Analysis of Time Delayed Systems via Lambert Functions

通过Lambert函数对时滞系统的分析.G.Ulsoy和P.W.Nelson密歇根大学，Ann Arbor，MI 48109延迟是许多物理、生物、经济和工程系统所固有的。特别是，纯延迟通常被用来理想地表示传输、传输和惯性现象的影响。时滞微分方程组(DDE)构成了这种真实现象的基本数学模型。研究DDES的主要困难在于其特殊的先验性。延迟问题总是导致无限的频率谱。因此，通常使用数值方法、渐近解、近似(例如，PADE)和图形方法来求解它们。在本项目中，发展并推广了一种新的基于矩阵Lambert函数的解析法，用于求解线性常系数常微分方程组的完全解。通过与数值积分的比较，验证了该方法的稳定性、自由响应和强迫响应。该方法还被应用于一个延迟很大的工程问题：车床加工过程中的再生颤振。基于矩阵Lambert函数的偏微分方程组的求解方法类似于使用矩阵指数来求解线性常系数常微分方程组的自由解和强迫解。建议的研究将基于我们最近的结果，寻求将该方法扩展到更一般的时滞系统。具体地说，我们建议研究具有多时滞、时变系数和特定非线性的系统。具有多个时滞和非线性的系统在工程和生物学中是很自然的，然而对它们的分析却很少受到关注。我们的方法应该为其他人提供一个框架，用于研究这些复杂的系统。这一建议的智能优点在于潜在地发展了基于矩阵Lambert函数方法的专门方法，用于解决时滞微分方程组中的重要问题(例如，可观测性和可控性准则、控制器和观测器设计、多时滞、时变系数或非线性)，从而有助于分析以此类方程为特征的动力系统。本项目中开发的新方法将通过应用于重大科学和工程问题，例如具有延迟的HIV的动态建模和铣削过程中的再生颤振来验证和验证。利用矩阵Lambert函数研究时滞微分方程解析解问题具有广泛的数学、工程和科学界研究价值。艾滋病毒的应用不仅有利于相关领域的研究人员，还将使接受医疗护理的患者受益。例如，拟议的方法将用于为艾滋病毒治疗建立适当的实验室测试和药物治疗程序。同样，颤振稳定性的结果也将有利于制造业。这些结果将使制造商能够为他们的机器确定适当的主轴速度和切割深度，以实现无抖动的高生产率操作。该项目将作为跨学科研究生的博士论文题目，他们将由机械工程和数学系的两名PI共同指导，还将有一名本科生参与，他们来自科学和工程领域代表性较低的群体，使用本研究中开发的方法开发实例和软件。

ABSTRACTANALYSIS OF TIME DELAYED SYSTEMS VIA LAMBERT FUNCTIONSA.G. Ulsoy and P.W. NelsonUniversity of Michigan, Ann Arbor, MI 48109Delays 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.