# An Optimal Fuzzy PID Controller Design Based on Conventional PID Control and Nonlinear Factors
## 1 Introduction  
The well-known proportional-integral-derivative (PID) controllers are still widely employed in industrial process control even though many kinds of control theories have been developed. The popularity of a PID controller can be attributed to its good performance and functional simplicity. The three-mode controller contains a proportional (P), an integral (I), and a derivative (D) term to make a system yield a desirable response in settling time, steady-state error, and overshoot. An engineer can efficiently tune the three gains through experience or some simple principles, such as the classical tuning rules proposed by [PERSON] [1]. Moreover, a simplified PI or PD controller is also popular for a multitude of practical applications. A fuzzy logic controller (FLC) is based on fuzzy rules and fuzzy inference. The fuzzy rules can reflect human experience or knowledge, and exhibit nonlinearity to control more complex plants, which can be linear or nonlinear. Like conventional PI or PD controllers, FLCs also have PI-type or PD-type controllers. Essentially, a FLC design includes the type of FLC, the number and shape of membership functions (MFs), and the fuzzy rules [2]. The genetic algorithm (GA) is employed to determine the optimal parameters of a system. Since the design techniques of conventional linear PID controllers have matured, it is advantageous to use the GA to optimize fuzzy PID controller design. Researchers have introduced an analytical design for an optimal fuzzy PID controller [3], which has a simple structure, but uses complicated procedures. Another optimal fuzzy PID controller combining a fuzzy PI controller with a fuzzy D controller was proposed [4], but this device is actually a conventional PID controller with an adaptive control capability with complicated analytic formulas. An optimal fuzzy PID controller can also be built by combining a fuzzy PI controller and a fuzzy PD controller in parallelism [5] with optimal tuning of scaling factors and MFs. The conventional PID controller can also be directly put in the optimal design for fuzzy controllers [6], where the PID control is the master controller, and the fuzzy control is the slave control to enhance the master one. The controller structure should be the primary consideration for a fuzzy PID controller design. As for the fuzzy control rules, in principle, they should follow conventional PID control. Then the problem of tuning the MFs in order to improve system performance must be solved [7]. The shape of MFs can be defined by chromosome bits and optimized by the GA [8; 9] to improve the system responses, such as speed and precision of control [9; 10]. On the other hand, each fuzzy variable MF is usually set to a symmetrical shape. The adjustment of a MF from symmetrical [11] to asymmetrical can also obtain improvements in system performance [12]. Moreover, some researchers use scaling factors to normalize the operating ranges and tune scaling factors to finish optimization [13]. In this paper, we choose to tune the operating ranges, which are important parameters for defining the equivalent FLC from a conventional PID controller. In recent decades, many evolutionary algorithms have been developed, such as the particle swarm algorithm (PSO), cuckoo search (CS), and so on. Evolutionary programming (EP) claims that the traditional GA will not only have a premature convergence, but may also be trapped in the local optima. A fuzzy PID controller design using a novel PSO-EP based hybrid algorithm has been found in [14]. Furthermore, it is shown that a FLC + EP based PID controller provides a more rapid response than a FLC + GA based PID controller [15]. In this paper, we will still apply GA, and give each optimized parameter its own crossover point in the GA process to enhance the GA's efficiency. As most optimal fuzzy PID controller designs with complex structures or a large number of tuning parameters, this study developed an optimal fuzzy PID controller with less parameters and a concise controller structure.