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
–
=
=
=
