Robust Stability and Robust Stabilizability

Abstract

Most real systems cannot be represented by linear dynamics, but sometimes, under some assumptions, it is possible to model the dynamical behavior of practical systems with a linear model having some uncertainties. The presence of these uncertainties in the dynamics requires the establishment of robust conditions that can guarantee the stability and/or the stabilizability of the practical system under study. This topic has in fact dominated the research effort of the control community during the last two decades. Among the contribution on this area of research we quote the work of [77, 105, 114] on robust stability and the work of [49, 108, 110, 118, 128, 155, 189, 190, 206, 207, 215] on robust stabilizability. This chapter will deal with the robust stability and robust stabilizability of the class of uncertain continuous-time linear time delay systems. Our results are mainly based on the Lyapunov second method. In some sense, given an uncertain dynamical system with time delay, we answer the following questions: How can we check whether the unforced nominal system with time delay is stable or not?How can we check if the unforced uncertain dynamical system with time delay is robust stable for all admissible uncertainties or not?When the unforced uncertain dynamical system with time delay is unstable, how can we design a memoryless state feedback, or memory state feedback or output feedback controller to stabilize the system and guarantee that it will remain stable for all admissible uncertainties?