Author : Hamideh Hamidian, Ali Akbar Jalali

International Journal of Scientific & Engineering Research, IJSER - Volume 2, Issue 3, March-2011

ISSN 2229-5518

Download Full Paper : PDF**Abstract**— In this paper, a numerical approach for the fractional order proportional-integral-derivative controller (FO-PID) design for the unstable first order time delay system is proposed. The controller design is based on the system time delay. In order to obtain the relation between the controller parameters and the time delay, for several amounts of the plant time delay and the fractional derivative and integral orders, the ranges of stabilizing controller parameters are determined. First, for a typical time delay plant and the fractional order controller, the D-decomposition technique is used to plot the stability region(s). The controller derivative gain has been considered as one. By changing the fractional derivative and integral orders, a small amount in each stage, some ranges of proportional and integral gains are achieved which stabilize the system, independent of the fractional , orders. Therefore a set of different controllers for any specified time delay system is obtained. This trend for several various systems with different values of time delay has been done and the proportional and integral gains of the stabilizing controller have been calculated. Then we have fitted these values to the exponential functions and the proportional and integral gains have been obtained in terms of the system time delay. Using these relations, we can specify some ranges of the proportional and integral gains and obtain a set of stabilizing controllers for any given system with certain time delay. In these relations, fractional derivative and integral orders haven’t part, and therefore can be applied to any fractional order controller design (for ). Thus we have reached a numerical approach from the graphical D-decomposition method. In this method, there is freedom in choosing the values of and (they can fall in the range of [0.1, 0.9] ), and there is no need to plot the stability boundaries and check the different regions to determine the stable one. This numerical method does not offer the complete set of the stabilizing controllers. Whenever the system time delay is more, the specified range of proportional and integral gains will be smaller. In other words, the extent of obtained stability region is inversely proportional to the system time delay. Finally, the introduced numerical approach is used for stabilizing an unstable first order time delay system.

**Index Terms**—Fractional order PID controller, numerical approach, time delay.

**Introduction**Although great advances have been achieved in the control science, the proportional-integral-derivative controller is still the most used industrial controller.

According to the Japan Electric Measuring Instrument Manufacturers’ Association in 1989, PID controller is used in more than 90% of control loops [1], [2]. As an example for the the application of PID controllers in industry, slow industrial processes can be pointed, low percentage overshoot and small settling time can be obtained by using this controller [1]. Widespread application of the PID controller is due to the simple and implementable structure and its robust performance in the wide range of the working conditions [3], [4]. This controller provides feedback, it has the ability to eliminate steady-state offsets through integral action, and it can anticipate the future through derivative action. The mentioned benefits have caused widespread use of the PID controllers. The derivative action in the control loop will improve the damping, and therefore by accelerating the transient response, a higher proportional gain can be obtained. Precise attention must be paid to setting the derivative gain because it can amplify high-frequency noise. In this paper, for the fractional order PID controller design, the derivative gain ( ) is set 1, that will result in design simplicity. Most available commercial PID controllers have a limitation on the de-rivative gain [2]. During the past half century, many theoretical and industrial studies have been done in PID controller setting rules and stabilizing methods [3]. So far several different techniques have been proposed to obtain PID controller parameters and the research still continues to improve the system performance and increase the control quality. Ziegler and Nichols in 1942 proposed a method to set the PID controller parameters. Hagglund and Astrom in 1995, and Cheng- Ching in 1999, introduced other techniques [5]. By generalizing the derivative and integral orders, from the integer field to non-integer numbers, the fractional order PID controller is obtained. In fractional order PID controller design, there is more freedom in selecting the parameters and more flexibility in their setting . This is due to posse of choice -both integer and non-integer numbers- for integral and derivative orders. Therefore control requirements will be easier to comply [6], [7].

