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Reseach Article

Stabilizing Controller Design for a Special Class of PWA Systems using Discontinuous Piecewise Quadratic Lyapunov Functions

by Hesam Sajjadi, Reyhaneh Kardehi Moghaddam, Najmeh Eghbal
International Journal of Computer Applications
Foundation of Computer Science (FCS), NY, USA
Volume 113 - Number 8
Year of Publication: 2015
Authors: Hesam Sajjadi, Reyhaneh Kardehi Moghaddam, Najmeh Eghbal
10.5120/19848-1714

Hesam Sajjadi, Reyhaneh Kardehi Moghaddam, Najmeh Eghbal . Stabilizing Controller Design for a Special Class of PWA Systems using Discontinuous Piecewise Quadratic Lyapunov Functions. International Journal of Computer Applications. 113, 8 ( March 2015), 26-31. DOI=10.5120/19848-1714

@article{ 10.5120/19848-1714,
author = { Hesam Sajjadi, Reyhaneh Kardehi Moghaddam, Najmeh Eghbal },
title = { Stabilizing Controller Design for a Special Class of PWA Systems using Discontinuous Piecewise Quadratic Lyapunov Functions },
journal = { International Journal of Computer Applications },
issue_date = { March 2015 },
volume = { 113 },
number = { 8 },
month = { March },
year = { 2015 },
issn = { 0975-8887 },
pages = { 26-31 },
numpages = {9},
url = { https://ijcaonline.org/archives/volume113/number8/19848-1714/ },
doi = { 10.5120/19848-1714 },
publisher = {Foundation of Computer Science (FCS), NY, USA},
address = {New York, USA}
}
%0 Journal Article
%1 2024-02-06T22:50:26.672816+05:30
%A Hesam Sajjadi
%A Reyhaneh Kardehi Moghaddam
%A Najmeh Eghbal
%T Stabilizing Controller Design for a Special Class of PWA Systems using Discontinuous Piecewise Quadratic Lyapunov Functions
%J International Journal of Computer Applications
%@ 0975-8887
%V 113
%N 8
%P 26-31
%D 2015
%I Foundation of Computer Science (FCS), NY, USA
Abstract

In this paper a new controller is proposed to stabilize an especial class of hybrid piecewise affine systems. In this study, for the first time, the stabilizing controller is designed based on discontinuous piecewise quadratic Lyapunov functions which decrease the conservation and propose a wider class of applicable Lyapunov functions as it omits the continuity condition in boundary points compared to continuous piecewise quadratic Lyapunov functions. In addition the stability conditions are formulated in the form of Bilinear Matrix Inequalities (BMI) problem. To solve the proposed problem, BMI is defined in the form of a multi-objective nonlinear optimization problem which has been solved through using genetic Algorithm (GA).

