The transfer-function approach to travelling waves in path graphs

The paper presents a novel approach for the analysis and control of distributed systems with an underlying path-graph topology based on the so-called wave-based control perspective. With the help of irrational wave transfer functions, the approach allows us to identify and describe the travelling waves in heterogenous path graphs. The main advantage of this approach is that it shows how a change in a system’s state propagates along the graph and how it is affected by graph heterogeneities. A graph heterogeneity acts as a boundary for wave propagation. Undesired effects of the boundary, such as amplification/attenuation, long transient or string instability, can be compensated by the feedback controllers introduced in this paper. Two types of path-graph boundaries, the soft and hard boundaries, are mathematically investigated. The type of the boundary determines the character of wave propagation in a heterogenous path graph. Each boundary is described by four boundary transfer functions, expressed in terms of the wave transfer functions. The properties of the boundary transfer functions are influenced by those of the wave transfer functions, for instance, the boundedness of the DC gains.