User`s guide

3 Linear Model Identification
The exact relations hip between K and

K
is com plicated. However, for short
sampling intervals T, the following approximation works w el l:
KeKd
F
T
=
∫
τ
τ

0
State-Space Representation of Transfer Functions
For line ar models, the general symbolic model des cription is given by:
yGuHe=+
G is a transfer function that takes the input u to the output y. H is a transfer
function that describes the properties of the additive o u tput noise model.
The discrete-time state-space representation is given by the following
equation:
x kT T Ax kT Bu kT Ke kT
ykT CxkT DukT ekT
x
( ) () () ()
() () () ()
()
+= + +
=+ +
=0 xx0
where T is the sampling interval, u(kT) is the input at time instant kT,and
y(kT) is the output at time instant kT.
The relationships between the transfer functions and the discrete-time
state-space matrices are given by the following equations:
Gq CqI A B D
Hq CqI A K I
nx
nx ny
() ( )
() ( )
=−+
=−+
−
−
1
1
where I
nx
is the nx-by-nx identity matrix, I
ny
is the nx-by-nx identity matrix,
and ny is the dimension of y and e.
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