Robust H∞ control of uncertain switched systems defined on polyhedral sets with Filippov solutions.

This paper considers the control problem of a class of uncertain switched systems defined on polyhedral sets known as piecewise linear systems where, instead of the conventional Carathéodory solutions, Filippov solutions are studied. In other words, in contrast to the previous studies, solutions with infinite switching in finite time along the facets and on faces of arbitrary dimensions are also taken into account. Firstly, established upon previous studies, a set of linear matrix inequalities are brought forward which determines the asymptotic stability of piecewise linear systems with Filippov solutions. Subsequently, bilinear matrix inequality conditions for synthesizing a robust controller with a guaranteed H(∞) performance are presented. Furthermore, these results has been generalized to the case of piecewise affine systems. Finally, a V-K iteration algorithm is proposed to deal with the aforementioned bilinear matrix inequalities. The validity of the proposed method is verified through the analysis of two simulation examples.

[1]  Jonathan P. How,et al.  Parametric robust H/sub /spl infin// controller synthesis: comparison and convergence analysis , 1998, Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207).

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

[3]  Discontinuous control systems , 2008 .

[4]  Munther A. Dahleh,et al.  Global analysis of piecewise linear systems using impact maps and surface Lyapunov functions , 2003, IEEE Trans. Autom. Control..

[5]  Yuzo Ohta,et al.  Stability analysis of uncertain piecewise linear systems using Piecewise Quadratic Lyapunov Functions , 2010, 2010 IEEE International Symposium on Intelligent Control.

[6]  M. Chen,et al.  Linear-matrix-inequality–based approach to H∞ controller synthesis of uncertain continuous-time piecewise linear systems , 2004 .

[7]  J. Cortés.  Discontinuous dynamical systems , 2008, IEEE Control Systems.

[8]  Kenji Kashima,et al.  A linear multi-agent systems approach to diffusively coupled piecewise affine systems: Delay robustness , 2011, IEEE Conference on Decision and Control and European Control Conference.

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

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

[11]  Amir G. Aghdam,et al.  Stability of uncertain piecewise affine systems with time-delay , 2009, 2009 American Control Conference.

[12]  Rafael Wisniewski,et al.  Robust Stability Analysis of Nonlinear Switched Systems with Filippov Solutions , 2012, ROCOND.

[13]  R. Braatz,et al.  A tutorial on linear and bilinear matrix inequalities , 2000 .

[14]  Stephen P. Boyd,et al.  Quadratic stabilization and control of piecewise-linear systems , 1998, Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207).

[15]  M Maarten Steinbuch,et al.  Robust control of piecewise linear systems: A case study in sheet flow control , 2008 .

[16]  N Sukavanam,et al.  Simulation and stability analysis of neural network based control scheme for switched linear systems. , 2012, ISA transactions.

[17]  Anders Rantzer,et al.  Computation of piecewise quadratic Lyapunov functions for hybrid systems , 1997, 1997 European Control Conference (ECC).

[18]  Fernando Paganini,et al.  A Course in Robust Control Theory , 2000 .

[19]  J. Aubin,et al.  Differential inclusions set-valued maps and viability theory , 1984 .

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

[21]  Ali Khaki Sedigh,et al.  Sufficient condition for stabilization of linear time invariant fractional order switched systems and variable structure control stabilizers. , 2012, ISA transactions.

[22]  Shouchuan Hu.  Differential equations with discontinuous right-hand sides☆ , 1991 .

[23]  J. Leth,et al.  On formalism and stability of switched systems , 2012, Journal of Control Theory and Applications.

[24]  Jun-ichi Imura,et al.  Controllability Analysis of Biosystems Based on Piecewise-Affine Systems Approach , 2008, IEEE Transactions on Automatic Control.

[25]  Christopher Edwards,et al.  Advances in variable structure and sliding mode control , 2006 .

[26]  Tamás Terlaky,et al.  A Survey of the S-Lemma , 2007, SIAM Rev..