【正文】 nsation item as a process 13 disturbance. Although the scaling gains? and? are coupled, a procedure is used to decouple them on the basis of the sliding mode control. The stability analysis shows that the control system is globally asymptotically stable. Fuzzy PID controllers tuned by the proposed method are more robust than the conventional PID controller. The simulation results show that fuzzy PID controllers tuned by the proposed method achieve better control performance in both the transient and steady states and are more robust than conventional PID controllers. Literature Cited (1) Sugeno. M. Industrial Applications of Fuzzy Control。 approximation, the delay time is approximated as follows: s21s21e s LLL???? (11) Therefore, the P? (s) can be factorized as P? (s) ) P? +(s)P? (s),其中 ? ?? ? ?????? ???? s211ss~LTKP (12) We can achieve ? ? ? ?? ?st1 s211ssc??????? ??? KLTC IM C （ 13） The bandwidth of the fuzzy PID at the kth level can be controlled by adjusting R. A small value of R gives wide bandwidth and fast response。 Green, A. W. Fuzzy Variable Structure Control. IEEE Trans. Syst., Man, Cyberics, Part B 1997, 27 (2), 306–312. 7 Effective Tuning Method for Fuzzy PID with Internal Model Control XiaoGang Duan, HanXiong Li, and Hua Deng School of Mechanical and Electrical Engineering, Central South UniVersity, Changsha 410083, China, and Department of Manufacturing Engineering and Engineering Management, City UniVersity of Hong Kong, Hong Kong An internal model control (IMC) based tuning method is proposed to auto tune the fuzzy proportional integral derivative (PID) controller in this paper. An analytical model of the fuzzy PID controller is first derived, which consists of a linear PID controller and a nonlinear pensation item. The nonlinear pensation item can be considered as a process disturbance, and then parameters of the fuzzy PID controller can be analytically determined on the basis of the IMC structure. The stability of the fuzzy PID control system is analyzed using the Lyapunov stability theory. The simulation results demonstrate the effectiveness of the proposed tuning method. 1. Introduction Generally speaking, conventional proportional integral derivative (PID) controllers may not perform well for the plex process, such as the highorder and time delay systems. Under this plex environment, it is wellknown that the fuzzy controller can have a better performance due to its inherent robustness. Thus, over the past three decades, fuzzy controllers, especially, fuzzy PID controllers have been widely used for industrial processes due to their heuristic natures associated with simplicity and effectiveness for both linear and nonlinear There are too many variations of fuzzy PID controllers,such as, oneinput, twoinput, and threeinput PID type fuzzy controllers. In general, there is no standard benchmark. The oneinput may miss more information on the derivative action, and the threeinput fuzzy PID controllers may cause exponential growth of rules. The twoinput fuzzy PID, as we used in the paper, has a proper structure and the most practical use, and thus is the most popular type of fuzzy PID used in various research and application. Despite the fact that industry shows greater and greater interest in the applications of fuzzy PID, it is still a highly controversial topic from the point of view of the mainstream control engineering munity. One reason is that the fundamental theory for the analytical tuning methods of fuzzy PID is still missing. Therefore, fuzzy PID controllers had to be tuned qualitatively by twolevel tuning. At a lower level, the tuning is performed by adjusting the scaling gains to obtain overall linear control performance. At a higher level, the tuning is performed by changing the knowledge base parameters to enhance the control performance. However, it is difficult to tune the knowledge base parameters. Moreover, it is hard to improve the transient response by changing the member the knowledge base conveys a general control policy, it is preferred to keep 8 the member function unchanged and to leave the design and tuning exercises to scaling gains. However, the tuning mechanism of scaling factors and the stability analysis are still difficult tasks due to the plexity of the nonlinear control surface that is generated by fuzzy PID controllers. If the nonlinearity can be suitably utilized, fuzzy PID controllers may pose the potential to achieve better system performance than conventional PID controllers. Some nonanalytical tuning methods were Although the nonlinearity was considered on the basis of gain margin and phase margin specifications, the fuzzy PID controller may produce higher gains than conventional PID controllers due to the nonlinear factor. A high gain could deteriorate the stability of the control The conventional PID controller is easy to implement, and lots of tuning rules are available to cover a wide range of process specifications. Among tuning methods of the conventional PID controller, the internal model control (IMC) based tuning is one of the popular methods in mercial PID software packages because only one tuning parameter is required and better set point response can be An analytical tuning method based on IMC to tune fuzzy PID controllers is proposed in this paper. The fuzzy PID controller is first deposed as a linear PID controller plus an onlinear pensation item. When the nonlinear pensation item is approximated as a process disturbance, the fuzzy PID scaling parameters can then be analytically designed using the IMC scheme. The stability analysis of the f