User`s guide

4 Nonlinear Black-Box Model Identification

The input signal passes through the ﬁrst nonlinear block, a linear block,

and a second nonlinear block to produce the output signal, as shown in the

following ﬁgure.

Input

u(t)

Output

y(t)

Input

Nonlinearity

Linear

Block

Output

Nonlinearity

The followi ng general equation describes the Hammerstein-W iener structure:

wt f ut

yt hxt

() ( ())

()

() ( ())

which contai ns the fo ll owing variable s :

• u(t) and y(t) are the inputs and outputs for the system, respectively.

• f and h are nonlinear functions that correspond to the input and output

nonlinearities, respectively.

For multiple inputs and multiple outputs, f and h are deﬁned independently

for each input and output channel.

• w(t) and x(t) are internal variables that deﬁne the input and output of the

linear block, respectively.

w(t) has the same dimension as u(t). x(t) has the same dimension as y(t).

• B(q) and F(q) in the linear dynamic block are linear functions, which a re

similar to the polynom ial in an Output-Error model, as described in “What

Are Black-Box Polynomial Models?” on page 3-41.

For ny outputs and nu inputs, the linear block is a transfer function matrix

containing entries in the following form:

()

where j = 1,2,...,ny and i = 1,2,.. .,nu.

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