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

时间尺度上的动态方程（DETS）范式是一种新兴的理论，它弥补了离散时间信号和系统与连续时间信号和系统之间的差距，提出了对工程动态系统进行大幅度改进的可能性。这些应用包括广泛应用于航空航天和汽车工业的分布式控制网络，以及用于金融分析的切换系统和条件持续时间模型。考虑到这些应用程序，我们建议应用DETS开发技术，用于设计一个给定的动态系统演变的时域（或“时间尺度”）。这样的设计技术涉及动态地改变一个名为（“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，其中整个时间尺度提前计算，和实时设计，其中嵌入式智能或控制器，适应时间尺度响应因果实时信息。例如，我们小组的工作表明，基本的实时自适应采样可以节省宝贵的带宽比传统的均匀采样的分布式控制网络中，高优先级的非周期性进程共享带宽与周期伺服进程，同时仍然满足系统的稳定性和性能标准。为了支持拟议的活动，研究小组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标存在定理，时标李雅普诺夫理论的工作，一个新的拉普拉斯正逆变换对，以及第一个MATLAB时标工具箱。 时标上的动力学方程范式揭示了对时域上的动力系统的新的和重要的见解，这些动力系统在性质上既不是纯连续的，也不是一致离散的。拟议的工作将促进一代变革性的数学和工程成果，并立即应用。同样重要的是，所提出的工作与许多相关领域正在进行的活动非常吻合，包括网络调度，具有未知延迟的实时控制以及时标本身的数学。成功的影响将是广泛的。实时网络在大多数现代车辆中，以及越来越多的医疗、航空航天和自动化/机器人技术中都有应用。成功和简单的方法来建模，分析和表征网络动态系统，演变自己的时域将有即时的效用和可能的直接经济影响，由于该理论适用的行业规模。高质量的研究将对贝勒大学工程和数学的学术基础设施产生深远而直接的影响，既提供了丰富的论文和论文主题来源，又加强了既定的、独特的和持续的跨学科合作。此外，我们还建议在时间尺度工程应用方面成立一个特别兴趣小组，并在适当选择的会议上举办一些特别会议。

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.