A.G. Ulsoy and P.W. Nelson University of Michigan, Ann Arbor, MI 48109
Delays are inherent in many physical, biological, economic and engineering systems. In particular, pure delays are often used to ideally represent the effects of transmission, transportation, and inertial phenomena. Delay differential equations (DDEs) constitute basic mathematical models for such real phenomena. The principal difficulty in studying DDEs lies in their special transcendental character. Delay problems always lead to an infinite spectrum of frequencies. Hence, they are often solved using numerical methods, asymptotic solutions, approximations (e.g., Pade) and graphical approaches. In this project a new analytic approach, based on the matrix Lambert function, for the complete solution of a system of linear constant coefficient DDEs is developed and expanded. The method is validated, for stability, free and forced response, by comparison to numerical integration. The method is also applied to an engineering problem where delay is significant: regenerative chatter in a machining operation on a lathe. The matrix Lambert function based solution approach for DDEs is analogous to the use of the matrix exponential for the free and forced solution of linear constant coefficient ordinary differential equations. The proposed research will seek extensions of the method, based on our recent results, to more general time-delayed systems. Specifically we propose to study systems with multiple delays, time-varying coefficients, and specific nonlinearities. Systems with multiple time delays and nonlinearities arise quite naturally in engineering and biology and yet little attention has been paid to their analyses. Our method should provide a framework for others to use in studying these complicated systems. The intellectual merit of this proposal lies in the potential development of specialized methods, based upon the matrix Lambert function approach, for solutions to important problems in systems of delay differential equations (e.g., observability and controllability criteria, controller and observer design, multiple delays, time-varying coefficients, or nonlinearities) that would facilitate the analysis of dynamical systems characterized by such equations. The new method developed in this project will be demonstrated and validated by application to significant problems in science and engineering, for example, the dynamic modeling of HIV with delay and to regenerative chatter in the milling process.
The proposed research on the analytical solution of delay differential equations using the matrix Lambert function promises to be of wide interest to the mathematics, engineering and science communities. The application to HIV will be of benefit not only to researchers in related fields, but will also benefit patients under medical care. For example, the proposed method will be used to establish appropriate lab testing and drug therapy procedures for HIV treatment. Similarly, the chatter stability results will be of benefit to the manufacturing industry. Those results will enable manufacturers to determine the appropriate spindle speeds and depth-of-cut for their machines for chatter-free high-productivity operation. The project will serve as a doctoral thesis topic for an interdisciplinary graduate student, who will be co-advised by the two PI's, who are faculty in Mechanical Engineering and Mathematics departments respectively, and will also involve an undergraduate student, from an underrepresented group in science and engineering, to develop examples and software using the methods developed in this research.
Time delays occur in many natural and engineered systems, such as HIV infections or teleoperation of robots. Such delays can lead to difficulties in understanding how the behavior of such systems develop with time, and even to instabilities (e.g., adjusting water temperature in a shower with delay in the hot water lines). Such time delay systems (TDS) are difficult to quantitatively analyze and control, and are often handled using approximate and numerical methods. This research has developed a new closed-form solution method for the analysis and control of TDS using a matrix version of the classical Lambert W function. This new method enables scientists and engineers to analyze and control TDS in a manner very similar to systems without time delays. The research has also trained two graduate students, one (Dr. Sun Yi) is now an Assistant Professor at North Carolina A&T State University (an historically black university in Greensboro, NC). The results of the research are summarized in a book Time Delay Systems: Analysis and Control Using the Lambert W Function by Sun Yi, Patrick W. Nelson and A. Galip Ulsoy (World Scientific, 2010) and numerous journal and conference articles. Additional problems and software, to assist instructors and students, can be found at the web site http://umich.edu/~ulsoy/TDS_Supplement.htm
