Ayrık zamanlı PI, PID ve PIR kontrolörler ile baskın kutup bölgesi atama

Yıl 2022, Cilt: 28 Sayı: 5, 692 - 700, 31.10.2022

Ayşe Duman Mammadov Emre Dincel Mehmet Turan Söylemez

Öz

Bu çalışmada, kapalı çevrim sistemin zaman özelliklerinin istenen aralıkta kalması için baskın kutup bölgesi atama yöntemiyle ayrık zamanlı PI, PID ve PIR kontrolörlerin tasarlanması amaçlanmıştır. Öncelikle, kapalı çevrim sistemin baskın ve baskın olmayan kutuplarının konumlanmaları istenen bölgeler için sınır fonksiyonlarının belirlenmesi anlatılmıştır. Burada, sınır fonksiyonları için, baskın kutupların konumlanması istenen bölge ayrık zaman düzleminde sabit yarıçaplı iki çember ve sabit bir sönüm oranı eğrisi, kalan kutupların konumlanması istenen bölge ise sabit yarıçaplı bir çember kullanılır. Daha sonra, baskın kutup bölgesi atama probleminin çözüm yöntemi ayrık PI kontrolör için verilmiştir. Önerilen yöntem, kontrolörün bir parametresini sabitleyerek (𝐾𝑝 = 𝑘𝑝 ∗ ) ayrık PID ve PIR kontrolörler için genişletilmiştir. PIR kontrolörde ek olarak gecikme parametresi ℎ’nin pozitif bir tamsayı olarak seçilmesi ile tasarıma başlanır. Önerilen yöntem, iki sistem üzerinden ayrık PI, PID ve PIR kontrolörler için anlatılmıştır.

Anahtar Kelimeler

Kutup atama, Baskın kutup bölgesi, PI, PID, PIR, Ayrık zaman düzlemi

Dominant pole region assignment with discrete time PI, PID and PIR controllers

Öz

In this study, it is aimed to design discrete time PI, PID and PIR controllers with the dominant pole region assignment method in order to have time domain characteristics of the closed loop system in the desired interval. First of all, determination of the boundary functions for the regions where the dominant and non-dominant poles of the closedloop system are desired to be located are explained. Here, for the boundary functions, the region where the dominant poles are desired to be located are two circles of constant radius and a constant damping ratio curve in the discrete time domain, and the region where the remaining poles are desired to be located is a circle of constant radius. Then, solution method of dominant pole region assignment problem is given for discrete PI controller. The proposed method is extended for discrete PID and PIR controllers by fixing a parameter of the controller (𝐾𝑝 = 𝑘𝑝 ∗ ). In addition, the design starts with selecting the delay parameter ℎ as a positive integer in the PIR controller. The proposed method is demonstrated for discrete PI, PID and PIR controllers via two systems.

