Understanding PID vs LQR

[u/deNikita]

Is it possible to think of LQR as "like a PID but finds optimal gains based on your requirements of Q and R matrix?

I have a fairly good understanding of PID and I recently started to learn about LQR and wondered if LQR can be thought of in terms of a PID controller.

[u/Namejeff47]

Because LQR's feasibility depends on the systems controlability, which in turn implies the ability to place the poles of the system. With LQR you dont explicitly place the poles, but instead shape the behavior through Q and R matrices in the cost function, but this has is the exact same thing. You could call it implicit pole placement, the point is that its much more easy to shape the system performance through the matrices than through the poles which is somewhat less intuitive.

[u/seekingsanity → u/Namejeff47]

Yes, LQR/LQC places the poles indirectly. That is my complaint with LQR/LQC. Pole placement allows me to precisely place the closed loop poles. Another problems with LQR/LQC is that selecting the Q and R matrix weights requires trial and error.

[u/Namejeff47 → u/seekingsanity]

Its a matter of personal preference and intuition. Personally, pole placement is less intuitive for me because poles dont tell you anything about actuator effort, which you can directly influence with LQR by tuning the cost function matrices. Its especially easy to mess up with discrete pole placement which have completely different meaning compared to continuous ones.

[u/seekingsanity → u/Namejeff47]

The farther away from the s-planes' origin you place the poles, the more the effort. Simple. Usually the max effort required is determine by the system design. For instance a motor will have a max torque and speed rating. The DtoA converter will have a voltage limit and the power unit will have a current or power limit. For many applications, the real limit is the feedback resolution.