References
  1. Hamed Molla Ahmadian, Ali Karimpour, Naser Pariz , "Stabilization and Control of Switched Linear Systems with State-Input Logic Constrained: LMI Approach" , Journal of Control , Vol. 5, No. 2, Summer 2011.
  2. N. Eghbal, N. Pariz, N. Karimpour, "Discontinuous piecewise quadratic Lyapunov functions for planar piecewise affine systems", Elsevier Journal of Mathematical Analysis and Applications , Volume 399, Issue 2 , Pages 586–593 , 2012.
  3. Najme Eghbal, Modelling class of parametric nonlinear systems as nondefinite section, Ph. D. thesis, Ferdosi University of Mashhad, 110 pages, February 2012.
  4. P. Julian, A. Desages, O. Agamennoni, "High -level canonical piecewise linear representation using a simplicial partition", IEEE Trans. Circuits Syst. I: Fund. Theory Appl. 46 (4) (1999) 463–480.
  5. P. Julian, A. Desages, B. D'Amico, "Orthonormal high level canonical PWL functions with applications to model reduction ," IEEE Tran sactions on Circuits and Systems I, vol. 47, no. 5, pp 702–712 , 2000.
  6. M. S. Branicky, "Studies in Hybrid Systems : Modeling , Analysis and Control ", Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, PHD thesis , June 1995.
  7. J. N. Lin, R. Unbehauen, "Canonical piecewise-linear approximations", IEEE Trans. Circuits Syst. I: Fund. Theory Appl. Vol 39, no 8, pp 697– 699, 1992 .
  8. M. Storace, O. D. Feo, "PWL approximation of nonlinear dynamical systems, part I: structural stability", J. Phys. Conf. Ser, Vol 22, pp 208–221, 2005.
  9. P. Julian, A. Desages, O. Agamennoni ," High-level canonical piecewise linear representation using a simplicial partition", IEEE Trans. Circuits Syst. I:Fund . Theory App l, vol. 46, issue 4, pp 463–480, 1999.
  10. N. Eghbal, N. Pariz, N. Karimpour, " Uniform modeling of parameter dependent nonlinear systems", Journal of Zhejiang University SCIENCE . vol. 13, issue11, pp 850-858 , nov 2012 .
  11. W. P. M. H. Heemels, B. De Schutter, A. Bemporad, "Equivalence of hybrid dynamical models", Automatica , Delft University of Technology, vol 37, no 7, pp 1085–1091 , july 2001.
  12. A. Bemporad , " Efficient conversion of mixed logical dynamical systems into an equivalent piecewise affine form", IEEE Trans Automat Control , vol 49, no 5 , pp 832–838 , 2004 .
  13. M. Johansson, A. Rantzer , "Computation of piecewise quadratic Lyapunov functions for hybrid systems", IEEE Transactions on Automatic Control, vol 43, no 4, pp 555–559, 1998 .
  14. L. Rodrigues,"Dynamic output feedback controller synthesis for piecewise affine systems," Ph. D thesis , Stanford University, Jun 2002.
  15. B. Samadi, "Stability Analysis and Controller Synthesis for a Class of Piecewise Smooth Systems", Ph. D thesis , Concordia University , 2008 .
  16. Boyd s and Ghaoui L. E and Feron E and Balakrishnan V," linear matrix inequalities in system and control theory ", Society for Industrial and Applied Mathematics, University City Science Center, Philadelphia , vol 15, 1994
  17. Kai liu , jianghai hu , yu yao , baoqing yang , xin huo,"Stability analysis of discrete-time piecewise-linear systems: a generating function approach", international journal of control ,automation and systems, volume12, issue5, pp 1005-1010, 30Aug 2014 .
  18. K. S. Narendra and J. Balakrishnan . "A Common Lyapunov Function for Stable Systems with Commuting A Matrices". IEEE Transactions on Automatic Control , vol 39, issue 12, pages 2469-2471 , Dec 1994 .
  19. Guisheng zhai, ryuuen kou, joe imae, tomoaki kobayashi ,"stability analysis and design for switched descriptor systems", international journal of control , automation and systems , volume7 , issue3 , pp 349-355 , 30 may 2009 .
  20. S. Pettersson and B. Lennartson , "Stability and robustness for hybrid systems", Proceedings of the 35th Conference on Decision and Control, Kobe, Japan , vol 2, pp 1202 - 1207 , 1996.
  21. Stephen Boyd, E. F," History of Linear Matrix Inequalities in Control Theory ", in American control conference: Maryland , June 1994 .
  22. M. Johansson , "Analysis of piecewise linear systems via convex optimization: a unifying approach," IFAC World Congress, Beijing, China, July 1999.
  23. J. Xu, and L. Xie, "Homogeneous polynomial Lyapunov functions for piecewise affine systems," Proceedings of the IEEE American Control Conference , vol 1 , pp 581 - 586 , June 2005 .
  24. D. Mignone, G. Ferrari-Trecate, and M. Morari. "Stability and Stabilization of Piecewise Affine and Hybrid Systems: An LMI Approach". Technical Report AUT00-12 , Automatic Control Laboratory , ETH Zurich, Switzerland , vol 1, pp 504- 509 , Dec 2000.
  25. M. Johansson ," Piecewise Linear Control " Systems , Springer , Berlin , vol 284 , 2003 .
  26. Liu Kai, Yao Yu, Sund engfeng, Venkataramanan Balakrishnan,"PWA State Feedback Controller Synthesis for Piecewise Linear Systems", Chinese control conference , pp 1252 – 1257 , July 2011.
  27. Liu Kai,Yao Yu, Sun Dengfeng,Venkataramanan Balakrishnan ," Improved State Feedback Controller Synthesis For Piecewise Linear Systems ", International Journal of Innovative Computing , Information and Control , Volume 8 , Number 9 , september 2012 .
  28. Ya-quan ma , Xin Jin , Ben Niu , yanyan liu , "robust tracking control of switched nonlinear systems via the multiple lyapunov function approach",IEEE Chinese control and Decision conference (CCDC) ,pp 2113-2118 ,May 2013.
Index Terms

Computer Science
Information Sciences

Keywords

Bilinear matrix inequalities discontinuous piecewise quadratic lyapunov function multiple lyapunov function piecewise affine