gram - Controllability and observability gramians

Syntax

gram

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] . 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 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

to compute the gramians of a continuous or discrete system. The LTI model sys must be 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, 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).

References

[1] Kailath, T., Linear Systems, Prentice-Hall, 1980.