A. A. Agrachev and D. Liberzon, Lie-Algebraic Stability Criteria for Switched Systems, SIAM Journal on Control and Optimization, vol.40, issue.1, pp.253-269, 2001.
DOI : 10.1137/S0363012999365704

URL : http://black.csl.uiuc.edu/~liberzon/research/agrachev.ps

M. Alwan, X. Liu, and B. Ingalls, Exponential stability of singularly perturbed switched systems with time delay, Nonlinear Analysis : Hybrid Systems, pp.913-921, 2008.
DOI : 10.1016/j.nahs.2008.03.003

J. Aubin and A. Cellina, Differential inclusions Set-valued maps and viability theory, 1984.

M. Balde and U. Boscain, Stability of planar switched systems : the nondiagonalizable case, Commun. Pure Appl. Anal, vol.7, issue.27, pp.1-21, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00105866

M. Balde, U. Boscain, and P. Mason, A note on stability conditions for planar switched systems, International Journal of Control, vol.47, issue.10, pp.1882-1888, 2009.
DOI : 10.1137/040613147

URL : https://hal.archives-ouvertes.fr/hal-00323682

G. Blankenship, Singularly perturbed difference equations in optimal control problems, IEEE Transactions on Automatic Control, vol.26, issue.4, pp.911-917, 1981.
DOI : 10.1109/TAC.1981.1102741

V. Blondel and J. Theys, On the relations between discrete and continuous time stability for switched linear systems, 16th International Symposium on Mathematical Theory of Networks and Systems, p.60, 2004.

V. D. Blondel, Y. Nesterov, and J. Theys, On the accuracy of the ellipsoid norm approximation of the joint spectral radius, Linear Algebra and its Applications, vol.394, pp.91-107, 2005.
DOI : 10.1016/j.laa.2004.06.024

V. D. Blondel and J. N. Tsitsiklis, NP-Hardness of Some Linear Control Design Problems, SIAM Journal on Control and Optimization, vol.35, issue.6, pp.2118-2127, 1997.
DOI : 10.1137/S0363012994272630

V. D. Blondel and J. N. Tsitsiklis, Complexity of stability and controllability of elementary hybrid systems, Automatica, vol.35, issue.3, pp.479-489, 1999.
DOI : 10.1016/S0005-1098(98)00175-7

U. Boscain, Stability of Planar Switched Systems: The Linear Single Input Case, SIAM Journal on Control and Optimization, vol.41, issue.1, pp.89-112, 2002.
DOI : 10.1137/S0363012900382837

U. Boscain, G. Charlot, and M. Sigalotti, Stability of planar nonlinear switched systems, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475), pp.415-432, 2006.
DOI : 10.1109/CDC.2003.1271823

URL : https://hal.archives-ouvertes.fr/hal-00004272

S. Boyd, L. Ghaoui, E. Féron, and V. Balakrishnan, Linear matrix inequalities in system and control theory, SIAM Studies in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), vol.15, issue.23, p.46, 1994.
DOI : 10.1137/1.9781611970777

B. Chen and C. Lin, Maximal stability bounds of singularly perturbed systems, Journal of the Franklin Institute, vol.336, issue.8, pp.1265-1270, 1990.
DOI : 10.1016/S0016-0032(99)00036-8

J. Daafouz, P. Riedinger, and C. Iung, Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach, IEEE Transactions on Automatic Control, vol.47, issue.11, pp.47227-234, 2002.
DOI : 10.1109/TAC.2002.804474

W. P. Dayawansa and C. F. Martin, Dynamical systems which undergo switching, IEEE Transactions on Automatic Control, vol.44, issue.4, pp.751-760, 1999.
DOI : 10.1109/9.754812

R. A. Decarlo, M. S. Branicky, S. Pettersson, and B. Lennartson, Perspectives and results on the stability and stabilizability of hybrid systems, Proceedings of the IEEE : Special Issue Hybrid Systems, pp.1069-1082, 2000.
DOI : 10.1109/5.871309

A. Dontchev, T. Donchev, and I. Slavov, A Tikhonov-type theorem for singularly perturbed differential inclusions, Nonlinear Analysis: Theory, Methods & Applications, vol.26, issue.9, pp.1547-1554, 1996.
DOI : 10.1016/0362-546X(95)00003-E

L. Fang, H. Lin, and P. J. Antsaklis, Stabilization and performance analysis for a class of switched systems, 43rd IEEE Conference on Decision and Control, pp.3265-3270, 2004.

