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Formulation and Solution of a Generalized Chebyshev Problem: Second Part

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Abstract

This study is a continuation of the article “Formulation and solution of a generalized Chebyshev problem: First Part,” in which a generalized Chebyshev problem was formulated and two theories of motion for nonholonomic systems with higher 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 third-order linear nonholonomic constraint). In this 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 the specified time period. The application of the theory is demonstrated by solving the 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 motionless state to a new motionless state in the specified time period with the cart’s horizontal displacement S (the vibration damping problem is considered). This approach is called the first method for solving the control problem. It is shown that a (2s + 4)-order linear nonholonomic constraint is continuously fulfilled. This suggests applying the second theory of motion for nonholonomic systems with higher order constraints to the same problem (see the previous article), the theory developed at the Department of Theoretical and Applied Mechanics of the Faculty of Mathematics and Mechanics of St. Petersburg State University. This approach is called the second method for solving the problem. Calculations are performed for the case s = 2 and show that the results of both methods are practically identical for a short motion of the system but they differ substantially for a long motion. This is due to fact that the control determined using the first method contains harmonics with the system’s natural frequencies, which tends to bring the system into resonance. For a short motion this is not noticeable, and for a long motion there are large fluctuations in the system. In contrast, when the second method is used, the control is polynomial in time, which provides a relatively smooth motion of the system. In addition, in order to eliminate the jumps in the control force at the beginning and end of the motion, it is proposed to solve the generalized boundary value problem. Some special cases occurring when using the second method for solving the boundary value problem are discussed.

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  1. St. Petersburg University, 199034, St. Petersburg, Russia

    M. P. Yushkov

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Translated by E. Oborin

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Yushkov, M.P. Formulation and Solution of a Generalized Chebyshev Problem: Second Part. Vestnik St.Petersb. Univ.Math. 53, 459–472 (2020). https://doi.org/10.1134/S1063454120040111

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