Analogue Implementation of a Fractional-PI^λ Controller for DC Motor Speed Control

Abstract

In this paper, an approach to design a fractionalorder integral operator s  where-1 < λ< 0, using an analogue technique, is presented. The integrator with a constant phase angle-80.1 degree (i.e. order l =-0.89), bandwidth greater than 3 decades, and maximum relative phase error 1.38% is designed by cascade connection of first-order bilinear transfer segments and first-order low-pass filter. The performance of suggested realization is demonstrated in a fractional-order proportionalintegral (FOPI ) controller described with proportional constant 1.37 and integration constant 2.28. The design specification corresponds to a speed control system of an armature controlled DC motor, which is often used in mechatronic and other fields of control theory. The behavior of both proposed analogue circuits employing two-stage Op-Amps is confirmed by SPICE simulations using TSMC 0.18 μm level-7 LO EPI SCN018 CMOS process parameters with ±0.9 V supply voltages.

Integer-and Fractional-Order Integral and Derivative Two-Port Summations: Practical Design Considerations

“…No [5] integrator No CMOS OTAs active Medium Yes <1 kHz N/A - [6] both No CFOAs active Medium Yes <100 kHz N/A - [7] both [10] integrator No OAs passive Low Yes <10 kHz I controller - [11] both No OAs passive Low Yes <100 Hz PID No [12] both N/A FPAA (OAs) active High No <100 Hz PID No [13] both [18] both No CMOS OAs active High Yes <1 kHz PI controller No [19] integrator No CMOS OTAs active High Yes <10 MHz N/A - [20] both No CFOAs active Medium Yes <100 kHz N/A - [21] integrator No OTAs active Medium Yes <100 kHz N/A - D. Mondal et al [2] brought a solution of the lossless integrator using a fractional-order passive element (FOE), known also as constant phase element (CPE) as a part of feedback loop of operational amplifier (OA) followed by inverter. However, adjustability and other features (two different slopes in magnitude frequency responses, various starting and final phase shifts in observed bandwidth, etc.)…”

“…Frequency responses influenced by the fractional-order behavior of used components are studied more frequently in recent years [1]. Many works in this field focus on novel solutions of integral and derivative two-ports (for example [2][3][4][5][6][7][8][9]), proportional integral and derivative controllers (for example [10][11][12][13][14][15][16][17][18] and so-called bilinear two ports [8,9,[19][20][21] serving for various purposes. Two ports, known as integrators and differentiators, have started to be interesting for designers of fractional-order systems, and especially for proportional, integral and derivative controllers (PIDs).…”

Comparative Study of Op-Amp-based Integrators Suitable for Fractional -Order Controller Design

“…1 and its transfer function (TF) can be expressed as: which is given by E(s) = R(s) í Y(s). An implementation of a control system used to control the speed and position of an armature controlled DC motor is shown in [16]. The system is composed of an analogue implementation of a FO proportional-integral (FOPI) controller (C(s)), while G(s) is the mathematical model of a DC motor -the controlled plant [7].…”

“…Considering e.g. the setup [16], the speed of a DC motor can be controlled using FOPI, which TF has a form C(s) = U(s)/E(s) = K P + K I s -λ , where the particular three independent parameters are: K P = 1.37 (proportional constant), K I = 2.28 (integration constant), and Ȝ = 0.89 (fractional order of a non-inverting integrator in Laplace domain). Therefore, the next subchapter aims to compare the behavior of 0.89-order integral operator implementations designed using two different techniques.…”

“…Fractional-order capacitors (FOCs) are key components for analogue FOPID design [3]− [7]. Different approaches for emulation of an FOC [8]− [13] and FO integral operator s -λ [14]− [16] are available in literature. Recursive algorithm by Valsa with its network structure, which is used in this work, is among them.…”