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Soft Computing

, Volume 17, Issue 9, pp 1553–1561 | Cite as

Limit-cycle analysis of dynamic fuzzy control systems

  • Jau-Woei PerngEmail author
Methodologies and Application

Abstract

The main purpose of this study is to predict limit cycles of a dynamic fuzzy control system by combining a stability equation, describing function and parameter plane. The stability of a linearized dynamic fuzzy control system is then analyzed using stability equations and the parameter plane method, with the assistance of a describing function method. This procedure identifies the amplitude and frequency of limit cycles that are clearly formed by the dynamic fuzzy controller in the parameter plane. Moreover, the suppression of the limit cycle by adjusting control parameters is proposed. Continuous and sampled-data systems are addressed, and the proposed approach can easily be extended to a fuzzy control system with multiple nonlinearities. Simulations are performed to demonstrate the effectiveness of the proposed scheme.

Keywords

Fuzzy control Limit cycle Describing function Parameter plane 

References

  1. Abdelnour G, Cheung TY, Chang CH, Tinetti G (1993) Steady-state analysis of a three-term fuzzy controller. IEEE Trans Syst Man Cybern 23:607–610zbMATHCrossRefGoogle Scholar
  2. Ackermann J, Bunte T (1997) Actuator rate limits in robust car steering control. In: Proceedings of the IEEE conference on decision and control, pp 4726–4731Google Scholar
  3. Chang CH, Chang MK (1994) Analysis of gain margins and phase margins of a nonlinear reactor control system. IEEE Trans Nucl Sci 41:1686–1691CrossRefGoogle Scholar
  4. Chang W, Park JB, Joo YH, Chen G (2003) Static output-feedback fuzzy controller for Chen’s chaotic system with uncertainties. Inf Sci 151:227–244MathSciNetzbMATHCrossRefGoogle Scholar
  5. Chen YL, Han KW (1970) Stability analysis of a nonlinear reactor control system. IEEE Trans Nucl Sci NS-17:18–25CrossRefGoogle Scholar
  6. Cheng CC, Huang CH (1998) On the limit cycle of the underwater vehicle control system. In: Proceedings of the international symposium on underwater technology, pp 461–465Google Scholar
  7. Chu YC, Dowling AP, Glover K (1998) Robust control of combustion oscillations. In: Proceedings of IEEE international conference on control applications, pp 1165–1169Google Scholar
  8. Cox CS, French IG (1986) Limit cycle prediction conditions for hydraulic control system. ASME J Dyn Syst Meas Control 108:17–23zbMATHCrossRefGoogle Scholar
  9. D’Amico MB, Moiola JL, Paolini EE (2002) Hopf bifurcation for maps: a frequency-domain approach. IEEE Trans Circuits Syst I 49:281–288MathSciNetCrossRefGoogle Scholar
  10. Genesio R, Tesi A (1988) On limit cycles in feedback polynomial systems. IEEE Trans Circuits Syst 35:1523–1528MathSciNetzbMATHCrossRefGoogle Scholar
  11. Genesio R, Tesi A, Villoresi F (1993) A frequency approach for analyzing and controlling chaos in nonlinear circuits. IEEE Trans Circuits Syst II 40:819–828zbMATHCrossRefGoogle Scholar
  12. Gordillo F, Aracil J, Alamo T (1993) Determining limit cycles in fuzzy control systems. In: Proceedings of IEEE international conference on fuzzy systems, pp 193–198Google Scholar
  13. Han KW (1977) Nonlinear control systems—some practical methods. Academic Cultural Company, CaliforniazbMATHGoogle Scholar
  14. Han KW (1979) Digital and sampled-data control systems. Kuang-Mei Publishing Co., LungtanGoogle Scholar
  15. Han KW, Thaler GJ (1966) Control system analysis and design using a parameter space method. IEEE Trans Autom Control 11:560–563CrossRefGoogle Scholar
