Abstract
Using duality, complementarity ideas, and Z-transformations, in this chapter we discuss equivalent ways of describing the existence of common linear/quadratic Lyapunov functions for switched linear systems. In particular, we extend a recent result of Mason–Shorten on positive switched system with two constituent linear time-invariant systems to an arbitrary finite system.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Akar, A. Paul, M.G. Safonov, and U. Mitra, Conditions on the stability of a class of second-order switched systems, IEEE Transactions on Automatic Control, 51 (2006) 338–340.
A. Berman and R.J. Plemmons, Nonnegative matrices in the mathematical sciences, SIAM, Philadelphia, PA, 1994.
A. Berman and N. Shaked-Monderer, Completely positive matrices, World Scientific, Singapore, 2003.
R.W. Cottle, J.S. Pang, and R.E. Stone, The linear complementarity problem, Academic Press, Boston, 1992.
L. Elsner, Quasimonotonie und Ungleischungen in halbgeordneten Räumen, Linear Algebra and its Applications, 8 (1974) 249–261.
J. Faraut and A. Koranyi, Analysis on symmetric cones, Oxford University Press, Oxford 1994.
L. Farina and S. Rinaldi, Positive linear systems: Theory and applications, Wiley, New York 2000.
M.S. Gowda and T. Parthasarathy, Complementarity forms of theorems of Lyapunov and Stein, and related results, Linear Algebra and its Applications, 320 (2000) 131–144.
M.S. Gowda and Y. Song, Some new results for the semidefinite linear complementarity problems, SIAM Journal on Matrix Analysis, 24 (2002) 25–39.
M.S. Gowda and R. Sznajder, The generalized order linear complementarity problem, SIAM Journal on Matrix Analysis, 15 (1994) 779–795.
M.S. Gowda, R. Sznajder, and J. Tao, Some P-properties for linear transformations on Euclidean Jordan algebras, Linear Algebra and its Applications, 393 (2004) 203–232.
M.S. Gowda and J. Tao, Z-transformations on proper and symmetric cones, Mathematical Programming, Series B, 117 (2009) 195–221.
L Gurvits, R. Shorten, and O. Mason, On the stability of switched positive linear systems, IEEE Transactions on Automatic Control, 52 (2007) 1099–1103.
V.A. Kamenetskiy and Ye.S. Pyatnitskiy, An iterative method of Lyapunov function construction for differential inclusions, Systems and Control Letters, 8 (1987) 445–451.
C. King and M. Nathanson, On the existence of a common quadratic Lyapunov function for a rank one difference, Linear Algebra and its Applications, 419 (2006) 400–416.
D. Liberzon, Switching in systems and control, Birkhäuser, Boston, 2003.
D. Liberzon, J.P. Hespanha, and A.S. Morse, Stability of switched systems: a Lie-algebraic condition, Systems and Control Letters, 37 (1999) 117–122.
M. Mason and R. Shorten, A conjecture on the existence of common quadratic Lyapunov functions for positive linear systems, Proceedings of American Control Conference, 2003.
M. Mason and R. Shorten, On common quadratic Lyapunov functions for stable discrete-time LTI systems, IMA Journal of Applied Mathematics, 69 (2004) 271–283.
M. Mason and R. Shorten, On linear copositive Lyapunov functions and the stability of positive switched linear systems, IEEE Transactions on Automatic Control, 52 (2007) 1346–1349.
Y. Mori, T. Mori, and Y. Kuroe, A solution to the common Lyapunov function problem for continuous-time systems, Decision and Control, 4 (1997) 3530–3531.
K.S. Narendra and J. Balakrishnan, A common Lyapunov function for stable LTI systems with commuting A-matrices, Automatic Control, IEEE Transactions, 39 (1994) 2469–2471.
H. Schneider and M. Vidyasagar, Cross-positive matrices, SIAM Journal on Numerical Analysis, 7 (1970) 508–519.
R. Shorten, O. Mason, and C. King, An alternative proof of the Barker, Berman, Plemmons (BBP) result on diagonal stability and extensions, Linear Algebra and its Applications, 430 (2009) 34–40.
R.N. Shorten and K.S. Narendra, On the stability and existence of common Lyapunov functions for stable linear switching systems, Decision and Control, 4 (1998) 3723–3724.
R.N. Shorten and K.S. Narendra, Necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for M stable second order linear time-invariant systems, American Control Conference, Chicago, 2000.
R. Shorten, F. Wirth, O. Mason, K. Wulff, and C. King, Stability criteria for switched and hybrid systems, SIAM Review, 49 (2007) 545–592.
Y. Song, M.S. Gowda, and G. Ravindran, On some properties of P-matrix sets, Linear Algebra and its Applications, 290 (1999) 237–246.
R. Sznajder and M.S. Gowda, Generalizations of P 0- and P-properties; extended vertical and horizontal linear complementarity problems, Linear Algebra and its Applications, 223/224 (1995) 695–715.
Author information
Authors and Affiliations
Corresponding authors
Editor information
Editors and Affiliations
Additional information
Dedicated to the memory of Professor George Isac
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Moldovan, M.M., Gowda, M.S. (2010). On Common Linear/Quadratic Lyapunov Functions for Switched Linear Systems. In: Pardalos, P., Rassias, T., Khan, A. (eds) Nonlinear Analysis and Variational Problems. Springer Optimization and Its Applications, vol 35. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-0158-3_25
Download citation
DOI: https://doi.org/10.1007/978-1-4419-0158-3_25
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-0157-6
Online ISBN: 978-1-4419-0158-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)