PDE Control: Designs and Applications

Abstract

In addition to classical physical applications such as fluid flows in engines, thermal dynamics in buildings, flexible wings of aircraft, electrochemistry in batteries, or plasmas in lasers and tokamaks, PDEs are effective in modeling large multi-agent systems as continua of networked agents, with applications ranging from vehicle formations to opinion dynamics. In its early period PDE control focused on replicating linear control methods (pole placement, LQG, H-infinity, etc) in infinite dimension. Over the last 15 years, a continuum version of the "backstepping" method has given rise to control design tools for nonlinear PDEs and PDEs with unknown functional coefficients. Backstepping designs now exist for each of the major PDE classes (parabolic, hyperbolic, real- and complex-valued, and of various orders in time and space). As a special case, continuum backstepping compensates delays of arbitrary length and dependence on time in general nonlinear ODE control systems. I will present a few design ideas and several applications, including deep oil drilling (where a large parametric uncertainty occurs), extruders in 3D printing (where a large delay is a nonlinear function of the value of the state), and deployment of 2D meshes of agents in 3D space (where deployment into complex surfaces necessarily gives rise to coupled unstable PDEs).