This book presents the recently introduced and already widely referred semi-discretization method for the stability analysis of delayed dynamical systems. Delay differential equations often come up in different fields of engineering, like feedback control systems, machine tool vibrations, balancing/stabilization with reflex delay. The behavior of such systems is often counter-intuitive and closed form analytical formulas can rarely be given even for the linear stability conditions. If parametric excitation is coupled with the delay effect, then the governing equation is a delay differential equation with time periodic coefficients, and the stability properties are even more intriguing. The semi-discretization method is a simple but efficient method that is based on the discretization with respect to the delayed term and the periodic coefficients only. The method can effectively be used to construct stability diagrams in the space of system parameters.

This volume is the first of the new series Advances in Dynamics and Delays. It offers the latest advances in the research of analyzing and controlling dynamical systems with delays, which arise in many real-world problems. The contributions in this series are a collection across various disciplines, encompassing engineering, physics, biology, and economics, and some are extensions of those presented at the IFAC (International Federation of Automatic Control) conferences since 2011. The series is categorized in five parts covering the main themes of the contributions: · Stability Analysis and Control Design · Networks and Graphs · Time Delay and Sampled-Data Systems · Computational and Software Tools · Applications This volume will become a good reference point for researchers and PhD students in the field of delay systems, and for those willing to learn more about the field, and it will also be a resource for control engineers, who will find innovative control methodologies for relevant applications, from both theory and numerical analysis perspectives.

This volume collects contributions related to selected presentations from the 12th IFAC Workshop on Time Delay Systems, Ann Arbor, June 28-30, 2015. The included papers present novel techniques and new results of delayed dynamical systems. The topical spectrum covers control theory, numerical analysis, engineering and biological applications as well as experiments and case studies. The target audience primarily comprises research experts in the field of time delay systems, but the book may also be beneficial for graduate students alike.

This book is a self-contained presentation of the background and progress of the study of time-delay systems, a subject with broad applications to a number of areas.

1. Introduction. 1.1. Motivation. 1.2. Background. 1.3. Scope of this document. 1.4. Original contributions -- 2. Solutions of systems of DDEs via the matrix Lambert W function. 2.1. Introduction. 2.2. Free systems of DDEs. 2.3. Forced systems. 2.4. Approach using the Laplace transformation. 2.5. Concluding remarks -- 3. Stability of systems of DDEs via the Lambert W function with application to machine tool chatter. 3.1. Introduction. 3.1. The Chatter equation in the turning process. 3.3. Solving DDEs and stability. 3.4. Concluding remarks -- 4. Controllability and observability of systems of linear delay differential equations via the matrix Lambert W function. 4.1. Introduction. 4.2. Controllability. 4.3. Observability. 4.4. Illustrative example. 4.5. Conclusions and future work -- 5. Eigenvalue assignment via the Lambert W function for control of time-delay systems. 5.1. Introduction. 5.2. Eigenvalue assignment for time-delay systems. 5.3. Design of a feedback Controller. 5.4. Conclusions -- 6. Robust control and time-domain specifications for systems of delay differential equations via eigenvalue assignment. 6.1. Introduction. 6.2. Robust feedback. 6.3. Time-domain specifications. 6.4. Concluding remarks -- 7. Design of observer-based feedback control for time-delay systems with application to automotive powertrain control. 7.1. Introduction. 7.2. Problem formulation. 7.3. Design of observer-based feedback controller. 7.4. Application to diesel engine control. 7.5. Conclusions -- 8. Eigenvalues and sensitivity analysis for a model of HIV pathogenesis with an intracellular delay. 8.1. Introduction. 8.2. HIV pathogenesis dynamic model with an intracellular delay. 8.3. Rightmost eigenvalue analysis. 8.4. Sensitivity analysis. 8.5. Concluding remarks and future work

Most practical processes such as chemical reactor, industrial furnace, heat exchanger, etc., are nonlinear stochastic systems, which makes their con trol in general a hard problem. Currently, there is no successful design method for this class of systems in the literature. One common alterna tive consists of linearizing the nonlinear dynamical stochastic system in the neighborhood of an operating point and then using the techniques for linear systems to design the controller. The resulting model is in general an approximation of the real behavior of a dynamical system. The inclusion of the uncertainties in the model is therefore necessary and will certainly improve the performance of the dynamical system we want to control. The control of uncertain systems has attracted a lot of researchers from the control community. This topic has in fact dominated the research effort of the control community during the last two decades, and many contributions have been reported in the literature. Some practical dynamical systems have time delay in their dynamics, which makes their control a complicated task even in the deterministic case. Recently, the class ofuncertain dynamical deterministic systems with time delay has attracted some researchers, and some interesting results have been reported in both deterministic and stochastic cases. But wecan't claim that the control problem ofthis class ofsystems is completely solved; more work must be done for this class of systems.