Input-to-state stability analysis for interconnected difference equations with delay

Open Access
Original Article


Input-to-state stability (ISS) of interconnected systems with each subsystem described by a difference equation subject to an external disturbance is considered. Furthermore, special attention is given to time delay, which gives rise to two relevant problems: (i) ISS of interconnected systems with interconnection delays, which arise in the paths connecting the subsystems, and (ii) ISS of interconnected systems with local delays, which arise in the dynamics of the subsystems. The fact that a difference equation with delay is equivalent to an interconnected system without delay is the crux of the proposed framework. Based on this fact and small-gain arguments, it is demonstrated that interconnection delays do not affect the stability of an interconnected system if a delay-independent small-gain condition holds. Furthermore, also using small-gain arguments, ISS for interconnected systems with local delays is established via the Razumikhin method as well as the Krasovskii approach. A combination of the results for interconnected systems with interconnection delays and local delays, respectively, provides a framework for ISS analysis of general interconnected systems with delay. Thus, a scalable ISS analysis method is obtained for large-scale interconnections of difference equations with delay.


Large-scale systems Time delay Difference equations Lyapunov methods Small-gain theorem 


  1. 1.
    Dashkovskiy SN, Naujok L (2010) Lyapunov–Razumikhin and Lyapunov–Krasovskii theorems for interconnected ISS time-delay systems. In: 19th International symposium on mathematical theory of networks and systems, Budapest, Hungary, pp 1179–1184Google Scholar
  2. 2.
    Dashkovskiy SN, Rüffer BS, Wirth FR (2010) Small gain theorems for large scale systems and construction of ISS Lyapunov functions. SIAM J Control Optim 48(6): 4089–4118MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Gielen RH, Lazar M (2011) Stabilization of polytopic delay difference inclusions via the Razumikhin approach. Automatica 47(12): 2562–2570CrossRefMATHGoogle Scholar
  4. 4.
    Gielen RH, Lazar M, Kolmanovsky IV (2012) Lyapunov methods for time-invariant delay difference inclusions. SIAM J Control Optim 50(1): 110–132CrossRefGoogle Scholar
  5. 5.
    Gu K, Kharitonov VL, Chen J (2003) Stability of time-delay systems. Birkhäuser, BostonCrossRefMATHGoogle Scholar
  6. 6.
    Hetel L, Daafouz J, Iung C (2008) Equivalence between the Lyapunov–Krasovskii functionals approach for discrete delay systems and that of the stability conditions for switched systems. Nonlinear Anal Hybrid Syst 2(3): 697–705MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Ito H, Pepe P, Jiang ZP (2009) Construction of Lyapunov–Krasovskii functionals for interconnection of retarded dynamic and static systems via a small-gain condition. In: Proceedings of the 48th IEEE conference on decision and control, Shanghai, China, pp 1310–1316Google Scholar
  8. 8.
    Ito H, Pepe P, Jiang ZP (2010) A small-gain condition for iISS of interconnected retarded systems based on Lyapunov-Krasovskii functionals. Automatica 46(10): 1646–1656CrossRefMATHGoogle Scholar
  9. 9.
    Ito H, Jiang ZP, Pepe P (2011) A small-gain methodology for networks of iISS retarded systems based on Lyapunov–Krasovskii functionals. In: Proceedings of the 18th IFAC world congress, Milano, Italy, pp 5100–5105Google Scholar
  10. 10.
    Jiang ZP, Wang Y (2001) Input-to-state stability for discrete-time nonlinear systems. Automatica 37: 857–869MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Jiang ZP, Lin Y, Wang Y (2008) Nonlinear small-gain theorems for discrete-time large-scale systems. In: Proceedings of the 27th Chinese control conference, Kunming, China, pp 704–708Google Scholar
  12. 12.
    Karafyllis I, Jiang ZP (2011) A vector small-gain theorem for general nonlinear control systems. IMA J Math Control Inf 28(3): 309–344MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kolmanovskii V, Myshkis A (1999) Introduction to the theory and applications of functional differential equations. Kluwer Academic Publishers, DordrechtMATHGoogle Scholar