  16. Hauksdottir AS, Sigurdaraottir G (1993) On the use of robust design methods in vehicle longitudinal controller design. ASME J Dyn Syst Meas Control 115:166–172CrossRefGoogle Scholar
  17. Itovich GR, Moiola JL (2006) On period doubling bifurcations of cycles and the harmonic balance method. Chaos Solitons Fractals 27:647–665MathSciNetzbMATHCrossRefGoogle Scholar
  18. Kim E, Lee H, Park M (2000) Limit-cycle prediction of a fuzzy control system based on describing function method. IEEE Trans Fuzzy Syst 8:11–21CrossRefGoogle Scholar
  19. Li J, Wang HO, Niemann D, Tanaka K (2000) Dynamic parallel distributed compensation for Takagi–Sugeno fuzzy systems: an LMI approach. Inf Sci 123:201–221MathSciNetzbMATHCrossRefGoogle Scholar
  20. Marco MD, Forti M, Tesi A (2002) Existence and characterization of limit cycles in nearly symmetric neural networks. IEEE Trans Circuits Syst I 49:979–992CrossRefGoogle Scholar
  21. Nagi F, Ahmed SK, Zainul Abidin AA, Nordin FH (2010) Fuzzy bang–bang relay controller for satellite attitude control system. Fuzzy Sets Syst 161:2104–2125MathSciNetzbMATHCrossRefGoogle Scholar
  22. Nguang SK (2006) Robust H-infinity output feedback control design for fuzzy dynamic systems with quadratic D stability constraints: an LMI approach. Inf Sci 176:2161–2191MathSciNetzbMATHCrossRefGoogle Scholar
  23. Olsson H, Astrom KJ (2001) Friction generated limit cycles, IEEE Trans. Control Syst Technol 9:629–636CrossRefGoogle Scholar
  24. Padin MS, Robbio FI, Moiola JL, Chen G (2005) On limit cycle approximations in the van der Pol oscillator. Chaos Solitons Fractals 23:207–220zbMATHCrossRefGoogle Scholar
  25. Perng JW (2012) Describing function analysis of uncertain fuzzy vehicle control systems. Neural Comput Appl 21:555–563CrossRefGoogle Scholar
  26. Precup RE, David RC, Petriu EM, Preitl S, Paul AS (2011) Gravitational search algorithm based tuning of fuzzy control systems with a reduced parametric sensitivity. Soft Computing in Industrial Applications, Springer, Berlin, pp 141–150Google Scholar
  27. Sadeghi-Tehran P, Cara AB, Angelov P, Pomares H, Rojas I, Prieto A (2012) Self-evolving parameter-free rule-based controller. In: Proceedings of IEEE conference on fuzzy systems, pp 754–761Google Scholar
  28. Shenton AT (1999) Parameter space design of PID limit cycle controllers. In: Proceedings of American control conference, pp 3342–3346Google Scholar
  29. Siljak DD (1964) Analysis and synthesis of feedback control systems in the parameter plane. IEEE Trans Ind Appl 83:466–473CrossRefGoogle Scholar
  30. Siljak DD (1969) Nonlinear systems—the parameter analysis and design. Wiley, New YorkzbMATHGoogle Scholar
  31. Siljak DD (1989) Parameter space methods for robust control design: a guide tour. IEEE Trans Autom Control 34:674–688MathSciNetzbMATHCrossRefGoogle Scholar
  32. Tanaka K, Manabu S (1993) Fuzzy stability criterion of a class of nonlinear systems. Inf Sci 71:3–26zbMATHCrossRefGoogle Scholar
  33. Tao CW, Taur JS, Chang CW, Chang YH (2012) Simplified type-2 fuzzy sliding controller for wing rock system. Fuzzy Sets Syst 207:111–129MathSciNetzbMATHCrossRefGoogle Scholar
  34. Ting CS (2006) Stability analysis and design of Takagi–Sugeno fuzzy systems. Inf Sci 176:2817–2845MathSciNetzbMATHCrossRefGoogle Scholar
  35. Xing J, Shi Z, Dai G (1996) Using describing function to analyze wriggling phenomenon of fuzzy control systems. In: Proceedings of IEEE conference on fuzzy systems, pp 1198–1204Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Mechanical and Electro-Mechanical EngineeringNational Sun Yat-sen UniversityKaohsiungTaiwan, ROC

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