Travelling waves in boundary-controlled, non-uniform, cascaded lumped systems

A companion paper in
this conference considers travelling and standing waves in cascaded, lumped,
mass-spring systems, controlled by two boundary actuators, one at each end,
when the system is uniform. It first proposes definitions of waves in finite
lumped systems. It then shows how to control the actuators to establish desired
waves from rest, and maintain them despite disturbances. The present paper
extends this work to the more general, non-uniform case, when mass and spring
values are arbitrary. A special "bi-uniform" case is first studied, consisting
of two different uniform cascaded systems in series, with an obvious,
uncontrolled, impedance mismatch where they meet. The paper shows how boundary
actuator control systems can be designed to establish, and robustly maintain,
apparently pure travelling waves of constant amplitude in either the first or
the second uniform section, in each case with an appropriate standing wave
pattern in the other section. Then a more general non-uniform case is studied.
A definition of a "pure travelling wave" in non-uniform systems is proposed.
Curiously, it does not imply constant amplitude motion. It does however yield
maximum power transfer between boundary actuators. The definition, and its
implementation in a control system, involves extending the notions of "pure"
travelling waves, standing waves, and input and output impedances of sources
and loads, when applied to non-uniform lumped systems. Practical, robust
control strategies are presented for all cases.