Multi-domain optimization of the eigenstructure of controlled underactuated vibrating systems
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INDUSTRIAL APPLICATION PAPER
Multi-domain optimization of the eigenstructure of controlled underactuated vibrating systems Roberto Belotti 1 & Dario Richiedei 2
&
Alberto Trevisani 2
Received: 22 April 2020 / Revised: 18 June 2020 / Accepted: 31 July 2020 # The Author(s) 2020
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. Keywords Optimal design . Eigenstructure assignment . Structural modification . Active control . Rank minimization
1 Introduction 1.1 Performance optimization through concurrent mechanical and control design Dynamic structural optimization in mechanisms and structures, often denoted as dynamic response topology optimization, aims at finding the optimal mass and stiffness Responsible Editor: Somanath Nagendra * Dario Richiedei [email protected] 1
Faculty of Science and Technology, Free University of Bozen-Bolzano, Piazza Università 5, Bolzano, Italy
2
Dipartimento di Tecnica e Gestione dei Sistemi Industriali, Università degli Studi di Padova, Stradella S. Nicola 3, 36100 Vicenza, Italy
distribution for obtaining the desired dynamic performances, by optimizing a cost function while satisfying constraints on the feasible parameters. The crux is defining such a cost function, or performance index, that properly rep
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