Control System Toolbox™

gram - Controllability and observability grammians

Syntax

gram

Description

gram calculates controllability and observability grammians. You can use grammians 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 grammian is defined by

and the observability grammian by

The discrete-time counterparts are

The controllability grammian is positive definite if and only if is controllable. Similarly, the observability grammian is positive definite if and only if is observable.

Use the commands

Wc = gram(sys,'c')     % controllability grammian
Wo = gram(sys,'o')     % observability grammian

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).

References