Abstract and Applied Analysis

Stability of Switched Feedback Time-Varying Dynamic Systems Based on the Properties of the Gap Metric for Operators

M. De la Sen

Full-text: Open access

Abstract

The stabilization of dynamic switched control systems is focused on and based on an operator-based formulation. It is assumed that the controlled object and the controller are described by sequences of closed operator pairs (L,C) on a Hilbert space H of the input and output spaces and it is related to the existence of the inverse of the resulting input-output operator being admissible and bounded. The technical mechanism addressed to get the results is the appropriate use of the fact that closed operators being sufficiently close to bounded operators, in terms of the gap metric, are also bounded. That philosophy is followed for the operators describing the input-output relations in switched feedback control systems so as to guarantee the closed-loop stabilization.

Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 612198, 17 pages.

Dates
First available in Project Euclid: 5 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1365168227

Digital Object Identifier
doi:10.1155/2012/612198

Mathematical Reviews number (MathSciNet)
MR2999874

Zentralblatt MATH identifier
1255.93115

Citation

De la Sen, M. Stability of Switched Feedback Time-Varying Dynamic Systems Based on the Properties of the Gap Metric for Operators. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 612198, 17 pages. doi:10.1155/2012/612198. https://projecteuclid.org/euclid.aaa/1365168227


