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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?
- \$\begingroup\$ I just stumbled across this, but am having the same problem. \$\endgroup\$ – Sam Spade Nov 21, 2017 at 14:41
- 1 \$\begingroup\$ I think the following conference paper provides a solution, ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=760858 \$\endgroup\$ – Hazem Dec 14, 2017 at 17:29
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.
- \$\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, 2017 at 16:44
- \$\begingroup\$ Someone should fix this answer as the general structure was enlightening to me, so I would like to see the details ironed out. After reading this, I come to the conclusion that LQR and PPR are similar in that they use state observers (unlike PID), and that PID and PPR are similar in that they think in frequency domain in terms of a transfer function (as opposed to LQR). Is this a fair way to compare them? \$\endgroup\$ – samlaf Apr 2, 2020 at 14:30
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Comparison of LQR and Pole Placement Design Controllers for Controlling the Inverted Pendulum
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.
… Journal of Robotics and Automation (IJRA)
Inverted pendulum control is one of the fundamental but interesting problems in the field of control theory. This paper describes the steps to design various controllers for a rotary motion inverted pendulum which is operated by a rotary servo plant, SRV 02 Series. In this paper some classical and modern control techniques are analyzed to design the control systems. Firstly, the most popular Single Input Single Output (SISO) system, is applied including 2DOF Proportional-Integral-Derivative (PID) compensator design. Here the common Root Locus Method is described step by step to design the two compensators of PID controller. Designing the control system using 2DOF PID is quiet challenging task for the rotary inverted pendulum because of its highly nonlinear and open-loop unstable characteristics. Secondly, the paper describes the two Modern Control techniques that include Full State Feedback (FSF) and Linear Quadratic Regulator (LQR). Here FSF and LQR control systems are tested both for the Upright and Swing-Up mode of the Pendulum. Finally, experimental and MATLAB based simulation results are described and compared based on the three control systems which are designed to control the Rotary Inverted Pendulum.
Applied Mathematics and Information Sciences
Prof. Dr. Saad Mekhilef
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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This paper presents a combination of a series active power filter and a shunt passive filter. The shunt passive filter is connected in parallel with a load and suppresses the harmonic current produced by the load, whereas the active filter connected in series to a source acts as a harmonic isolator between the source and load. For active filter control, Sinusoidal Pulse Width Modulation (SPWM) is developed and the modulation index is selected by calculating the DC bus voltage of the active filter. The PSpice®, Matlab/Simulink® and MAX PLUS II® softwares are used for simulation and hardware implementation. © 2005 IEEE. ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1619823
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The acrobot is an underactuated two-link planar robot that mimics the human acrobat who hangs from a bar and tries to swing up to a perfectly balanced upside-down position with his/her hands still on the bar. In this paper we develop intelligent controllers for swing-up and balancing of the acrobot. In particular, we first develop classical, fuzzy, and adaptive fuzzy controllers to balance the acrobot in its inverted unstable equilibrium region. Next, a proportional-derivative (PD) controller with inner-loop partial feedback linearization, a state-feedback, and a fuzzy controller are developed to swing up the acrobot from its stable equilibrium position to the inverted region, where we use a balancing controller to ‘catch’ and balance it. At the same time, we develop two genetic algorithms for tuning the balancing and swing-up controllers, and show how these can be used to help optimize the performance of the controllers. Overall, this paper provides (i) a case study of the development of a variety of intelligent controllers for a challenging application, (ii) a comparative analysis of intelligent vs. conventional control methods (including the linear quadratic regulator and feedback linearization) for this application, and (iii) a case study of the development of genetic algorithms for off-line computer-aided-design of both conventional and intelligent control systems.
In this paper, we propose a nonlinear control approach for balancing underactuated legged robots. For the balancing task, the robot is modeled as a generalized version of a Segway. The control design is based on the State-Dependent Riccati Equation (SDRE) approach. The domain of attraction of the SDRE controller is compared to the domain of attraction of a linear quadratic controller. Using a simulation example of a four-legged robot balancing on its hind legs, we show that the SDRE controller gives a reasonably large domain of attraction, even with level realistic level constraints on the control input, while the linear quadratic controller is unable to stabilize the system.
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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.
Linear Quadratic Regulator is an optimal technique of pole placement method which defines the optimal pole location based on a definite cost function.
Linear Quadratic Regulator (LQR) is the optimal theory of pole placement method. LQR algorithm defines the optimal pole location based on two cost function.
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
Two types of regulators are studied, the State-Regulator using Pole Placement, and the Linear-Quadratic regulator (LQR). The LQR is obtained by resolving
pendulum in the vertical position. After linearization, we have: V.
Linear Quadratic Regulator is an optimal technique of pole placement method which defines the optimal pole location based on a definite cost function. The
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
some conclusions are given in Section V. ... feedback gain matrix of pole placement.
To stabilizes and control the position of rolling Ball on the beam, Pole Placement Technique and Linear Quadratic Regulator (LQR) Controller has been applied on
امهو نيتيمزراوخ. Simulated Annealing (SA) optimization. و. Ant Colony (AC) optimization. ترهظا . ةقيرط قوفت جئاتنلا. LQR. رطيسم ميمصتل state
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