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**Variable universe stable adaptive fuzzy control of a nonlinear system.**
*(English)*
Zbl 1045.93026

Summary: A kind of stable adaptive fuzzy control of a nonlinear system is implemented based on the variable universe method proposed first by H.-X. Li [“The mathematical essence of fuzzy controls and fine fuzzy controllers”, in: P. P. Wang (ed.), Advances in machine intelligence and soft computing. Vol. IV. Durham: Bookwrights Press, 55–74 (1997)]. First of all, the basic structure of variable universe adaptive fuzzy controllers is briefly introduced. Then the contraction-expansion factor, which is a key tool of the variable universe method, is defined by means of the integral regulation idea, and then a kind of adaptive fuzzy controller is designed by using such a contraction-expansion factor. The simulation on the first-order nonlinear system is done and, as a result, its simulation effect is quite good in comparison with the corresponding result in the literature. Second, it is proved that the variable universe adaptive fuzzy control is asymptotically stable by use of Lyapunov theory. The simulation on a second-order nonlinear system shows that its simulation effect is also quite good in comparison with the corresponding result of L.-X. Wang [“Stable adaptive fuzzy control of nonlinear systems”, IEEE Trans. Fuzzy Syst. 1, No. 2, 146–155 (1993)]. Besides, a useful tool, called symbolic factor, is proposed, which may be of universal significance. It can greatly reduce the setting time and enhance the robustness of the system.

### MSC:

93C42 | Fuzzy control/observation systems |

93C10 | Nonlinear systems in control theory |

93D20 | Asymptotic stability in control theory |

### Keywords:

Nonlinear systems; Variable universe adaptive fuzzy control; Contraction-expansion factor; Symbolic factor; Interpolation mechanism of fuzzy control
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\textit{H.-X. Li} et al., Comput. Math. Appl. 44, No. 5--6, 799--815 (2002; Zbl 1045.93026)

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### References:

[1] | Li, H.-X., (), 55-74 |

[2] | Wang, L.-X., Stable adaptive fuzzy control of nonlinear systems, IEEE transactions on fuzzy systems, 1, 2, 146-155, (1993) |

[3] | Tsay, D.-L.; Chung, H.-Y.; Lee, C.-J., The adaptive control of nonlinear systems using the sugeno-type of fuzzy logic, IEEE transactions on fuzzy systems, 7, 2, 225-229, (1999) |

[4] | Li, H.-X., To see the success of fuzzy logic from mathematical essence of fuzzy control (in Chinese), Fuzzy systems and mathematics, 9, 4, 1-14, (1995) |

[5] | Li, H.-X., Adaptive fuzzy controllers based on variable universe, Science in China, ser. E, 42, 1, 10-20, (1999) · Zbl 0935.93042 |

[6] | Li, H.-X.; Chen, C.L.P., The equivalence between fuzzy logic systems and feedforward neural networks, IEEE transactions on neural networks, 11, 2, 356-365, (2000) |

[7] | Slotine, J.E.; Li, W., (), 150-163 |

[8] | Li, H.-X.; Chen, C.L.P.; Huang, H.-P., Fuzzy neural intelligent systems, (2000), CRC Press Boca Raton, FL |

[9] | Li, H.-X.; Yen, V.C., Fuzzy sets and fuzzy decision-making, (1995), CRC Press Boca Raton, FL · Zbl 0864.90069 |

[10] | Wang, G., (), 1-22 |

[11] | Ogata, K., Modern control engineering, (1999), Prentice-Hall Englewood Cliffs, NJ |

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