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.

中文翻译：

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广义切比雪夫问题的表述和解决方案：第二部分

摘要

这项研究是“广义切比雪夫问题的形成与解决方案：第一部分”文章的延续，其中阐述了广义切比雪夫问题，并提出了具有更高阶约束的非完整系统的两种运动理论。这些理论用于研究固定固定卫星加速度的幅度时的地球卫星运动（这相当于施加三阶线性非完整约束）。在本文中，基于广义高斯原理的应用的第二种理论被用来解决控制理论中最重要的问题之一：找到最佳的控制力，该最优的控制力可转换具有有限个自由度的机械系统在指定的时间段内从一种相态转换为另一种相态。的数学摆。最初，通过应用Pontryagin最大原理解决了该问题，该原理使所需水平控制力的平方的函数最小化，从而将机械系统在指定时间段内随着推车的水平方向从静止状态转移到新的静止状态位移S（考虑减振问题）。这种方法称为解决控制问题的第一种方法。结果表明a（2 s+ 4）阶线性非完整约束被连续满足。这表明将具有更高阶约束的非完整系统的第二运动理论应用于相同的问题（请参见前一篇文章），该理论是在圣彼得堡国立大学数学和力学系理论与应用力学系开发的。这种方法称为解决问题的第二种方法。针对情况s进行计算= 2并表明两种方法的结果对于系统的短运动实际上是相同的，但是对于长运动它们却有很大不同。这是由于这样的事实：使用第一种方法确定的控制包含系统固有频率的谐波，这容易使系统共振。对于短距离运动，这并不明显；对于长距离运动，系统中的波动很大。相反，当使用第二种方法时，控制在时间上是多项式的，这提供了系统的相对平滑的运动。另外，为了消除运动开始和结束时控制力的跳跃，提出了解决广义边值问题的方法。讨论了使用第二种方法解决边值问题时发生的一些特殊情况。