An Extension of the Invariance Principle for Switched Affine System
DOI:
https://doi.org/10.5540/tema.2020.021.01.171Keywords:
Switched affine system, invariance principle, dwell-time, attractor set.Abstract
In this paper, an approach to investigate switched affine system via matrix inequalities is presented. Particularly, an extension of LaSalle’s invariance principle for this class of systems under arbitrary dwell-time switching signal is presented. The proposed results employ a common auxiliary scalar function and also multiple auxiliary scalar functions to study the asymptotic behavior of switched solutions and estimate their attractors for any dwell-time switching signal. A specific feature of these results is that the derivative of the auxiliary scalar functions can assume positive values in some bounded sets. Moreover, a problem of constrained optimization is formulated to numerically determine the auxiliary scalar functions and minimize the volume of the estimated attractor. Numerical examples show the potential of the theoretical results in providing information on the asymptotic behavior of solutions of the switched affine systems under arbitrary dwell-time switching signals.
References
H. Lin and P. J. Antsaklis, “Stability and stabilizability of switched linear
systems: A survey of recent results,” IEEE Transactions on Automatic Control,vol. 54, pp. 308–322, Feb 2009.
M. Branicky, “Multiple lyapunov functions and other analysis tools for switched and hybrid systems,” Automatic Control, IEEE Transactions on, vol. 43,pp. 475–482, Apr 1998.
D. Liberzon and A. Morse, “Basic problems in stability and design of switched systems,” Control Systems, IEEE, vol. 19, pp. 59–70, Oct 1999.
A. Bacciotti and F. Ceragioli, “Stability and stabilization of discontinuous systems and nonsmooth lyapunov functions,” ESAIM: Control Optimisation and Calculus of Variations 4, 1999.
R. Kuiava, R. A. Ramos, H. R. Pota, and L. F. Alberto, “Practical stability
of switched systems without a common equilibria and governed by a timedependent switching signal,” European Journal of Control, vol. 19, no.3, pp. 206 – 213, 2013.
H. Rodrigues, L. Alberto, and N. Bretas, “On the invariance principle: generalizations and applications to synchronization,” Circuits and Systems I:Fundamental Theory and Applications, IEEE Transactions on, vol. 47, no. 5,pp. 730–739, 2000.
L. F. C. . B. N. G. RODRIGUES, Hildebrando Munhoz ; ALBERTO, “Uniform invariance principle and synchronization. robustness with respect to parameter variation.,” Journal of Differential Equations, vol. 169, no. 1, pp. 228–254, 2001.
L. Alberto, T. Calliero, and A. Martins, “An invariance principle for nonlinear discrete autonomous dynamical systems,” Automatic Control, IEEE Transactions on, vol. 52, no. 4, pp. 692–697, 2007.
W. Raffa and L. Alberto, “A uniform invariance principle for periodic systems with applications to synchronization,” Systems & Control Letters, vol. 97, pp. 48 – 54, 2016.
M. Valentino, V. Oliveira, L. Alberto, and D. Azevedo, “An extension of the invariance principle for dwell-time switched nonlinear systems,” Systems & Control Letters, vol. 61, no. 4, pp. 580 – 586, 2012.
A. Bacciotti and L. Mazzi, “An invariance principle for nonlinear switched systems,” Systems & Control Letters, vol. 54, no. 11, pp. 1109 – 1119, 2005.
T. Pinto, L. Alberto, and M. Valentino, “Uma extensão do princípio de invariancia para sitemas chaveados afins,” Anais do XXXVI Congresso Nacional de Matemática Aplicada e Computacional, 2017.
T. Pinto, L. Alberto, and M. Valentino, “Uma extensão do princípio de invariancia para sitemas chaveados afins via múltiplas funções auxiliares,” Anais do XXXVI Congresso Nacional de Matemática Aplicada e Computacional, 2018.
D. Liberzon, Switching in Systems and Control. Birkhäuser Basel, 2003.
T. Pinto, L. Alberto, and M. Valentino, “Uma extensão do princípio de invariancia para sitemas chaveados afins via múltiplas funções auxiliares,” Anais do XXXVII Congresso Nacional de Matemática Aplicada e Computacional ,
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish in this journal agree to the following terms:
Authors retain copyright and grant the journal the right of first publication, with the work simultaneously licensed under the Creative Commons Attribution License that allows the sharing of the work with acknowledgment of authorship and initial publication in this journal.
Authors are authorized to assume additional contracts separately, for non-exclusive distribution of the version of the work published in this journal (eg, publish in an institutional repository or as a book chapter), with acknowledgment of authorship and initial publication in this journal.
Authors are allowed and encouraged to publish and distribute their work online (eg, in institutional repositories or on their personal page) at any point before or during the editorial process, as this can generate productive changes as well as increase impact and the citation of the published work (See The effect of open access).
This is an open access journal which means that all content is freely available without charge to the user or his/her institution. Users are allowed to read, download, copy, distribute, print, search, or link to the full texts of the articles, or use them for any other lawful purpose, without asking prior permission from the publisher or the
author. This is in accordance with the BOAI definition of open access
Intellectual Property
All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License under attribution BY.