Abstract

Lumped-distributed control systems are collections of interacting sub-systems, some of which have finite dimensional vector state spaces (comprising ‘lumped’ components) and some of which have infinite dimensional vector state spaces (comprising ‘distributed’ components). Lumped-distributed control systems are encountered, for example, in models of thermal or distributed mechanical devices under boundary control, when we take account of control actuator dynamics or certain kinds of dynamic loading effects. This talk will focus on an important class of (possibly non-linear) lumped-distributed control systems, in which the control action directly affects only the lumped sub-systems and the output is a function of the lumped state variables alone. We give examples of such systems, including a temperature-controlled test bed for measuring semiconductor material properties under changing temperature conditions and robot arms with flexible links. A key observation is that there exists an exact representation of the mapping from control inputs to outputs, in terms of a finite dimensional control system with memory. (We call it the reduced system representation). The reduced system representation can be seen as a time-domain analogue of frequency response descriptions involving the transfer function from input to output. It is however much broader because, in contrast to frequency response descriptions, the reduced system representation allows non-linear dynamics, hard constraints on controls and outputs and non-zero initial data. We shall report on recent case studies that illustrate the computational advantages of the reduced system representation. We show that, for related output tracking improved tracking and reduction in computation time, as compared with traditional methods, based on the approximation of infinite dimensional state spaces by high dimensional linear subspaces.