Input-to-state stability analysis for interconnected difference equations with delay Original Article First Online: 04 March 2012 Received: 06 May 2011 Accepted: 16 February 2012 DOI :
10.1007/s00498-012-0080-4

Cite this article as: Gielen, R.H., Lazar, M. & Teel, A.R. Math. Control Signals Syst. (2012) 24: 33. doi:10.1007/s00498-012-0080-4
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Abstract 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.

Keywords Large-scale systems Time delay Difference equations Lyapunov methods Small-gain theorem This paper was partially presented at the 18th IFAC World Congress, Milano, Italy, 2011.

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References 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 1179–1184

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): 4089–4118

MathSciNet CrossRef MATH Google Scholar 3.

Gielen RH, Lazar M (2011) Stabilization of polytopic delay difference inclusions via the Razumikhin approach. Automatica 47(12): 2562–2570

CrossRef MATH Google Scholar 4.

Gielen RH, Lazar M, Kolmanovsky IV (2012) Lyapunov methods for time-invariant delay difference inclusions. SIAM J Control Optim 50(1): 110–132

CrossRef Google Scholar 5.

Gu K, Kharitonov VL, Chen J (2003) Stability of time-delay systems. Birkhäuser, Boston

CrossRef MATH Google Scholar 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): 697–705

MathSciNet CrossRef MATH Google Scholar 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 1310–1316

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): 1646–1656

CrossRef MATH Google Scholar 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 5100–5105

10.

Jiang ZP, Wang Y (2001) Input-to-state stability for discrete-time nonlinear systems. Automatica 37: 857–869

MathSciNet CrossRef MATH Google Scholar 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 704–708

12.

Karafyllis I, Jiang ZP (2011) A vector small-gain theorem for general nonlinear control systems. IMA J Math Control Inf 28(3): 309–344

MathSciNet CrossRef MATH Google Scholar 13.

Kolmanovskii V, Myshkis A (1999) Introduction to the theory and applications of functional differential equations. Kluwer Academic Publishers, Dordrecht

MATH Google Scholar 14.

Kreyszig E (1989) Introductory functional analysis with applications. Wiley, New York

MATH Google Scholar 15.

Laila DS, Nes̆ić D (2003) Discrete-time Lyapunov-based small-gain theorem for parameterized interconnected ISS systems. IEEE Trans Autom Control 48(10): 1783–1788

CrossRef Google Scholar 16.

Lakshmikantham V, Matrosov VM, Sivasundaram S (1991) Vector Lyapunov functions and stability analysis of nonlinear systems. Kluwer Academic Publishers, Dordrecht

MATH Google Scholar 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): 797–803

MathSciNet CrossRef MATH Google Scholar 18.

Liu B, Hill DJ (2009) Input-to-state stability for discrete time-delay systems via the Razumikhin technique. Syst Control Lett 58: 567–575

MathSciNet CrossRef MATH Google Scholar 19.

Liu B, Marquez HJ (2007) Razumikhin-type stability theorems for discrete delay systems. Automatica 43(7): 1219–1225

MathSciNet CrossRef MATH Google Scholar 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 568–573

21.

Michel AN, Miller RK (1977) Qualatitive analysis of large scale dynamical systems, Mathematics in Science and Engineering, vol 134. Academic Press, Inc., New York

Google Scholar 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): 501–518

CrossRef Google Scholar 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): 404–409

MathSciNet CrossRef Google Scholar 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: 1651–1667

MathSciNet CrossRef MATH Google Scholar 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): 1722–1725

CrossRef Google Scholar 26.

Teel AR (1996) A nonlinear small gain theorem for the analysis of control systems with saturation. IEEE Trans Autom Control 41(9): 1256–1270

MathSciNet CrossRef MATH Google Scholar 27.

Teel AR (1998) Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem. IEEE Trans Autom Control 43(7): 960–964

MathSciNet CrossRef MATH Google Scholar 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 7407–7412

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 7204–7209

30.

Vidyasagar M (1981) Input–output analysis of large-scale interconnected systems. Lecture notes in control and information sciences, vol 29. Springer, Berlin

Google Scholar 31.

Šiljak DD (1978) Large-scale dynamic systems: stability and structure. North-Holland, Amsterdam

MATH Google Scholar 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): 1333–1340

CrossRef Google Scholar 33.

Willems JC (1972) Dissipative dynamical systems. Arch Ration Mech Anal 45: 321–393

MathSciNet CrossRef MATH Google Scholar Authors and Affiliations 1. Electrical Engineering Department Eindhoven University of Technology Eindhoven The Netherlands 2. Electrical and Computer Engineering Department University of California Santa Barbara USA