Anahtar Kelimeler

Pole assignment, Dominant pole region, PI, PID, PIR, Discrete time domain

Kaynakça

• [1] Åström KJ, Murray RM. Feedback Systems: An İntroduction For Scientists And Engineers. New Jersey, USA, Princeton University, 2010.
• [2] Halder K, Das S, Gupta A. “Time delay handling in dominant pole placement with PID controllers to obtain stability regions using random sampling”. International Journal of Control, 94(12), 3384-3405, 2020.
• [3] Wang H, Han QL, Liu J, He D. “Discrete-time filter proportional-integral-derivative controller design for linear time-invariant systems”. Automatica, 116, 1-15, 2020.
• [4] Das S, Halder K, Gupta A. “Delay handling method in dominant pole placement based PID controller design”. IEEE Transactions on Industrial Informatics, 16(2), 980-991, 2019.
• [5] Dincel E, Mutlu İ, Schrödel F, Söylemez MT. “Further results on dominant pole placement via stability mapping approach”. IFAC-PapersOnLine, 51(4), 918–923, 2018.
• [6] Das S, Halder K, Gupta A. “Performance analysis of robust stable PID controllers using dominant pole placement for SOPTD process models”. Knowledge-Based Systems, 146, 12-43, 2018.
• [7] Srivastava S, Misra A, Sarkar A. “Analysis design of robust pıd controller with dominant pole placement approach”. 2019 6th International Conference on Signal Processing and Integrated Networks (SPIN), Noida, India, 7-8 March 2019.
• [8] Wang QG, Zhang Z, Astrom KJ, Zhang Y, Zhang Y. “Guaranteed dominant pole placement with PID controllers”. IFAC Proceedings Volumes, 41(2), 5842-5845, 2008.
• [9] Dincel E, Söylemez MT. “Limitations on dominant pole pair selection with continuous PI and PID controllers”. 2016 International Conference on Control, Decision and Information Technologies (CoDIT), Saint Julian's, Malta, 20 October 2016.
• [10] Ang KH, Chong G, Li Y. “PID control system analysis, design, and technology”. IEEE Transactions on Control Sysems Technology, 13(4), 559-576, 2005.
• [11] Hägglund T. “The one-third rule for PI controller tuning”. Computers & Chemical Engineering, 127, 25-30, 2019.
• [12] Grimholt C, Skogestad S. “Optimal PI and PID control of first-order plus delay processes and evaluation of the original and improved SIMC rules”. Journal of Process Control, 70, 36-46, 2018.
• [13] O’Dwyer A. Handbook of PI and PID Controller Tuning Rules. 3rd ed. London, UK, Imperial College Press, 2009.
• [14] Nise NS. Control systems engineering. 8th ed. Pomona, USA, John Wiley & Sons, 2020.
• [15] Ribeiro JMS, Santos MF, Carmo MJ, Silva MF. “Comparison of PID controller tuning methods: analytical/classical techniques versus optimization algorithms”. In 2017 18th international Carpathian control conference (ICCC), Sinaia, Romania, 7 July 2017.
• [16] Bharat S, Ganguly A, Chatterjee R, Basak B, Sheet DK, Ganguly A. “A Review on tuning methods for PID controller”. Asian Journal For Convergence In Technology (AJCT), 5(1), 1-4, 2019.
• [17] Bucz Š, Kozáková A. Advanced Methods of PID Controller Tuning for Specified Performance. Editor: Shamsuzzoha M. PID Control Industrial Process, 73-119, London, UK, IntechOpen, 2018.
• [18] Kharitonov VL, Niculescu SI, Moreno J, Michiels W. “Static output feedback stabilization: Necessary conditions for multiple delay controllers”. IEEE Transactions on Automatic Control, 50(1), 82-86, 2005.
• [19] Niculescu SI, Michiels W.“Stabilizing a chain of integrators using multiple delays”. IEEE Transactions on Automatic Control, 49(5), 802-807, 2004.
• [20] Swisher GM, Tenqchen S. “Design of proportional-minusdelay action feedback controllers for second-and thirdorder systems”. In 1988 American Control Conference, Atlanta, GA, USA, 15-17 June 1988.
• [21] Galip Ulsoy A. “Time-Delayed control of sıso systems for ımproved stability margins”. Journal of Dynamic Systems, Measurement, and Control, 137(4), 1-12, 2015.
• [22] Villafuerte R, Mondie S, Garrido R. “Tuning of proportional retarded controllers: Theory and experiments”. IEEE Transactions on Control Systems Technology, 21(3), 983-990, 2013.
• [23] Ramírez A, Mondié S, Garrido R. “Proportional Integral Retarded control of second order linear systems”. In 52nd IEEE Conference on Decision and Control, Firenze, Italy, 10-13 December 2013.
• [24] Ramírez A, Garrido R, Mondié S. “Velocity control of servo systems using an integral retarded algorithm”. ISA Transactions, 58, 357-366, 2015.
• [25] Ramírez A, Mondié S, Garrido R, Sipahi R. “Design of proportional-integral-retarded (PIR) controllers for second-order LTI systems”. IEEE Transactions on Automatic Control, 61(6), 1688-1693, 2015.
• [26] López K, Mondié S, Garrido R. “A tuning procedure for the Cascade Proportional Integral Retarded Controller”. IFAC-PapersOnLine, 51(14), 61-65, 2018.
• [27] Koh M, Ramírez A, Sipahi R. “Single-Delay proportionalretarded (PR) protocols for fast consensus in a multiagent system”. IFAC-PapersOnLine, 51(14), 31-36, 2018.
• [28] Lpez K, Garrido R, Mondi S. “Position control of servodrives using a Cascade Proportional Integral Retarded controller”. In 2017 4th International Conference on Control, Decision and Information Technologies (CoDIT), Barcelona, Spain, 5-7 April 2017.
• [29] Ramirez A, Sipahi R. “Fast Consensus Against Noise in a Large-Scale Multi-Agent System with Distributed Proportional-Retarded (PR) Controllers”. In 2018 Annual American Control Conference (ACC), Milwaukee, WI, USA, 27-29 June 2018.
• [30] Kang HIl. “Design of dominant pole region assignment with PID controllers”. In 2010 International Conference on Intelligent Computation Technology and Automation, Changsha, China, 11-12 MAy 2010.
• [31] Dincel E, Söylemez MT. “Dominant pole region assignment with continuous PI and PID controllers”. In 2017 10th International Conference on Electrical and Electronics Engineering (ELECO), Bursa, Turkey, 30 November2 December 2017.