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 Order editor handles options to fix K, D and x0 to zero in the black box case.

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 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.

The initial state vector x(0) can either be estimated or fixed to zero.

The D matrix can either be estimated or fixed to zero.

Help topics.

Estimation Methods for 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 follows a default canonical form.

  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 an auxiliary order (like a prediction horizon). This auxiliary order can be given within parentheses after the order in the Orders: edit box (e.g., 4 (9)). If no auxiliary order is given, a default value is used.

  If the auxiliary order is given as a vector (e.g. 4 (5:17)), models for all these auxiliary orders are computed and evaluated. The model that gives the best fit between model and measured output is chosen. A figure is opened that illustrates the fit as a function of the auxiliary order. The fit is either a simulation error fit if K=0 is chosen, or a one-step ahead prediction error fit.

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

Help topics.

Arbitrary State-Space Model Structures The SITB supports user-defined linear state-space models of arbitrary structure. Using the command MS2TH 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 MF2TH allows you to use a completely arbitrary structure, defined by an M-file. FIXPAR and UNFIXPAR can be used to manipulate structures.

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.

Help topics.

(file iduiss.htm)