Controllability Gramian

I have a system of the form xdot= Ax + Bu. I have my A and B matrix defined. But I'm having a bit of trouble using the controllability gramian. The syntax just says

   Wc = gram(sys,'c')     % controllability gramian

But I don't know how to define 'sys'. Any help? Thanks!

Answer 1

[Fangjun Jiang]

use the state space ss() function.

help ss

 SS  Constructs state-space model or converts model to state space.

   Construction:
     SYS = SS(A,B,C,D) creates an object SYS of class @ss representing 
     the continuous-time state-space model
          dx/dt = Ax(t) + Bu(t)
           y(t) = Cx(t) + Du(t)

[Zach]

Thanks! Do you know by any chance if there's a way to do the gramian from some initial time t0 to some final time tf instead of doing it from 0 to infinity?

[Fangjun Jiang]

Sorry, I have no idea.

[Meena Farag]

I know this is several years late, but for anyone else reading this wondering the same thing, MATLAB 2016a has the option of computing the controllability gramian over some finite time. If your system is stable you can use the following:

opt = gramOptions('TimeInterval', [t0 tf]);
Wc = gram(sys,'c',opt);