Does the order (index) of inputs and outputs matter in MIMO system identification?

I have a simple MISO system identification code, where "GenVolt" is the output; "Pressure" and "MagVolt" are the input signals. I noticed if we swap the order of inputs when we define the iddata structure, the result would be totally different! As can be seen in the following code, I just swapped the order of "Pressure" and "MagVolt" as the inputs, and I get totally diffent estimations. I was wondering why this happens?

clc
clear

% load data
load SteamEng
Pressure = 0.1*MagVolt + 0.2;

% Estimating the transfer function
steam1 = iddata([GenVolt],[Pressure,MagVolt],0.05);
steam1.InputName  = {'Pressure1';'MagVolt1'};
steam1.OutputName = {'GenVolt1'};
sys1 = tfest(steam1,2,1)

% Swapping the order of the inputs
steam2 = iddata([GenVolt],[MagVolt,Pressure],0.05);
steam2.InputName  = {'MagVolt2';'Pressure2'};
steam2.OutputName = {'GenVolt2'};
sys2 = tfest(steam2,2,1)

The result is

sys1 =
  %From input "Pressure1" to output "GenVolt1":
    -4.961 s + 3.082
  ---------------------
  s^2 + 29.89 s + 11.06
 
  %From input "MagVolt1" to output "GenVolt1":
     17.97 s + 48.5
  ---------------------
  s^2 + 41.93 s + 120.6
  
  
  sys2 =
  %From input "MagVolt2" to output "GenVolt2":
     17.59 s + 54.35
  ---------------------
  s^2 + 42.73 s + 138.3
 
  %From input "Pressure2" to output "GenVolt2":
     -9.183 s + 43.3
  ---------------------
  s^2 + 422.1 s + 154.5

Answer 1

[Rajiv Singh]

Yes the order matters since within a given noise level, there are many models that can explain the data. Settings related to search method, search options and initialization method all matter. In this example, an initial model is first generated using instrument variables. This model is different when the inputs are swapped. You can enforce a closer agreement by using a different initializer, but there is not gaurantee, or a way of enforcing, a strict match. The following seems to work better here:

steam2.InputName  = {'MagVolt1';'Pressure1'};
steam2.OutputName = {'GenVolt1'};
m1=ssest(steam1,2);
m2=ssest(steam2,2);
sys1 = tfest(steam1,m1);
sys2 = tfest(steam2,m2);
figure
bode(sys1,sys2(1,[2 1]))
figure
compare(steam1,sys1,sys2)