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**Sliding order and sliding accuracy in sliding mode control.**
*(English)*
Zbl 0789.93063

Summary: The synthesis of a control algorithm that stirs a nonlinear system to a given manifold and keeps it within this constraint is considered. Usually, what is called sliding mode is employed in such synthesis. This sliding mode is characterized, in practice, by a high-frequency switching of the control. It turns out that the deviation of the system from its prescribed constraints (sliding accuracy) is proportional to the switching time delay. A new class of sliding modes and algorithms is presented and the concept of sliding mode order is introduced. These algorithms feature a bounded control continuously depending on time, with discontinuities only in the control derivative. It is also shown that the sliding accuracy is proportional to the square of the switching time delay.

### MSC:

93C15 | Control/observation systems governed by ordinary differential equations |

93B50 | Synthesis problems |

93C10 | Nonlinear systems in control theory |

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\textit{A. Levant}, Int. J. Control 58, No. 6, 1247--1263 (1993; Zbl 0789.93063)

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### References:

[1] | AIZERMAN M. A., Automation and Remote Control 35 pp 1066– (1974) |

[2] | DOI: 10.1016/0022-247X(86)90289-1 · Zbl 0622.93026 |

[3] | DOI: 10.1109/5.4400 |

[4] | DOI: 10.1080/00207178608933583 · Zbl 0596.93036 |

[5] | EMELYANOV S. V., Binary Systems of Automatic Control (1984) |

[6] | EMELYANOV S. V., Soviet Physics, Doklady 26 pp 562– (1981) |

[7] | EMELYANOV S. V., Soviet Physics, Doklady 31 pp 291– (1986) |

[8] | EMELYANOV S. V., Theory of Variable Structure Systems (1970) |

[9] | FIUPPOV A. F., Mathematical Sbornik 51 pp 99– (1960) |

[10] | ITKIS U., Control Systems of Variable Structure (1976) · Zbl 0256.93033 |

[11] | LEVANTOVSKY L. V., Dynamics of Heterogeneous Systems. Materials of the Seminar pp 32– (1985) |

[12] | DOI: 10.1093/imamci/1.3.223 · Zbl 0662.93059 |

[13] | DOI: 10.1016/0005-1098(84)90044-X · Zbl 0532.93002 |

[14] | DOI: 10.1109/TAC.1977.1101446 · Zbl 0382.93036 |

[15] | DOI: 10.1109/TAC.1977.1101661 · Zbl 0382.49029 |

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.