MATLAB: Integral of controllability gramian

controllability gramian control systems integrate matrix exponential

I am having trouble finding a way to integrate the controllability gramian in Matlab. My system is unstable so I can't use the built in function. My system also has eigenvalues on the imaginary axis so I can't use a function I found online for unstable systems. So i'm trying to integrate the controllability gramian for some finite time interval on MATLAB, but it seems impossible. Here's what the integral looks like.
Where A is a 6×6 matrix and B is a 6×1 matrix. The matrix exponential in the equation is what's causing me the most trouble.
Any ideas?

Best Answer

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.
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