Design of integer order PID controller for fractional order single pole system model

Fractional Calculus deals with cases in which there are non-integer order derivative and non-integer fold integral. Non-integer order and fold can be a rational number, an irrational number or even a complex number. For Grunwald-Letnikov definition of fractional order operator, general formula of nth order derivative is obtained and is generalized to n fold multiple integral under certain constraints. Similarly, general formula of n fold multiple integral can also be obtained and generalized to nth order derivative under certain constraints, which corresponds to Riemann-Liouville and Caputo definition. The difference between Riemann-Liouville and Caputo definitions is that the initial condition of Laplace transformation according to Caputo definition is more meaningful than that of obtained according to Riemann-Liouville definition because of the integer order derivative of initial conditions. These definitions are commonly used to define fractional order derivative and integral operators in the literature. The physical interpretation on fractional order integral is that ¡§time flowing equally¡¨ for integer order integral corresponds to ¡§time not flowing equally¡¨ for fractional order integral. The time not flowing equally is defined as cosmic time. The geometric interpretation is that three dimensional space is formed as time, cosmic time and function axes, the projection of resulting three dimensional curve on function-time plane shows integer order integral and also that on function-cosmic time plane indicates fractional order integral. The frequency responses of the fractional order derivative and integral are highly similar to those of the integer order derivative and integral. The amplitude curve of the first order derivative increases with the slope of 20 dB/decade and the phase curve of it is added 90¢X while the amplitude curve of fractional order (£\) derivative increases with the slope of 20£\ dB/decade and the phase curve of it is added 90£\¢X. The amplitude curve of fractional order (£\) integral decreases with the slope of 20£\ dB/decade and the phase curve of it is added -90£\¢X, which is quite similar to the frequency response of one fold integral. The mostly used approximation for fractional order operator to be realized is Oustaloup approximation in the literature. In this approximation, an integer high order transfer function is considered with a frequency response similar to the frequency response of the fractional order operator within a frequency interval obtained by choosing lower and upper bound frequencies. The order of this approximation is mostly chosen as 11 since the frequency response of the high order transfer function in the chosen frequency interval fits highly to that of the fractional order operator. The applications in engineering areas of fractional calculus are a quite new topic although fractional calculus is a continuing topic for more than 300 years. The first application of the fractional calculus was made by Abel in 1823 for the solution of the integral equation for teutochrone problem. Later in the nineteenth century, Heaveside showed that fractional calculus can solve certain problems in electromagnetic theory. The application of fractional calculus to control areas is a more current issue and there are numerous applications. The fractional calculus is used for modeling and control of thermal system, liquid level system, control of autonomous underwater vehicle, control of velocity of servo system etc. The concept of fractional calculus in control engineering is first used by Manabe in 1961. He obtained the transient state response and frequency response of non-integer integral and used these concepts for control systems. In automatic control, Oustaloup used the fractional control algorithm to control the dynamic systems and he demonstrated that CRONE (Commande Robuste d¡¦Ordre Non Entier) method outperforms integer order PID controllers. In 1999, Podlubny proposed the generalization of integer order PID controllers called fractional order PID controllers (PIƒÜDƒİ). He also showed that these controllers have better performance than integer order PID controllers for fractional order dynamic control systems. FOPID controllers provide more flexibility in controller design compared to classical IOPID controllers because of having five parameters instead of three. However, these two more parameters cause more complexity in tuning procedure of the controller. Fractional order system models and controllers are generalizations of integer order system models and controllers in control engineering. There are four configurations in control area for controllers and systems. These configurations are integer order controllers (IOCs) for integer order systems (IOSs), FOCs for integer order systems (IOSs), integer order controllers (IOCs) for fractional order systems (FOSs), and fractional order controllers (FOCs) for fractional order systems (FOSs). There are various controller tuning procedures for fractional order controllers in the literature. These procedures are mostly the generalization of the procedures presented for integer order controllers. The most widely used fractional controllers are fractional order PID controllers. Fractional order PID controller tuning methods are divided into three parts: analytical methods, numerical methods and rule based methods. In analytical methods, there are fractional order PI controller design methods according to three performance measures such as phase margin, gain crossover frequency and flat phase and fractional order PID controller design method based on internal model control. In numerical methods, there are fractional order PID controller design methods according to five performance measures such as phase margin, gain crossover frequency, flat phase, good output disturbance rejection and high frequency noise rejection. In rule based methods, fractional order controller is obtained by numerical methods and each parameter of the controller is fitted to a function in terms of system model parameters. In addition, nonlinear fractional order PID controllers can also be seen in the literature. In this type of controller design method, system models are obtained from various operating points for nonlinear system by the help of numerical methods and the corresponding controllers are blended in terms of the position of the output. Another method is nonlinerization of error signal. In these researches, the most widely studied system is fractional order single pole model. In this thesis, we dealt with the fractional order single pole system model which is the simplest and the most widely studied fractional order model in literature. This simple fractional order structure performs a better modelling performance than integer order models for modelling many nonlinear systems. Firstly, we observed the differences of integer order representations of the fractional order single pole model using different order Oustaloup expansions for the fractional operator. Based on this review, we proposed different integer order inverse controller structures for the control of fractional order system and examined the impact of the controller order on various performance measures. Integer high order inverse controllers are expected to be more satisfying than low order inverse controllers according to the defined performance criteria. Here, we observed that low order inverse controllers that depend on low order expansions can also compensate the effects of numerous poles and zeros that are thought to be in the structure of fractional order systems. As a result, extremely simple integer order controller structures are obtained. Moreover, we determined the parameters of these controllers in terms of the fractional system model parameters and frequency interval that is chosen for integer order expansion of the fractional order operator of the system model. It is a well-known fact that integer order PID controllers are commonly used in industrial applications. In the next phase, we proposed integer order PID controllers by neglecting the non-dominant zeros and poles of the integer low order inverse controller. The parameters of the proposed PID controller are also expressed in terms of the fractional order system parameters and the frequency interval used for the integer order expansion of the fractional order operator. We considered the gain of the controller as the only design parameter. Finally, we compared the proposed controllers with integer order and fractional order PID controllers the parameters of which are determined optimally using a search algorithm according to various performance measures. As a result we observed that the controllers we proposed perform equally to or better than the controllers compared. This proposed integer order PID controller design has been implemented over two tank liquid level system. For this purpose, a fractional order single pole system model is fitted after the step response data are obtained from a certain operation area. Moreover, The gain is added to the controller for the design to make more flexibility. The design parameter is found by means of Big Bang-Big Crunch search algorithm based on performans criteria which is sum of integral square error and total variation of control signal. Satisfactory results are obtained.