Scale Free Bounds on the Amplification of Disturbances in Mass Chains
Kaoru Yamamoto and I have posted a manuscript on the topic of disturbance amplification in mass chains.
Fig 1: Scale free design can be conducted using a graphical method based on the function illustrated above.
The importance of a scale free design method for a mass chain stems from the prevalence of modelling network control problems with mechanical analogues. Examples include both frequency and voltage stability problems in electrical power systems, vehicle platooning, and flocking and consensus phenomena. These applications typically involve very large numbers of subsystems, and the numbers of subsystems is often subject to change. The advantage of a scale free method is that it is easily applied independently of problem size, and any design remains valid even if the number of subsystems changes.
Fig 2: A chain of N masses interconnected by mechanical impedances.
In this paper we develop a method for designing impedances in a mechanical network with the simplest possible topology, that of a string. The key feature of our method is that it can be used to give a single, fixed, design, with provable performance guarantees in mass chains of any length. The design itself can be conducted using a graphical method aided by Fig. 1. We illustrate the approach by designing a bidirectional control law in a vehicle platoon in a manner that is independent of the number of vehicles in the platoon.