Scale-invariant Controller Synthesis in Mass Chains

This paper studies the disturbance suppression problem for a homogeneous mass chain, i.e., a chain of identical point masses interconnected by identical mechanical impedances. The particular focus is placed on whether the disturbance attenuation level of the chain of arbitrary length can be uniformly bounded with a size-independent controller. We explicitly represent the scalar transfer functions from the disturbances to a given intermass displacement as a function of the number of masses. This is an extension of the previous work by the author that established a boundedness of the H-infinity norm when one end of the chain is perturbed. We propose a new method that drastically simplifies its derivation process and provide the complete forms of all the transfer functions of our interest.