Automation and Remote Control

, Volume 79, Issue 12, pp 2114–2127 | Cite as

On Diagonal Stability of Positive Systems with Switches and Delays

  • A. Yu. AleksandrovEmail author
  • O. Mason
Nonlinear Systems


We consider linear positive systems with delay and switchings of operation modes. We establish conditions under which it is possible to construct a common Lyapunov–Krasovskii diagonal functional for the family of subsystems corresponding to the system with switchings in consideration. These conditions are formulated in terms of the feasibility of auxiliary systems of linear algebraic inequalities. In addition, we study the problem of the existence of a diagonal functional of a special form. We also show that our results can be used to analyze the stability of some classes of nonlinear positive systems with delay.


switching systems delays diagonal stability positive system linear inequalities 


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  1. 1.
    Blanchini, F., Colaneri, P., and Valcher, M.E., Switched Positive Linear Systems, Foundat. Trends Syst. Control, 2015, vol. 2, no. 2, pp. 101–273.CrossRefGoogle Scholar
  2. 2.
    Rantzer, A., Scalable Control of Positive Systems, Eur. J. Control, 2015, vol. 24, pp. 72–80.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Zhang, J., Huang, J., and Zhao, X., Further Results on Stability and Stabilisation of Switched Positive Systems, IET Control Theory Appl., 2015, vol. 9, no. 14, pp. 2132–2139.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Valcher, M.E. and Zorzan, I., On the Consensus of Homogeneous Multiagent Systems with Positivity Constraints, IEEE Trans. Autom. Control, 2017, vol. 62, no. 10, pp. 5096–5110.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Farina, L. and Rinaldi, S., Positive Linear Systems: Theory and Applications, New York: Wiley, 2000.CrossRefzbMATHGoogle Scholar
  6. 6.
    Berman, A. and Plemmons, R.J., Nonnegative Matrices in the Mathematical Sciences, Philadelphia: SIAM, 1994.CrossRefzbMATHGoogle Scholar
  7. 7.
    Kazkurewicz, E. and Bhaya, A., Matrix Diagonal Stability in Systems and Computation, Boston: Birkhauser, 1999.Google Scholar
  8. 8.
    Shorten, R.N., Wirth, F., and Leith, D., A Positive Systems Model of TCP-Like Congestion Control, IEEE Trans. Networking, 2006, vol. 14, no. 3, pp. 616–629.CrossRefGoogle Scholar
  9. 9.
    Metod vektornykh funktsii Lyapunova v teorii ustoichivosti (Method of Vector Lyapunov Functions in Stability Theory), Voronov, A.A. and Matrosov, V.M., Eds., Moscow: Nauka, 1987.Google Scholar
  10. 10.
    Tkhai, V.N., Model with Coupled Subsystems, Autom. Remote Control, 2013, vol. 74, no. 6, pp. 919–931.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Aleksandrov, A.Yu., Chen, Y., Platonov, A.V., and Zhang, L., Stability Analysis and Uniform Ultimate Boundedness Control Synthesis for a Class of Nonlinear Switched Difference Systems, J. Differ. Equat. Appl., 2012, vol. 18, no. 9, pp. 1545–1561.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Mason, O., Diagonal Riccati Stability and Positive Time-Delay Systems, Syst. Control Lett., 2012, vol. 61, no. 1, pp. 6–10.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Aleksandrov, A.Yu. and Platonov, A.V., On Absolute Stability of One Class of Nonlinear Switched Systems, Autom. Remote Control, 2008, vol. 69, no. 7, pp. 1101–1116.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Aleksandrov, A. and Mason, O., Absolute Stability and Lyapunov–Krasovskii Functionals for Switched Nonlinear Systems with Time-Delay, J. Franklin Inst., 2014, vol. 351, pp. 4381–4394.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Pastravanu, O.C. and Matcovschi, M.-H., Max-Type Copositive Lyapunov Functions for Switching Positive Linear Systems, Automatica, 2014, vol. 50, no. 12, pp. 3323–3327.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Liberzon, D., Switching in Systems and Control, Boston: Birkhauser, 2003.CrossRefzbMATHGoogle Scholar
  17. 17.
    Shorten, R., Wirth, F., Mason, O., Wulf, K., and King, C., Stability Criteria for Switched and Hybrid Systems, SIAM Rev., 2007, vol. 49, no. 4, pp. 545–592.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Vassilyev, S.N. and Kosov, A.A., Analysis of Hybrid Systems’ Dynamics using the Common Lyapunov Functions and Multiple Homomorphisms, Autom. Remote Control, 2011, vol. 72, no. 6, pp. 1163–1183.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Krasovskii, N.N., On Applications of the Second Lyapunov Method for Equations with Time Delay, Prikl. Mat. Mekh., 1956, vol. 20, no. 3, pp. 315–327.Google Scholar
  20. 20.
    Aleksandrov, A. and Mason, O., Diagonal Riccati Stability and Applications, Linear Algebra Appl., 2016, vol. 492, pp. 38–51.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Aleksandrov, A. and Mason, O., Diagonal Lyapunov–Krasovskii Functionals for Discrete-Time Positive Systems with Delay, Syst. Control Lett., 2014, vol. 63, no. 1, pp. 63–67.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Narendra, K.S. and Balakrishnan, J., A Common Lyapunov Function for Stable LTI Systems with Commuting A-Matrices, IEEE Transact. Autom. Control, 1994, vol. 39, no. 12, pp. 2469–2471.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Liberzon, D., Morse, A.S., and Hespanha, J., Stability of Switched Systems: A Lie Algebraic Condition, Syst. Control Lett., 1999, vol. 37, pp. 117–122.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ebihara, Y., Peaucelle, D., and Arzelier, D., LMI Approach to Linear Positive System Analysis and Synthesis, Syst. Control Lett., 2014, vol. 63, pp. 50–56.MathSciNetCrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Inc. 2018

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia
  2. 2.ITMO University (National Research University of Information Technologies, Mechanics and Optics)St. PetersburgRussia
  3. 3.National University of IrelandMaynoothIreland
  4. 4.Irish Software Research Centre LeroLimerickIreland

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