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Chapter 6 State-Space Design
Xmath Control Design Module 6-34 ni.com
For discrete-time systems, the integrals in the W
c
and W
o
equations are
replaced by summation signs and the grammians are obtained as the
solutions of the discrete-time Lyapunov equations:
(6-19)
The controllability grammian must be full-rank for the system to be
completely controllable; similarly, the observability grammian must be
full-rank for the system to be completely observable (refer to [Kai80]). The
condition number of W
c
reflects how well conditioned the system model is
with regard to pointwise state control.
The condition number of W
o
reflects the condition of the model with regard
to zero-input state-observation.
A linear transformation T of the system {A,B,C} also results in a linear
transformation of the grammians. If the state vector is transformed as
the system and grammian transformations in the following
equations:
Although the poles of the system (of the eigenvalues of A) do not change
under the transformation, the singular values (eigenvalues of the
grammians) do. However, the eigenvalues of the product of the grammians
are invariant under transformation, and these are the singular values of the
system input-to-state and state-to-output maps [LHPW87].
The system is defined as being internally balanced if for some
transformation T,
where
AW
c
A' BB'+ W
c
=
A'W
o
AC'C+ W
o
=
xTx
ˆ
,=
A
ˆ
T
1–
AT=
W
ˆ
c
T
1–
W
c
T'()
1–
=
C
ˆ
CT=
D
ˆ
D=
B
ˆ
T
1–
B=
W
ˆ
o
T'W
o
T=
W
ˆ
c
2
W
ˆ
o
2
Σ
2
==
Σ
2
diagonal σ
1
2
σ
2
2
…σ
2
2
,,,[]()=