Minimum norm partial quadratic eigenvalue assignment with time delay in vibrating structures using the receptance and the system matrices

Abstract

a b s t r a c tThe partial quadratic eigenvalue assignment problem (PQEVAP) concerns reassigning a few undesired eigenvalues of a quadratic matrix pencil to suitably chosen locations and keeping the other large number of eigenvalues and eigenvectors unchanged (no spillover). The problem naturally arises in controlling dangerous vibrations in structures by means of active feedback control design. For practical viability, the design must be robust, which requires that the norms of the feedback matrices and the condition number of the closed-loop eigenvectors are as small as possible. The problem of computing feedback matrices that satisfy the above two practical requirements is known as the Robust Partial Quadratic Eigenvalue Assignment Problem (RPQEVAP). In this paper, we formulate the RPQEVAP as an unconstrained minimization problem with the cost function involving the condition number of the closed-loop eigenvector matrix and two feedback norms. Since only a small number of eigenvalues of the open-loop quadratic pencil are computable using the state-of-the-art matrix computational techniques and/or measurable in a vibration laboratory, it is imperative that the problem is solved using these small number of eigenvalues and the corresponding eigenvectors. To this end, a class of the feedback matrices are obtained in parametric form, parameterized by a single parametric matrix, and the cost function and the required gradient formulas for the optimization problem are developed in terms of the small number of eigenvalues that are reassigned and their corresponding eigenvectors. The problem is solved directly in quadratic setting without transforming it to a standard first-order control problem and most importantly, the significant ''no spill-over property'' of the closed-loop eigenvalues and eigenvectors is established by means of a mathematical result. These features make the proposed method practically applicable even for very large structures. Results on numerical experiments show that the proposed method considerably reduces both feedback norms and the sensitivity of the closed-loop eigenvalues. A study on robustness of the system responses of the method under small perturbations show that the responses of the perturbed closed-loop system are compatible with perturbations.& 2009 Published by Elsevier Ltd.

Numerical approach for partial eigenstructure assignment problems in singular vibrating structure using active control

“…That is, for different X 1 , the obtained solutions are not unique. To reduce energy loss in vibrating structure, the minimum norm solutions are often considered (Bai et al, 2010, 2013, 2018; Liu et al, 2019). In fact, the obtained solutions using Algorithm 1 can be of minimum norm for some appropriate initial value matrices.…”

“…Several different active control laws are used to investigate the partial quadratic eigenvalues assignment problems for second-order vibrating structure. The velocity-displacement active control is mainly used for vibrating structure in the earlier studies because of some successful applications in practice (Bai et al, 2010(Bai et al, , 2013Cai et al, 2011;Liu et al, 2019). Recently, the acceleration-velocity active control has attracted a great interesting in second-order vibration system on account of use of accelerometers which can obtain the acceleration signals precisely (Abdelaziz, 2014(Abdelaziz, , 2015Araujo et al, 2016).…”

A Receptance-Based Optimization Approach for Minimum Norm and Robust Partial Quadratic Eigenvalue Assignment

“…Recently, some receptance-based methods have been proposed for the pole assignment problem and the PQEAP (see for instance [4, 22-24, 27-29, 32, 33, 37]). The robust and minimum norm PQEAP is also considered based on the measured receptances and system matrices [1,3,5,34,37].…”

Partial Frequency Assignment for Torsional Vibration Control of Complex Marine Propulsion Shafting Systems

“…Many researchers [21][22][23][24][25] studied this problem based on the stiffness matrix perturbation, the receptance measurements, or a combination of these two. Bai et al [26][27][28] presented hybrid optimization methods for the robust and minimum norm partial quadratic eigenvalue assignment problems, which are suitable for systems with and without time delays. Based on [28], Liu et al [29] proposed a modified conjugate gradient method to solve the problem only using receptance.…”