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

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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.