The Hermite-Biehler Theorem and its Generalization -- PI Stabilization of Delay-Free Linear Time-Invariant Systems -- PID Stabilization of Delay-Free Linear Time-Invariant Systems -- Preliminary Results for Analyzing Systems with Time Delay -- Stabilization of Time-Delay Systems using a Constant Gain Feedback Controller -- PI Stabilization of First-Order Systems with Time Delay -- PID Stabilization of First-Order Systems with Time Delay -- Control System Design Using the PID Controller -- Analysis of Some PID Tuning Techniques -- PID Stabilization of Arbitrary Linear Time-Invariant Systems with Time Delay -- Algorithms for Real and Complex PID Stabilization.The Proportional-Integral-Derivative (PID) controller operates the majority of modern control systems and has applications in many industries; thus any improvement in its design methodology has the potential to have a significant engineering and economic impact. Despite the existence of numerous methods for setting the parameters of PID controllers, the stability analysis of time-delay systems that use PID controllers remains extremely difficult, and there are very few existing results on PID controller synthesis. Filling a gap in the literature, this book is a presentation of recent results in the field of PID controllers, including their design, analysis, and synthesis. The focus is on linear time-invariant plants, which may contain a time-delay in the feedback loop---a setting that captures many real-world practical and industrial situations. Emphasis is placed on the efficient computation of the entire set of PID controllers achieving stability and various performance specifications---both classical (gain and phase margin) and modern (H-infinity norms of closed-loop transfer functions)---enabling realistic design with several different criteria. Efficiency is important for the development of future software design packages, as well as further capabilities such as adaptive PID design and online implementation. Additional topics and features include: * generalization and use of results—due to Pontryagin and others—to analyze time-delay systems * treatment of robust and nonfragile designs that tolerate perturbations * examination of optimum design, allowing practitioners to find optimal PID controllers with respect to an index * study and comparison of tuning techniques with respect to their resilience to controller parameter perturbation * a final chapter summarizing the main results and their corresponding proposed algorithms The results presented here are timely given the resurgence of interest in PID controllers and will find widespread application, specifically in the development of computationally efficient tools for PID controller design and analysis. Serving as a catalyst to bridge the theory--practice gap in the control field as well as the classical--modern gap, this monograph is an excellent resource for control, electrical, chemical, and mechanical engineers, as well as researchers in the field of PID controllers.