Journal of Applied Mathematics

Input-to-State Stability of Singularly Perturbed Control Systems with Delays

Abstract

We study the input-to-state stability of singularly perturbed control systems with delays. By using the generalized Halanay inequality and Lyapunov functions, we derive the input-to-state stability of some classes of linear and nonlinear singularly perturbed control systems with delays.

Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 401572, 8 pages.

Dates
First available in Project Euclid: 14 March 2014

https://projecteuclid.org/euclid.jam/1394808016

Digital Object Identifier
doi:10.1155/2013/401572

Mathematical Reviews number (MathSciNet)
MR3064957

Zentralblatt MATH identifier
1271.93113

Citation

Zhao, Yongxiang; Xiao, Aiguo; Li, Li. Input-to-State Stability of Singularly Perturbed Control Systems with Delays. J. Appl. Math. 2013 (2013), Article ID 401572, 8 pages. doi:10.1155/2013/401572. https://projecteuclid.org/euclid.jam/1394808016

References

• E. D. Sontag, “Smooth stabilization implies coprime factorization,” IEEE Transactions on Automatic Control, vol. 34, no. 4, pp. 435–443, 1989.
• A. R. Teel, L. Moreau, and D. Nešić, “A unified framework for input-to-state stability in systems with two time scales,” IEEE Transactions on Automatic Control, vol. 48, no. 9, pp. 1526–1544, 2003.
• D. Kazakos and J. Tsinias, “The input to state stability condition and global stabilization of discrete-time systems,” IEEE Transactions on Automatic Control, vol. 39, no. 10, pp. 2111–2113, 1994.
• E. D. Sontag, “Further facts about input to state stabilization,” IEElE Transactions on Automatic Control, vol. 35, no. 4, pp. 473–476, 1990.
• E. D. Sontag and Y. Wang, “New characterizations of input-to-state stability,” IEEE Transactions on Automatic Control, vol. 41, no. 9, pp. 1283–1294, 1996.
• E. D. Sontag, “Input-to-state stability: basic concepts and results,” in Nonlinear and Optimal Control Theory, P. Nistri and G. Stefani, Eds., pp. 163–220, Springer, Berlin, Germany, 2007.
• E. D. Sontag, Mathematical Control Theory Deterministic Finite-Dimensional Systems, vol. 6, Springer, New York, NY, USA, 2nd edition, 1998.
• G.-D. Hu and M. Liu, “Input-to-state stability of Runge-Kutta methods for nonlinear control systems,” Journal of Computational and Applied Mathematics, vol. 205, no. 1, pp. 633–639, 2007.
• I. Karafyllis and J. Tsinias, “Nonuniform in time input-to-state stability and the small-gain theorem,” IEEE Transactions on Automatic Control, vol. 49, no. 2, pp. 196–216, 2004.
• N. Yeganefar, P. Pepe, and M. Dambrine, “Input-to-state stability of time-delay systems: a link with exponential stability,” IEEE Transactions on Automatic Control, vol. 53, no. 6, pp. 1526–1531, 2008.
• P. D. Christofides and A. R. Teel, “Singular perturbations and input-to-state stability,” IEEE Transactions on Automatic Control, vol. 41, no. 11, pp. 1645–1650, 1996.
• Z. -P. Jing, E. Sontag, and Y. Wang, “Input-to-state stablility for discrete-time nonlinear control systems,” Automatica, vol. 37, pp. 857–869, 1999.
• A. Saberi and H. Khalil, “Quadratic-type Lyapunov functions for singularly perturbed systems,” IEEE Transactions on Automatic Control, vol. 29, no. 6, pp. 542–550, 1984.
• M. Corless and L. Glielmo, “On the exponential stability of singularly perturbed systems,” SIAM Journal on Control and Optimization, vol. 30, no. 6, pp. 1338–1360, 1992.
• X. Liu, X. Shen, and Y. Zhang, “Exponential stability of singularly perturbed systems with time delay,” Applicable Analysis, vol. 82, no. 2, pp. 117–130, 2003.
• H.-J. Tian, “Dissipativity and exponential stability of $\theta$-methods for singularly perturbed delay differential equations with a bounded lag,” Journal of Computational Mathematics, vol. 21, no. 6, pp. 715–726, 2003.
• H. J. Tian, “The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag,” Journal of Mathematical Analysis and Applications, vol. 270, no. 1, pp. 143–149, 2002.
• Y. X. Yu, “Input-to-state stablility of one-leg methods for nonlinear control systems,” Journal of System Simulation, vol. 20, no. 1, pp. 19–20, 2008.
• Y. X. Yu and X. Z. Tian, “Input-to-state stablility of muti-step Runge-Kutta methods for nonlinear control systems,” Journal of System Simulation, vol. 6, no. 2, pp. 1573–1574, 2008, vol. 37, pp. 857–869, 2001.
• P. V. Kokotovic, H. K. Khalil, and J. Orelly, Singular Perturbation Methods in Control: Analysis and Design, Academic Press, London, UK, 1986.
• X. X. Liao, Theory Methods and Application of Stability, Huazhong University of Science and Technology Press, Wuhan, China, 1999.