User`s guide

gram
11-92
tocompute thegramians ofa continuousor discretesystem.The LTImodel sys
must b e in state-space form.
Algorithm The controllability gramian is obtained by solving the continuous-time
Lyapunov equation
or its discrete-time counterpart
Similarly, the observability gramian solves the Lyapunov equation
in continuous time, a nd the Lyapunov equation
in discrete time.
Limitations The matrix must be stable (all eigenvalues have negative real part in
continuous time, and magnitude strictly less than one in discrete time).
See Also balreal Gramian-based balancing of state-space realizations
ctrb Controllability matrix
lyap, dlyap Lyapunov equation solvers
obsv Observability matrix
References [1] K ailath, T., Linear Systems, Prentice-Hall, 1980.
W
c
AW
c
W
c
A
T
BB
T
++ 0=
AW
c
A
T
W
c
– BB
T
+ 0=
W
o
A
T
W
o
W
o
AC
T
C++ 0=
A
T
W
o
AW
o
– C
T
C+ 0=
A