title: Formulation and solution of a generalized Chebyshev problem. II
metadata.name: dc.contributor.author
metadata.value: Yushkov, Mikhail P.
metadata.lang: -
metadata.name: dc.date.accessioned
metadata.value: 2020-12-21T19:47:59Z
metadata.lang: -
metadata.name: dc.date.available
metadata.value: 2020-12-21T19:47:59Z
metadata.lang: -
metadata.name: dc.date.issued
metadata.value: 2020-12
metadata.lang: -
metadata.name: dc.identifier.citation
metadata.value: Yushkov M. P. Formulation and solution of a generalized Chebyshev problem. II. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 2020, vol. 7 (65), issue 4, pp. 714–733.
metadata.lang: en_GB
metadata.name: dc.identifier.other
metadata.value: https://doi.org/10.21638/spbu01.2020.412
metadata.lang: -
metadata.name: dc.identifier.uri
metadata.value: http://hdl.handle.net/11701/21921
metadata.lang: -
metadata.name: dc.description.abstract
metadata.value: This work is a continuation of the article “Formulation and solution of a generalized Chebyshev problem. I”, in which a generalized Chebyshev problem was formulated, two theories of motion for non-holonomic systems with high-order constraints were presented for its
  solution. These theories were used to study the motion of the Earth’s satellite when fixing
  the magnitude of its acceleration (this was equivalent to imposing a linear non-holonomic
  constraint of the third order). In the offered article, the second theory, based on the application of the generalized Gauss principle, is used to solve one of the most important
  problems of control theory: finding the optimal control force that translates a mechanical
  system with a finite number of degrees of freedom from one phase state to another in a
  specified time. The application of the theory is demonstrated by solving a model problem of
  controlling the horizontal motion of a cart bearing the axes of s mathematical pendulums.
  Initially, the problem is solved by applying the Pontryagin maximum principle, which minimizes the functional of the square of the desired horizontal control force, which transfers
  the mechanical system from a state of rest to a new state of rest in the specified time with
  the horizontal displacement of the cart by S (that is, the problem of vibration damping
  is considered). Let’s call this approach the first method of solving the control problem.
  It is shown that a linear non-holonomic constraint of the order (2s + 4) is continuously
  performed. This suggests applying the second theory of motion for non-holonomic systems
  with high-order constraints to the same problem (see the previous article), developed at
  the Department of Theoretical and Applied Mechanics of the Faculty of Mathematics and
  Mechanics of Saint Petersburg State University. Let’s call this approach the second method
  of solving the problem. Calculations performed for the case of s = 2 showed that the results obtained by the both methods are practically the same for a short-term motion of the
  system, while they differ sharply for a long-term motion. This is because the control found
  using the first method contains harmonics with the system’s natural frequencies, which
  tends to bring the system into resonance. With a short-term motion, this is not noticeable,
  and with a long-term motion, there are large fluctuations in the system. In contrast, when
  using the second method, the control is in the form of a polynomial in time, which provides
  a relatively smooth motion of the system. In addition, in order to eliminate the control
  force jumps at the beginning and end of the motion, we propose to solve a generalized
  boundary value problem and discuss some special cases that sometimes occur when using
  the second method for solving the boundary value problem.
metadata.lang: en_GB
metadata.name: dc.language.iso
metadata.value: ru
metadata.lang: en_GB
metadata.name: dc.publisher
metadata.value: St Petersburg State University
metadata.lang: en_GB
metadata.name: dc.relation.ispartofseries
metadata.value: Vestnik of St Petersburg University. Mathematics. Mechanics. Astronomy;Volume 7 (65); Issue 4
metadata.lang: -
metadata.name: dc.subject
metadata.value: non-holonomic mechanics
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: high-order constraints
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: Pontryagin maximum principle
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: generalized Gauss principle
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: control
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: suppression of oscillation
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: generalized boundary problem
metadata.lang: en_GB
metadata.name: dc.title
metadata.value: Formulation and solution of a generalized Chebyshev problem. II
metadata.lang: en_GB
metadata.name: dc.type
metadata.value: Article
metadata.lang: en_GB
resources.filename: 714-733.pdf
resources.url: https://dspace.spbu.ru/bitstream/11701/21921/1/714-733.pdf
resources.format: Adobe PDF
resources.size: 618,12 kB