G. Goodwin, R. Middleton, and H. Poor, High-speed digital signal processing and control, Proceedings of the IEEE, pp.240-259, 1992.
DOI : 10.1109/5.123294

Y. Granjon, Automatique -Systèmes linéaires, non linéaires, temps continu, temps discret, représentation d'état. Dunod Collection Sciences Sup, 2007.

J. Hespanha and A. Morse, Stability of switched systems with average dwell-time, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304), pp.2655-2660, 1999.
DOI : 10.1109/CDC.1999.831330

R. Husson, C. Iung, J. Aubry, J. Daafouz, and D. Wolf, Automatique : du cahier des charges à la réalisation de systèmes. Dunod Collection Sciences Sup, 2007.

M. Johansson and A. Rantzer, Computation of piecewise quadratic Lyapunov functions for hybrid systems, IEEE Transactions on Automatic Control, vol.43, issue.4, pp.555-559, 1998.
DOI : 10.1109/9.664157

R. Jungers, The joint spectral radius, volume 385 of Lecture Notes in Control and Information Sciences, Theory and applications, vol.26, issue.66, p.67, 2009.

H. Khalil, Nonlinear systems, 1996.

P. V. Kokotovi?, H. K. Khalil, and J. O. Reilly, Singular perturbation methods in control : analysis and design, pp.14-70, 1986.
DOI : 10.1137/1.9781611971118

T. S. Li, J. Chiou, and F. Kung, Stability bounds of singularly perturbed discrete systems, IEEE Transactions on Automatic Control, vol.44, issue.10, pp.1934-1938, 1999.

T. S. Li and J. Li, Stabilization bound of discrete two-time-scale systems, Systems & Control Letters, vol.18, issue.6, pp.479-489, 1992.
DOI : 10.1016/0167-6911(92)90052-T

D. Liberzon, Switching in systems and control. Systems & Control : Foundations & Applications, Birkhäuser Boston Inc, vol.17, pp.16-24, 2003.

D. Liberzon, J. P. Hespanha, and A. S. Morse, Stability of switched systems: a Lie-algebraic condition, Systems & Control Letters, vol.37, issue.3, pp.117-122, 1999.
DOI : 10.1016/S0167-6911(99)00012-2

D. Liberzon and A. S. Morse, Basic problems in stability and design of switched systems, IEEE Control Systems Magazine, vol.19, issue.5, pp.59-70, 1999.
DOI : 10.1109/37.793443

D. Liberzon and R. Tempo, Common Lyapunov Functions and Gradient Algorithms, IEEE Transactions on Automatic Control, vol.49, issue.6, pp.990-994, 2004.
DOI : 10.1109/TAC.2004.829632

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, issue.2, pp.308-322, 2009.
DOI : 10.1109/TAC.2008.2012009

B. Litkouhi and H. Khalil, Multirate and composite control of two-time-scale discrete-time systems, IEEE Transactions on Automatic Control, vol.30, issue.7, pp.645-651, 1985.
DOI : 10.1109/TAC.1985.1104024

I. Malloci, Two time scale switched systems : An application to steering control in hot strip mills, pp.47-50, 2009.
URL : https://hal.archives-ouvertes.fr/tel-00439457

I. Malloci, J. Daafouz, and C. Iung, Stabilization of continuous-time singularly perturbed switched systems, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference, pp.6371-6376, 2009.
DOI : 10.1109/CDC.2009.5399876

URL : https://hal.archives-ouvertes.fr/hal-00439265

I. Malloci, J. Daafouz, C. Iung, R. Bonidal, and P. Szczepanski, Switched system modeling and robust steering control of the tail end phase in a hot strip mill, Nonlinear Analysis: Hybrid Systems, vol.3, issue.3, pp.239-250, 2009.
DOI : 10.1016/j.nahs.2009.01.007

URL : https://hal.archives-ouvertes.fr/hal-00414112

M. Margaliot, Stability analysis of switched systems using variational principles: An introduction, Automatica, vol.42, issue.12, pp.2059-2077, 2006.
DOI : 10.1016/j.automatica.2006.06.020

M. Margaliot and M. S. Branicky, Nice reachability for planar bilinear control systems with applications to planar linear switched systems [corrected], IEEE Transactions on Automatic Control, vol.54, issue.6, pp.1430-1435, 2009.
DOI : 10.1109/TAC.2009.2022905

