National Instruments 370757C-01 User Manual
Chapter 3
System Evaluation
© National Instruments Corporation
3-11
Where
SysC=system(Ac,Bc,Cc,Dc)
,
Sys=system(A,B,C,D)
, and
nz
is the dimension of
z
and
nw
is the dimension of
w
:
Given the above,
SysCL
is calculated as shown in Figure 3-7.
Figure 3-7. Calculation of the Closed Loop System (SysCL)
The closed-loop system is assumed to be well-posed—(I – D
c
D
yu
) must
be invertible). A well-posed closed-loop system assures that if two given
systems,
systems,
Sys
and
SysC
, are proper (only proper transfer functions can be
represented in state space), then the resulting closed-loop system,
SysCL
,
also is proper and therefore realizable in state space.
Figure 3-8 is an example of an ill-posed feedback system, where the
closed-loop transfer function is
closed-loop transfer function is
s+1
, which cannot be represented as
a state-space system.
B is
B
w
B
u
C is C
z
,C
y
D is
D
zw
D
zu
D
yw
D
yu
nz
nw
nw
nz
A
CL
A+B
u
I D
c
D
yu
–
(
)
1
–
D
c
C
y
B
u
I D
c
D
yu
–
(
)
1
–
C
c
B
c
C
y
B
c
D
yu
I D
c
D
yu
–
(
)
1
–
D
c
C
y
+
A
c
B
c
D
yu
I D
c
D
yu
–
(
)
1
–
C
c
+
=
B
CL
B
w
B
u
I D
c
D
yu
–
(
)
1
–
D
c
D
yw
+
B
c
D
yw
B
c
D
yu
I D
c
D
yu
–
(
)
1
–
D
c
D
yw
+
=
D
CL
D
zw
D
zu
I D
c
D
yu
–
(
)
1
–
D
c
D
yw
+
=
C
CL
C
z
D
zu
I D
c
D
yu
–
(
)
1
–
D
c
C
y
+
D
zu
I D
c
D
yu
–
(
)
1
–
C
c
=