Structural analysis of fuzzy controllers with nonlinear input fuzzy sets in relation to nonlinear PID control with variable gains. (English) Zbl 1055.93524

Summary: The popular linear PID controller is mostly effective for linear or nearly linear control problems. Nonlinear PID controllers, however, are needed in order to satisfactorily control (highly) nonlinear plants, time-varying plants, or plants with significant time delay. This paper extends our previous papers in which we show rigorously that some fuzzy controllers are actually nonlinear PI, PD, and PID controllers with variable gains that can outperform their linear counterparts. In the present paper, we study the analytical structure of an important class of two- and three-dimensional fuzzy controllers. We link the entire class, as opposed to one controller at a time, to nonlinear PI, PD, and PID controllers with variable gains by establishing the conditions for the former to structurally become the latter. Unlike the results in the literature, which are exclusively for the fuzzy controllers using linear fuzzy sets for the input variables, this class of fuzzy controllers employs nonlinear input fuzzy sets of arbitrary types. Our structural results are thus more general and contain the existing ones as special cases. Two concrete examples are provided to illustrate the usefulness of the new results.


93C42 Fuzzy control/observation systems
93B51 Design techniques (robust design, computer-aided design, etc.)
Full Text: DOI


[2] Åström, K. J.; Hägglund, T., The future of PID control, Control Engineering Practice, 9, 1163-1175 (2001)
[3] Bennett, S., The past of PID controllers, Annual Reviews in Control, 25, 43-53 (2001)
[4] Bohn, C.; Atherton, D. P., An analysis package comparing PID anti-windup strategies, IEEE Control Systems Magazine, 15, 34-40 (1995)
[5] Chen, C.-L; Chang, F.-Y, Design and analysis of neural/fuzzy variable structural PID control systems, IEE Proceedings—Control Theory and Applications, 143, 200-208 (1996) · Zbl 0872.93032
[6] Chen, C.-L; Wang, S.-N; Hsieh, C.-T; Chang, F.-Y, Theoretical analysis of crisp-type fuzzy logic controllers using various t-norm sum-gravity inference methods, IEEE Transactions on Fuzzy Systems, 6, 122-136 (1998)
[8] Hajjaji, A. E.; Rachid, A., Explicit formulas for fuzzy controller, Fuzzy Sets and Systems, 62, 135-141 (1994)
[11] Lewis, F. L.; Liu, K., Towards a paradigm for fuzzy logic control, Automatica, 32, 167-181 (1996) · Zbl 0845.93048
[12] Li, H. X.; Gatland, H. B., A new methodology for designing a fuzzy logic controller, IEEE Transactions on Systems, Man and Cybernetics, 25, 505-512 (1995)
[13] Mann, G. K.I; Hu, B.-G; Gosine, R. G., Analysis of direct action fuzzy PID controller structures, IEEE Transactions on Systems, Man and Cybernetics (Part B), 29, 371-388 (1999)
[17] Sheppard, L. C., Computer control of the infusion of vasoactive drugs, Annual of Biomedical Engineering, 8, 431-444 (1980)
[19] Wong, C. C.; Chou, C. H.; Mon, D. L., Studies on the output of fuzzy controller with multiple inputs, Fuzzy Sets and Systems, 57, 149-158 (1993) · Zbl 0792.93082
[20] Yager, R. R.; Filev, D. P., Essentials of fuzzy modeling and control (1994), Wiley: Wiley New York
[22] Ying, H., The simplest fuzzy controllers using different inference methods are different nonlinear proportional-integral controllers with variable gains, Automatica, 29, 1579-1589 (1993) · Zbl 0790.93089
[23] Ying, H., General analytical structure of typical fuzzy controllers and their limiting structure theorems, Automatica, 29, 1139-1143 (1993) · Zbl 0782.93062
[24] Ying, H., Practical design of nonlinear fuzzy controllers with stability analysis for regulating processes with unknown mathematical models, Automatica, 30, 1185-1195 (1994) · Zbl 0800.93711
[25] Ying, H., Constructing nonlinear variable gain controllers via the Takagi-Sugeno fuzzy control, IEEE Transactions on Fuzzy Systems, 6, 226-234 (1998)
[26] Ying, H., The Takagi-Sugeno fuzzy controllers using the simplified linear control rules are nonlinear variable gain controllers, Automatica, 34, 157-167 (1998) · Zbl 0989.93053
[27] Ying, H., Theory and application of a novel Takagi-Sugeno fuzzy PID controller, Information Sciences, 123, 281-292 (2000)
[28] Ying, H., Fuzzy control and modeling: Analytical foundations and applications (2000), IEEE Press: IEEE Press New York
[29] Ying, H., A general technique for deriving analytical structure of fuzzy controllers that use arbitrary trapezoidal/triangular input fuzzy sets and Zadeh fuzzy logic AND operator, Automatica, 39, 1171-1184 (2003) · Zbl 1058.93036
[30] Ying, H.; McEachern, M.; Eddleman, D.; Sheppard, L. C., Fuzzy control of mean arterial pressure in postsurgical patients with sodium nitroprusside infusion, IEEE Transactions on Biomedical Engineering, 39, 1060-1070 (1992)
[31] Ying, H.; Siler, W.; Buckley, J. J., Fuzzy control theory: A nonlinear case, Automatica, 26, 513-520 (1990) · Zbl 0713.93036
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