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.