Before using the fractional order controllers in design, an introduction to fractional calculus is required. Over 300 years have passed since the fractional calculus has been introduced. The first time, calculus generalization to fractional, was proposed by Leibniz and Hopital for the first time and afterwards, the systematic studies in this field by many researchers such as Liouville (1832), Holmgren (1864) and Riemann (1953) were performed [8]. Fractional calculus is used in many fields such as electrical transmission losses systems and the analysis of the mechatronic systems. Some controller design techniques are based on the classic PID control theory generalization [7]. Due to the recent advances in the fractional calculus field and the emergence of fractance electrical element, the fractional order controller implementation has become more feasible [6], [9], [10]. Consequently, fractional order PID controller analysis and synthesis have received more attention [11], [12], [13], [14], [15], [16]. Results obtained from various articles published in this field, indicate that the fractional order PID controllers enhance the stability, performance and robustness of the feedback control system [6], [11], [12]. Maiti, Biswas and Konar [1] have significantly re-duced the overshoot percentage, the rise and settling times, compared to classic PID controller, using the fractional order PID controller. Applying the fractional order PID controller ( ), the system dynamic cha-racteristics can be adjusted better [17]. Many dynamic processes can be described by a first order time delay transfer function [18]. The need to control time delay processes can be found in different industries such as rolling mills. Varying time delay process control becomes difficult using classical control methods [19]. Simple formulas are available for setting the PID controller parameters for the stable first order time delay system, but when the system is unstable, the problem will be more difficult and therefore the unstable systems control requires more attention. Many attempts have been made in field of their stabilization [20], [21], [22], [23], [24]. So far, various design techniques have been suggested for the fractional order controller design [13], [14], [25], [26]. It has been shown that fractional order PID controllers have a better performance comparing to integer order ones, for both integer and fractional order control systems.

In the controller design for an unstable system, the most important design issue is stabilizing the closed-loop system [6]. As an example of previous research in stabilizing the unstable processes, we can point to De Paor and O’Malley research in 1989, which discussed unstable open loop system stabilization with a PID or PD controller [23]. Hamaci [3] has concluded that fractional order PID controller has a better response than classic one. In this paper, a numerical method is introduced to design the fractional order controllers for any unstable first order system with specified time delay.

**2 THE FRACTIONAL ORDER PID CONTROLLER DESIGN****2.1 A Review to Design Methods**Hamamci and Koksal [4] have designed the fractional order PD controller to stabilize the integration time delay system, which result that stability region extent is reversed with the system time delay. Maiti, Biswas, and Konar, in 2008, could significantly reduce the overshoot percentage, the rise time, and the settling time by using fractional order PID controllers. They introduced PSO (particle swarm optimization) optimization technique for the fractional order PID controller design. In their me-thod, the controller has been designed based on required maximum overshoot and the rise time. In the mentioned technique, the closed loop system characteristic equation is minimized in order to get an optimal set of the control-ler parameters [1]

One of the methods to obtain the complete set of stabi-lizing PID controllers is plotting the global stability re-gions in the -space, which is called the D-decomposition technique [3], [4], [6], [8]. This technique is used in both fractional and integer order systems anal-ysis and design.

Cheng and Hwang [6] has designed the fractional order proportional - derivative controller to stabilize the unsta-ble first order time delay system and D- decomposition method has been used. The graphical D- decomposition technique results for such systems are simple.

The D- decomposition technique can be used for frac-tional order time delay systems and fractional order chaos systems. In this method, the stability region boundaries are obtained, which are described by real root boundary (RRB), infinite root boundary (IRB), and complex root boundary (CRB). By crossing these boundaries in the -space, several regions will be achieved. By choosing an arbitrary point from each region and checking its stability, the region’s stability is tested. If the selected point is stable, the region including that point would be stable, and if the selected point is not stable then the region would be unstable. By obtaining the stability boundaries and plotting the stability regions, a complete set of stabilizing fractional order controller parameters is obtained. The mentioned algorithm is simple and effective.

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