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

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?

[Paul]

What are "x values"?

[DB]

@Paul x is a vector of state variables in a general system $ \dot{x} = Ax + B$

[Star Strider]

It likely depends on the size of ‘A’ however the behaviour is certainly reproducable (although with only positive numbers here) —

N = 4;
A = randi([0 100], N)

A = 4×4

35 34 62 31 14 71 79 90 15 36 50 84 53 100 21 85

B = randi(10,N);

Co = ctrb(A,B)

Co = 4×16

8 5 3 9 1519 767 867 845 317514 153876 174639 162365 71836162 33474631 39134051 35978253 10 1 6 1 2422 1155 1381 1132 546282 255118 297142 272557 125373385 58158833 68392413 62494952 10 6 7 5 1736 915 947 925 398293 182919 214735 199805 91642064 42516222 50080679 45651097 9 6 4 6 2399 1001 1246 1192 563078 260451 309848 278730 127682225 59646862 69816582 63749000

format shortE
minC0 = min(Co(:))

minC0 = 
     1

maxC0 = max(Co(:))

maxC0 = 
   127682225

.

[DB]

@Star Strider That is a nice insight, thanks. Indeed the higher order of A would result in the controllability matrix having high values on its far-right values. So would you say the controllability matrix does not give any information on the system, except for its controllability?

[Star Strider]

‘So would you say the controllability matrix does not give any information on the system, except for its controllability?’

Yes. That’s all it’s designed to do.

[Paul]

@DB

I don't understand how one can compare the elements of Co to the "x values." Co is just a matrix a numbers. The x values will be whatever they are for a given input and initial condition. In fact, if the system is unstable the at leastt on x value will migrate off to infinity. So I'm still not clear on how to compare the entries of Co to the "x values."

[DB]

Well someone pointed out that the controllability matrix has hugely varying values, and therefore it may invalidate my system. I wanted to check if that was correct, and it seems that isn't so! Thanks @Star Strider

[Star Strider]

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]

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.