  14. 14.
    Kreyszig E (1989) Introductory functional analysis with applications. Wiley, New YorkMATHGoogle Scholar
  15. 15.
    Laila DS, Nes̆ić D (2003) Discrete-time Lyapunov-based small-gain theorem for parameterized interconnected ISS systems. IEEE Trans Autom Control 48(10): 1783–1788CrossRefGoogle Scholar
  16. 16.
    Lakshmikantham V, Matrosov VM, Sivasundaram S (1991) Vector Lyapunov functions and stability analysis of nonlinear systems. Kluwer Academic Publishers, DordrechtMATHGoogle Scholar
  17. 17.
    Limon D, Alamo T, Salas F, Camacho EF (2006) Input to state stability of min–max MPC controllers for nonlinear systems with bounded uncertainties. Automatica 42(5): 797–803MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Liu B, Hill DJ (2009) Input-to-state stability for discrete time-delay systems via the Razumikhin technique. Syst Control Lett 58: 567–575MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Liu B, Marquez HJ (2007) Razumikhin-type stability theorems for discrete delay systems. Automatica 43(7): 1219–1225MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Liu T, Hill DJ, Jiang ZP (2010) Lyapunov formulation of ISS cyclic-small-gain in discrete-time dynamical networks. In: Proceedings of the 8th WCICA, Jinan, China, pp 568–573Google Scholar
  21. 21.
    Michel AN, Miller RK (1977) Qualatitive analysis of large scale dynamical systems, Mathematics in Science and Engineering, vol 134. Academic Press, Inc., New YorkGoogle Scholar
  22. 22.
    Orero SO, Irving MR (1998) A genetic algorithm modelling framework and solution technique for short term optimal hydrothermal scheduling. IEEE Trans Power Syst 13(2): 501–518CrossRefGoogle Scholar
  23. 23.
    Polushin I, Marquez HJ, Tayebi A, Liu PX (2009) A multichannel IOS small gain theorem for systems with multiple time-varying communication delays. IEEE Trans Autom Control 54(2): 404–409MathSciNetCrossRefGoogle Scholar
  24. 24.
    Raimondo DM, Magni L, Scattolini R (2007) Decentralized MPC of nonlinear systems: an input-to-state stability approach. Int J Robust Nonlinear Control 17: 1651–1667MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Rüffer BS, Sailer R, Wirth FR (2010) Comments on “a multichannel ios small gain theorem for systems with multiple time-varying communication delays”. IEEE Trans Autom Control 55(7): 1722–1725CrossRefGoogle Scholar
  26. 26.
    Teel AR (1996) A nonlinear small gain theorem for the analysis of control systems with saturation. IEEE Trans Autom Control 41(9): 1256–1270MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Teel AR (1998) Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem. IEEE Trans Autom Control 43(7): 960–964MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Tiwari S, Wang Y (2010) Razumikhin-type small-gain theorems for large-scale systems with delays. In: Proceedings of the 49th IEEE conference on decision and control, Atlanta, GA, pp 7407–7412Google Scholar
  29. 29.
    Tiwari S, Wang Y, Jiang ZP (2009) A nonlinear small-gain theorem for large-scale time delay systems. In: Proceedings of the 48th IEEE conference on decision and control, Shanghai, China, pp 7204–7209Google Scholar
  30. 30.
    Vidyasagar M (1981) Input–output analysis of large-scale interconnected systems. Lecture notes in control and information sciences, vol 29. Springer, BerlinGoogle Scholar
  31. 31.
    Šiljak DD (1978) Large-scale dynamic systems: stability and structure. North-Holland, AmsterdamMATHGoogle Scholar
  32. 32.
    Wang C, Shahidehpour SM (1993) Power generation scheduling for multi-area hydro-thermal systems with tie line constraints, cascaded reservoirs and uncertain data. IEEE Trans Power Syst 8(3): 1333–1340CrossRefGoogle Scholar
  33. 33.
    Willems JC (1972) Dissipative dynamical systems. Arch Ration Mech Anal 45: 321–393MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.Electrical Engineering DepartmentEindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.Electrical and Computer Engineering DepartmentUniversity of CaliforniaSanta BarbaraUSA

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