 'auto'
           InputOffset: []
          OutputOffset: []
    EstimateCovariance: 1
          OutputWeight: []
                 Focus: 'prediction'
       WeightingFilter: []
      EnforceStability: 0
          SearchMethod: 'auto'
         SearchOptions: '<Optimization options set>'
        Regularization: [1x1 struct]
              Advanced: [1x1 struct]

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

opt = ssestOptions;
opt.InitialState = "backcast";
opt.Display = "on";

References

[1] Larimore, Wallace 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, Michel. "Identification of the deterministic part of MIMO state space models given in innovations form from input-output data." Automatica, Vol. 30, No. 1, 1994, pp. 61–74. https://doi.org/10.1016/0005-1098(94)90229-1

[3] 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.

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

[5] McKelvey, Tomas, 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, Magnus. “Subspace identification and ARX modeling.” 13th IFAC Symposium on System Identification , Rotterdam, The Netherlands, 2003.

[7] Ozdemir, Ahmet Arda, 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.

Version History

Introduced in R2012a

R2022b: InputInterSample option allows intersample behavior specification for continuous models estimated from timetables or matrices.

iddata objects contain an InterSample property that describes the behavior of the signal between sample points. The InputInterSample option implements a version of that property in ssestOptions so that intersample behavior can be specified also when estimation data is stored in timetables or matrices.

R2018a: Renaming of Estimation and Analysis Options

The names of some estimation and analysis options were changed in R2018a. Prior names still work.