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I am trying to control a wheeled inverted pendulum with a PID controller. I already designed a linear quadratic regulator (LQR) and a pole placement regulator (PPR). I would like to design a PID controller, but it seems almost impossible to tune the PID gains. I wanted to check if I could stabilize the system by using the solution that I obtained from the LQR/PPR.

Is there any method or reference that is linking the design of a PID controller to the solution obtained by a LQR or pole placement regulator?

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LQR or PPR regulators are both state-space regulators, which use the states of the system in order to bring the poles of the closed loop system to some desired locations. They do this by multiplying the states with a constant matrix of gains and feeding the result back as an actuating value, i.e. u = -Kx. Technically, LQR and PPR regulator is the same regulator, but different methods have been used to calculate the gain matrix K. As long as the system is observable and controllable, it is possible to place all closed loop poles to a specific location.

PID controllers however use the output measurement of the plant (instead of the states) and since the controller structure is fixed, it can only provide restricted dynamics, i.e. it is generally not possible to directly place poles of the closed loop system to desired locations.

That means, in general, there is no way of getting a PID out of a state space controller or vice versa. You need to use other PID designing techniques in order to stabilize your system.

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  • \$\begingroup\$ -1: I think there are some points in your answer that are not right. I don't think that LQR & PPR are the same regulators. LQR gives some guaranteed stability margins whereas PPR does not provide them. They both use state feedback but they are not the same. I also do not think that a system needs to be observable in order to place the poles arbitrarily but it needs to be controllable. The last paragraph is also wrong as this article provides a method how to use the LQR gains for a PID design (waset.org/publications/9999411/…). \$\endgroup\$ – MrYouMath Dec 14 '17 at 16:44

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