# Equivalence between Fuzzy PID Controllers and Conventional PID Controllers
## 1 Introduction  
Proportional-Integral-Derivative (PID) controllers are widely used in industrial process control. The three-mode controller contains a proportional, an integral, and a derivative term. The popularity of a PID controller can be attributed to its good performance and functional simplicity, which allows engineers to operate it in a simple and straightforward manner. For example, the three controller gains can be chosen independently by an engineer, based on one's experience or through some simple selection methods such as the classical tuning rules proposed by [PERSON] [1]. For simplicity of the controller design, a PI or PD controller are also popular for practical applications. A PI controller can add damping to a system and reduce steady-state error, but yields penalized rise time and settling time. A PD controller also adds damping and reliably predicts and reduces large overshoots, but does not improve the steady-state error. Thus, for complete design considerations, a PID controller should be employed to obtain a desirable system response in settling time, steady-state error, and overshoot. On the other hand, since [PERSON] rediscovered and promoted fuzziness in 1965, the subsequent two fuzzy inference techniques proposed by [PERSON] [2] and [PERSON] [3] have inspired research in fuzzy logic controllers (FLC). The heuristic fuzzy rules, which reflect the experience of human experts, can be applied to plants that are difficult to model mathematically. The most common FLCs are PI-type or PD-type controllers [4; 5; 6], which possess the same characteristics as traditional PI or PD controllers, respectively. Moreover, they exhibit superior applicability compared with traditional PI or PD controllers [7]. The FLC commonly outperforms the corresponding PI, PD, or PID controller because a FLC is a nonlinear controller, while a PI, PD, or PID controller is linear. This raises the equivalence problem between a fuzzy PID controller and a conventional PID controller. It is well known that the design of fuzzy rules for a FLC requires expert knowledge, and those who are skilled in conventional PID controller design are then qualified as the experts. For a well-tuned conventional PID controller, design work is saved by replacing it with an equivalent linear FLC, then improving the performance over a conventional PID controller by slightly modifying the fuzzy rules. [PERSON] [8] revealed that when a PI controller is given, an FLC output is identical to that of the PI controller by using specified fuzzy logic operations. However, [PERSON]'s design is limited to PI controllers, and the design procedure is not clear enough. Several studies have investigated fuzzy PID controller structures, by taking different combinations of the fuzzy PID structural elements [9; 10; 11]. This involves a large number of parameters in defining the fuzzy rule base. [PERSON] et al. [12] presented a design for an equivalent fuzzy PID controller from the conventional PID controller, but the tuning procedure was too complicated and the resulting FLC was not purely linear according to the control surface view of the study. Therefore, the equivalence problem between different systems is very crucial to many research fields [13]. The objective of the study is to extend the significant results derived by [PERSON] [8] by examining the equivalence relationship and design procedure between a traditional PID controller and its corresponding equivalent FLC. This research proposes an equivalent fuzzy PID controller which has a simple PID structure design with a 3-dimensional fuzzy rule table, instead of the combination of different fuzzy PID structural elements or a hybrid controller structure [14]. Moreover, to achieve optimal control performance for a FLC, some artificial intelligent techniques such as Genetic Algorithm and Neural Network are efficient approaches [15; 16]. This inspires us in the future to propose nonlinear factors for tuning the membership functions to develop an optimal fuzzy PID controller design with less parameters.