I am trying to design a control system for an unstable MIMO system (4 inputs, 4 outputs, 8 states).
I use the full state feedback to place the poles such that the closed loop is stable. But the outputs still need to track some reference inputs. So I added an integral controller. But I am having a hard time finding the gain matrix for the integrator.
I tried to augment the system and include 4 additional states corresponding to 4 integrators, so that I can find the gains for both stability and reference tracking.
But the augmented system turns out to be uncontrollable, which prevents me from placing the poles.
Any help would be very much appreciated.
Perhaps you are trying to integrate a rate? That would lead to pole/zero cancellation and loss of controlability.
Thank you James Whidborne !
Indeed I was trying to integrate a rate. That solves the problem with controllability.
However the algorithm dosn't work. And I don't know why.
As shown in the attached picture, I use the augmented A and B matrices to find the gain K that would place the poles at some locations. And I'm splitting the gain matrix K in two parts. In my case, the first 4 columns are Ki and the last 8 columns are Kx.
Kx is the gain matrix supposed to stabilize my original system. But it doesn't.
The entire K stabilizes the augmented system though.
I do not see that there is any particular reason why (A-B K_x) would be stable.
If you wanted it to be stable, you would need to solve your problem in 2 steps, but you would not have full state feedback for the 2nd integrating loop step, so you could not use Acker to design the integrating loop.
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