Inputtostate stability analysis for interconnected difference equations with delay
 Rob H. Gielen,
 Mircea Lazar,
 Andrew R. Teel
 … show all 3 hide
Abstract
Inputtostate 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 smallgain arguments, it is demonstrated that interconnection delays do not affect the stability of an interconnected system if a delayindependent smallgain condition holds. Furthermore, also using smallgain 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 largescale interconnections of difference equations with delay.
 Dashkovskiy SN, Naujok L (2010) Lyapunov–Razumikhin and Lyapunov–Krasovskii theorems for interconnected ISS timedelay systems. In: 19th International symposium on mathematical theory of networks and systems, Budapest, Hungary, pp 1179–1184
 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: pp. 40894118 CrossRef
 Gielen, RH, Lazar, M (2011) Stabilization of polytopic delay difference inclusions via the Razumikhin approach. Automatica 47: pp. 25622570 CrossRef
 Gielen, RH, Lazar, M, Kolmanovsky, IV (2012) Lyapunov methods for timeinvariant delay difference inclusions. SIAM J Control Optim 50: pp. 110132 CrossRef
 Gu, K, Kharitonov, VL, Chen, J (2003) Stability of timedelay systems. Birkhäuser, Boston CrossRef
 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: pp. 697705 CrossRef
 Ito H, Pepe P, Jiang ZP (2009) Construction of Lyapunov–Krasovskii functionals for interconnection of retarded dynamic and static systems via a smallgain condition. In: Proceedings of the 48th IEEE conference on decision and control, Shanghai, China, pp 1310–1316
 Ito, H, Pepe, P, Jiang, ZP (2010) A smallgain condition for iISS of interconnected retarded systems based on LyapunovKrasovskii functionals. Automatica 46: pp. 16461656 CrossRef
 Ito H, Jiang ZP, Pepe P (2011) A smallgain 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
 Jiang, ZP, Wang, Y (2001) Inputtostate stability for discretetime nonlinear systems. Automatica 37: pp. 857869 CrossRef
 Jiang ZP, Lin Y, Wang Y (2008) Nonlinear smallgain theorems for discretetime largescale systems. In: Proceedings of the 27th Chinese control conference, Kunming, China, pp 704–708
 Karafyllis, I, Jiang, ZP (2011) A vector smallgain theorem for general nonlinear control systems. IMA J Math Control Inf 28: pp. 309344 CrossRef
 Kolmanovskii, V, Myshkis, A (1999) Introduction to the theory and applications of functional differential equations. Kluwer Academic Publishers, Dordrecht
 Kreyszig, E (1989) Introductory functional analysis with applications. Wiley, New York
 Laila, DS, Nes̆ić, D (2003) Discretetime Lyapunovbased smallgain theorem for parameterized interconnected ISS systems. IEEE Trans Autom Control 48: pp. 17831788 CrossRef
 Lakshmikantham, V, Matrosov, VM, Sivasundaram, S (1991) Vector Lyapunov functions and stability analysis of nonlinear systems. Kluwer Academic Publishers, Dordrecht
 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: pp. 797803 CrossRef
 Liu, B, Hill, DJ (2009) Inputtostate stability for discrete timedelay systems via the Razumikhin technique. Syst Control Lett 58: pp. 567575 CrossRef
 Liu, B, Marquez, HJ (2007) Razumikhintype stability theorems for discrete delay systems. Automatica 43: pp. 12191225 CrossRef
 Liu T, Hill DJ, Jiang ZP (2010) Lyapunov formulation of ISS cyclicsmallgain in discretetime dynamical networks. In: Proceedings of the 8th WCICA, Jinan, China, pp 568–573
 Michel, AN, Miller, RK (1977) Qualatitive analysis of large scale dynamical systems, Mathematics in Science and Engineering, vol 134. Academic Press, Inc., New York
 Orero, SO, Irving, MR (1998) A genetic algorithm modelling framework and solution technique for short term optimal hydrothermal scheduling. IEEE Trans Power Syst 13: pp. 501518 CrossRef
 Polushin, I, Marquez, HJ, Tayebi, A, Liu, PX (2009) A multichannel IOS small gain theorem for systems with multiple timevarying communication delays. IEEE Trans Autom Control 54: pp. 404409 CrossRef
 Raimondo, DM, Magni, L, Scattolini, R (2007) Decentralized MPC of nonlinear systems: an inputtostate stability approach. Int J Robust Nonlinear Control 17: pp. 16511667 CrossRef
 Rüffer, BS, Sailer, R, Wirth, FR (2010) Comments on “a multichannel ios small gain theorem for systems with multiple timevarying communication delays”. IEEE Trans Autom Control 55: pp. 17221725 CrossRef
 Teel, AR (1996) A nonlinear small gain theorem for the analysis of control systems with saturation. IEEE Trans Autom Control 41: pp. 12561270 CrossRef
 Teel, AR (1998) Connections between Razumikhintype theorems and the ISS nonlinear small gain theorem. IEEE Trans Autom Control 43: pp. 960964 CrossRef
 Tiwari S, Wang Y (2010) Razumikhintype smallgain theorems for largescale systems with delays. In: Proceedings of the 49th IEEE conference on decision and control, Atlanta, GA, pp 7407–7412
 Tiwari S, Wang Y, Jiang ZP (2009) A nonlinear smallgain theorem for largescale time delay systems. In: Proceedings of the 48th IEEE conference on decision and control, Shanghai, China, pp 7204–7209
 Vidyasagar, M (1981) Input–output analysis of largescale interconnected systems. Lecture notes in control and information sciences, vol 29. Springer, Berlin
 Šiljak, DD (1978) Largescale dynamic systems: stability and structure. NorthHolland, Amsterdam
 Wang, C, Shahidehpour, SM (1993) Power generation scheduling for multiarea hydrothermal systems with tie line constraints, cascaded reservoirs and uncertain data. IEEE Trans Power Syst 8: pp. 13331340 CrossRef
 Willems, JC (1972) Dissipative dynamical systems. Arch Ration Mech Anal 45: pp. 321393 CrossRef
 Title
 Inputtostate stability analysis for interconnected difference equations with delay
 Open Access
 Available under Open Access This content is freely available online to anyone, anywhere at any time.
 Journal

Mathematics of Control, Signals, and Systems
Volume 24, Issue 12 , pp 3354
 Cover Date
 20120401
 DOI
 10.1007/s0049801200804
 Print ISSN
 09324194
 Online ISSN
 1435568X
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Largescale systems
 Time delay
 Difference equations
 Lyapunov methods
 Smallgain theorem
 Industry Sectors
 Authors

 Rob H. Gielen ^{(1)}
 Mircea Lazar ^{(1)}
 Andrew R. Teel ^{(2)}
 Author Affiliations

 1. Electrical Engineering Department, Eindhoven University of Technology, Eindhoven, The Netherlands
 2. Electrical and Computer Engineering Department, University of California, Santa Barbara, USA