New Results on Time-Delay Systems Towards Model-Based Design

This thesis is devoted to the stability study of time-delay systems, a subject that has been vigorously pursued by such learned societies as diversely represented in mathematics, science, engineering, and economics. Time-delay systems, which are also sometimes known as hereditary systems, systems with memory, after effects and time-lag, represent a class of infinite-dimensional systems used to describe, among other types of systems, propagation and transport phenomena, population dynamics, economic systems, communication networks, and neural network models. The aim of the present thesis is to develop techniques and tools that may help to study the stability of commensurate time-delay systems. Stability analysis methods are developed based on the corresponding characteristic equation following a frequency sweeping test and constant matrix tests. Rewriting the quasi-polynomial equation, the coefficients can be found and then the roots of the characteristic equation can be plotted in a complex plane. If the roots cross the imaginary axis of the system, it is said to be Delay-Dependent Stable System (DDSS). A controller design procedure for Commensurate Multiple Time-Delay Systems (CMTDSs) is developed, able to transform the system into a Delay Independent Stable System (DISS). The controller based on a single parameter is used to make the system DISS. It may be determined by adopting different strategies, either analytical or graphical. Based on this theorem, a stability chart is partitioned into two regions, that are DDSS and DISS. As an application, it is demonstrated that model-based design can be used to design systems with time delays. The stability analysis methods developed in this thesis are tailored and applied to find if the system is DDSS and to transform systems from DDSS to DISS.