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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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Cart Inverted Pendulum (CIP) system is a benchmark problem in nonlinear automatic control. It is commonly used to verify the robustness of any proposed nonlinear controller. CIP is mostly represented by two second order differential equations to avoid complexity due to the DC motor dynamics. This representation is not practical for the real CIP dynamics and might lead to instability. Therefore in this paper, two third-order differential equations were derived to combine the pendulum system and DC motor dynamics to have a more realistic mathematical model. Friction between the cart and rail was included in the system equations through a nonlinear friction model. To stabilize the obtained nonlinear electromechanical CIP model, a third-order Fuzzy Sliding Mode Controller (FSMC) was designed. The chattering of the control signal was eliminated using general bill shape membership functions for the Fuzzy controller. Simulation results proved the robustness of the proposed FSMC over Linear Quadratic Regulator Controller (LQRC). For instant, the overshoot in the cart position response was reduced by 300%. © 2013 NSP. Natural Sciences Publishing Cor. htwww.naturalspublishing.com/files/published/2g71k419ni9c46.pdf

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In industrial environments, the heat exchanger is a \r\nnecessary component to any strategy of energy conversion. Much of \r\nthermal energy used in industrial processes passes at least one times \r\nby a heat exchanger, and methods systems rec\ overing thermal \r\nenergy. \r\nThis survey paper tries to presents in a systemic way an sample \r\ncontrol of a heat e\ xchanger by comparison between three controllers \r\nLQR \(linear quadratic regulator\), PID \(proportional, integrator and \r\nderivate\) and Pole Placement. All of these controllers are used mainly \r\nin industrial sectors \(chemicals, petrochemical\ s, steel, food \r\nprocessing, energy production, etc…\) of transportation \(automotive, \r\naeronautics\), but also i\ n the residential sector and tertiary \(heating, air \r\nconditioning, etc...\) The choice of a heat exchanger, for a given \r\napplication depends on many parameters: field temperature and \r\npressure of fluids, and physical properties of aggressi\ ve fluids, \r\nmaintenance and space. It is clear that the fact of having an \r\nexchanger appropriate, well-sized, well m\ ade and well used allows \r\ngain efficiency and energy processes.

IMAGES

  1. The Pole Placement Region

    pole placement vs lqr

  2. Pole Placement

    pole placement vs lqr

  3. Pole placement by er. sanyam s. saini (me reg)

    pole placement vs lqr

  4. Pole Placement in Digital Control

    pole placement vs lqr

  5. Pole placement configuration for full state feedback.

    pole placement vs lqr

  6. Region of pole placement.

    pole placement vs lqr

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COMMENTS

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    Linear Quadratic Regulator is an optimal technique of pole placement method which defines the optimal pole location based on a definite cost function.

  2. Pole-Placement Method LQR

    Linear Quadratic Regulator (LQR) is the optimal theory of pole placement method. LQR algorithm defines the optimal pole location based on two cost function.

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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

  4. Comparison of Pole Placement and LQR Applied to Single Link

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    pole/placement LQR technique. Fig 4. Plant with state feedback. V. POLE PLACEMENT DESIGN. Pole placement method is a controller design method in which the

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    some conclusions are given in Section V. ... feedback gain matrix of pole placement.

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    Or, even if we can change the pole locations. • Where do we put the poles? — Linear Quadratic Regulator. — Symmetric Root Locus. • How well does this approach