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Chapter 3 System Evaluation
© National Instruments Corporation 3-11 MATRIXx Xmath Robust Control Module
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,
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
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
ID
c
D
yu
–()
1–
D
c
C
y
B
u
ID
c
D
yu
–()
1–
C
c
B
c
C
y
B
c
D
yu
ID
c
D
yu
–()
1–
D
c
C
y
+ A
c
B
c
D
yu
ID
c
D
yu
–()
1–
C
c
+
=
B
CL
B
w
B
u
ID
c
D
yu
–()
1–
D
c
D
yw
+
B
c
D
yw
B
c
D
yu
ID
c
D
yu
–()
1–
D
c
D
yw
+
=
D
CL
D
zw
D
zu
ID
c
D
yu
–()
1–
D
c
D
yw
+=
C
CL
C
z
D
zu
ID
c
D
yu
–()
1–
D
c
C
y
+ D
zu
ID
c
D
yu
–()
1–
C
c
=