System Identification Toolbox

State-Space Models

The Model Structure

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 System Identification Toolbox supports two kinds of parametrizations of state-space models: black-box, free parametrizations, and parametrizations tailor-made to the application. The latter is discussed below under the heading User Defined Model Structures. First we will discuss the black-box case.

Entering Black-Box State-Space Model Structures

The most important structure index is the model order; i.e., the dimension of the state vector x.

Use the pop-up menu in the Order Editor to choose the model order, or enter it directly into the Orders edit box in the Parametric Models window. Using the other pop-up menus in the Order Editor, you can further affect the chosen model structure:

• Fixing K to zero gives an Output-Error method; i.e., the difference between the model's simulated output and the measured one is minimized. Formally, this corresponds to an assumption that the output disturbance is white noise.

The delays from the input can be chosen independently for each input. It will be a row vector nk, with nu entries. When the delay is larger than or equal to one, the D-matrix in the discrete time model is fixed to zero. For physical systems, without a pure time delay, that are driven by piece-wise constant inputs, nk = 1 is a natural assumption This is also the default. Note also that the delays can be directly entered into the Orders edit box.

Estimating Many Models Simultaneously

By entering a vector for the model order, using MATLAB's colon notation, (such as "1:10") all indicated orders will be computed using a preliminary method. You can then enter models of different orders into the Model Board by clicking in a special graph that contains information about the models.

Estimation Methods

There are two basic methods for the estimation:

PEM: Is a standard prediction error/maximum likelihood method, based on iterative minimization of a criterion. The iterations are started up at parameter values that are computed from n4sid. The parametrization of the matrices A, B, C, D, and K is free. The search for minimum is controlled by a number of options. These are accessed from the Option button in the Iteration Control window.

N4SID: Is a subspace-based method that does not use iterative search. The quality of the resulting estimates may significantly depend some options called N4Weight and N4Horizon. These options can be chosen in the Order Editor window. If N4Horizon is entered with several rows, the models corresponding to the horizons in each row are examined separately using the Working data. The best model in terms of prediction (or simulation, if K = 0) performance is selected. A figure is shown that illustrates the fit as a function of the horizon. If the N4Horizon box is left empty, a default choice is made.

See n4sid and pem in the "Command Reference" chapter for more information.

ARMAX, Output-Error and Box-Jenkins Models

User Defined Model Structures