【正文】 Gatland, H. B.。 Kiam, H. A.。 Liu, C.。 Nikhil, R. P. A SelfTuning Fuzzy PI Controller. Fuzzy Sets Syst. 2021, 115, 327–338. (12) Yesil, E.。 Lin, J. J. A PID Type Fuzzy Controller with SelfTuning Scaling Factors. Fuzzy Sets Syst. 2021, 115, 321–326. (9) Vega, P.。 Gatland, H. Conventional Fuzzy Logic Control and Its Enhancement. IEEE Trans. Syst., Man, Cyberics 1996, 26 (10), 791–797. (7) Gee, K. I. M.。 Sheikhzadeh, M.。 Elsevier: Amsterdam, The Netherlands, 1985. (2) Manel, A.。 r and d are the set point and the disturbance, and y and yk are the outputs of the plant and its nominal model, respectively. The IMC structure is equivalent to the classical singleloop feedback controller shown in Figure 1b. If the singleloop controller CIMC is given by ? ? ? ?? ? ? ?ss1 ss ~PCCC IMC ?? （ 3） with ? ?? ? ? ?sfsp 1s ~_?C （ 4） where P? (s)=P? (s)P? +(s), P? (s) is the minimum phase part of the plant model, P? +(s) contains any time delays and righthalf plane zeros, and f(s) is a lowpass filter with a steadystate gain of one, which typically has the form: ? ? ? ?ncst11sf ?? （ 5） The tuning parameter tc is the desired closedloop time constant, and n is a positive integer to be determined. Figure Figure 3 FuzzyPID controller structure Model of Fuzzy PID Controller The fuzzy PID controller, as shown in Figure 2, is described as follows: ???????? ?? 10 p1u KKU PID (6) .eK.eK eK Ki Kd ? Rule Base s .e E R u e PIDU MICC P r + + e + + _ + + + u + + d + + y y 10 with ? ?u k 1 BBSA??? ? ? γ is a nonlinear time varying parameter( 132 ??? ), A and B are half of the spread of each input and out member function, respectively. The fuzzy PID control actually has two levels of The scaling gains (Ke, Kd, K0, and K1) are at the lower level. The tuning of these scaling gains will affect the gains of fuzzy PID The fuzzy PID control actually has two levels of The scaling gains (Ke, Kd, K0, and K1) are at the lower level. The tuning of these scaling gains will affect the gains of fuzzy PID controllers, resulting in the changing of the control performance. As the control actions are fuzzily coupled, the contribution of each Ke, Kd, K0, and K1 to different control actions is still not very clear, which makes the practical design and tuning process rather difficult. 3 Tuning Fuzzy PID Based on the IMC To tune the fuzzy PID controller based on the IMC method,an analytical model of the fuzzy PID controller is obtained first by simple derivation. Then, the parameters of the fuzzy PID controller can be determined on the basis of the IMC principle. Suppose that an industrial process can be modeled by a first order plus delay time (FOPDT) structure that has the transfer function as follows: ? ? LST KP ??? e1ss~ (7) where K, T, and L are the steadystate gain, the time constant, and the time delay, respectively. 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). 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。 Chun, D. W.。 Hang, C. C.。 Eksin, I. Self Tuning Fuzzy PID Type Load and Frequency Controller. Energy ConVers. Manage. 2021, 45, 377–390. (13) Xu, J. X.。 Aleixander, V. SelfTuning Predictive PID Controller. IEE Pro. D 1991, 138 (3), 303–311. (10) Rajani, K. M.。 Raymond, G. G. TwoLevel Tuning of Fuzzy PID Cotrollers. IEEE Trans. Syst., Man, Cyberics, Part B 2021, 31 (2), 263–269. (8) Woo, Z. W.。 Hu, B. G.。 Jordi, A.。仿 真結(jié)果表明，模糊PID 控制器 通過(guò)此種 整 定方法，與 傳統(tǒng)的 PID 控制器 相比在動(dòng)態(tài)和靜態(tài)上都 實(shí)現(xiàn)更好的控制性能 和更好的魯棒性。在 內(nèi)?？刂?方法 基礎(chǔ)上 ， 模糊 PID 控制器 的 參數(shù) 可由過(guò) 程干擾 的補(bǔ) 償 部分來(lái) 分析 確定。 此外 ， 由圖8可以看 出 模糊 PID 控制器增益低于常規(guī) PID控制器 。 圖 4 范例 1中 模糊 PID控制 （實(shí)線） 和常規(guī) 圖 5 延遲時(shí)間增加至 L= ， 模糊 PID控 PID控制 （虛線）性能比較 制 （實(shí)線） 和常規(guī) PID控制 （虛線） 性能比較 范例 2 假設(shè)一工業(yè)過(guò)成描述如下 ： ? ? ? ?8as 1s ??P （ 16） 其中 a=1，假設(shè)不存在 建模誤差 ， 在階躍響應(yīng)和奈奎斯特工業(yè)過(guò)程曲線基礎(chǔ)上 可獲得 逼近模型如下 ： ? ? ~ 1s ???P （ 17） 如圖 7所示， 常規(guī) PID控制和模糊 PID控制 差異不大。 小延遲時(shí)間意味著弱非線性特性。 由于滑?？刂剖且环N魯棒控制 所以 模糊 PID控制 是力的比傳統(tǒng)的 PID