gapmetric for MIMO LTI systems with unequal input and output dimensions

I have two LTI systems with identical structure (same dimensions). I want to find the distance between these systems using the vinnicombe gap metric.

The transfer function matrices for both the systems have a dimension of 4x6. In the documentation of gapmetric, the following is mentioned for the input arguments-

" Input systems, specified as dynamic system models. P1 and P2 must have the same input and output dimensions. If P1 or P2 is a generalized state-space model (genss or uss) then gapmetric uses the current or nominal value of all control design blocks. "

I have different input and output dimensions (2 and 4 respectively) and I don't have a generalized state space model, how can I calculate the gap metric for such LTI systems?

Something I tried:

I tried calculating the gapmetric between two LTI systems each having 1 output and 2 inputs. I don't understand how is it calculating the gap metric when the input and output dimensions are not the same as mandated by the documentation. If this is valid, I could use the same for my 2-inputs 4-outputs LTI systems.

A = 1;
B = [1 2];
C = 1;

s = tf('s');

G = C*inv(s*eye(1) - A)*B;  % Transfer Function for LTI system: x_dot = x + [1 2][u1 ; u2] and y = x

A = 3;
B = [2 0];
C = 2;

H = C*inv(s*eye(1) - A)*B;  % Transfer Function for LTI system: x_dot = 3x + [2 0][u1 ; u2] and y = 2x

[gap,nugap] = gapmetric(G,H)

gap = 0.9117

nugap = 0.8206

Answer 1

[Paul]

Hi Shivam,

My interpretation of the doc page gapmetric is that the number of inputs to P1 has to be same as the number of inputs to P2, and the number of outputs from P1 has to be the same as the number of outputs from P2. The language could be much more clear. As you've shown, gapmetric is happy to accept non-square system inputs, as long as P1 and P2 have the same size. Furthermore, if the inputs do not meet that criteria, gapmetric throws an error

try
    gapmetric(rss(3,3,2),rss(3,2,3))
catch ME
    ME.identifier
    ME.message
end

ans = 'Robust:analysis:gapmetric3'

ans = 'P1 and P2 must have the same numbers of inputs and outputs.'

though the error message could be less ambiguous (though the use of the plural "numbers" seems more consistent with my interpretation). Presumably, gapmetric would throw an error for any case of P1 and P2 that don't meet the requirements of the function.

However, I don't know enough about the gap metric to say definitively that gapmetric is giving the correct result for non-square inputs.

Perhaps you can find a reference with a gap metric example for non-square systems and test if gapmetric yields the same result.

At least a simple case for a non-square systrem seems to give a reasonable result

P1 = rss(3,3,2);
gapmetric(P1,P1)

ans = 1.8114e-13

The More About section of the doc page speaks to P1 and P2 being defined in terms of right normalized coprime factorizations, and RNCFs don't need to be square systems.

Additionally, if you click one of the stars at the bottom of the doc page under How useful was this information? you should get a pop-up that allows you to provide feedback. If you ask for clarification on that clause, you'll probably get a response from TMW, and, if appropriate based on that response, you can make a suggestion to improve clarity.

[Shivam]

Thank You for your elaborate response. I think your interpretation might be correct given that your example non-square system is returning a meaningful response. I didn't know about the clarification thing. I'll see if I can get clarification on your interpretation of the clause using your suggestion. Thank You!