### Abstract

We propose a novel method of determining the control force that brings a mechanical system with finite degrees of freedom from one phase state into another one in a given time. When a system is brought from an available phase state into a prescribed state of rest we can speak about the oscillation suppression. Horizontal motion of a cart with s mathematical pendula is studied as an example. At first, a control by the Pontryagin maximum principle that minimizes the functional of the squared control force is suggested. This approach results in a nonholonomic constraint of order 2s+4. In order to develop a control we propose to employ the generalized Gauss principle which underlies the theory of motion of nonholonomic systems with high-order constraints. In this case, the motion is more smooth than the Pontryagin maximum principle suggests. In addition, by increasing the order of the generalized Gauss principle (when solving the extended boundary-value problem) one manages to get rid of jumps in the control at the beginning and end of motion which are characteristic for the Pontryagin maximum principle. We also discuss the singular points of solutions that appear when solving the extended boundary-value problems.

Original language | English |
---|---|

Pages (from-to) | 781 - 788 |

Number of pages | 8 |

Journal | ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik |

Volume | 98 |

Issue number | 5 |

DOIs | |

Publication status | Published - May 2018 |

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### Scopus subject areas

- Mathematics(all)
- Applied Mathematics
- Computational Mechanics

### Cite this

*ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik*,

*98*(5), 781 - 788. https://doi.org/10.1002/zamm.201700005

}

*ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik*, vol. 98, no. 5, pp. 781 - 788. https://doi.org/10.1002/zamm.201700005

**A novel approach to suppression of oscillations.** / Zegzhda, S.; Yushkov, M.; Soltakhanov, Sh.; Naumova, N.; Shugaylo, T.

Research output

TY - JOUR

T1 - A novel approach to suppression of oscillations

AU - Zegzhda, S.

AU - Yushkov, M.

AU - Soltakhanov, Sh.

AU - Naumova, N.

AU - Shugaylo, T.

PY - 2018/5

Y1 - 2018/5

N2 - We propose a novel method of determining the control force that brings a mechanical system with finite degrees of freedom from one phase state into another one in a given time. When a system is brought from an available phase state into a prescribed state of rest we can speak about the oscillation suppression. Horizontal motion of a cart with s mathematical pendula is studied as an example. At first, a control by the Pontryagin maximum principle that minimizes the functional of the squared control force is suggested. This approach results in a nonholonomic constraint of order 2s+4. In order to develop a control we propose to employ the generalized Gauss principle which underlies the theory of motion of nonholonomic systems with high-order constraints. In this case, the motion is more smooth than the Pontryagin maximum principle suggests. In addition, by increasing the order of the generalized Gauss principle (when solving the extended boundary-value problem) one manages to get rid of jumps in the control at the beginning and end of motion which are characteristic for the Pontryagin maximum principle. We also discuss the singular points of solutions that appear when solving the extended boundary-value problems.

AB - We propose a novel method of determining the control force that brings a mechanical system with finite degrees of freedom from one phase state into another one in a given time. When a system is brought from an available phase state into a prescribed state of rest we can speak about the oscillation suppression. Horizontal motion of a cart with s mathematical pendula is studied as an example. At first, a control by the Pontryagin maximum principle that minimizes the functional of the squared control force is suggested. This approach results in a nonholonomic constraint of order 2s+4. In order to develop a control we propose to employ the generalized Gauss principle which underlies the theory of motion of nonholonomic systems with high-order constraints. In this case, the motion is more smooth than the Pontryagin maximum principle suggests. In addition, by increasing the order of the generalized Gauss principle (when solving the extended boundary-value problem) one manages to get rid of jumps in the control at the beginning and end of motion which are characteristic for the Pontryagin maximum principle. We also discuss the singular points of solutions that appear when solving the extended boundary-value problems.

KW - control force

KW - damping of oscillations

KW - generalized Gauss principle

KW - Pontryagin maximum principle

UR - http://www.scopus.com/inward/record.url?scp=85039560152&partnerID=8YFLogxK

U2 - 10.1002/zamm.201700005

DO - 10.1002/zamm.201700005

M3 - Article

VL - 98

SP - 781

EP - 788

JO - ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik

JF - ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik

SN - 0044-2267

IS - 5

ER -