MATLAB: Controllability Gramian

Control System Toolboxgramian

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!

Best Answer

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)

Related Solutions

MATLAB: Conversion from State Space to TF and then back to State Space

Some points:

Don't do "algebra" with tf variables, use the functions series, parallel and feedback.

sys_mimo1 = ss(sys_mimo) is going to convert the tf model to a ss model, I don't think there is a need for the extra step with ssdata.

This one here

sys_mimo_tf(1,2) = sys_mimo_tf(1,2)*feedback(sys_mimo_tf(1,2),C);

is probably not doing what you want it to, it will give you the transfer function

sys_mimo_tf(1,2)^2 /  1 + sys_mimo_tf(1,2)*C

so that's probably where many extra states come from. You should expect 2 extra states per added PID.

Lastly, look at the Hankel Singular Values or a similar measure to figure out whether there might be a lot of almost unobservable/almost uncontrollable states in your resulting model. That would indicate numerical issues.

MATLAB: Integral of controllability gramian

Try this with your matrix and vector:

A = rand(6);                                        % Create Data

B = rand(6, 1);                                     % Create Data
f = @(tau) expm(A*tau)*B*B'*expm(A'*tau);           % Integrand
W = @(t) integral(f, 0, t, 'ArrayValued',1);        % Controllability Gramian
Wt = W(1)

The integral function was introduced in R2012a. Before that, I believe the appropriate function is quadv.

Best Answer

Related Solutions

MATLAB: Conversion from State Space to TF and then back to State Space

MATLAB: Integral of controllability gramian

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