National Instruments Xmath Interactive Control Design Module ICDM User Manual page 89

© National Instruments Corporation
The standard feedback system has two vector input signals, r and d
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:
e
=
u
y
The entries of this block matrix, that is, the transfer functions from r and
d to e, u, and y, have standard names and interpretations (which agree
act
with the standard SISO notation):
• The sensitivity transfer function is denoted S and given by
–1
S = (I + PC) . 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)
the transfer function from r to y. T can be expressed in several other
ways, for example:
T PC I
=
• The actuator effort transfer function C(I + PC)
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
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
–1
(I + PC) P. Notice that it is "complementary" to the transfer function
described just above, that is, C(I + PC)
transfer functions can be obtained from each other by swapping P
and C.
• The transfer function from d
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)
• The transfer function from d
called the actuator-referred closed-loop transfer function.
Chapter 11
1 –
I PC P I
+ +
1 –
C I PC CP I
+
1 –
PC I PC P I
+ –
1 –
CP I PC
+ = +
to e, P(I + CP)
act
to u, CP(I + CP)
act
–1
P.
to y, (–P)(I +CP)
act
11-3 Xmath Interactive Control Design Module
Introduction to MIMO Design
1 –
CP
r
1 –
CP
+
d
act
1 –
CP
+
1 –
PC I S
= –
–1
is the transfer
–1
, is denoted S
–1
, in the sense that the two
–1
, is called the
–1
, is denoted T
, and
act
–1
. T is
and
act
and
act