Stability analysis of conewise linear systems with sliding modes

In this paper cone-copositive piecewise quadratic Lyapunov functions (PWQ-LFs) for the stability analysis of conewise linear systems with the possible presence of sliding modes are proposed. The existence of a PWQ-LF is formulated as the feasibility of a cone-copositive programming problem which is represented in terms of linear matrix inequalities with equality constraints. An algorithm for the construction of a PWQ-LF is provided. Examples show the effectiveness of the approach in the presence of stable and unstable sliding modes.

[1]  Gabriele Eichfelder,et al.  Foundations of Set-Semidefinite Optimization , 2010 .

[2]  Reiner Horst,et al.  On generalized bisection of n-simplices , 1997, Math. Comput..

[3]  Zhendong Sun,et al.  Stability of piecewise linear systems revisited , 2010, Annu. Rev. Control..

[4]  Franco Blanchini,et al.  Stabilizability of switched linear systems does not imply the existence of convex Lyapunov functions , 2006, CDC.

[5]  Vadim I. Utkin,et al.  Sliding Modes in Control and Optimization , 1992, Communications and Control Engineering Series.

[6]  M. Branicky.  Multiple Lyapunov functions and other analysis tools for switched and hybrid systems , 1998, IEEE Trans. Autom. Control..

[7]  Antonis Papachristodoulou,et al.  Robust Stability Analysis of Nonlinear Hybrid Systems , 2009, IEEE Transactions on Automatic Control.

[8]  M. Kanat Camlibel,et al.  Conewise Linear Systems: Non-Zenoness and Observability , 2006, SIAM J. Control. Optim..

[9]  Luigi Iannelli,et al.  A cone-copositive approach for the stability of piecewise linear differential inclusions , 2011, IEEE Conference on Decision and Control and European Control Conference.

[10]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[11]  Mirjam Dür,et al.  Copositive Lyapunov functions for switched systems over cones , 2009, Syst. Control. Lett..

[12]  Mirjam Dür,et al.  An improved algorithm to test copositivity , 2012, J. Glob. Optim..

[13]  Morten Hovd,et al.  Relaxing PWQ Lyapunov stability criteria for PWA systems , 2013, Autom..

[14]  W. P. M. H. Heemels,et al.  Input-to-state stability and interconnections of discontinuous dynamical systems , 2008, Autom..

[15]  A. Rantzer,et al.  Piecewise linear quadratic optimal control , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

[16]  Francesco Vasca,et al.  Cone-copositivity for absolute stability of Lur'e systems , 2014, 53rd IEEE Conference on Decision and Control.

[17]  Stephen P. Boyd,et al.  Graph Implementations for Nonsmooth Convex Programs , 2008, Recent Advances in Learning and Control.

[18]  Alexandre Trofino,et al.  Stability analysis of piecewise affine systems with sliding modes , 2014, 2014 American Control Conference.

[19]  Emanuele Garone,et al.  Piecewise quadratic Lyapunov functions over conical partitions for robust stability analysis , 2015 .

[20]  Hai Lin,et al.  Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results , 2009, IEEE Transactions on Automatic Control.

[21]  Javad Mohammadpour,et al.  Control of linear parameter varying systems with applications , 2012 .

[22]  Luis Rodrigues,et al.  A unified dissipativity approach for stability analysis of piecewise smooth systems , 2011, Autom..

[23]  M. Johansson,et al.  Piecewise Linear Control Systems , 2003 .

[24]  Luigi Iannelli,et al.  Cone-Copositive Piecewise Quadratic Lyapunov Functions for Conewise Linear Systems , 2015, IEEE Transactions on Automatic Control.