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Connection between PID, Pole placement and LQR

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?

MrYouMath's user avatar

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.

David G.'s user avatar

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Comparison of LQR and Pole Placement Design Controllers for Controlling the Inverted Pendulum

Profile image of Navid Razmjooy

In this paper an inverted pendulum is modeled firstly by using Euler – Lagrange energy equation for stabilization of the pendulum. To control the modeled system, both full-state feedback and Linear Quadratic Regulator controller methods are applied and the results are compared. After that, a pre-compensator is implemented to eliminate the steady-state error. Linear Quadratic Regulator is an optimal technique of pole placement method which defines the optimal pole location based on a definite cost function. The investigated system develops classical inverted pendulum by forming two moving masses. The motion of two masses in the pendulum which slide along the horizontal plane is controllable.

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