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1

Fractional order PIλDµA controller design based on Bode’s ideal function

Bettou Khalfa, Charef Abdelfatah

Archives of Control Sciences

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2023

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Vol. 33, No. 2

425--458

EN

The fractional order proportional, integral, derivative and acceleration (PIλDµA) controller is an extension of the classical PIDA controller with real rather than integer integration action order λ and differentiation action order µ. Because the orders λ and µ are real numbers, they will provide more flexibility in the feedback control design for a large range of control systems. The Bode’s ideal transfer function is largely adopted function in fractional control systems because of its iso-damping property which is an essential robustness factor. In this paper an analytical design technique of a fractional order PIλDµA controller is presented to achieve a desired closed loop system whose transfer function is the Bode’s ideal function. In this design method, the values of the six parameters of the fractional order PIλDµA controllers are calculated using only the measured step response of the process to be controlled. Some simulation examples for different third order motor models are presented to illustrate the benefits, the effectiveness and the usefulness of the proposed fractional order PIλDµA controller tuning technique. The simulation results of the closed loop system obtained by the fractional order PIλDµA controller are compared to those obtained by the classical PIDA controller with different design methods found in the literature. The simulation results also show a significant improvement in the closed loop system performances and robustness using the proposed fractional order PIλDµA controller design.

2

Note on describing function analysis of fractional order nonlinear control systems

Yeroglu C., Tan N.

Control and Cybernetics

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2015

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Vol. 44, no. 2

233--255

EN

This paper presents extensions of some results, obtained for the analysis of classical nonlinear control systems, to the nonlinear fractional order systems. It is shown that the results related to limit cycle prediction using describing function method can be applied to the fractional order plants. The frequency and the amplitude of the limit cycle are used for auto-tuning of the PID controller for nonlinear control systems with fractional order transfer functions. Fractional order control system with parametric uncertainty is also considered for the nonlinear case. On the other hand, a New method is provided for stability margin computation for fractional order nonlinear control system with parametric uncertainty structure using the Nyquist envelopes of the fractional order uncertain plant and the describing function that represents the nonlinearity of the system. Maximum perturbation bounds of the parameters of the fractional order plant are computed. Numerical examples are included to illustrate the methods presented.

3

Regulator ułamkowy PIα w sterowaniu ruchem podłużnym samolotu w warunkach zewnętrznych zakłóceń

Żur K. K.

Modelowanie Inżynierskie

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2013

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T. 15, nr 46

141--149

EN

Angle of pitch control system of airplane disturbedby model Drydena in longitudinal movement was shown in the paper. Unconventional fractional order controller PI in control system was designed in order auto-pilot. All simulation investigations was made by using Matlab-Simulink software. Finally, advantages and disadvantages above controller was shown in automatic control system angle of pitch.

PL

W artykule przedstawiono układ sterowania kątem pochylenia samolotu, odbywającego lot podłużny, zakłócony modelem turbulencji Drydena. Rolę automatycznego pilota w układzie sterowania spełnia regulator ułamkowy PIα . Model układu sterowania oraz badania symulacyjne wykonano w środowisku programu Matlab-Simulink. Przedstawiono wady i zalety regulatora ułamkowego PI oraz porównano go z klasycznym regulatorem PI na przykładzie układu sterowania lotem podłużnym samolotu w warunkach zewnętrznych zakłóceń.

4

Analiza sterowania ułamkowego PIλDμ mocą reaktora jądrowego

Puchalski B., Duzinkiewicz K., Rutkowski T.

Informatyka, Automatyka, Pomiary w Gospodarce i Ochronie Środowiska

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2013

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nr 4

63--68

PL

W artykule przedstawiono syntezę regulatora PIλDμ niecałkowitego rzędu dla potrzeb sterowania mocą reaktora jądrowego lekko wodnego określanego, jako typu PWR (Pressurized Water Reactor). W tym celu wykorzystano nieliniowy model matematyczny reaktora PWR o parametrach skupionych obejmujący procesy generacji i wymiany ciepła oraz termicznych efektów reaktywnościowych. Nastawy regulatora PIλDμ niecałkowitego rzędu dobrano w sposób optymalny, minimalizując klasyczne całkowe wskaźniki jakości określane na bazie odpowiedzi skokowej. Na przykładzie najpierw układu oscylacyjnego drugiego rzędu, a potem na przykładzie liniowego modelu reaktora PWR otrzymanego z linearyzacji modelu nieliniowego w określonym punkcie pracy wykazano, że proponowany ułamkowy regulator PIλDμ może stać się alternatywą dla powszechnie stosowanego klasycznego regulatora PID.

EN

In this paper, the synthesis of fractional–order PIλDμ power controller for the PWR (Pressurized Water Reactor) nuclear reactor is presented. For this purpose the point kinetics and heat transfer nonlinear model of the PWR nuclear reactor with thermal feedback mechanisms from lumped fuel and coolant temperature was used. The parameters of the fractional-order PIλDμ controller were tuned due to optimization of the standard integral performance indexes determined on the basis of the step response. Firstly – the typical second order oscillatory dynamic system and secondly – the linear PWR nuclear reactor model derived from nonlinear model was used to show that the fractional order PIλDμ controller may become a alternative to standard PID controller.