Chapter 3 SystemEvaluation
© 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–DcDyu) 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.
Figure3 -8 is an example of an ill-posed feedback system, where the
closed-loop transfer function is s+1, which cannot be represented as
astate-space system.
Bis Bw
Bu
Cis Cz,CyDis Dzw Dzu
Dyw Dyu
nz
nw
nw

nz

ACL
A+BuID
cDyu
–()
1–DcCyBuID
cDyu
–()
1–Cc
BcCyBcDyu ID
cDyu
–()
1–DcCy
+AcBcDyu ID
cDyu
–()
1–Cc
+
=
BCL
BwBuID
cDyu
–()
1–DcDyw
+
BcDyw BcDyu ID
cDyu
–()
1–DcDyw
+
=
DCL Dzw Dzu ID
cDyu
–()
1–DcDyw
+=
CCL CzDzu ID
cDyu
–()
1–DcCy
+Dzu ID
cDyu
–()
1–Cc
=