Traveling Wave and Scattering Approaches to Control of Chains

In this talk we introduce a powerful and convenient framework for modeling, analysis and control synthesis of interconnected lumped (physical) dynamical systems. Although we restrict ourselves to a chain type of an interconnection, the framework can extend to general interconnection topologies. The motivation for this research comes from the physical domains as diverse as robotics, electronics, hydraulics, and even vehicular platooning. We borrow from the well developed areas of distributed parameter systems such as transmission lines, flexible mechanical structures and (thermo)acoustic systems and show that numerous phenomena in chains of lumped systems can be modeled by superposition of incident and reflected traveling waves. The goal of control design can then be formulated as finding the matching impedance. We will carefully discuss various concepts of impedance matching. We will also show how the scattering description well mastered by electrical engineers allows us to to transition from physical variables to wave variables. This, in turn, prepares the ground for the popular H-infinity optimal control framework. Although all the above mentioned concepts are classical, it may be useful to see them formulated in the language of control theory and it may come as a surprise to see that they can be applied in control application domains such as vehicular platooning. In the talk we also compare these results with the relatively new framework based on the so-called wave transfer functions as invented and promoted by W. O'Connor in the field of flexible robotic manipulators. In fact, we will show that these are equivalent.