What do you mean LQR is suited for non linear systems? This is all misinformation. LQR is a linear systems tool.
The nice thing about linear systems is that almost every dynamical system is locally linear near some operating point.
The nice thing about linear systems is that almost every dynamical system is locally linear near some operating point.
Maybe it's explained by the state space containing more terms than PID controller, so the transfer function can be linear for each individual term?
Given a linear plant model and a PID controller, you can compute all the classic control metrics like phase and gain margins, settling time, and even how the controller would perform according to the LQR metric.
That's the theory. All linear. Theoretically it's as appropriate to apply LQR to a nonlinear plant as it is to apply PID.
If you tweak the four LQR matrices (say, for a second-order system), and couple the gain matrix with the output of a linear observer (Luenberger), that combination system should be able to generate any PID controller. It's an over-parameterization, however (which is why I think PID controllers are ubiquitous. Not many parameters). Many settings will produce the same controller.
If anyone knows a reference that discussed PID and LQR like this, I'd love to see it
But as noted in my comment above, linear controllers can be used to control nonlinear systems under certain circumstances.
p.s. btw, having more states in the model does not linearize a nonlinear system per se. Only nonlinear transformations like log-transforms are capable of linearizing a system.