On the Relationship between the Absolute Vibration Suppression and Back-Stepping Methods in Control of the Wave Equation with Possibly Unstable Boundary Conditions

In this paper we compare two methods in control of the wave equation with a possibly destabilizing boundary force. Non-collocated actuation and measurement of boundary position is assumed. The Absolute Vibration Suppression (AVS) controller is based on an exact transfer function model and exploits the system traveling wave nature. It eliminates the wave reflections in the system and employs a dead time compensator that shifts the resulting poles to any desired location. The BackStepping (BS) controller is designed in time domain. Its general principle is using dedicated transformations to map the original unstable system to a stable target system. For the system at hand the target system is a wave equation with stable and adjustable boundary conditions. We show that for a certain choice of BS control parameters the methods coincide.