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How do you design a PID controller for MIMO systems in which the outputs outnumber the inputs?

It's typical to design a PID controller for MIMO industrial process with the same number of inputs and outputs, or with more inputs than outputs. And very few references talk about systems with less inputs than outputs. However, there are lots of chemical process where this is normal.

Is there any analytical method except (square down ans nrga) to design a PID controller for system with more outputs than inputs? I've considered a lot of references but haven't found anything!

Mimo Sys

Where Mimo sys matrixes are: (Matlab defiend programm)

A = [-6.932e-2,17.41,-36.75,0,0,0,-6.0660,-31.54,0;
    -1.435e-4,2.719e-2,-1.411e-3,3.467e-1,0,-9.380e-1,7.139e-2,-1.691e-2,0;
    -4.537e-4,1.870e-3,-2.025e-1,0,1,0,-4.688e-2,7.563e-3,0;
    -1.304e-4,-7.179,-4.916e-1,-6.172e-1,-3.689e-2,7.631e-1,0,0,0;
    2.297e-5,0,-8.667e-1,4.393e-2,-1.947e-1,-2.026e-2,0,0,0;
    1.964e-5,4.263e-2,-1.329e-2,1.233e-3,1.579e-2,-1.600e-1,0,0,0;
    0,0,0,1,1.941e-1,2.771e-1,0,6.258e-2,0;
    0,0,0,0,8.192e-1,-5.736e-1,-5.612e-2,0,0;
    0,0,0,0,6.055e-1,8.648e-1,0,2.006e-2,0]


B = [0,0,-7.560,9.067e-4;
     -6.952e-3,1.293e-2,0,0;
     0,0,-3.425e-2,-9.577e-7;
     4.249,5.989e-1,0,0;
     0,0,-1.796,0;
     -7.287e-2,-2.877e-1,0,0;
     0,0,0,0;
     0,0,0,0;
     0,0,0,0]


C = [0,-5.758e-1,0,0,0,0,0,0,0;
     0,0,0,1,0,0,0,0,0;
     0,0,0,0,0,1,0,0,0;
     0,0,1,0,0,0,0,0,0;
     0,1,0,0,0,0,0,0,0;
     0,0,0,0,1,0,0,0,0;
     0,2.719e-2,-1.411e-3,3.467e-1,0,-9.380e-1,7.139e-2,0,0]


D = [-1.298e-1,-1.610e-1,0,0;
     0,0,0,0;
     0,0,0,0;
     0,0,0,0;
     0,0,0,0;
     0,0,0,0;
     -6.952e-3,1.293e-2,0,0]

 Gss = ss(A,B,C,D)
 G = tf(Gss)

The forth input of Mimo sys is Constant.

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    $\begingroup$ Did you search for sequentual loopshaping, see alexandria.tue.nl/repository/books/633242.pdf? Furthermore, the method of choosing would also depent on which input controls which output(s). $\endgroup$
    – WG-
    Mar 7 '16 at 0:01
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    $\begingroup$ Your state space model is not minimal, namely the system is not fully observable for the mode corresponding to the eigenvalue of $A$, $\lambda=0$. Since this unobservable eigenvalue does not lie in the open left half plane, this state space model is not detectable. So there will be no output based feedback controller which can bring all states to zero. $\endgroup$
    – fibonatic
    Oct 21 '16 at 23:24
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It really depends on the logic required for your system. I don't think there is a hard fast mathematical solution like there is for a standard one-input one-output PID. In many cases a PID may not even be the best logic for your system; especially if the outputs have a complex relationship with the input like a chemical reaction. PID's are attractive academically because you put a number in and get a number out regardless of the process. However the real world often benefits from controllers that consider the physics involved in the process.

A control system with more outputs than inputs is going to be reliant on some internal equations and assumptions that tell it the physical rules it can anticipate. The controller could also be relying on open loop outputs in addition to the closed loop outputs.

Perhaps you could give an example of one you have seen and we can assess it.

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  • $\begingroup$ Although I agree that modeling the physics/dynamics of a process improves performance and using PID for MIMO systems is hard. I disagree that academia is the one "[attracted to PIDs] because you put a number in and get a number out regardless of the process" since 97% of the controllers in industry are PID (at least in 2001 loopscout.com/Info/cpc2001desboroughandmiller.pdf). Also, PID is not a big research area nowadays (look it up and see how it fares with different control topics in the last 10 years, specially in journals and major conferences). $\endgroup$
    – jDAQ
    Feb 16 at 18:22

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