# What do the (high) values in a controllability matrix mean?

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DB on 11 Aug 2022
Commented: Paul on 11 Aug 2022
In a linear state space system, I used the function
Co = ctrb(A,B)
To check the controllability of the system. I get a full rank matrix and thus, the system is controllable.
However, the matrix Co has values that range from -8.0220e+05 until 1.0672e+06, which seems quite extreme, as my x values lie in the range of 0-100.
My question is then:
What do these high values in a controllability matrix mean, and should I worry about these values being this far from the range of x values?
Star Strider on 11 Aug 2022
My pleasure!
Check the rank of the controllabililty matrix with respect to the size (in one dimension) of ‘A’. If they¹re essentially equal, there are no problems. If the controllability matrix drops rank, the system is likely not controllable.
Paul on 11 Aug 2022
Not sure what that someone meant by "invalidate my system?" How can a system be invalidated?
Offhand, the only thing I can think of related to the actual values in Co is the difference between the theoretical rank and the computed rank. For example, consider a second order system with
format short e
A = [0 0;1e-20 0];
B = [1e20; 1];
C = [1 1];
D = 0;
sys = ss(A,B,C,D);
The controllability matrix is
Co = ctrb(sys)
Co = 2×2
1.0e+00 * 1.0000e+20 0 1.0000e+00 1.0000e+00
Clearly full rank and therefore controllable. But
rank(Co)
ans =
1
In this case, the numerical rank test (with the default tolerances) yields a wrong conclusion.
Maybe that was the concern?
Or maybe they were referring to a wide range in the entries of the state space matrices themselves? That actually can be a problem.