Algebraic Solution to H2 Control Problems. I. The Scalar Case

A frequently encountered problem in industrial control systems is how to control a plant with significant time delay. Dealing with this problem serves then as a starting point of most design methods, regardless of their configurations. In this paper, we give a treatment for the optimal design problem of control systems with time delay in a quadratic cost setting. This is accomplished by exploiting a simple, yet effective, parametrization of all stabilizing controllers, which allows us to compute the optimal controller for both stable and unstable plants with time delays. Differing from most of other methods, the proposed design procedure is conducted analytically, within the framework of algebra theory. The robustness of the closed-loop system is also discussed. A simple quantitative tuning procedure is developed, which permits the tradeoff between conflict indices. Examples are given to illustrate the proposed method.