Analysis of Time Delayed Systems via Lambert Functions

通过朗伯函数分析时滞系统

• 批准号: 0555765
• 负责人: A. Galip Ulsoy
• 金额: --
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-08-01 至 2010-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0555765&HistoricalAwards=false
• 关键词: Analysis; Time; Delayed; Systems; via

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.

用Lambert函数分析时滞系统Ulsoy和P.W.纳尔逊密歇根大学，安阿伯，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，纯延迟通常用于理想地表示传输，运输和惯性现象的影响。延迟微分方程（DDE）构成了这种真实的现象的基本数学模型。 研究动态微分方程的主要困难在于其特殊的先验性质。 延迟问题总是导致无限的频谱。 因此，它们通常使用数值方法、渐近解、近似（例如，Pade）和图形方法。 在这个项目中，一个新的分析方法，基于矩阵朗伯函数，为完整的解决方案的线性常系数微分方程组的发展和扩展。 通过与数值积分的比较，验证了该方法的稳定性、自由和强迫响应。 该方法也适用于延迟是显着的工程问题：再生颤振在车床上的加工操作。 基于矩阵Lambert函数的求解方法类似于使用矩阵指数求解线性常系数常微分方程的自由解和强迫解。 建议的研究将寻求扩展的方法，根据我们最近的研究结果，更一般的时滞系统。具体来说，我们建议研究系统的多个延迟，时变系数，和特定的非线性。具有多时滞和非线性的系统在工程和生物学中很自然地出现，但很少有人关注它们的分析。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。这一建议的智力价值在于基于矩阵Lambert函数方法的专门方法的潜在发展，用于解决延迟微分方程系统中的重要问题（例如，可观测性和可控制性准则、控制器和观测器设计、多延迟、时变系数或非线性），这将有助于分析以这些方程为特征的动力系统。 本项目开发的新方法将通过应用于科学和工程中的重大问题进行演示和验证，例如，具有延迟的HIV的动态建模和铣削过程中的再生颤振。 提出的研究延迟微分方程的分析解决方案，使用矩阵Lambert函数的承诺是广泛的兴趣，数学，工程和科学界。 将其应用于艾滋病毒将不仅有益于相关领域的研究人员，而且也将有益于接受医疗护理的患者。 例如，拟议的方法将用于建立适当的实验室测试和药物治疗程序，用于艾滋病毒治疗。 同样，颤振稳定性的结果将有利于制造业。 这些结果将使制造商能够确定合适的主轴速度和切削深度，为他们的机器无颤振的高生产率操作。 该项目将作为跨学科研究生的博士论文主题，他们将由两名PI（分别是机械工程和数学系的教师）共同提供咨询，还将涉及一名本科生，来自科学和工程领域代表性不足的群体，使用本研究中开发的方法开发示例和软件。