ODEs with Heat PDE Actuator Dynamics

Abstract

In this chapter we use the backstepping approach from Chapter 2 to expand the scope of predictor feedback and build a much broader paradigm for the design of control laws for systems with infinite-dimensional actuator dynamics, as well as for observer design for systems with infinite-dimensional sensor dynamics.

In this chapter we address the problems of compensating for the actuator and sensor dynamics dominated by diffusion, i.e., modeled by the heat equation. Purely convective/first-order hyperbolic PDE dynamics (i.e., transport equation or, simply, delay) and diffusive/parabolic PDE dynamics (i.e., heat equation) introduce different problems with respect to controllability and stabilization. On the elementary level, the convective dynamics have a constant-magnitude response at all frequencies but are limited by a finite speed of propagation. The diffusive dynamics, when control enters through one boundary of a 1D domain and exits (to feed the ODE) through the other, are not limited in the speed of propagation but introduce an infinite relative degree, with the associated significant roll-off of the magnitude response at high frequencies.