# Why does the index based connection for a MIMO system with a unity feedback loop differ in response with a signal name based system when using the function "connect"?

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I am trying to create an equivalent feedback loop using index based connections for the MIMO system "T" designed with named connections in this example:
My script for the corresponding index based system is as follows:
%% With connection indices
blksys = append(C,G);
connections = [1 -3;2 -4;3 1;4 2];
T_conn = connect(blksys,connections,[1 2],[3 4]);
bode(T_conn,'g--')
However, the bode plot of the system T_conn differs from that of the system T (in red) at very low frequencies. Is this the right way to model index based interconnections and if so, why is the response different?

MathWorks Support Team on 13 Oct 2021
Edited: MathWorks Support Team on 13 Oct 2021
Inspection reveals that the only difference between T and T_conn is the state ordering. This can be traced back to:
blksys = append(C,G);
This puts the states of G last whereas they appears first in the system T. To eliminate the discrepancy, you make the following alterations to the connection index based system:
blksys = append(G,C);
connections = [1 3;2 4;3 -1;4 -2];
T_conn = connect(blksys,connections,[3 4],[1 2]);
While this produces the same response at lower frequencies, T_conn is no worse than T. To see a difference, we are observing the behavior at extremely low frequencies, but it turns out that both Tand T_conn have no accuracy at all at such frequencies, as evidenced by the following in say, the frequency band [1e-20 1e5]:
prescale(T)
This is due to the pole/zero cancellation at s=0 in G*C and the closed-loop system: You could try "minreal(T)" to get rid of this cancellation. Such cancellations give rise to a  0/0 expression near the zero frequency, which is extremely sensitive to rounding errors.
To summarize, there is nothing wrong with T_conn as computed, as the observed discrepancy is just a manifestation of the fundamental limit of accuracy for any finite-precision computation. You can safely proceed with the T_conn formula in the script.

R2018a

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