Basic Predictor Feedback

Abstract

In this chapter we introduce the basic idea of a PDE backstepping design for systems with input delay.We treat the input delay as a transport PDE, an elementary first–order hyperbolic PDE. Our design yields a classical control formula obtained through various other approaches–modified Smith predictor (mSP), finite spectrum assignment (FSA), and the Artstein–Kwon–Pierson “reduction“ approach. The backstepping approach is distinct because it provides a construction of an infinitedimensional transformation of the actuator state, which yields a cascade system of transformed stable actuator dynamics and stabilized plant dynamics. Our design results in the construction of an explicit Lyapunov–Krasovskii functional and an explicit exponential stability estimate.