Robust Feedback Control and Analysis of Queueing Systems

排队系统的鲁棒反馈控制与分析

• 批准号: 0102266
• 负责人: Martin Day
• 金额: $ 11万
• 依托单位: Virginia Polytechnic Institute and State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-09-01 至 2005-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0102266&HistoricalAwards=false
• 关键词: Robust; Feedback; Control; Analysis; Queueing

This project develops mathematical techniques for the analysis of optimal robust control (service) strategies for queuing systems. Fluid models (deterministic, continuous state and time variables) are considered which describe queuing network configurations motivated by applications in manufacturing, communications and vehicular traffic control. Optimal control problems are formulated using the general approach of nonlinear robust control, resulting in an appropriate Hamilton-Jacobi-Isaacs (HJI) partial differential equation which determines the value function of a dynamic game. Solutions to this HJI are constructed by methods similar to classical techniques involving (bi)characteristics or a field of extremals. While this approach is well understood in general, it has had little development in applications to queuing networks. One reason for this is that the precise mathematical description of queuing systems often involves a Skorokhod problem formulation to account for the changes to systems dynamics when some of the queues become empty. A major goal of the project is to develop this approach for a variety of examples (e.g. different network structures and cost criteria) in order to discern the mathematical structure of solutions to the HJI, the form of the optimal policies themselves, and how the unique features of the Skorokhod problem formulation can be treated in this approach. Using the understanding of solutions to the HJI gained through these case studies, numerical methods are developed to analyze larger and more complex examples. The mathematical tools produced will contribute to the theory of robust control in general, to problems involving Skorokhod dynamics in particular, and will introduce robust control techniques to the area of traffic control.The flow of "traffic" through a network, whether vehicles on a system of intersecting roads, products in a multistage manufacturing process, or "packets" of information in an electronic network, is an increasingly important aspect of our commercial and public services infrastructure. In the familiar example of vehicles on a system of roads the network consists of several intersections connected by roads of various sizes. The different streams of traffic that meet at an intersection must take turns or share the limited capacity of the intersection. The efficiency of the network depends on the strategy which controls the traffic streams that are allowed to use the various intersections at each moment of time. (In the most familiar vehicular setting this simply means the scheme which governs the sequence and timing of traffic lights.) Now that sensors are available to detect the number of vehicles waiting or traveling on each of the roads in a network, the design intelligent signal light control strategies to manage the network with optimal efficiency is a natural goal. The same general issue of service allocation strategy is present in manufacturing or communication systems, although the terminology and specific features of the networks are different in those settings. Traffic and network engineers have developed various ways to study the performance of network service control strategies "experimentally" and to adjust those strategies to current network conditions in an adaptive way. However, with only a few exceptions, past research has not developed systematic tools to identify service control strategies whose performance is optimal in a precise mathematical sense. This project develops mathematical ideas and tools for this purpose. The new understanding and mathematical techniques which result should be a valuable contribution to the management and design of high performance networks.

该项目开发了用于分析排队系统的最优鲁棒控制(服务)策略的数学技术。考虑了流体模型(确定性、连续状态和时间变量)，这些模型描述了在制造、通信和车辆交通控制中的应用所驱动的排队网络结构。利用非线性鲁棒控制的一般方法建立了最优控制问题，得到了确定动态博弈价值函数的适当的Hamilton-Jacobi-Isaacs(HJI)偏微分方程式。这种HJI的解是用类似于涉及(双)特征或极值域的经典技术的方法来构造的。虽然这种方法在一般情况下被很好地理解，但它在排队网络的应用中几乎没有发展。其中一个原因是，排队系统的精确数学描述通常涉及Skorokhod问题公式，以说明当一些队列变空时系统动力学的变化。该项目的一个主要目标是为各种实例(例如，不同的网络结构和成本标准)开发这种方法，以辨别HJI解决方案的数学结构、最优策略本身的形式，以及如何在这种方法中处理Skorokhod问题公式的独特特征。利用通过这些案例研究获得的对HJI解的理解，发展了数值方法来分析更大和更复杂的例子。所产生的数学工具将有助于一般的稳健控制理论，特别是涉及Skorokhod动力学的问题，并将稳健控制技术引入交通控制领域。通过网络的交通流量，无论是交叉道路系统上的车辆，多阶段制造过程中的产品，还是电子网络中的信息分组，都是我们商业和公共服务基础设施的一个日益重要的方面。在常见的车辆在道路系统上的例子中，网络由几个由不同大小的道路连接的交叉口组成。在交叉口相遇的不同车流必须轮流通行或共享交叉口有限的通行能力。网络的效率取决于控制允许在每个时刻使用各种交叉口的交通流的策略。(在最熟悉的车辆设置中，这只是指管理交通灯顺序和定时的方案。)由于传感器可以用来检测网络中每条道路上等待或行驶的车辆数量，设计智能信号灯控制策略以最佳效率管理网络是一个自然的目标。在制造或通信系统中也存在相同的服务分配策略的一般问题，尽管这些网络的术语和具体特征在这些环境中不同。流量和网络工程师已经开发了各种方法来研究网络服务控制策略的性能，并以自适应的方式调整这些策略以适应当前的网络条件。然而，除了少数例外，过去的研究没有开发出系统的工具来确定在精确的数学意义上性能最优的服务控制策略。该项目为此目的开发了数学思想和工具。新的理解和数学技术将对高性能网络的管理和设计做出有价值的贡献。