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.
Objective of this research. Engineers rely heavily on mathematical theory and computing. Much of the mathematics engineers learn in college can be divided into roughly two kinds: differential equations (which describe the behavior of natural physical phenomena in "continuous time") and difference equations (which describe the behavior of computers in "discrete time"). These two bodies of mathematical theory are taught separately, but together they are the foundation for most of the modern practice of engineering. Relatively recently, a new theory has emerged that bridges the gap between the continuous and the discrete, known as the theory of dynamic equations on time scales. Dynamic equations can behave on one extreme like differential equations, or on the other like difference equations. Most interestingly, they can exhibit unique behaviors not common to either traditional body of mathematical theory and are therefore more generalized, and potentially of greater analytical power. This research is focused on understanding the charaterstics of dynamic equations, so that engineers of the future will have the mathematical tools for new advances, just as engineers today rely on differential and difference equations. Specifically, our team focused on two areas of study: understanding how to quantify the stability of difference equations, and developing a tool known as the Laplace Transform. Intellecutal Merit. This research produced 19 peer-reviewed, archived publications detailing all of our findings. We did not attempt to guess what specific engineering fields or applications would most benefit from our work, but rather focused on creating what we believe will be the most useful mathematical tools and results that will be needed in the future. We took our cue from the worlds of differential and difference equations; we focused mainly on the development of the time scale Laplace Transform, solved a new types of time scale Lypunov equation, and developed stability criteria for switched linear systems on time scales. Broader Impact. Throughout the period of the project, we worked with five graduate students in mathematics and engineering, and two undergraduate students in engineering. One Ph.D. was awarded from this research. These students have graduated (or will shortly) and are bringing this expertise to their industries or universities. The faculty supervising the research also taught several courses in which students outside of the research group could learn about dynamic equations and about our research. In addition to giving talks and lectures, we also organized a special meeting at the American Mathemtical Society's regional conference so discuss dynamic equations with other researchers from around the world. Lastly, this work led to a partnership with a local area company that identified a project where dynamic equations may prove helpful.