Export citation

References

  • V.-M. Popov, Hyperstability of Control Systems, Springer, Berlin, Germany, 1973.
  • Y. D. Landau, Adaptive Control: The Model Reference Approach, Marcel Dekker, New York, NY, USA, 1979.
  • M. De la Sen and I. D. Landau, “An on-line method for improvement of adaptation transients in adaptive control,” in Adaptive Systems in Control and Signal Processing,, Pergamon Press, New York, NY, USA, 1984.
  • K. S. Narendra and J. H. Taylor, Frequency Domain Criteria for Absolute Stability, Academic Press, New York, NY, USA, 1973.
  • M. De la Sen, “Asymptotic hyperstability under unstructured and structured modeling deviations from the linear behavior,” Nonlinear Analysis: Real World Applications, vol. 7, no. 2, pp. 248–264, 2006.
  • M. De la Sen, “Absolute stability of feedback systems independent of internal point delays,” IEE Proceedings-Control Theory and Applications, vol. 152, no. 5, pp. 567–574, 2005.
  • M. De la Sen, “On the asymptotic hyperstability of dynamic systems with point delays,” IEEE Transactions on Circuits and Systems, vol. 50, no. 11, pp. 1486–1488, 2003.
  • M. De La Sen, “Preserving positive realness through discretization,” Positivity, vol. 6, no. 1, pp. 31–45, 2002.
  • M. De La Sen, “A method for general design of positive real functions,” IEEE Transactions on Circuits and Systems, vol. 45, no. 7, pp. 764–769, 1998.
  • E. Braverman and S. Zhukovskiy, “Absolute and delay-dependent stability of equations with a distributed delay,” Discrete and Continuous Dynamical Systems A, vol. 32, no. 6, pp. 2041–2061, 2012.
  • Z. Tai, “Absolute mean square exponential stability of Lur'e stochastic distributed parameter control systems,” Applied Mathematics Letters, vol. 25, no. 4, pp. 712–716, 2012.
  • Z. Tai and S. Lun, “Absolutely exponential stability of Lur'e distributed parameter control systems,” Applied Mathematics Letters, vol. 25, no. 3, pp. 232–236, 2012.
  • Y. Chen, W. Bi, and W. Li, “New delay-dependent absolute stability criteria for Lur'e systems with time-varying delay,” International Journal of Systems Science, vol. 42, no. 7, pp. 1105–1113, 2011.
  • J. T. Zhou, Q. K. Song, and J. X. Yang, “Stochastic passivity of uncertain networks with time-varying delays,” Abstract and Applied Analysis, vol. 2009, Article ID 725846, 16 pages, 2009.
  • Q. Song and J. Cao, “Global dissipativity analysis on uncertain neural networks with mixed time-varying delays,” Chaos, vol. 18, no. 4, Article ID 043126, 10 pages, 2008.
  • D. Liu, X. Liu, and S. Zhong, “Delay-dependent robust stability and control synthesis for uncertain switched neutral systems with mixed delays,” Applied Mathematics and Computation, vol. 202, no. 2, pp. 828–839, 2008.
  • A. Ibeas and M. De la Sen, “Robustly stable adaptive control of a tandem of master-slave robotic manipulators with force reflection by using multiestimation scheme,” IEEE Transactions on Systems, Man and Cybernetics Part B-Cybernetics, vol. 36, no. 5, pp. 1162–1179, 2006.
  • W. P. Dayawansa and C. F. Martin, “A converse Lyapunov theorem for a class of dynamical systems which undergo switching,” IEEE Transactions on Automatic Control, vol. 44, no. 4, pp. 751–760, 1999.
  • A. A. Agrachev and D. Liberzon, “Lie-algebraic stability criteria for switched systems,” SIAM Journal on Control and Optimization, vol. 40, no. 1, pp. 253–269, 2001.
  • Z. Li, Y. Soh, and C. Wen, Switched and Impulsive Systems: Analysis, Design, and Applications, vol. 313 of Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, 2005.
  • M. De La Sen and A. Ibeas, “Stability results for switched linear systems with constant discrete delays,” Mathematical Problems in Engineering, vol. 2008, Article ID 543145, 28 pages, 2008.
  • M. De la Sen and A. Ibeas, “Stability results of a class of hybrid systems under switched continuous-time and discrete-time control,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 315713, 28 pages, 2009.
  • M. De la Sen, “On the characterization of Hankel and Toeplitz operators describing switched linear dynamic systems with point delays,” Abstract and Applied Analysis, vol. 2009, Article ID 670314, 34 pages, 2009.
  • M. De La Sen and A. Ibeas, “On the global asymptotic stability of switched linear time-varying systems with constant point delays,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 231710, 31 pages, 2008.
  • M. De La Sen and A. Ibeas, “Stability results for switched linear systems with constant discrete delays,” Mathematical Problems in Engineering, vol. 2008, Article ID 543145, 28 pages, 2008.
  • V. Covachev, H. Akça, and M. Sarr, “Discrete-time counterparts of impulsive Cohen-Grossberg neural networks of neutral type,” Neural, Parallel & Scientific Computations, vol. 19, no. 3-4, pp. 345–359, 2011.
  • G. T. Stamov and J. O. Alzabut, “Almost periodic solutions in the PC space for uncertain impulsive dynamical systems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 14, pp. 4653–4659, 2011.
  • Z. Sun and S. S. Ge, Switched Linear Systems: Control and Design, Springer, London, UK, 2005.
  • D. Liberzon, Switching in Systems and Control, Systems & Control: Foundations & Applications, Birkhäauser, Boston, Mass, USA, 2003.
  • A. Ibeas and M. De La Sen, “Exponential stability of simultaneously triangularizable switched systems with explicit calculation of a common Lyapunov function,” Applied Mathematics Letters, vol. 22, no. 10, pp. 1549–1555, 2009.
  • R. N. Shorten and F. O. 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, no. 3, pp. 341–353, 2001.
  • M. De la Sen and A. Ibeas, “On the stability properties of linear dynamic time-varying unforced systems involving switches between parameterizations from topologic considerations via graph theory,” Discrete Applied Mathematics, vol. 155, no. 1, pp. 7–25, 2007.
  • M. De La Sen, “On the necessary and sufficient condition for a set of matrices to commute and some further linked results,” Mathematical Problems in Engineering, vol. 2009, Article ID 650970, 24 pages, 2009.
  • H. Ishii and B. A. Francis, “Stabilizing a linear system by switching control with dwell time,” in Proceedings of the American Control Conference, vol. 3, pp. 1876–1881, 2001.
  • 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, no. 2, pp. 308–322, 2009.
  • H. Yang, G. Xie, T. Chu, and L. Wang, “Commuting and stable feedback design for switched linear systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 2, pp. 197–216, 2006.
  • M. Duarte-Mermoud, R. Ordonez-Hurtado, and P. Zagalak, “A method for determining the non-existence of a common quadratic Lyapunov function for switched linear systems based on particle swarm optimization,” International Journal of Systems Science, vol. 43, no. 11, pp. 2015–2029, 2012.
  • A. Feintuch, Robust Control Theory in Hilbert Space, vol. 130 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1998.
  • G. Vinnicombe, Uncertainty and Feedback. H-infinity Loop-Shaping and the $\nu $-Gap Metric, Imperial College Press, London, UK, 2001.