Influence of the Numerical Method Sampling on the Digital Pid-controller Behavior

: pp. 35 - 45
Lviv Polytechnic National University
Lviv Polytechnic National University

Modern digital control systems make it possible to implement quite complex control strategy, the complexity of which is limited by the hardware capabilities, the available software and the implemented control algorithm. An important component of such an algorithm is the numerical method, which allows to discretize the control rule on the basis of a continuous prototype. Such application example is the classic PID controller, which has become the basis for the development of digital control systems. Two mathematical operations are performed in such a controller: integration and differentiation, which in a digital system obtain discrete equivalents in the corresponding recurrent equations form.

The article considers a digital PID controller as a digital filter, which with the use of the frequency characteristics (Bode diagrams) allowed to determine its most “narrow” place — the high-frequency diagram region that correspond to the differentiating part of the controller. This made it possible to focus the research on the practical implementation of the differential part of the digital PID controller. It is shown that the traditional way of the differential operation performing by the simple method of finite differences has some of disadvantages that make it fundamentally impracticable, as illustrated by the corresponding graphs.

To eliminate the limitations of the traditional differential operation by the finite difference method, two variants of structural schemes of a real differentiator are proposed. The first variant of the real differentiator proposed to build on the structural scheme in feedback with the integrator. The second variant of the real differentiator is proposed to build according to the structural scheme with a parallel connection of the proportional and the first order blocks. The application of explicit numerical Adams integrators (also known as Adams-Bashforth rule) from the first to the fourth order under the conditions of physical realization is proposed to perform sampling. A study of both their frequency characteristics and their behavior performed on a noisy signal for these structural schemes.

All research in the article was conducted using the Control System Toolbox library of the mathematical application MATLAB. It is shown that the use of the proposed real differen- tiation methods allows simple and efficient implementation of digital PID controllers.

  1. Michael A. Johnson and Mohammad H. Moradi (Editors). PID Control: New Identification and Design Methods. © Springer-Verlag London Limited, 2005. 544 p. [ISBN-10: 1-85233-702-8; ISBN-13: 978-1-85233-702-5]
  2. Moroz V. Chyslovi intehratory v tsyfrovykh systemakh keruvannya. Visnyk Natsionalʹnoho universytetu “Lʹvivsʹka politekhnika” “Elektroenerhetychni ta elektromekhanichni systemy”. 2006. No 563. S. 99–104. (Ukr)
  3. Elijah I. Jury. Theory and Application of the Z-Transform Method. Krieger Pub Co, 1973. [ISBN 0- 88275-122-0]
  4. Katsuhiko Ogata. Discrete-Time Control Systems, 2nd edition. Published by Pearson, 1995. [ISBN-13: 9780130342812]
  5. MATLAB Environment. © 1994-2020 The MathWorks, Inc. URL: ducts/matlab.html
  6. Control System Toolbox: Design and analyze control systems. © 1994-2020 The MathWorks, Inc. URL:
  7. E. Hairer, S. Nørsett, G. Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems. 2nd Edition. Springer, 2008. [ISBN 978-3-540-56670-0]
  8. V. Moroz, V. Oksentyuk, I. Snitkov. Realizatsiya operatsiyi dyferentsiyuvannya u mikrokontrolerakh. Matematychne ta kompʺyuterne modelyuvannya. Seriya: Tekhnichni nauky : zb. nauk. pratsʹ. Instytut kibernetyky im. V. M. Hlushkova NAN Ukrayiny, Kamʺyanetsʹ-Podilʹsʹkyy nats. universytet im. I. Ohiyenka. Kamʺyanetsʹ-Podilʹsʹkyy : Kam'yanetsʹ-Podilʹsʹkyy natsionalʹnyy universytet im. I. Ohiyenka, 2010. Vyp. 3. [232 s.]. S. 154–159. (Ukr)
  9. Moroz V. Pohlyad inzhenera-elektryka na chyslovi metody rozv'yazuvannya zvychaynykh dyferentsialʹnykh rivnyanʹ. Visnyk Natsionalʹnoho universytetu “Lʹvivsʹka politekhnika” “Elektroenerhetychni ta elektromekhanichni systemy”, No 485, 2003, s. 208-213. (Ukr)
  10. Moroz V. Analiz ratsionalʹnoho poryadku aproksymatsiyi dlya vidnovlennya informatsiyi za yiyi dyskretnymy vidlikamy. RIU (Radioelektronika. Informatyka. Upravlinnya). 2008. No 1 (19). S. 74–78. (Ukr)
  11. Cleve Moler. Numerical Computing with MATLAB. Copyright 2004, Cleve  B.  Moler,  MathWorks. SIAM, 2004. 336 Pp. ISBN: 0-89871-560-1