Specifications

gram
16-92
16gram
Purpose Compute controllability and observability state-space
models:descriptor;state-space models:quick data retrievalgrammians
Syntax Wc = gram(sys,'c')
Wo = gram(sys,'o')
Description gram calculates controllability and observability grammians. You can use
grammians to study the controllability and observability properties of
state-space models and for model reduction [1,2]. They have better numerical
properties than the controllability and observability matrices formed by
ctrb
and obsv.
Given the continuous-time state-space model
the controllability grammian is defined by
and the observability grammian by
The discrete-time counterparts are
The controllability grammian is positive definite if and only if is
controllable. Similarly, the observability grammian is positive definite if and
only if is observable.
Use the commands
Wc = gram(sys,'c') % controllability grammian
Wo = gram(sys,'o') % observability grammian
x
·
Ax Bu+=
yCxDu+=
W
c
e
A
τ
BB
T
e
A
T
τ
τ
d
0
∞
ò
=
W
o
e
A
T
τ
C
T
Ce
A
τ
τ
d
0
∞
ò
=
W
c
A
k
BB
T
A
T
()
k
,
k 0=
∞

= W
o
A
T
()
k
C
T
CA
k
k 0=
∞

=
AB
,()
CA
,()