National Instruments Xmath Interactive Control Design Module ICDM User Manual
Page 89
Chapter 11
Introduction to MIMO Design
© National Instruments Corporation
11-3
Xmath Interactive Control Design Module
The standard feedback system has two vector input signals, r and d
act
, and
three vector output signals, e, u, and y. It can therefore be described by the
3
× 2 block matrix that relates the three output vector signals to the two
input vector signals:
The entries of this block matrix, that is, the transfer functions from r and
d
act
to e, u, and y, have standard names and interpretations (which agree
with the standard SISO notation):
•
The sensitivity transfer function is denoted S and given by
S = (I + PC)
–1
. The sensitivity transfer function is the transfer function
from reference input r to the error signal e.
•
The closed-loop transfer function T is given by T = PC(I + PC)
–1
. T is
the transfer function from r to y. T can be expressed in several other
ways, for example:
•
The actuator effort transfer function C(I + PC)
–1
is the transfer
function from r to u, and so is related to the actuator effort required.
For example, its step response matrix shows the closed-loop step
responses from each reference input signal to each actuator signal.
•
The transfer function from d
act
to e, P(I + CP)
–1
, is denoted S
act
and
called the actuator-referred sensitivity transfer function. The
actuator-referred sensitivity transfer function determines the errors
generated by actuator-referred disturbances. It also can be expressed as
(I + PC)
–1
P. Notice that it is “complementary” to the transfer function
described just above, that is, C(I + PC)
–1
, in the sense that the two
transfer functions can be obtained from each other by swapping P
and C.
•
The transfer function from d
act
to u, CP(I + CP)
–1
, is called the
actuator-referred actuator effort transfer function. Notice that it is
related to the closed-loop transfer function by swapping P and C. It can
also be expressed as C(I + PC)
–1
P.
•
The transfer function from d
act
to y, (–P)(I +CP)
–1
, is denoted T
act
and
called the actuator-referred closed-loop transfer function.
e
u
y
I PC
+
(
)
1
–
P I CP
+
(
)
1
–
C I PC
+
(
)
1
–
CP I CP
+
(
)
1
–
PC I PC
+
(
)
1
–
P
–
I CP
+
(
)
1
–
=
r
d
act
T
PC I CP
+
(
)
1
–
I PC
+
(
)
1
–
PC
I S
–
=
=
=