Luca Fenzi
PhD in Engineering Science
Expert in Computational Mathematics



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Stabilization of Uncertain Time-Delay Systems Polynomial (Chaos) Approximation

Stability and Stabilization of Periodic Time-Delay Systems


Stability optimization of uncertain time-delay systems
MATLAB package

  • Author
  • The software was written by Fenzi Luca under supervision of Wim Michiels.


  • Corresponding articles
  • L. Fenzi and W. Michiels, Robust stability optimization for linear delay systems in a probabilistic framework, Linear Algebra and its Application, 526:1-26 (2017).
    (Preprint.)L. Fenzi and W. Michiels, Robust stability optimization for linear delay systems in a probabilistic framework, Internal report TW 671, Department of Computer Science, KU Leuven, August 2016.

  • Download
  • Download the software. All the codes are commented and there are several examples, including a model of an experimental heat-exchanger. The software is released under the GNU GPL v3.0 license.

  • All scientific publications, for which the Matlab package has been used, must refer to the above publications.

  • Description of the method
  • This eigenvalue-based stabilization method tunes the controller parameters (static or dynamic feedbacks) in order to improve the stability properties of a time-delay system, which can be affected by uncertainties, modeled by the realizations of a random vector. The closed-loop system is described by a delay differential algebraic equations (DDAE) of retarded type, in this way we can take into account integral controller and distributed terms. The time-delay system may non-linearly depend on the controller and uncertain parameters.

    In order to take into account the uncertainty, the stabilization minimizes an objective function consisting of the mean of the spectral abscissa with a variance penalty. For every realization of the uncertainties, the spectral abscissa (real part of the rightmost eigenvalue) is computed with the Infinitesimal Generator Approach, and then corrected with Newton's method. The objective function and its gradient are numerically evaluated by computing integrals using quasi-Monte Carlo methods. The minimization of the objective function relies on the software HANSO (Hybrid Algorithm for Non Smooth Optimization).

  • Acknowledgments
  • Special thanks goes to Dan Pilbauer for his careful tests and constructive comments.

    Funding: This work was supported by the project C14/17/072 of the KU Leuven Research Council, by the project G0A5317N of the Research Foundation-Flanders (FWO - Vlaanderen), and by the project UCoCoS, funded by the European Unions Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Grant Agreement No 675080.



Polynomial (chaos) approximation of maximum eigenvalue functions
MATLAB tutorial

  • Authors
  • The software was written by Fenzi Luca under supervision of Wim Michiels.


  • Corresponding articles
  • L. Fenzi and W. Michiels. Polynomial (chaos) approximation of maximum eigenvalue functions: efficiency and limitations, Numerical Algorithms, 82:1143–1169 (2019).
    (Preprint) L. Fenzi and W. Michiels. Polynomial (chaos) approximation of maximum eigenvalue functions: efficiency and limitations, arXiv: 1804.03881 (2018).

  • Download
  • Download the tutorial. This tutorial is also available at TW internal reports KU Leuven.
    Download the Matlab script. This script generates the latex tutorial by the Matlab code publish.

  • All scientific publications, for which the tutorial has been used, must refer to the tutorial, and the above publications.

  • Abstract of the Tutorial
  • This tutorial reviews the numerical experiments contained in the article, Fenzi & Michiels (2019) "Polynomial (chaos) approximation of maximum eigenvalue functions: efficiency and limitations", providing a template that can be modified for explorations of your own.

    The tutorial explores the polynomial approximation of smooth, non-differentiable and non-Lipschitz continuous functions in the univariate and bivariate cases. The analyzed functions arise from parameter eigenvalue problems; in particular, they are the real part of the rightmost eigenvalue (the so-called spectral abscissa).

    The polynomial approximations are obtained by Galerkin and collocation approaches. In the Galerkin approach, the numerical approximation of the coefficients in the univariate case is achieved by extended (or composite) Trapezoidal and Simpson's rules or by Gauss and Clenshaw-Curtis quadrature rules. For the bivariate case, the coefficients are approximated by tensorial and non-tensorial Clenshaw-Curtis cubature rules, based on tensor-product Chebyshev grid and Padua points, respectively. The collocation approach interpolates the function on Chebyshev points in the univariate case, while for the bivariate case the interpolant nodes are given by tensor-product Chebyshev grid and Padua points.

  • Funding: This work was supported by the project C14/17/072 of the KU Leuven Research Council, by the project G0A5317N of the Research Foundation-Flanders (FWO - Vlaanderen), and by the project UCoCoS, funded by the European Unions Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Grant Agreement No 675080.



Stability and Stabilization of Periodic Time-Delay Systems
MATLAB package

  • Authors
  • The software was written by Fenzi Luca under supervision of Wim Michiels.


  • Corresponding articles
  • W. Michiels and L. Fenzi. Spectrum-based stability analysis and stabilization of a class of time-periodic time delay systems, arXiv: 1804.03881 (2019).

  • Download
  • Download the software. All the codes are commented and there are several examples.
    The software is released under the GNU GPL v3.0 license.

  • All scientific publications, for which the Matlab package has been used, must refer to the above publications.

  • Description of the software
  • The software deals with spectrum-based stability assessment and stabilization methods for periodic linear time-delay systems, where the delays and the period of the system matrices are commensurate. The software relies on the dual interpretation of the Floquet multipliers, which can be characterized as non-zero eigenvalues either of the monodromy operor or of a finite-dimensional nonlinear eigenvalue problem, whose solution is obtained by solving an initial value problem.

    The stability assessment of a periodic time-delay system is determined by the largest in modulus Floquet multiplier. This Floquet multiplier is first approximated by a spectral collocation of the monodromy operator eigenproblem. Hence, its precision is refined up to machine precision by Broyden's method, considering the finite-dimensional nonlinear eigenvalue problem.

    The derivate of the Floquet multiplier with respect to system parameter can also be computed. The derivative computation is used in the stabilization method, which designs targeted parameters of the time-delay systems in order to minimize the modulus of the largest Floquet multipliers. The stabilization method requires the usage of the software HANSO (Hybrid Algorithm for Non Smooth Optimization).

    The package presents additional routines for the stability and stabilization of large-scale periodic time-delay systems. These large-scale routines solve with Arnoldi's method the discretized aigenvalue problem arising from the monodromy operator. Moreover, they consider a dual time-systems for the Floquet multiplier derivative computation.

  • Funding: This work was supported by the project C14/17/072 of the KU Leuven Research Council, by the project G0A5317N of the Research Foundation-Flanders (FWO - Vlaanderen), and by the project UCoCoS, funded by the European Unions Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Grant Agreement No 675080.