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

A companion paper 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 to maintain them despite disturbances. The present paper
extends this work to the more general, non-uniform case, when mass and spring
values can be 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, partial, 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, of standing waves, and of input and output impedances
of sources and loads, when applied to non-uniform lumped systems. Practical,
robust control strategies are presented for all cases.