'iv' — Instrument Variable approach.

'svf' — State Variable Filters approach.

'gpmf' — Generalized Poisson Moment Functions approach.

'n4sid' — Subspace state-space 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.

N4Weight — Calculates the weighting matrices used in the singular-value decomposition step of the 'n4sid' algorithm. Applicable when InitializeMethod is 'n4sid'.

N4Weight is specified as one of the following values:

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 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 [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-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 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:

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

'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.

[] — 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:

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.

'on' — Information on model structure and estimation results are displayed in a progress-viewer window.

'off' — No progress or results information is displayed.

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.

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.

'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.

Positive semidefinite symmetric matrix (W) — Minimize the trace of the weighted prediction error matrix trace(E'*E*W/N) where:

[] — The software chooses between the 'noise' or using the identity matrix for W.

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

'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 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^TJ. 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 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, 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 (Optimization Toolbox). 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 may result in improved estimation results in the following scenarios:

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 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 value.

Default: 250000

StabilityThreshold — Specifies thresholds for stability tests.

StabilityThreshold is a structure with the following fields:

AutoInitThreshold — Specifies when to automatically estimate the initial conditions.

The initial condition is estimated when

Applicable when InitialCondition is 'auto'.

Default: 1.05