【正文】 spectively. The estimation of these parameters using the step response method, frequency response, and closedloop relay feedback, etc., is welldescribed. The FOPDT model is one of the most mon and adequate ones used, especially in the process control One obtains from(6): ? ? ???????? ??? 10 1 KpKsABU P ID ? （ 8） ? ? ? ? ? ?sususU NP IDP ID ?? （ 9） ? ? ? ?ssABKsu N ?? ?????? ?? 10 （ 10） with δ(s) being a nonlinear term without an explicit analytical expression. Obviously, the fuzzy PID control can be considered as a conventional PID with a nonlinear pensation. The conventional PID control term is uPID(s) and the nonlinear pensation is uN(s). 11 Tuning of Fuzzy PID Controller Based on IMC. If we consider the nonlinear pensation uN as a process disturbance and set Gf(s) )=CIMC(s), which is shown in Figure 3, the IMCbased tuning for fuzzy PID controllers can be simplified as follows. By the firstorder Pade180。 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。 otherwise, it gives a low bandwidthand sluggish response. To improve the rise time, the value of R should be small. Therefore, the two parameters ? and? can be determined. Remark: The fuzzyPID control (11) is actually a conventional PID control uPID plus a pseudosliding mode control δ. Because the sliding mode control is a robust control, the fuzzy PID control is more robust than a conventional PID control. 4 Control Simulations In this section, the control performance of fuzzy PID tuned by the proposed method is pared with that of conventional PID control. Quantitative criteria for measuring the performance are chosen as IAE and ITAE. Smaller numbers imply better performance. dtetdte ?? ?? IT AEIE A ， (14) In all control simulations, parameters of conventional PID control are determined by IMCbased method and the parameters of fuzzy PID control are determined by the proposed tuning method. Example an industrial process that is approximately described by a firstorder rational transfer function model with a delay time as follows: ? ? 1s ???P (15) The linear part is the dominant process. The small delay time implies weak nonlinear features. As shown in Figure 5, little difference is observed between the conventional PID control and fuzzy PID control due to the small delay time. However, when the delay time is increased to L=, there will be large model error caused by approximating the delay time with a firstorder Pade180。 approximation in (15). As shown in Figure 6, fuzzy PID control achieves better control performance than conventional PID control. Morever, the gain of the fuzzy 12 PID controller is lower than that of conventional PID controller. Figure 4 Control performance of fuzzy PID Figure 5 Performance of fuzzy PID and PID and PID for example 1, fuzzy PID (solid line), for delay L= , fuzzy PID (solid line), and conventional PID (dotted line). and conventional PID (dotted line) . Example 2. Assume that an industrial process is described by ? ? ? ?8as 1s ??P (16) where a=1, Suppose that there is no modeling error in the process . On the basis of step response and Nyquist curves of the industrial process , the approximation model can be obtained as follows: ? ? ~ 1s ???P (17) As shown in Figure 7, little difference isobserved between the conventional PID control and fuzzy PIDcontrol because the model is , suppose that there is modeling error and the practical value of the parameter a is .. As shown in Figure 8, fuzzy PID control achieves better control performance than conventional PID control. Morever, the gain of the fuzzy PID controller is lower than that of the conventional PID controller, which is shown in Figure 8. Figure 6. Control performance of fuzzy PID Figure 7. Control performance of fuzzy PID and PID and PID for a ) 1. Fuzzy PID (solid line) and for process a = PID conventional PID (dotted line). (solid line) and conventional PID (dotted line) 5 Conclusion An effective tuning method for fuzzy PID controllers based on IMC is presented in this paper. An analytical model is first developed for the tuning of fuzzy PID controllers. The analytical model includes a linear PID control and a nonlinear pensation item. On the basis of the IMC method, the parameters of fuzzy PID controller can be analytically determined by regarding the pensation 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。 Elsevier: Amsterdam, The Netherlands, 1985. (2) Ma