Robust control of hyperbolic PDE systems

Abstract

This work concerns systems of quasi-linear first-order hyperbolic partial differential equations (PDEs) with uncertain variables and unmodeled dynamics. For systems with uncertain tain variables, the problem of complete elimination of the effect of uncertainty on the output via distributed feedback (uncertainty decoupling) is initially considered; a necessary and sufficient condition for its solvability as well as explicit controller synthesis formulas are derived. Then, the problem of synthesizing a distributed robust controller that achieves asymptotic output tracking with arbitrary degree of attenuation of the effect of uncertain variables on the output of the closed-loop system is addressed and solved. Robustness with respect to unmodeled dynamics is studied within a singular perturbation framework. It is established that controllers which are synthesized on the basis of a reduced-order slow model, and achieve uncertainty decoupling or uncertainty attenuation, continue to enforce these objectives in the presence of unmodeled dynamics, provided that they are stable and sufficiently fast. The developed controller synthesis results are successfully implemented through simulations on a fixed-bed reactor, modeled by two quasi-linear first-order hyperbolic PDEs, where the reactant wave propagates through the bed with significantly larger speed than the heat wave, and the heat of reaction is unknown and time varying.