Trajectory Planning and Coordinated Control

这里提出的问题与应用控制理论的一个基本问题有关：为受控系统设计轨迹的问题。这些问题，正如这里所讨论的，是为了应对空中交通管制的需要而产生的，但比任何一个应用程序都更基本。最基本的问题是给出一个受控系统，测量y被认为是测量系统的位置。假定给出了工厂必须达到的(yi，ti)形式的一组路点。问题是要找到一个控制律，使系统以y(Ti)=yi的方式移动。对于大多数系统来说，没有唯一的方法来解决这个问题，因此有必要施加其他条件以确保唯一性。对于空中交通管制问题，有d/dt x=f(X)ug(X)，初始数据x(O)=x-O，测量函数y=h(X)。明显限制了飞机的动力性和乘客的舒适性。应该满足的一个明显条件是x(T)be的动力学应该是光滑的。对于乘客来说，速度或加速度的突然变化是非常明显的，物理现实要求x(T)是连续的。强加的附加条件对系统的最终行为产生了很大的影响，系统本身的性质也是如此。这项研究的目标是开发一种工具，允许在一般控制设计的背景下设计满足需要的轨迹。为了做到这一点，我们必须重新审查上面提出的基本问题，并将其扩展到其他更复杂的任务。这里提出的基本问题与数值样条线的构造密切相关。为了将其扩展到控制理论中更一般的问题，有必要进入混杂系统领域。这里的问题涉及到更一般的Birkhoff样条的数学问题，以及我们选择称为协调控制问题的一组新问题。在第二节中，我们将阐述我们强加于该制度的基本假设，并将描述已获得的基本问题的解决办法。这里概述了解决这一整套问题的基本但重要的基本程序，并将证明所做的许多假设是正确的。这里还将描述已经发表的关于这个问题的几篇论文的内容，这些论文将与正在进行的关于弹道设计问题的其他工作有关。在第三节中，将介绍一组与Birkhoff样条的经典问题有关的新问题。在飞机控制中，当希望能够设置指令速度、加速度、角速度等时，这些问题就会出现。因此，实际上我们有一组测量值Yk(T)，在不同的时间点，其中一些(但不是全部)必须具有规定值，并且必须设计控制以达到所需的值。在这里，问题更基本，而不是独特性，它与最基本的问题--存在--有关。在第4节中，将讨论什么是真正的新领域。在许多复杂的系统中，系统的运动由影响特定控制途径的子系统来管理。这项研究的基本例子是人体的肌肉控制。在大多数情况下，肌肉是成对出现的，在任何给定的时间，一对肌肉中只有一对是活跃的。为了执行一个动作，肌肉收缩并导致一个特定的动作。为了执行一个复杂的动作，肌肉按照某种更高的控制规律被开启和关闭，身体做出平稳的协调运动。你只需要看一位专业体操运动员就可以看到，身体必须保持出色的协调性，才能在复杂的动作中平稳而快速地移动，例如，你会在鞍马上看到这一点。目前还没有为这种复杂的机动开发控制策略的计划，但将开发出执行其他更简单任务的控制策略。最后，在第5节中，说明了稳定性方面的基本问题，这些问题中的每一个都必须加以考虑。正在切换的系统的稳定性问题相对较新。这里讨论了研究的现状，并描述了一系列必须解决的问题，以确保所提出的控制方案是可实施的。

The problems that will be proposed here have to do with one of the basic problem of applied control theory: the problem of designing trajectories for controlled systems. These problems, as are discussed here, arose in response to the needs of air traffic control, but are more basic than any one application. The most basic of the problems is to be given a controlled system, The measurement y is to be thought of as measuring the position of the system. A set of way-points in the form (yi, ti) that must be attained by the plant is assumed to be given. The problem is to find a control law that will move the system in such a way that y(ti)=yi. For most systems there is no unique way of solving this problem so it is necessary to impose other conditions in order to ensure uniqueness. For the air traffic control problem there are d/dt x=f(x)+ug(x) with initial data x(O)=x-O and a measurement function y=h(x). obvious constraints concerning the dynamics of the aircraft and passenger comfort. One obvious condition that should be met is that the dynamics of x(t) be should be smooth. For passengers abrupt changes in velocity or acceleration are very noticeable and physically reality requires that x(t) be continuous. The additional conditions that are imposed make a great deal of difference on the final behavior of the system as does the nature of the system itself. The goal of this research is to develop a tool that will allow the design of trajectories that meet the needs within the context of general control design. In order to do this we must reexamine the basic problem as posed above and extend it to other more complicated tasks. The basic problem as posed here is intimately related to the construction of numerical splines. To extend this to more general problems in control theory it is necessary to enter the area of hybrid systems. Here the problem is related to the more general mathematical problem of Birkhoff splines and to a set of new problems which we choose to call problems of coordinated control. in Section 2 the basic assumptions which we impose on the system will be set forth and the solutions that have been obtained to the basic problem will be described. Here is outlined an elementary, but important basic procedure for solving this entire set of problems and will justify many of the assumptions which have been made. Here will also be described the contents of the several papers that have been published on this problem and they will be related to other work that is ongoing on the problem of trajectory design. In Section 3 there will be presented a set of new problems that will be related to classical problems of Birkhoff splines. These problems arise in aircraft control when it is desired to be able to set commanded velocities, accelerations, angular velocities etc. So in fact we have a set of measurements yk(t) and at various points in time some, but not all, of these must have prescribed values and the control must be designed in order to attain the required values. Here the problem is more basic than uniqueness and is related to the most basic of problems-existence. In Section 4 will be discussed what is really new ground. In many complicated systems the movement of the system is governed by subsystems that effect particular avenues of control. The basic example of this research is the muscular control of the human body. For the most part muscles occur in opposing pairs and only one of a pair can be active at any given time. In order to execute a movement a muscle contracts and causes one particular movement. To execute a complicated movement muscles are switched on and off in accordance with some higher control law and the body makes a smooth coordinated movement. One only need watch a professional gymnast to see the superb coordination that must be maintained for the body to move smoothly and quickly through a complicated maneuver as, for example, one would see on the pommel horse. There are no current plans to develop a control strategy for such a complicated maneuver, but control strategies will be developed that will perform other simpler tasks. Finally in Section 5 it is shown that there are basic problems in stability that must be considered with each of these problems. The problem of stability for systems which are being switched is relatively new. Here is discussed the current state of research and described a series of problems that must be solved to insure that the controls schemes that are being proposed are implementable.