Multi-domain optimization of the eigenstructure of controlled underactuated vibrating systems

Abstract

The paper proposes a multi-domain approach to the optimization of the dynamic response of an underactuated vibrating linear system through eigenstructure assignment, by exploiting the concurrent design of the mechanical properties, the regulator and state observers. The approach relies on handling simultaneously mechanical design and controller synthesis in order to enlarge the set of the achievable performances. The underlying novel idea is that structural properties of controlled mechanical systems should be designed considering the presence of the controller through a concurrent approach: this can considerably improve the optimization possibilities. The method is, first, developed theoretically. Starting from the definition of the set of feasible system responses, defined through the feasible mode shapes, an original formulation of the optimality criterion is proposed to properly shape the allowable subspace through the optimal modification of the design variables. A proper choice of the modifications of the elastic and inertial parameters, indeed, changes the space of the allowable eigenvectors that can be achieved through active control and allows obtaining the desired performances. The problem is then solved through a rank-minimization with constraints on the design variables: a convex optimization problem is formulated through the “semidefinite embedding lemma” and the “trace heuristics”. Finally, experimental validation is provided through the assignment of a mode shape and of the related eigenfrequency to a cantilever beam controlled by a piezoelectric actuator, in order to obtain a region of the beam with negligible oscillations and the other one with large oscillations. The results prove the effectiveness of the proposed approach that outperforms active control and mechanical design when used alone.

Integrated Inverse Dynamics and Optimized Mechanical Design in Underactuated Linear Vibratory Feeders Under Periodic Excitation

“…It should be noted that also the use of feedback control is not effective in imposing the correct shape of vibration because of the limitation in assigning eigenvectors, and hence the vibrational mode shape, in underactuated systems [31,32].…”

A comprehensive review of educational articles on structural and multidisciplinary optimization

“…The design variables are determined by the problems on hand, e.g., height and width of cross sections for a cantilever beam. Belotti et al (2021) proposed a multi-domain approach to optimize the dynamic response of vibrating linear systems under actuation by varying the values of lumped masses. The corresponding MATLAB implementations are provided.…”

Robust Assignment of Natural Frequencies and Antiresonances in Vibrating Systems through Dynamic Structural Modification

“…EA has been tackled through passive, active, and hybrid approaches (see, e.g., [2][3][4]). Active approaches exploit the energy introduced in the system by external actuators based on the information of some sensors to perform the EA.…”