System Identification Toolbox

Optional Variables

The estimation functions accept a list of property name/property value pairs that may affect both the model structure and the estimation algorithm. For complete lists of these properties, see algorithm properties, idarx, idmodel, idpoly, idss, and idgrey in the Function Reference. Some of them are listed here. Note that any property, as well as values that are strings, can be entered as any unambiguous, case-insensitive abbreviation, as is

• m = pem(Data,mi,'fo','si').
  

Note    Algorithm is a property of idmodel. Any algorithm property can be separately set as above. Also, if you have a standard algorithm set up that you prefer, you can set those properties simultaneously as in m = pem(Data,mi,'alg',myalg)

Note    The algorithm properties, like all other model properties, will be inherited by the resulting model m. If you continue the estimation using m as initial model, all previously set algorithm features will thus apply, unless you specify otherwise.

Applying to All Estimation Methods

The following properties apply to all estimation methods:

• Focus
• MaxSize
• FixedParameter

Focus: This property affects the weighting applied to the fit between the model and the data. It can be used to assure that the model approximates the true system well over certain frequency intervals. Focus can assume the following values:

• Prediction: This is the default and means that the model is determined by minimizing the prediction errors. It corresponds to a frequency weighting that is given by the input spectrum times the inverse noise model. Typically, this favors a good fit at high frequencies. From a statistical variance point of view, this is the optimal weighting, but then the approximation aspects (bias) of the fit are neglected.
• Simulation: This means that frequency weighting of the transfer function fit is given by the input spectrum. Frequency ranges where the input has considerable power will thus be better described by the model. In other words, the model approximation is such that the model will produce as good simulations as possible, when applied to inputs with the same spectra as used for the estimation. For models that have no disturbance model (A=C=D for idpoly models and K=0 for idss models) there is no difference between the Simulation and Prediction values. For models with a disturbance description, this is estimated by a prediction error method, keeping the estimated transfer function from input to output fixed. The resulting model is guaranteed to be stable.
• Stability: The algorithm is modified so that a stable model is guaranteed, but the weighing still correspond to prediction.
• Any SISO linear filter. Then the transfer function from input to output is determined by a frequency fit with this filter times the input spectrum as weighting function. The noise model is determined by a prediction error method, keeping the transfer function estimate fixed. To obtain a good model fit over a special frequency range, the filter should thus be chosen with a passband over this range. For a model with no disturbance model, the result is the same as first applying prefiltering to data using idfilt. The filter can be specified in a few different ways:
• as any single-input-single-output idmodel
• as a ss, tf, or zpk model from the Control System Toolbox
• as {A,B,C,D} with the state-space matrices for the filter
• as {numerator, denominator} with the transfer function numerator/denominator of the filter

MaxSize: No matrix with more than MaxSize elements is formed by the algorithm. Instead, for loops will be used. MaxSize will thus decide the memory/speed trade-off, and can prevent slow use of virtual memory. MaxSize can be any positive integer, but it is of course required that the input-output data themselves contain less than MaxSize elements. The default value of MaxSize is Auto, which means that the value is determined in the M-file idmsize. This file may be edited by the user to optimize speed on a particular computer. See also Memory/Speed Trade-Offs.

FixedParameter: A list of parameters that will be kept fixed to the nominal/initial values and not estimated. This is a vector of integers containing the indices of the fixed parameters, or a cell array of parameter names. If names are used, wildcard entries apply, which may be convenient if you have groups of parameters in your model. See the reference page of Algorithm properties.

Algorithm Properties That Apply to n4sid, Estimating State-Space Models

The properties that apply to sub-space model estimation are:

• N4Weight
• N4Horizon

These properties then also apply to pem for estimating black-box state-space models, since pem is then initialized by the n4sid estimate

N4Weight: This property determines some weighting matrices used in the singular-value decomposition that is a central step in the algorithm. Two choices are offered: moesp that corresponds to the MOESP algorithm by Verhaegen and cva which is the canonical variable algorithm by Larimore. The default value is N4Weight = Auto, which gives an automatic choice between the two options.

N4Horison: Determines the prediction horizons forward and backward, used by the algorithm. This is a row vector with three elements: N4Horison =[r sy su], where r is the maximum forward prediction horizon, i.e., the algorithms 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 Ljung (1999), pages 345-348. These numbers may have a substantial influence of 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 N4Horison will be tried out, and the value that gives the best (prediction) fit to data will be selected. If you specify only one column in N4Horizon, the interpretation is r=sy=su. The default choice is
N4Horizon = Auto, which uses the Akaike Information Criterion (AIC) for the selection of sy and su. See the reference page for n4sid for literature references.

Properties That Apply to Estimation Methods Using Iterative Search for Minimizing a Criterion

The properties that govern the iterative search are:

• Trace
• LimitError
• MaxIter
• Tolerance
• SearchDirection
• Advanced

These properties apply to armax, bj, oe, and pem

Trace: This property determines the information about the iterative search that is provided to the MATLAB command window:

• Trace = Off: No information is written to the screen
• Trace = On: Information about criterion values and the search process is given for each iteration.
• Trace= Full: The current parameter values and the search direction are also given (except in the "free" SSParameterization case for idss models)

LimitError: This variable determines how the criterion is modified from quadratic to one that gives linear weight to large errors. Errors larger than LimitError times the estimated standard deviation will carry a linear weight in the criteria. The default value of LimitError is 1.6. LimitError =0 disables the robustification and leads to a purely quadratic criterion. The standard deviation is estimated robustly as the median of the absolute deviations from the median, divided by 0.7. (See Eq. (15.9)-(15.10) in Ljung (1999).)

MaxIter: The maximum number of iterations performed during the search for minimum. The iterations will stop when MaxIter is reached, or some other stopping criterion is satisfied. The default value of MaxIter is 20. Setting MaxIter=0 will return the result of the start-up procedure. The actual number of used iterations is given by the property EstimationInfo.Iterations.

Tolerance: Based on the Gauss-Newton vector computed at the current parameter value, an estimate is made of the expected improvement of the criterion at the next iteration. When this expected improvement is less than Tolerance %, the iterations are stopped. Default value: 0.01.

SearchDirection: The direction along which a line search is performed to find a lower value of the criterion function. It may assume the following values:

• gn: The Gauss-Newton direction (inverse of the Hessian times the gradient direction) If no improvement is found along this direction, the gradient direction is also tried out.
• gns: A regularized version of the Gauss-Newton direction. Eigenvalues less than pinvtol of the Hessian are neglected, and the Gauss-Newton direction is computed in the remaining subspace. (pinvtol is part of the 'advanced' field: See Algorithm Properties in the "Command Reference".
• lm: The Levenberg-Marquard method is used. This means that the next parameter value is -pinv(H+d*I)*grad from the previous one, where H is the Hessian, I is the identity matrix, grad is the gradient. d is a number that is increased until a lower value of the criterion is found.
• Auto: A choice between the above is made in the algorithm. This is the default choice.

One property of the returned model is EstimationInfo. That is a structure that contains useful information about the estimation process. See EstimationInfo in "Command Reference" for a list of fields.

Another important option is InitialState. See Initial State.

For the spectral analysis estimate, you can compute the frequency functions at arbitrary frequencies. If the frequencies are specified in a row vector w, then

• g = spa(z,M,w)
  

results in g being computed at these frequencies. You can generate logarithmically spaced frequencies using the MATLAB logspace function. For example

• w = logspace(-3,pi,128)
  

State-Space Models

Defining Model Structures