Multi-objective optimization based fractional order PID controller design

Fractional calculus is related to situations where the order of the derivative and the integral are non-integer numbers. Even though the concepts of fractional order derivative and integral have existed for more than 300 years in the mathematical literature, the first meeting on this topic was held in New Haven, Connecticut, in 1974 [10]. At that time, this field only caught the attention of a few mathematicians and theoretical physicists. Nevertheless, the interest in fractional calculus grew considerably since then and currently there are countless studies dedicated to fractional calculus. This major growth in fractional calculus is mostly due to the applications in engineering fields, especially in control engineering. Control is an interdisciplinary field of engineering and mathematics that usually deals with controlling dynamic systems in order to achieve a certain behavior based on a set of specifications. To be able to properly modify a dynamic system, a mathematical model of the system is needed and the most common way to model dynamic systems (in linear systems) is by using integrals and derivatives. Therefore, extending the definitions of integrals and derivatives by using fractional calculus can improve their applicability. Proportional-integral-derivative (PID) controllers are without doubt the most popular controllers that are used in the industry due to their various benefits and the countless different tuning methods available. Recently, the use of fractional order PID (FOPID) controllers have started to increase as well. FOPID controllers have five free parameters (K_p,K_i,λ,K_d,μ) to select whereas PID controllers only have three (K_p,K_i,K_d), therefore this situation causes FOPID controllers to be much more complex and difficult to tune compared to PID controllers. On the other hand, due to the increased number of free parameters, FOPID controllers can be more flexible during the design process which suggests that by using FOPID controllers, it is more likely to obtain better results. A tuning rule for PID controllers was proposed in [15] by considering a trade-off between servo and regulation modes while constraining the maximum sensitivity value. First, a multi-objective optimization (MOO) problem is created in which the trade-off between servo and regulation modes is considered. The aim is to minimize the integrated square error (ISE) for both modes while keeping the maximum sensitivity value between 1.4 and 2.0. A MOO algorithm is used to generate a set of optimal solutions which is called the Pareto set. From this Pareto set, the non-dominated solutions are selected to create the Pareto front, which is a trade-off curve where no solution is better than the others. From this Pareto front, the Nash solution is selected to determine the tuning rule. The Nash solution is subtracted from the worst solutions for both objective functions and the multiplication of these results are maximized. Later, simulation examples are given in order to prove the effectiveness of the method presented. The aim of this study is to design and compare the results of a PID controller and a fractional order PID controller for different cases. In the first case, the control problem is stated as minimizing the ISE value for both the set-point following and the disturbance rejection responses. The first case is similar to the problem stated in [15], except for in this study the maximum sensitivity value is not considered and FOPID controllers are designed in addition. In the second case, the control problem is stated as minimizing the integrated square error (ITSE) value for both the set-point following and the disturbance rejection responses. Lastly, in the third case, the control problem is stated as minimizing the weighted sum of the rise time and the settling time values and the overshoot precentage. Since all of these objective functions are in conflict, the problems are defined as multi-objective optimization problems and solved accordingly by using genetic algorithm to obtain the coefficients of the integer order and fractional order PID controllers. As a result, a set of Pareto optimal solutions is obtained and among these solutions the Nash bargaining solution is selected in order to make comparisons. For the first two cases, a first order plus dead time system model is used due to its simplicity and ability to represent higher order systems. For the third case, a second order plus dead time system model is used in order to minimize the overshoot percentage. Satisfactory results were obtained.