Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 5478781, 9 pages

https://doi.org/10.1155/2018/5478781

## Fuzzy Fractional-Order PID Controller for Fractional Model of Pneumatic Pressure System

Correspondence should be addressed to K. S. Nisar

Received 22 June 2017; Revised 8 January 2018; Accepted 30 January 2018; Published 28 February 2018

Academic Editor: Hung-Yuan Chung

Copyright © 2018 M. Al-Dhaifallah et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This article presents a fuzzy fractional-order PID (FFOPID) controller scheme for a pneumatic pressure regulating system. The industrial pneumatic pressure systems are having strong dynamic and nonlinearity characteristics; further, these systems come across frequent load variations and external disturbances. Hence, for the smooth and trouble-free operation of the industrial pressure system, an effective control mechanism could be adopted. The objective of this work is to design an intelligent fuzzy-based fractional-order PID control scheme to ensure a robust performance with respect to load variation and external disturbances. A novel model of a pilot pressure regulating system is developed to validate the effectiveness of the proposed control scheme. Simulation studies are carried out in a delayed nonlinear pressure regulating system under different operating conditions using fractional-order PID (FOPID) controller with fuzzy online gain tuning mechanism. The results demonstrate the usefulness of the proposed strategy and confirm the performance improvement for the pneumatic pressure system. To highlight the advantages of the proposed scheme a comparative study with conventional PID and FOPID control schemes is made.

#### 1. Introduction

Pneumatic pressure is one among the vital variables used in industries like power plants, chemical reaction control, pneumatic position servo systems, well drilling, heating, ventilating and air conditioning systems, automobile, and so on. The dynamic characteristics of pneumatic pressure plants are highly nonlinear due to the compressibility of air, load variations, and external disturbances. Further, the industrial pneumatic pressure plants are usually interconnected and operating at different pressure level. Therefore, the precise control of pressure plant is complex due to the presence of uncertainties and nonlinearity. Hence, an efficient control strategy is needed for trouble-free operation of the pneumatic system in industries. The classical PI and PID controllers are widely used in industrial applications in the past because of its advantages. PID control is a simple and effective control method and it can be easily implemented for industrial control applications. However, the PID control algorithm is not advisable for the complex and nonlinear system. On the other hand, the fractional calculus is getting much more attention in the field of control system engineering due to its potential and significant importance [1–3]. The controllers, making use of fractional-order derivatives and integrals, give improved results compared to the classical controllers in terms of robustness [4–6]. Fractional-order (FO) controllers are usually expressed by fractional-order differential equations. The FO controllers are derived from the integer order by adding the fractional powers in integral and derivative terms. For example, in addition to the proportional , integral , and derivative parameters which comprise the integer-order PID, the FOPID controller has two more parameters an integrator order and a differentiator order . Adding the integral and derivative terms of fractional order will improve system frequency response to be better and leads to design an improved control system [7–10]. The FOPID control scheme has certain merits whereby it offers five parameters to be tuned. However, this control scheme has its own demerits as it makes the system more complex than the classical one.

Recent research trends in FO control are looking towards using fuzzy with FO control scheme to improve the control performances. The rule base fuzzy set theory provides more flexibility in designing complex industrial control system. In fuzzy set theory, linguistic notations are used to express the observations easily to form a control structure. The fuzzy logic controller (FLC) design is becoming simple, even for more complex and nonlinear industrial process without knowledge of the exact mathematical description of the system [11]. Further, FLC is combined with the FO controller for fine-tuning parametric gains and guarantee optimal performance owing to nonlinearities, load disturbances, and plant parameters variations [12–14]. The adaptive method provided by the fuzzy system will improve the dynamic performances of the FO controller through which the controller may respond quickly to parameter variation.

By considering all these aspects, a fuzzy-based FOPID control scheme is proposed for faster response and better control performances. To demonstrate the performances of proposed control technique, a novel pneumatic pressure system model is developed and the system performances are studied under load disturbances and changes in set-point conditions. This paper is organized as follows. In Section 2, the mathematical background of fractional calculus is discussed. Section 3 gives an overview of the experimental setup for pneumatic pressure control system. The modeling of the proposed system is discussed in Section 4. Section 5 describes the controller design for the proposed system. Section 6 presents the simulation results. The findings are given as a conclusion in Section 7, followed by the references.

#### 2. Mathematical Background of Fractional Calculus

Fractional-order calculus (FOC) is one of the popular and emerging mathematics branches that deals with differentiation and integration of real or complex order [1, 2, 15]. The fractional-order calculus provides efficient tools for many situations related to the fractal dimension, infinite memory, and chaotic behavior. Recently, FOC attracted various researchers due to its application in electronics, Bioengineering, control theory, and many more areas [7–10]. Fractional-order mathematical phenomena are very useful to describe and model real-time system more accurately than the conventional integer methods. The fractional-order differentiator can be denoted as a continuous differintegral operator [1, 2, 16–18], which is given bywhere is the order of the differintegration and is constant related to initial conditions.

The most commonly used definitions in FOC are Riemann-Liouville (e.g., in calculus), Caputo (e.g., in numerical integration and physics), and Grunwald-Letnikov (e.g., communications and control).

The definitions due to Riemann-Liouville and Caputo are, respectively, given by where and is the familiar gamma function defined byHere, in this paper the Grunwald-Letnikov definition is used and is expressed bywhere , is integer part, and , are the limits of operator. The binomial coefficient is evaluated by the gamma function that generalizes the factorial operator:The equation given in (4) is very useful to obtain a numerical solution of fractional differential equation [19].

#### 3. Pneumatic Pressure Control Experimental Setup

The experimental setup of the pneumatic pressure control system is shown in Figure 1. It consists of an air compressor, pressure transmitter, pressure regulator, electropneumatic control valve, pressure indicator, and controller interfacing units. Figure 2 depicts the scheme of the pressure control. The air compressor is attached to the pressure regulator, which supplies the air at constant pressure. An equal percentage electropneumatic control valve of 50 mm size at inlet regulates air flow to the pressure tank. A precise pressure transmitter (PT) attached to the pressure tank measures pressure at each sampling period and gives output as a current signal of 4 to 20 mA. The current signal is converted into a voltage signal in the range of 0 to 5 volts by current to voltage (*V* to* I*) converter. The controller block of this setup computes tank pressure using input voltage signal and by applying control algorithm the position of the control valve will be manipulated to keep the tank pressure at the desired level. A 32-bit advanced RISC architecture ARM7 (AT91M55800A) microcontroller is used in the controller part. The microcontroller output voltage magnitude is based on the measured and reference values. The controller output voltage is first converted into a current signal of 4 to 20 mA and then to pressure signal suitably to manipulate the control valve. A pressure indicator (PI) fixed at the top of the pressure tank is used to read the tank pressure manually. Based on the valve characteristics, it is obvious that the pneumatic system possesses inherent nonlinearity and parameters uncertainty because of air compressibility.