User`s guide

gram
11-91
11gram
Purpose Compute controllability and observability gramians
Syntax Wc = gram(sys,'c')
Wo = gram(sys,'o')
Description gram calculates controllability and observability gramians. You can use
gramians 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 t he continuous-time state-space model
the controllability gramian is defined by
and the observability gramian by
The discrete-time counterparts are
The controllability gramian is positive definite if and only if is
controllable. Similarly, the observability gramian is positive definite if and
only if is observable.
Use the commands
Wc = gram(sys,'c') % controllability gramian
Wo = gram(sys,'o') % observability gramian
x
·
Ax Bu+=
yCxDu+=
W
c
e
Aτ
BB
T
e
A
T
τ
τd
0
∞
∫
=
W
o
e
A
T
τ
C
T
Ce
Aτ
τd
0
∞
∫
=
W
c
A
k
BB
T
A
T
()
k
,
k0=
∞
∑
= W
o
A
T
()
k
C
T
CA
k
k 0=
∞
∑
=
AB
,()
CA
,()