Mu-Dynamics on Time Scales: Adaptive Time Domains for Dynamical Systems

时间尺度上的 Mu 动力学：动力系统的自适应时域

• 批准号: 0726996
• 负责人: Ian Gravagne
• 金额: $ 14.4万
• 依托单位: Baylor University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-09-15 至 2011-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0726996&HistoricalAwards=false
• 关键词: Mu; Dynamics; Time; Scales; Adaptive

The dynamic equations on time scales (DETS) paradigm, an emerging theory bridging the gap between discrete and continuous time signals and system, suggests the possibility of dramatic improvements to engineered dynamical systems in which the underlying time domain can be designed. Such applications include distributed control networks used widely in the aerospace and automotive industries, and switched systems, and conditional duration models used in financial analysis. With these applications in mind, we propose to apply DETS to develop techniques for designing the time domain (or "time scale") on which a given dynamical system evolves. Such design techniques involve dynamically changing a parameter named (the "graininess"), giving rise to the term -dynamics. Distributed networks and switched systems represent exemplars of so-called "explicit model" time scale design (versus implicit model design). We will study two types of explicit model time scale deign methodologies: a priori design, in which the entire time scale is calculated in advance, and real-time design, in which an embedded intelligence, or controller, adapts the time scale in response to causal real-time information. For example, work by our group suggests that rudimentary real-time adaptive sampling can save valuable bandwidth over traditional uniform sampling on a distributed control network in which high-priority aperiodic processes share bandwidth with periodic servo processes, while still meeting system stability and performance criteria. In support of the proposed activity, the research team brings a suite of recently developed tools including time scale existence theorems for certain classes of nonlinear systems, a body of work on time scale Lyapunov theory, a new Laplace forward and inverse transform pair, and the first MATLAB time scales toolbox. The dynamic equations on time scales paradigm reveals new and important insights into dynamical systems on time domains that are neither purely continuous nor uniformly discrete in nature. The proposed work will foster a generation of transformative mathematical and engineering results with immediate application. Just as importantly, the proposed work fits well with ongoing activity in a number of related areas, including network scheduling, real-time control with unknown delays, and the mathematics of time scales itself. The impact of success will be wide. Real-time networks are found in most modern vehicles, as well as a growing number of medical, aerospace, and automation/robotics technologies. Successful and straightforward methods to model, analyze and characterize networked dynamical systems that evolve their own time domain will have immediate utility and possibly direct economic impact due to the size of the industries for which the theory is applicable. High quality research will have a profound and immediate impact on the academic infrastructure in both engineering and mathematics at Baylor, both by providing a rich source of thesis and dissertation topics and by strengthening an established, unique and ongoing cross-disciplinary collaboration. We furthermore propose to initiate a special interest group in time scale engineering applications, as well as a number of special sessions at appropriately selected conferences.

时标动态方程 (DETS) 范式是一种弥合离散和连续时间信号与系统之间差距的新兴理论，它表明可以对工程动力系统进行显着改进，在该系统中可以设计底层时域。此类应用包括广泛用于航空航天和汽车行业的分布式控制网络、交换系统以及财务分析中使用的条件持续时间模型。考虑到这些应用，我们建议应用 DETS 来开发设计给定动态系统演化的时域（或“时间尺度”）的技术。这种设计技术涉及动态改变名为（“颗粒度”）的参数，从而产生了术语“动态”。分布式网络和交换系统代表了所谓的“显式模型”时间尺度设计（相对于隐式模型设计）的范例。我们将研究两种类型的显式模型时间尺度设计方法：先验设计（提前计算整个时间尺度）和实时设计（其中嵌入式智能或控制器根据因果实时信息调整时间尺度）。例如，我们小组的工作表明，与分布式控制网络上的传统均匀采样相比，基本的实时自适应采样可以节省宝贵的带宽，其中高优先级非周期性进程与周期性伺服进程共享带宽，同时仍然满足系统稳定性和性能标准。为了支持拟议的活动，研究团队带来了一套最近开发的工具，包括某些类别的非线性系统的时间尺度存在定理、时间尺度李雅普诺夫理论的工作主体、新的拉普拉斯正向和逆变换对以及第一个 MATLAB 时间尺度工具箱。 时标动力学方程范式揭示了对时域动力系统的新的重要见解，这些系统本质上既不是纯粹连续的也不是均匀离散的。拟议的工作将促进产生可立即应用的变革性数学和工程成果。同样重要的是，拟议的工作非常适合许多相关领域正在进行的活动，包括网络调度、未知延迟的实时控制以及时间尺度本身的数学。成功的影响将是广泛的。大多数现代车辆以及越来越多的医疗、航空航天和自动化/机器人技术中都采用了实时网络。由于该理论适用的行业规模，成功且直接的方法来建模、分析和表征在其自己的时域中演化的网络动力系统将具有直接的效用，并可能产生直接的经济影响。高质量的研究将对贝勒大学工程和数学方面的学术基础设施产生深远而直接的影响，既可以提供丰富的论文和论文主题来源，又可以加强已建立的、独特的和持续的跨学科合作。我们还建议在时间尺度工程应用方面发起一个特别兴趣小组，并在适当选择的会议上召开一些特别会议。