A Game Theoretic Approach for Automated PID Controller Parameter Tuning

Cristopher Luciano, Mandoye Ndoye, Gregory V. Murphy, Kennedy Aganah Tuskegee University

ABSTRACT THEORETICAL & • Proportional Integral Derivative (PID) controllers have the ability EXPERIMENTAL to eliminate Steady-State error and anticipate output changes in Control Systems. RESULTS • Improper values of 퐾푝 , 퐾푖 , and 퐾푑 can make the system • First order : response unstable. ퟓ 푮 = • A Game Theoretic approach to PID controllers parameters ퟏퟎ풔+ퟏ tuning is proposed. • Benefits: One can get a set of the best combinations values of • Case 1: Game Theory 퐾푝 , 퐾푖 , and 퐾푑 that maximize the Rise-Time, Settling-Time, • Case 2,3,4: pidTuner Fig 2: Experimental setup for speed control Overshoot and Steady-State Error of the system transient of a DC motor. response. PID CONTROLLER OVERVIEW • Eliminate Steady-State Fig 1: Rise-Time, Overshoot, Settling-Time and Error Steady-State Error [1]. • Ability to anticipate changes • Robustness and Simplicity • System inestability due to improper values of 퐾푝, 퐾푖, and 퐾푑 • Methods already proposed: Fig 3:Theoretical Step Response o Ziegler and Nichols. Drawback: High-percentage overshoot • Second order o Cohen-Coon. Drawback: Good approximation only for First Under Damped: Order Plus Dead Time systems (FOPDT) ퟏퟑퟏ. ퟏ 푮 = o Softwares: Matlab pidTuner, PITOPS PID, etc. 풔ퟐ + ퟏퟗ. ퟐퟑ풔 + ퟏퟏퟓ. ퟕ GAME THEORY - MODELING • Case 16, 24, 28: Game Theory • Interaction between independent rational-thinking players • Case 32, 33: pidTuner • Players. Decisions-makers: o 퐾푝: Proportional Constant o 퐾푖: Integral Constant o 퐾푑: Derivative Constant • Actions. Choices available to players. o The player’s actions is the set of values that each one of them (퐾푝, 퐾푖, and 퐾푑) can take when tunning the PID controller. Fig 5: Theoretical Step Response Fig 6: Experimental Step Response • Payoffs. Utility players receive from taking actions under given scenarios. Each player’s payoff is calculated based on three criterions; o Rise-Time, Settling-Time, Overshoot and Steady-State Error of the system with no PID controller o Contribution of each one of the players to the system transient response based on TABLE I. o System stability CONCLUSIONS • A Game Theoretic approach to PID controller parameters tuning TABLE I: Effects of independent P, I and D tuning on Closed-loop response [2]. was proposed. • Solutions are presented based on the Nash Equilibrium solution concept. • Approach is suitable for different types of systems transfer functions. • 푲풑 = 풓풕 + 풔풔풆 − ퟎ. ퟓ ퟏퟎퟎ − 풔풕 − ퟎ. ퟓ(ퟏퟎퟎ − 풐) REFERENCES • 푲풊 = ퟎ. ퟓ풓풕 + ퟏ. ퟓ ∗ 풔풔풆 − ퟎ. ퟓ ퟏퟎퟎ − 풔풕 − ퟎ. ퟓ ퟏퟎퟎ − 풐 • 푲 = ퟎ. ퟓ풓풕 + 풔풕 + 풐 • [1]: Zhong, Jinghua. "PID controller tuning: A short 풅 . 푟푡: Rise-Time tutorial." class lesson), Purdue University (2006). . 푠푠푒: Steady-State Error • [2]: Ang, K.H., Chong, G. and Li, Y., 2005. PID control system Where: . 푠푡: Settling-Time analysis, design, and technology. IEEE transactions on control . 표: Overshoot systems technology, 13(4), pp.559-576.

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