Controllability and observability Gramians
Wc = gram(sys,'c')
Wc = gram(sys,'o')
Wc = gram(___,opt)
calculates the controllability Gramian of the state-space (
Wc = gram(
calculates the observability Gramian of the
Wc = gram(
time-limited or frequency-limited Gramians.
Wc = gram(___,
an option set that specifies time or frequency intervals for the computation.
opt using the
You can use Gramians to study the controllability and observability
properties of state-space models and for model reduction  . They have better numerical properties
than the controllability and observability matrices formed by
Given the continuous-time state-space model
the controllability Gramian is defined by
The controllability Gramian is positive definite if and only if (A, B) is controllable.
The observability Gramian is defined by
The observability Gramian is positive definite if and only if (A, C) is observable.
The discrete-time counterparts of the controllability and observability Gramians are
Use time-limited or frequency-limited Gramians to examine the controllability or observability of states within particular time or frequency intervals. The definition of these Gramians is as described in .
Compute the controllability Gramian of the following state-space model. Focus the computation on the frequency interval with the most energy.
sys = ss([-.1 -1;1 0],[1;0],[0 1],0);
The model contains a peak at 1 rad/s. Use
gramOptions to specify an interval around that frequency.
opt = gramOptions('FreqIntervals',[0.8 1.2]); gc = gram(sys,'c',opt)
gc = 2×2 4.2132 -0.0000 -0.0000 4.2433
The A matrix must be stable (all eigenvalues have negative real part in continuous time, and magnitude strictly less than one in discrete time).
or its discrete-time counterpart
Similarly, the observability Gramian Wo solves the Lyapunov equation
in continuous time, and the Lyapunov equation
in discrete time.
The computation of time-limited and frequency-limited Gramians is as described in .
 Kailath, T., Linear Systems, Prentice-Hall, 1980.
 Gawronski, W. and J.N. Juang. “Model Reduction in Limited Time and Frequency Intervals.” International Journal of Systems Science. Vol. 21, Number 2, 1990, pp. 349–376.