Specifications

gram
16-93
to compute the grammians of a continuous or discrete system. The LTI model
sys must be in state-space form.
Algorithm The controllability grammian is obtained by solving the continuous-time
Lyapunov equation
or its discrete-time counterpart
Similarly, the observability grammian solves the Lyapunov equation
in continuous time, and 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 Grammian-based balancing of state-space realizations
ctrb Controllability matrix
lyap, dlyap Lyapunov equation solvers
obsv Observability matrix
References [1] Kailath, 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