ABSTRACT
In this chapter, we provide results on the robust passivity analysis and synthesis problems for classes of time-delay systems (TDS) and uncertain timedelay systems (UTDS). This is equally true for continuous-time systems as well as discrete-time systems. For analytical tractability, we consider in the sequel the uncertainties to be time-varying norm-bounded and the delay factor an unknown constant within a prescribed interval. In systems theory, positive real (passivity) theory has played a major role in stability and systems theory [197, 200-202, 205-208]. A summary of the properties of positive real systems is given in Appendix D. The primary motivation for designing strict positive real controllers is for applications to positive real plants. It is well-known that when a strict positive real system is connected to a positive real plant in a negative-feedback configuration, the closed-loop is guaranteed to be stable for arbitrary plant variations as long as the plant remains to be positive real. Although a passivity problem can be converted into a small gain by the so-called Cayley transform [201], direct treatment is often desirable, especially when the system under consideration is subject to uncertainty. A natural problem is to design an internally stabilizing feedback controller such that a given closed-loop system is passive. In the context of linear systems, this problem is referred to as a positive real control problem [202] where a complete solution to the extended strict positive real (ESPR) is developed via two AREs or ARIs. In the context of nonlinear systems, a geometric characterization is introduced in [225] for systems that can be passified by a state-feedback.
