Defining Shapeability in Eigenstate Specification for Linear Systems

Abstract

This paper introduces a metric called degree of shapeability to determine whether or not it is possible to assign a set of desired eigenstates to a particular linear dynamic system. Specifying an eigenstate, which consists of an eigenvalue and its associated eigenvector, can be essential in designing certain vibrating structures and undulating mechanisms. In general it may be desirable to specify more than one eigenstate. The number of eigenstates that can be specified for a particular linear system is not always immediately evident, however, because of physical and kinematic constraints, as well as actuator placement. To quantify the potential for active feedback and/or passive parameter tuning to influence a system’s eigenstates, we identify a metric, the degree of shapeability, that relates the structure of the linear system to the number of eigenstates that a designer may fully define. We investigate the effect of system equality and inequality constraints on the ability to specify eigenstates for linear dynamic systems.

Embedding Desired Eigenstates into Active and Passive Dynamics of a Linear, Underactuated Feedback System

“…In short, the codesign approach permits the underactuated system to access nearly the entire design space accessible by a fully actuated system. A corollary implication is that codesign could enable the identification and removal of nonessential actuators, allowing for highly articulated mechanical systems with lower cost, weight and complexity [30].…”

“…The algorithm sets all N eigenvalues. This step implicitly assumes that the system model is controllable in the sense of codesign, where controllable means that all eigenvalues can be set by tuning either feedback or physical parameters [30]. System stability is guaranteed (so long as the user specified eigenvalues k ii are all placed in the left-half plane).…”