Optimization methods for the quadratic eigenvalue assignment with bounded feedback controller

Mostafa, El-Sayed M.E.; Abdelkader, Shayma Y.; Anis, Mohamed A.

doi:10.21608/ejmaa.2025.399201.1365

Optimization methods for the quadratic eigenvalue assignment with bounded feedback controller

Document Type : Regular research papers

Authors

¹ Department of Mathematics and Computer Science, Faculty of Science, Alexandria University

² Teaching Assistant, German International University

10.21608/ejmaa.2025.399201.1365

Abstract

Abstract
This work considers the quadratic eigenvalue assignment problem for vibrating structures by state feedback. The problem in its original form of a linear time-invariant quadratic control system is restated as a linear control system. The considered minimization problem is the sum of the squares of the residuals of the eigenvalues of the closed-loop system matrix and the desired eigenvalues. This objective function of eigenvalues is characterized as semi-smooth. A Levenberg-Marquardt method that uses a nonmonotone trust region combined with a line search backtracking strategy is proposed to tackle the problem. Global convergence and local superlinear/quadratic convergence rates of the algorithm are established.
By incorporating an upper bound on the computed feedback controllers, a logarithmic barrier interior-point method is addressed to tackle the considered inequality constrained problem. Numerical results are given to demonstrate the performance of proposed methods. The article is concluded by some comments related to the considered methods. An appendix is given including the required derivatives of the objective function.

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