# Analysis of FLC with changing fuzzy variables in frequency domain

• Kyoung-woong Lee
• Hansoo Choi
Technical Notes and Correspondence

## Abstract

This paper discusses a simple method for analyzing FLC in frequency domain based on describing function. Since nonlinear characteristics of FLC make it difficult FLC analysis, it usually requires a big deal of trial-and-error procedures based on computer simulation. The proposed method is simple and easy to understand, because it is based on the Nyquist stability criterion used to analyze absolute and relative stability, phase and gain margin of a linear system. To linearize in frequency domain, a describing function for FLC is derived by using a piecewise linearization of the FLC response plot. This describing function is represented as a function of magnitude of input sinusoid and nonlinear parameters x 1 and x 2 which change consequence fuzzy variables and nonlinearity of FLC. The describing function is redefined without the magnitude of sinusoid input because maximum values of the describing function can explain the stability of the system. This redefined describing function is used to get minimum stability characteristic, an absolute stability, phase margin and gain margin, of FLC. Using this function, we can explicitly figure out various characteristic of FLC according to x 1 and x 2 in frequency domain. In this work, we suggest a minimum phase margin (MPM) and a minimum gain margin (MGM) for FLC which can be used to determine whether the system is stable or not and how stable it is. For simplicity, we use one-input FLC with three rules. For various nonlinear response of FLC, changing fuzzy variables of a consequence membership function is used. Simulation results show that these parameters are effective in analyzing FLC.

## Keywords

Describing function fuzzy logic control nonlinear system stability

## References

1. [1]
C. W. de Silva, Intelligent Control, Fuzzy Logic Applications, CRC, Boca Ration, FL, 1995.
2. [2]
D. Driankov, Hellendoorn, and M. Reinfrank, An Introduction to Fuzzy Control, 2nd ed., Springer-Verlag, New York, 1996.
3. [3]
C. C. Lee, “Fuzzy logic in control systems: Fuzzy logic controller-part1 and part 2,” IEEE Trans. Syst., Man, Cybern., vol. 20, pp. 404–435, 1990.
4. [4]
W. Silver and H. Ying, “Fuzzy control theory: the linear case,” Fuzzy Sets Syst., vol. 33, pp. 275–290, 1989.
5. [5]
H. Ying, W. Silver, and J. J. Buckley, “Fuzzy control theory: a nonlinear case,” Automatica, vol. 26, pp. 513–520, 1990.
6. [6]
S. Murakami and M. Maeda, “Automobile speed control system using a fuzzy logic controller,” Industrial Applications of Fuzzy Control, Amsterdam, North-Holland, The Netherlands, pp. 105–124, 1985.Google Scholar
7. [7]
B. Hu, G. K. I. Mann, and R. G. Gosine, “New methodology for analytical and optimal design of fuzzy PID controllers,” IEEE Trans. on Fuzzy Syst., vol. 7, 5. pp. 521–539, 1999.
8. [8]
B. Hu, G. K. I. Mann, and R. G. Gosine, “A systematic study of fuzzy PID controllers-functionbased evaluation approach,” IEEE Trans. on Fuzzy Syst., vol. 9, no. 5. pp. 699–712, 2001.
9. [9]
B. Hu, G. K. I. Mann, and R. G. Gosine, “Two-level tunning of fuzzy PID controller,” IEEE Trans. Syst., Man, Cybern., vol. 31,2, pp. 263–269, 2001.
10. [10]
Š. Tomislav, S. Tešnjak, Sejid, Kuljača Ognjen, “Stability analysis of fuzzy control system using describing function method,” Proc. of the 9th Mediterranean Conference on Control and Automation, Dubrovnik, Croatia, 2001.Google Scholar
11. [11]
M. Sugeno, “On stability of fuzzy systems expressed by fuzzy rules with singleton consequents,” IEEE Trans. on Fuzzy Systems, vol. 7, pp. 201–224, 1999.
12. [12]
J. Aracil, A. Ollero, and A. Garcia-Cerezo “Stability indices for the global analysis of expert control systems,” IEEE Trans. on Systems, Man, and Cybernetics, vol. 19, pp. 998–1007, 1989.
13. [13]
R.-E. Precup, S. Doboli, and S. Preitl “Stability analysis and development of a class of fuzzy control systems,” Engineering Applications of Artificial Intelligence, vol. 13, pp. 237–247, 2000.
14. [14]
H.-P. Opitz, “Fuzzy control and stability criteria,” Proc. of First EUFIT’93 European Congress, Aachen, Germany, vol. 1, pp. 130–136, 1993.Google Scholar
15. [15]
R.-E. Precup and S. Preitl “Popov-type stability analysis method for fuzzy control systems,” Proc. of Fifth EUFIT’97 European Congress, Aachen, Germany, vol. 2, pp. 1306–1310, 1997.Google Scholar
16. [16]
K. M. Passino and S. Yurkovich, Fuzzy Control, Addison Wesley Longman, Inc., Menlo Park, CA, 1998.Google Scholar
17. [17]
D. Driankov, H. Hellendoorn, and M. Reinfrank, An Introduction to Fuzzy Control, Springer Verlag, Berlin, Heidelberg, New York, 1993.
18. [18]
H. Kiendl, “Harmonic balance for fuzzy control systems,” Proc. of First EUFIT’93 European Congress, Aachen, Germany, vol. 1, pp. 137–141, 1993.Google Scholar
19. [19]
H. Ying, “Analytical structure of a two-input twooutput fuzzy controller and its relation to PI and multilevel relay controllers,” Fuzzy Sets and Systems, vol. 63, pp. 21–33, 1994.
20. [20]
F. Gordillo, J. Aracil, and T. Alamo “Determining limit cycles in fuzzy control systems,” Proc. of FUZZ-IEEE’97 Conference, Barcelona, Spain, pp. 193–198, 1997.Google Scholar
21. [21]
G. Calcev, “Some remarks on the stability of Mamdani fuzzy control systems,” IEEE Trans. on Fuzzy Systems, vol. 6, pp. 436–442, 1998.
22. [22]
C.-T. Chao and C.-C. Teng “A PD-like self-tuning fuzzy controller without steady-state error,” Fuzzy Sets and Systems, vol. 87, pp. 141–154, 1997.