M. Margaliot and G. Langholz, Necessary and sufficient conditions for absolute stability: the case of second-order systems, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol.50, issue.2, pp.227-234, 2003.
DOI : 10.1109/TCSI.2002.808219

P. Mason, U. Boscain, and Y. Chitour, Common Polynomial Lyapunov Functions for Linear Switched Systems, SIAM Journal on Control and Optimization, vol.45, issue.1, pp.226-245, 2006.
DOI : 10.1137/040613147

R. Middleton and G. Goodwin, Improved finite word length characteristics in digital control using delta operators, IEEE Transactions on Automatic Control, vol.31, issue.11, pp.311015-1021, 1986.
DOI : 10.1109/TAC.1986.1104162

R. Middleton and G. Goodwin, Digital Control and Estimation : A Unified Approach, p.74, 1990.

A. Molchanov and Y. Pyatnitskiy, Lyapunov functions that specify necessary and sufficient conditions of absolute stability of nonlinear nonstationary control systems iii. Automation and Remote Control, pp.620-630, 1986.

A. Molchanov and Y. Pyatnitskiy, Criteria of asymptotic stability of differential and difference inclusions encountered in control theory, Systems & Control Letters, vol.13, issue.1, pp.59-64, 1989.
DOI : 10.1016/0167-6911(89)90021-2

A. P. Molchanov and P. H. Bauer, Robust stability of linear time-varying delta-operator formulated discrete-time systems, IEEE Transactions on Automatic Control, vol.44, issue.2, pp.325-327, 1999.
DOI : 10.1109/9.746259

Y. Mori, T. Mori, and Y. Kuroe, A solution to the common Lyapunov function problem for continuous-time systems, Proceedings of the 36th IEEE Conference on Decision and Control, pp.3530-3531, 1997.
DOI : 10.1109/CDC.1997.652397

A. S. Morse, Supervisory control of families of linear set-point controllers - Part I. Exact matching, IEEE Transactions on Automatic Control, vol.41, issue.10, pp.411413-1431, 1996.
DOI : 10.1109/9.539424

D. S. Naidu, Singular perturbation methodology in control systems, IEE Control Engineering Series, vol.34, issue.51, p.63, 1988.
DOI : 10.1049/PBCE034E

D. S. Naidu, D. B. Price, and J. L. Hibey, Singular perturbations and time scales (SPaTS) in discrete control systems-an overview, 26th IEEE Conference on Decision and Control, pp.2096-2103, 1987.
DOI : 10.1109/CDC.1987.272924

K. Narendra and J. Balakrishnan, A common Lyapunov function for stable LTI systems with commuting A-matrices, IEEE Transactions on Automatic Control, vol.39, issue.12, pp.2469-2471, 1994.
DOI : 10.1109/9.362846

Y. Ohta, H. Imanishi, and H. Haneda, Computer generated Lyapunov functions for a class of nonlinear systems, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol.40, issue.5, pp.343-354, 1993.
DOI : 10.1109/81.232578

A. Pietrus and V. M. Veliov, On the discretization of switched linear systems, Systems & Control Letters, vol.58, issue.6, pp.395-399, 2009.
DOI : 10.1016/j.sysconle.2009.01.005

M. Quincampoix, Singular perturbations for differential equations and inclusions: An approach through constrained systems, Nonlinear Analysis: Theory, Methods & Applications, vol.32, issue.5, pp.697-710, 1998.
DOI : 10.1016/S0362-546X(97)00509-9

F. Rossi, P. Colaneri, and R. Shorten, Padé Discretization for Linear Systems With Polyhedral Lyapunov Functions, IEEE Transactions on Automatic Control, vol.56, issue.11, pp.2717-2722, 2011.
DOI : 10.1109/TAC.2011.2161028

R. Shorten and F. Ó. Cairbre, A proof of global attractivity for a class of switching systems using a non-quadratic Lyapunov approach, IMA Journal of Mathematical Control and Information, vol.18, issue.3, pp.341-353, 2001.
DOI : 10.1093/imamci/18.3.341

R. Shorten, M. Corless, S. Sajja, and S. Solmaz, On Pad?? approximations, quadratic stability and discretization of switched linear systems, Systems & Control Letters, vol.60, issue.9, pp.683-689, 2011.
DOI : 10.1016/j.sysconle.2011.04.024

R. Shorten, O. Mason, F. Cairbre, and P. Curran, A unifying framework for the SISO circle criterion and other quadratic stability criteria, International Journal of Control, vol.11, issue.1, pp.1-8, 2004.
DOI : 10.1109/TAC.1973.1100260

