Control and stabilization of decentralized systems

Abstract

The control and stabilization of decentralized systems attracted a great deal of attention in the late 1970s and 1980s, in part because the microprocessor revolution made it feasible to envision the control of large, complex systems. Most of the stability research undertaken during this era was analytic, focusing for example on various interpretations of weak or strong coupling between subsystems, the existence of multiple time-scales, etc. Motivated by the so-called ‘twin-lift’ problem, a problem of stabilizing the motion of two helicopters manoeuvring a single load, we develop a methodology for stabilizing classes of decentralized systems based on a more algebraic approach involving the external symmetries of decentralized systems. The study of the symmetry algebra of classes of such systems was initiated by Hazewinkel and Martin and in this language, we derive stabilizing, local feedback laws for any class of decentralized systems having a semi-simple algebra of symmetries. The twin-lift problem as well as certain problems involving stabilization of discretizations of distributed parameter problems have semi-simple algebras of symmetries.