User`s guide

Identifying Nonlinear ARX Models
The predicted output
ˆ
()yt
of a n onlinear mode l at time t is given by the
following general equation:
ˆ
() ( ())yt F xt=
where x(t) repres ents the regressors. F is a nonlinear regre ssion function,
which is approximated by a nonlin ea rity estimator, whi c h might be a binary
partition tree, a neural network, or a network based on w avelets. The
following figure shows how the predicted output of the model is formed from
the inputs and outputs.
Regressors
Nonlinear
Function
Predicted
Outputs
Inputs
u(t)
Outputs
y(t)
u1(t-1),u2(t-3),y1(t-1), ...
Linear
Function
The function F can include both linear and nonlinear functions of x(t),as
showninthepreviousdiagram. Youcan specify which regressors to use a
inputs to the nonlinear block.
The following equation provides a general description o f F:
Fx x
k
k
d
kk
()=−
()
()
=
∑
ακβ γ
1
where
κ
is the unit nonlinear function, d is the number of nonlinearity units,
and
α
k
,
β
k
,and
γ
k
are the parameters of the nonlinearity estimator.
You choose a nonlinear structure that independently combines linear and
nonlinear regressors and the structure of the nonlinearity itself, such as
treepartition or wave net.TheSystemIdentification Toolbox product uses
input/output data to find the linear and nonlinear mappings that give the best
predicted outputs of the nonlinear model.
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