# zbMATH — the first resource for mathematics

Fuzzy controller design based on stability criteria. (English) Zbl 1075.93522
The paper deals with stability of fuzzy controllers, namely Sugeno-Takagi ones. The latter are considered to be in the standardized form $f_i(y) = \frac {\sum _{\alpha }u_{\alpha }^{(i)}(y)\cdot a_{\alpha }^t(y)} {\sum _{\alpha } a_{\alpha }^t(y)}$ for each $$i$$-th output from the $$i$$-th rule (cf. the book [D. Driankov, H. Hellendoorn and M. Reinfrank, An introduction to fuzzy control, Springer-Verlag, Berlin (1993; Zbl 0789.93088)]).
The author finds a linearization formula for the above controller provided that specific (and quite common) assumptions for the membership functions are fulfilled. On the basis of this he proves existence of stability intervals for certain parameters of the fuzzy controller and accompanies the theory by a numerical example.
##### MSC:
 93C42 Fuzzy control/observation systems 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory 93C15 Control/observation systems governed by ordinary differential equations
Full Text:
##### References:
 [1] Bandemer H., Gottwald S. (1993). : Einführung in Fuzzy-Methoden. 4. edn, Akademie Verlag, Berlin. · Zbl 0771.94018 [2] Bretthauer G., Opitz H.-P. (1994). : Stability of fuzzy systems - a survey. Proc. of EUFIT ’94 Second European Congress on Intelligent Techniques & Soft Computing, Aachen, pp. 283-290. [3] Delgado-Romero J., Rojas-Estrada J. (1995). : New simple tests for Hurwitz and Schur stability of interval matrices. Proc. European Control Conference 95 pp. 3009-3014. [4] Driankov D., Hellendorn H., Reinfrank R. (1993). : An introduction to fuzzy control. Springer-Verlag, Berlin. [5] Hahn W. (1967). : Stability of motion. Springer-Verlag, New York. · Zbl 0189.38503 [6] Kiendl H. (1997). : Fuzzy Control methodenorientiert. Oldenbourg Verlag, München. [7] Mansour M. (1989). : Simplified sufficient conditions for the asymptotic stability of interval matrices. Int. J. Control 50: 443-444. · Zbl 0683.93064 · doi:10.1080/00207178908953374 [8] Möllers T. (1997). : On the linearization of a class of multivariable Sugeno fuzzy controllers. Proc. of EUFIT ’97 5th European Congress on Intelligent Techniques & Soft Computing, Verlag Mainz, Aachen, pp. 1433-1437. [9] Möllers T. (1998). : Stabilitat von Fuzzy-Reglern: Analysekriterien und explizite Syntheseverfahren. Shaker-Verlag, Aachen. [10] Möllers T., van Laak O. (1998). : On the global approximation and interpolation of locally described real valued functions. Applied Mathematics and Computation 93: 1-10. · Zbl 0943.65016 · doi:10.1016/S0096-3003(97)10052-2 [11] Rohn J. (1994). : Positive definitness and stability of interval matrices. SIAM J. Matrix Anal. Appl. 14: 175-184. · Zbl 0796.65065 · doi:10.1137/S0895479891219216 [12] Verbruggen H., Bruijn P. (1997). : Fuzzy Control and conventional control: What is (and can be) the real contribution of Fuzzy Systems?. Fuzzy Sets Syst. 90: 151-160. · Zbl 0924.93019 · doi:10.1016/S0165-0114(97)00081-X [13] Vidyasagar M. (1993). : Nonlinear system analysis. 2. edn, Prentice Hall, Englewood Cliffs. [14] Wang K., Michel A. (1993). : On sufficient conditions for the stability of interval matrices. Systems & Control Letters 20: 345-351. · Zbl 0772.93073 · doi:10.1016/0167-6911(93)90012-U [15] Zabczyk J. (1992). : Mathematical control theory: An introduction. 2. edn, Birkhauser, Boston. · Zbl 1123.93003 [16] Zadeh L. A. (1965). : Fuzzy Sets. Information and Control 8: 338-353. · Zbl 0942.00007 · doi:10.1016/S0019-9958(65)90241-X
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.