Tuning and Analysis of Fractional Order PID Controller

This paper presents the development of a new tuning method and performance of the fractional order PID controller includes the integer order PID controller parameter. The tuning of the PID controller is mostly done using Zeigler and Nichols tuning method. All the parameters of the controller, namely p K (Proportional gain), i K (integral gain), d K (derivative gain) can be determined by using Zeigler and Nichols method. Fractional order PID (FOPID) is a special kind of PID controller whose derivative and integral order are fractional rather than integer. To design FOPID controller is to determine the two important parameters λ (integrator order) and μ (derivative order).In this paper it is shown that the response and performance of FOPID controller is much better than integer order PID controller for the same system. Introduction PID controller is a well known controller which is used in the most application.PID controller becomes a most popular industrial controller due to its simplicity and the ability to tune a few parameters automatically. According to the Japan electric measuring instrument manufacture’s association in 1989, PID controller is used in more than 90% of the control loop. As an example for the application of PID controller in industry, slow industrial process can be pointed, low percentage overshoot and small settling time can be obtained by using this controller. This controller provides feedback, it has ability to eliminate steady state offsets through derivative action. The derivative action in the control loop will improve the damping and therefore by accelerating the transient response, a lighter proportional gain can be 12 Vineet Shekher et al obtained during the past half century, many theoretical and industrial studies have been done in PID controller setting rules Zeigler and Nichol’s in 1942 proposed a method to set the PID controller parameter Hagglund and Astrom in 1955 and chengching in 1999, introduced other technique. By generalizing the derivative and integer orders, from the integer field to non-integer numbers, the fractional order PID control is obtained. The performance of the PID controller can be improved by making the use of fractional order derivatives and integrals. This flexibility helps the design more robust system. Before using the fractional order controller in design an introduction to the fractional calculus is required. The first time, calculus generation to fractional, was proposed Leibniz and Hopital for the first time afterwords, the systematic studies in this field by many researchers such as Liouville (1832), Holmgren (1864) and Riemann (1953) were performed. Integer Order PID Controller The PID refers to the first letter of the names that make up the standard three term controller. These are P for the proportional term, I for the Integral term and D for the derivative term in the controller. Three term or PID controllers are probably used by most widely industrial controller. A PID controller is essentially a generic closed loop feedback mechanism. Figure 1: SISO unity feedback controller Controller monitors the error between a measured process variable and a desired set point; from this error, a corrective signal is computed and is eventually feedback to the input side to adjust the process accordingly. The differential equation for the PID controller is 0 ( ) ( ) ( ) ( ) (1 ) t p i p d d u t K e t T e t d t T e t d t d t = + + ∫ Thus, the PID controller algorithm is described by a weighted sum of the three time functions where the three distinct weights are: p K (Proportional gain) determines the influence of the present error value on the control mechanism, I (integral gain) decides the reaction based on the area under the error time curve up to Tuning and Analysis of Fractional Order PID Controller 13 the present point and Td (derivative gain) accounts for the extent of the reaction to the rate of change of the error with time. Tuning of Integer Order PID Controller Many tuning method are presented in literature that are based on a few structure of the process dynamics. Zeigler Nichols Process Reaction Curve Method In 1942, Zeigler and Nichols presented two classical methods to tune a PID controller. These methods are widely used, due to their simplicity. In the first method, the controller setting are based on two parameter θ and a of the process reaction curve. The proposed Zeigler Nichols setting is shown Table 1 The frequency domain method proposed by Zeigler and Nichols is based on the ultimate gain c K and the ultimate period u T . The controller setting is shown Table 2 14 Vineet Shekher et al Astrom and Haggland (1985) Astrom and Haggland recognized that the Zeigler-Nichols continuous cycling method actually identifies the point ( 1/ u K − , 0) on the Nyquist curve, and move it to a predefined point. With PID control, it is possible to move a given point on the Nyquist curve to an arbitrary position. By increasing the gain, the arbitrary point moves in the direction of ( ) G jω . By changing I and D action moves the point in the orthogonal direction. Brief Mathematical Background of Fractional calculus Orders of fractional calculus are real number. Many different definitions for general integrodifferentional operation can be found in the literature. Among them the most commonly used for general fractional Integrodifferential expressions are given by chauchy, Riemann-Liouville, Grunwald letnikov and Caputo. These definitions are required for realization of control algorithm. At first, we generalize the differential and Integral operators in to one fundamental operator a t D α where ( ) 0 1 ( ) 0