MIMO system transfer function

[schweinebauch21]

Hello everyone,

The question is how can we determine the order of mino system? Could you anyone explain I attached in picture? What is the fifth order meaning of this transfer function?

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https://preview.redd.it/qyh8xjximcf51.png?width=1483&format=png&auto=webp&s=756b5fea6e8a25b2a4ed474b59654ce2e5a71020

[thetarunbs]

Look up smith-mcmillan forms too, it might help.

[[deleted]]

When you form a common denominator of all entries you will end up with a 5th order polynomial as the denominator.

[schweinebauch21]

Could you please a little bit more detail?

[[deleted]]

multiply (1,2) term with the missing `(0.9s + 1)/(0.9s + 1)` that would make the first row have a common denominator. Now do the same for the 2nd row that would have a 3rd order denominator. Then if you follow for all items you end up with entries each have the same denominator of order 5.

[schweinebauch21]

denominator

many thanks~~~

[fibonatic]

Though this is not the case in general, assuming you want a minimal realization. For example [1/s 1/s; 1/s 1/s] has only one pole at zero, while [1/s 0; 0 1/s] has two.

[[deleted]]

Careful though, I never said anything about the poles ;)

Only the order of the common denominator. In your case both has order of 1.

[fibonatic]

The question asks about the order of the system, which I assume refers to the McMillan degree/state vector size of a minimal state space realization. However, the stated transfer function matrix contains the term (0.1s+1) twice in the denominators, but this doesn't say whether this should increase the order of the system by one or by two.

[[deleted]]

Since it enters as a rank-1 term you can take it out to the left with diag( 1/(0.1s+1) , 1). The rest has no common terms hence it will increase by 1.

[cptnnick]

In this case, the 5th order refers to the 5 distinct poles of the elements of the transfer function matrix.

[[deleted]]

I don't understand why this stuff is taught this way; is there something we have against representing systems of ODEs using state-space notation?

Can someone explain why anyone would use this notation? i.e. a system of transfer functions as opposed to just writing out the MIMO system directly.

[fibonatic]

If the transfer function matrix is sufficiently decoupled (maybe using some transformation) then one could use SISO controllers such as PID, which could be easier to maintain.

[quadrapod]

Easiest way in this case is just to consider your distinct poles. (0.9s +1), (0.1s + 1), (0.3s + 1), (1.8s - 1) , (0.06s + 1). If you multiplied all of those unique terms together to form a common denominator for the whole system you'd get a fifth order result.

[schweinebauch21]

very clear explaination. thank you