Time-Delay Systems: Analysis and Control using the Lambert W Function.

Delay Differential Equation;Lambert W Function;Human Immunodeficiency Virus;Machining Chatter;Diesel Engine Control;Time Delay System;Mechanical Engineering;Engineering;Mechanical Engineering

Yi, SunYounger, John G. ;

University of Michigan

关键词: Delay Differential Equation;    Lambert W Function;    Human Immunodeficiency Virus;    Machining Chatter;    Diesel Engine Control;    Time Delay System;    Mechanical Engineering;    Engineering;    Mechanical Engineering;   

Others  :  https://deepblue.lib.umich.edu/bitstream/handle/2027.42/64756/syjo_1.pdf?sequence=1&isAllowed=y

瑞士|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

Time-delay systems can arise due to inherent time-delays in the system or a deliberateintroduction of time-delays into the system for control purposes. Such systems frequentlyoccur in engineering and science. Time-delays can cause significant problems (e.g., instabilityand inaccuracy) and, thus, limit and degrade achievable performance. Time-delayterms lead to an infinite number of roots of the characteristic equation, and make analysisdifficult using classical methods, especially, in determining stability and designing stabilizingcontrollers. Thus, such problems have been addressed mainly by using approximate,numerical, and graphical methods. However, such approaches constitute limitations, forexample, on accuracy and robustness.The objective of this research is to develop an effective approach to analyze and controltime-delay systems. Using the LambertWfunction, free and forced analytical solutions todelay differential equations are derived. The main advantage of this solution approach liesin the fact that the solution has an analytical form expressed in terms of system parametersand, thus, one can explicitly determine how each parameter affects each eigenvalue andthe solution. Also, each eigenvalue in the infinite eigenspectrum is associated individuallywith a branch of the LambertWfunction. Solutions are obtained, for the first time, for systemsof delay differential equations using the matrix Lambert W function. The obtainedsolutions are used to analyze essential system properties, such as stability, controllabilityand observability, and to design controllers for stabilizing systems, improving robustnessand/or meeting time-domain specifications. Then, these methods are applied to biologicalsystems to analyze the immune system via eigenvalue sensitivity analysis, to automotivepowertrain systems to design feedback control with observers for improvements in fueleconomy and emissions, and to manufacturing processes to improve productivity via stabilityanalysis. The newly developed approach based on the matrix Lambert W functionprovides a tool for analysis and control, which is accurate (i.e., no approximation of timedelayterms), robust (i.e., no prediction of responses from models), and easy to implement(i.e., no need for complex nonlinear controllers).