R. Shorten and K. Narendra, On the stability and existence of common Lyapunov functions for stable linear switching systems, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171), pp.3723-3724, 1998.
DOI : 10.1109/CDC.1998.761788

R. Shorten and K. Narendra, Necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for M stable second order linear time-invariant systems, Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334), p.24, 2000.
DOI : 10.1109/ACC.2000.878913

R. Shorten and K. Narendra, Necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for a finite number of stable second order linear time-invariant systems, International Journal of Adaptive Control and Signal Processing, vol.38, issue.10, pp.709-728, 2002.
DOI : 10.1002/acs.719

R. Shorten and K. Narendra, A result on common quadratic Lyapunov functions, IEEE Transactions on Automatic Control, vol.48, issue.1, pp.110-113, 2003.
DOI : 10.1109/TAC.2002.806661

R. Shorten, F. Wirth, O. Mason, K. Wulff, and C. King, Stability Criteria for Switched and Hybrid Systems, SIAM Review, vol.49, issue.4, pp.545-592, 2005.
DOI : 10.1137/05063516X

G. V. Smirnov, Introduction to the theory of differential inclusions, Graduate Studies in Mathematics, vol.41, 1921.
DOI : 10.1090/gsm/041

S. Solmaz, R. Shorten, K. Wulff, and F. O. Cairbre, A design methodology for switched discrete time linear systems with applications to automotive roll dynamics control, Automatica, vol.44, issue.9, pp.2358-2363, 2008.
DOI : 10.1016/j.automatica.2008.01.014

J. Theys, Joint spectral radius : Theory and Approximations, p.53, 2005.

J. N. Tsitsiklis and V. D. Blondel, Lyapunov exponents of pairs of matrices, a correction, Mathematics of Control, Signals, and Systems, vol.10, issue.4, pp.381-408, 1997.
DOI : 10.1007/BF01211553

V. Veliov, A generalization of the Tikhonov theorem for singularly perturbed differential inclusions, Journal of Dynamical and Control Systems, vol.23, issue.8, pp.291-319, 1997.
DOI : 10.1007/BF02463254

F. Watbled, On singular perturbations for differential inclusions on the infinite interval, Journal of Mathematical Analysis and Applications, vol.310, issue.2, pp.362-378, 2005.
DOI : 10.1016/j.jmaa.2005.01.067

URL : https://hal.archives-ouvertes.fr/hal-00907502

M. Wicks, P. Peleties, and R. Decarlo, Construction of piece-wise Lyapunov functions for stabilizing switched systems, 33rd IEEE Conference on Decision and Control, pp.3492-3497, 1994.

M. Wicks, P. Peleties, and R. Decarlo, Switched Controller Synthesis for the Quadratic Stabilisation of a Pair of Unstable Linear Systems, European Journal of Control, vol.4, issue.2, pp.140-147, 1998.
DOI : 10.1016/S0947-3580(98)70108-6

K. Wulff, R. Shorten, and P. Curran, On the 45?? -Region and the uniform asymptotic stability of classes of second order parameter-varying and switched systems, International Journal of Control, vol.29, issue.11, pp.812-823, 2002.
DOI : 10.1049/cce:19980205

C. A. Yfoulis and R. Shorten, A numerical technique for stability analysis of linear switched systems, HSCC'04, pp.631-645, 2004.

A. Zappavigna, P. Colaneri, S. Kirkland, and R. Shorten, Essentially negative news about positive systems, Linear Algebra and its Applications, vol.436, issue.9, pp.3425-3442, 2012.
DOI : 10.1016/j.laa.2011.12.021

G. Zhai, B. Hu, K. Yasuda, and A. N. Michel, Stability analysis of switched systems with stable and unstable subsystems: An average dwell time approach, International Journal of Systems Science, vol.32, issue.8, pp.1055-1061, 2001.
DOI : 10.1080/00207720116692

G. Zhai and H. Lin, Controller failure time analysis for symmetric control systems, International Journal of Control, vol.36, issue.6, pp.598-605, 2004.
DOI : 10.1109/37.898794

G. Zhai, X. Xu, H. Lin, and A. N. Michel, Analysis and design of switched normal systems, Nonlinear Analysis: Theory, Methods & Applications, vol.65, issue.12, pp.2248-2259, 2006.
DOI : 10.1016/j.na.2006.01.034