# How to design a PID controller for MIMO systems in which there are more outputs than inputs?

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!

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.

• 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).
– WG-
Mar 7, 2016 at 0:01
• 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. Oct 21, 2016 at 23:24