Delay-Adaptive Observer-Based Control for Linear Systems with Unknown Input Delays

Abstract

Abstract Laurent Praly’s contributions to adaptive control and to state and parameter estimation are inestimable. Inspired by them, over the last several years we have developed adaptive and observer-based control designs for the stabilization of linear systems that have large and unknown delays at their inputs. In this chapter, we provide a tutorial introduction to this collection of results by presenting several of the most basic ones among them. Among the problems considered are some with measured and some with unmeasured states, some with known and some with unknown plant parameters, some with known and some with unknown delays, and some with measured and some with unmeasured actuator state under unknown delays. We have carefully chosen, for this chapter, several combinations among these challenges, in which estimation of a state (of the plant or of the actuator) and/or estimation of a parameter (of the plant or the delay) is being conducted and such estimates fed into a certainty-equivalence observer-based adaptive control law. The exposition progresses from designs that are relatively easy to those that are rather challenging. All the designs and stability analyses are Lyapunov based. The delay compensation is based on the predictor approach and the Lyapunov functionals are constructed using backstepping transformations and the underlying Volterra integral operators. The stability achieved is global, except when the delay is unknown and the actuator state is unmeasured, in which case stability is local.