gram - Controllability and observability gramians

Syntax

Description

Wc = gram(sys,'c') calculates the controllability gramian of the state-space (ss) model sys.

Wc = gram(sys,'o') calculates the observability gramian of the ss model sys.

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

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 (C, B) is observable.

The discrete-time counterparts of the controllability and observability gramians are

respectively.

Algorithms

The controllability gramian W_c is obtained by solving the continuous-time Lyapunov equation

or its discrete-time counterpart

Similarly, the observability gramian W_o solves the Lyapunov equation

in continuous time, and the Lyapunov equation

in discrete time.

Limitations

The A 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.

See Also

balreal | ctrb | dlyap | lyap | obsv

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