State-space Models

The basic state-space model in innovations form can be written

x(t+1) = A x(t) + B u(t) + K e(t)
y(t) = C x(t) + D u(t) + e(t)

The SITB supports two kinds of parametrizations of state-space models: Black box, free parametrizations and parametrizations tailor made to the application.

To estimate a state space model, select Parametric Models in the pop-up menu Estimate in the ident window. In the dialog that opens, choose State space as the model structure, and enter the desired order. The delays from the different inputs u can be entered within square brackets. The default values of the delays are all 1. This means that the D-matrix is fixed to zero. The estimation of the initial state x(0) is goverend by the pop-up menu Initial State in the Parametric Model window.

The Order editor handles options to fix K to zero in the black box case, as well as entering the delays from the input(s)

The order editor also gives the possibility to choose two variables that may have an influence on the quality of the N4SID estimated. This estimate is also used to initialize the PEM estimate. The N4sid Options are

N4Weight
This govern the pre- and post-weighting matrices that are used at an SVD step in the algorithm. The choices are 'Auto', 'MOESP', and 'CVA'. 'Auto' gives an automatic choice. MOESP is the method by Verhaegen and 'CVA' is the canonical variable algorithm by Akaike and Larimore.
N4Horizon
This vector determines the predictions horizons used. N4Horizon =[r,sy,su], where r is the maximum prediction horizon, sy is the number of past outputs used in the predictors, and su is the number of past inputs used in the predictors. Taking sy = 0 gives a method that does not take the noise influend into acount. There is no simple theory for hoe to choose N4Horizon. Setting it to 'Auto', which is the default, gives a procedure, based on the Akaike AIC criterion, that makes the choice guided by the esitimation data. If N4Horizon has several rows, each row will be tried. A plot will be shown that describes the fit as a function of the different choices.
You can enter any variable name in the N4Horizon field, that will be evaluated in the workspace to give the desired option.
Note that in the black box case, there is a special feature to select the model order by entering a vector (like 1:10) for the model order. You can push the Order selection button to fill out the model order field in this case.

Help topics.


Black Box State-Space Models

Black Box State-space Models

Use the pop-up menu to choose the model order, i.e. the dimension of the state/space vector, or enter it directly into the Orders: edit box in the Parametric Models dialog.

By entering a vector (e.g. 1:10), all orders will be computed using a preliminary method, and you will have to choose order(s) based on information in a special graph. You can also use the button Order selection to fill out the Model order field with a default model order range.

There are two basic methods for the estimation: PEM and N4SID

There are a number of structure options that can be reached in the Order Editor:

By fixing the matrix K to zero, an Output Error method is obtained.

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Estimation Methods for Black-box State-space Models

Estimating Black Box State-space Models

PEM:
A standard prediction error/maximum likelihood method, based on iterative minimization of a criterion. The iterations are started from parameter values that are computed from N4SID. The parametrization of the matrices A, B, C, D, and K is free an adjusted to be numerically well conditioned.

The search for a minimum is controlled by a number of options. These are accessed from the Options button in the Iteration control... dialog.

N4SID:
A subspace-based method based on projections, that does not use iterative search.

The quality of the resulting estimates may significantly depend on the N4sid Options N4Weight and N4Horizon

See the commands PEM and N4SID in the manual for more information.

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Arbitrary State-Space Model Structures

Tailor-made State-space Models

The SITB supports user-defined linear state-space models of arbitrary structure. Using the command IDSS known and unknown parameters in the A, B, C, D, K, and X0 matrices can be easily defined both for discrete and continuous-time models. The command IDGREY allows you to use a completely arbitrary structure, defined by an M-file. The properties of these model objects can be easlitymanipulated.

To use them in conjunction with ident, define the appropriate structure in the MATLAB command line and enter its variable name in the Orders: edit box of the Parametric Models dialog. If desired, select the appropriate iteration options for PEM by pressing the Iteration control... button.

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(file iduiss.htm)