This monograph presents our recent results on the proportional-integr- derivative (PID) controller and its design, analysis, and synthesis. The fo cus is on linear time-invariant plants that may contain a time delay in the feedback loop. This setting captures many real-world practical and in dustrial situations. The results given here include and complement those published in Structure and Synthesis of PID Controllers by Datta, Ho, and Bhattacharyya [10]. In [10] we mainly dealt with the delay-free case. The main contribution described here is the efficient computation of the entire set of PID controllers achieving stability and various performance specifications. The performance specifications that can be handled within our machinery are classical ones such as gain and phase margin as well as modern ones such as Hoo norms of closed-loop transfer functions. Finding the entire set is the key enabling step to realistic design with several design criteria. The computation is efficient because it reduces most often to lin ear programming with a sweeping parameter, which is typically the propor tional gain. This is achieved by developing some preliminary results on root counting, which generalize the classical Hermite-Biehler Theorem, and also by exploiting some fundamental results of Pontryagin on quasi-polynomials to extract useful information for controller synthesis. The efficiency is im portant for developing software design packages, which we are sure will be forthcoming in the near future, as well as the development of further capabilities such as adaptive PID design and online implementation.
