Application of tf function for large systems

Question

I'm looking at the transfer functions (transfer matrix) of a Multiple-input single-output (MISO) system. The system has 32 dynamic states, four inputs, and one output. The system A, B, C, and matrices are calculated in a Matlab code, and the state space model is created as sys=ss(A,B,C,D).

My question is that why are the transfer functions obtained by applying the "tf" function on "sys" (a 1*4 structure) different from those obtained by applying the "tf" function on individual system models "sys(1)", "sys(2)", "sys(3)", and "sys(4)", whereas the system matrices obtained by individuals "sys(1)" to "sys(4) completely match with the corresponding matrices and matrices' columns of the "sys"?

I tried the same thing for a simple 4th-order system, and they completely match. I also tried it for a 32th-state system (of same dimension of my original system), where all system matrices are generated by randn function. Then, I tried to find the transfer function coefficients by using cell2mat(T.den) and cell2mat(T.num) for sys and for sys(1) to sys(4). All the denominator coefficients match. Also, except for one of the transfer functions, the numerator coefficients match as well.

It should be mentioned that in the original example, the matrix A is singular, but in the synthetic example 2 (of dimension 32), the condition number of system matrix is around 120. You can find the code below. Your help is highly appreciated.

clear all;
clc;
%% Building the system matrices
A=randn(32,32);
B=randn(32,4);
C=randn(1,32);
D=randn(1,4);
sys=ss(A,B,C,D); % creating the state space model
TFF=tf(sys); % calculating the tranfer matrix

%% extracting the numerator and denominator coefficients of 4 transfer 
 %functions
for i=1:4
Ti=TFF(i);
Tin(i,:)=cell2mat(Ti.num); % numerator coefficients
Tid(i,:)=cell2mat(Ti.den); % denominator coefficients
clear Ti
end

%% calculatingthe numerator and denominator coefficients based on individual 
 %transfer functions
TF1=tf(sys(1));
T1n=cell2mat(TF1.num);
T1d=cell2mat(TF1.den);

TF2=tf(sys(2));
T2n=cell2mat(TF2.num);
T2d=cell2mat(TF2.den);

TF3=tf(sys(3));
T3n=cell2mat(TF3.num);
T3d=cell2mat(TF3.den);

TF4=tf(sys(4));
T4n=cell2mat(TF4.num);
T4d=cell2mat(TF4.den);

num2str([T1n.'-Tin(1,:).']) % the error between the numerator coefficients 
 % of the TF1 by 2 aproaches
num2str([T2n.'-Tin(2,:).'])
num2str([T3n.'-Tin(3,:).'])
num2str([T4n.'-Tin(4,:).'])

num2str([T1d.'-Tid(1,:).'])
num2str([T2d.'-Tid(2,:).'])
num2str([T3d.'-Tid(3,:).'])
num2str([T4d.'-Tid(4,:).'])

Answer 1

It is a mixture of a few things; because the resulting model is not guaranteed to be minimal per se after model switches you get some discrepancies but also a subsystem of the MIMO system is not guaranteed to agree with a SISO part of the system which can remove some of the poles of the other modes if input is not acting on them.

However, beyond 5,6 transfer matrices are numerically terrible to perform operations with. Hence try to avoid them. Check other properties of the subsystems such as Bode plots for comparison