##
**Robust control of uncertain systems with polynomial nonlinearity by output feedback.**
*(English)*
Zbl 1169.93322

Summary: The problem of global robust stabilization by output feedback is investigated for two classes of uncertain systems with polynomial nonlinearity – one is with controllable/observable linearization and the other is not. The uncertainties in the systems are assumed to be dominated by both lower- and higher-order nonlinearities multiplying by an output-dependent growth rate. There are two ingredients in this study. One is to exploit the idea of how to handle polynomial growth conditions via homogeneity and domination without introducing an observer gain updated law. The other is the development of a recursive design algorithm for the construction of reduced-order observers, which is not only interesting in its own right but also has a valid counterpart, capable of dealing with strongly nonlinear systems, even lack of uniform observability and smooth stabilizability.

### MSC:

93B35 | Sensitivity (robustness) |

93C10 | Nonlinear systems in control theory |

93C41 | Control/observation systems with incomplete information |

93D21 | Adaptive or robust stabilization |

PDF
BibTeX
XML
Cite

\textit{H. Lei} and \textit{W. Lin}, Int. J. Robust Nonlinear Control 19, No. 6, 692--723 (2009; Zbl 1169.93322)

Full Text:
DOI

### References:

[1] | Zubov, Mathematical Methods for the Study of Automatic Control Systems (1964) |

[2] | Kawski M. Geometric homogeneity and applications to stabilization. Proceedings of the 3rd IFAC NOLCOS, CA, 1995; 164-169. |

[3] | Bacciotti, Lyapunov Functions and Stability in Control Theory (2001) |

[4] | Kawski, Homogeneous stabilizing feedback laws, Control Theory and Advanced Technology 6 pp 497– (1990) |

[5] | Hermes, Nilpotent & high-order approximations of vector field systems, SIAM Review 33 pp 238– (1991) · Zbl 0733.93062 |

[6] | Tzamtzi, Explicit formulas of feedback stabilizers for a class of triangular systems with uncontrollable linearization, Systems and Control Letters 38 pp 115– (1999) · Zbl 1043.93548 |

[7] | Qian, A continuous feedback approach to global strong stabilization of nonlinear systems, IEEE Transactions on Automatic Control 47 pp 1061– (2001) · Zbl 1012.93053 |

[8] | Hermes, Differential Equation Stability and Control 109 pp 249– (1991) |

[9] | Rosier, Homogeneous Lyapunov function for homogeneous continuous vector field, Systems and Control Letters 19 pp 467– (1992) · Zbl 0762.34032 |

[10] | Dayawansa, Asymptotic stabilization of a class of smooth two dimensional systems, SIAM Journal on Control 28 pp 1321– (1990) · Zbl 0731.93076 |

[11] | Praly, Linear output feedback with dynamic high gain for nonlinear systems, Systems and Control Letters 53 pp 107– (2004) · Zbl 1157.93494 |

[12] | Qian, Output feedback control of a class of nonlinear systems: a non-separation principle diagram, IEEE Transactions on Automatic Control 47 pp 1710– (2002) |

[13] | Qian C. A homogeneous domination approach for global output feedback stabilization of a class of nonlinear systems. Proceedings of the 2005 American Control Conference, Portland, OR, 2005; 4708-4715. |

[14] | Andrieu V, Astolfi A, Praly L. Nonlinear output feedback design via domination and generalized weighted homogeneity. Proceedings of the 45th IEEE CDC, San Diego, 2006; 6391-6396. |

[15] | Lei H, Lin W. A universal control approach for output feedback stabilization of uncertain nonlinear systems. Proceedings of the 44th IEEE CDC, Seville, 2005; 802-807; Systems and Control Letters 2007; 56:529-537. |

[16] | Yang, Robust output feedback stabilization of uncertain nonlinear systems with uncontrollable and unobservable linearization, IEEE Transactions on Automatic Control 50 pp 619– (2005) · Zbl 1365.93457 |

[17] | Yang, Homogeneous observers, iterative design and global stabilization of high order nonlinear systems by smooth output feedback, IEEE Transactions on Automatic Control 49 pp 1069– (2004) · Zbl 1365.93209 |

[18] | Qian, Recursive observer design, homogeneous approximation, and nonsmooth output feedback stabilization of nonlinear systems, IEEE Transactions on Automatic Control 51 pp 1457– (2006) · Zbl 1366.93523 |

[19] | Mazenc, Global stabilization by output feedback: examples and counterexamples, Systems and Control Letters 23 pp 119– (1994) · Zbl 0816.93068 |

[20] | Polendo J, Qian C. A generalized framework for global output feedback stabilization of nonlinear systems. Proceedings of the 44th IEEE CDC, Seville, Spain, 2005; 2646-2651. |

[21] | Lin W, Lei H. Taking advantage of homogeneity: a unified framework for output feedback control of nonlinear systems. Proceedings of the 7th IFAC NOLCOS, Pretoria, South Africa, 2007; 27-38. |

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.