Input-to-state exponents and related ISS for delayed discrete-time systems with application to impulsive effects. (English) Zbl 1390.93710
Summary: This paper aims to investigate the input-to-state exponents (IS-\(e\)) and the related input-to-state stability (ISS) for delayed discrete-time systems (DDSs). By using the method of variation of parameters and introducing notions of uniform and weak uniform M-matrix, the estimates for 3 kinds of IS-\(e\) are derived for time-varying DDSs. The exponential ISS conditions with parts suitable for infinite delays are thus established, by which the difference from the time-invariant case is shown. The exponential stability of a time-varying DDS with zero external input cannot guarantee its ISS. Moreover, based on the IS-\(e\) estimates for DDSs, the exponential ISS under events criteria for DDSs with impulsive effects are obtained. The results are then applied in 1 example to test synchronization in the sense of ISS for a delayed discrete-time network, where the impulsive control is designed to stabilize such an asynchronous network to the synchronization.
MSC:
| 93D25 | Input-output approaches in control theory |
| 93C55 | Discrete-time control/observation systems |
| 93D20 | Asymptotic stability in control theory |
| 93C05 | Linear systems in control theory |
Keywords:
input-to-state exponent (IS-\(e\)); input-to-state stability (ISS); delayed discrete-time system (DDS); uniform M-matrix; time delayReferences:
| [1] | SontagED. Smooth stabilization implies coprime factorization. IEEE Trans Autom Control. 1989;34:435‐443. · Zbl 0682.93045 |
| [2] | SontagED, WangY. On characterization of the input‐to‐state stability property. Syst Control Lett. 1995;24:351‐359. · Zbl 0877.93121 |
| [3] | JiangZ‐P, TeelA, PralyL. Small‐gain theorem for ISS systems and applications. Math Control Signals Syst. 1994;7:95‐120. · Zbl 0836.93054 |
| [4] | JiangZ‐P, WangY. Input‐to‐state stability for discrete‐time nonlinear systems. Automatica. 2001;37:857‐869. · Zbl 0989.93082 |
| [5] | TeelAR. Connections between Razumikhin‐type theorems and the ISS nonlinear small gain theorem. IEEE Trans Autom Control. 1998;43:960‐964. · Zbl 0952.93121 |
| [6] | TeelAR, NešićD, KokotovićPV. A note on input‐to‐state stability of sampled‐data nonlinear systems. In: Proceeding of the 37th IEEE Conferences on Decision and Control. Tempa, Florida USA; 1998:2473‐2478. |
| [7] | LailaDS, AstolfiA. Input‐to‐state stability for discrete‐time time‐varying systems with applications to robust stabilization of systems in power form. Automatica. 2005;41:1891‐1903. · Zbl 1125.93475 |
| [8] | PepeP, JiangZ‐P. A Lyapunov‐Krasovskii methodology for ISS and iISS of time‐delay systems. Syst Control Lett. 2006;55(12):1006‐1014. · Zbl 1120.93361 |
| [9] | MazencF, MalisoffM, LinZ. On input‐to‐state stability for nonlinear systems with delayed feedbacks. Proc Autom Control Conf. 2007:4804‐4809. |
| [10] | YeganefarN, PepeP, DambrineM. Input‐to‐state stability of time‐delay systems: a link with exponential stability. IEEE Trans Autom Control. 2008;53:1526‐1531. · Zbl 1367.93284 |
| [11] | GielenRH, TeelAR, LazarM. Tractable Razumikhin‐type conditions for input‐to‐state stability analysis of delay difference inclusions. Automatica. 2013;49:619‐625. · Zbl 1259.93107 |
| [12] | GielenRH, LazarM, TeelAR. On input‐to‐state stability of delay difference equations. In: Proc of the 18th IFAC World Congress. Milano, Italy; 2011:3372‐3377. |
| [13] | LiuB, HillDJ. Input‐to‐state stability for discrete delay systems via Razumikhin technique. Syst Control Lett. 2009;58:567‐575. · Zbl 1166.93371 |
| [14] | HespanhaJP, LiberzonD, TeelAR. Lyapunov conditions for input‐to‐state stability of impulsive systems. Automatica. 2008;44:2735‐2744. · Zbl 1152.93050 |
| [15] | LiuB, MarquezHJ. Quasi‐exponential ISS for discrete impulsive hybrid systems. Intern J Control. 2007;80(4):540‐554. · Zbl 1117.93065 |
| [16] | CaiCH, TeelAR. Characterizations of input‐to‐state stability for hybrid systems. Syst Control Lett. 2009;58:47‐53. · Zbl 1154.93037 |
| [17] | DashkovskiySN, RüfferBS, WirthFR. An ISS small gain theorem for general networks. Math Control Signals Syst. 2007;19:93‐122. · Zbl 1125.93062 |
| [18] | DashkovskiyS, RüfferBS, WirthF. Small gain theorems for large scale systems and construction of ISS Lyapunov functions. SIAM J Control and Optim. 2010;48(6):4089‐4118. · Zbl 1202.93008 |
| [19] | DashkovskiyS, KosmykovM. Input‐to‐state stability of interconnected hybrid systems. Automatica. 2013;49(4):1068‐1074. · Zbl 1285.93086 |
| [20] | DashkovskiyS, MironchenkoA. Input‐to‐state stability of nonlinear impulsive systems. SIAM J Control Optim. 2013;51(3):1962‐1987. · Zbl 1271.34011 |
| [21] | LiuB, XuC, LiuD. ISS‐type comparison principles and ISS for discrete‐time dynamical networks with time‐delays. Int J Robust Nonlinear Control. 2013;23:450‐472. · Zbl 1271.93133 |
| [22] | LiuB, DouC, HillDJ. Robust exponential input‐to‐state stability of impulsive systems with an application in Micro‐Grids. Syst Control Lett. 2014;65:64‐73. · Zbl 1285.93087 |
| [23] | GrüneL, SontagED, WirthFR. 2000. On equivalence of exponential and asymptotic stability under changes of variables. In International Conference on Diferential Equations, vol. Vol. 1, 2 World Sci. Publishing: River Edge, NJ; 850‐852. · Zbl 0966.34045 |
| [24] | GielenRH, LazarM, KolmanovskyIV. Lyapunov methods for time‐invariant delay difference inclusions. SIAM J Control Optim. 2012;50(1):110‐132. · Zbl 1242.93101 |
| [25] | LiuB, MarquezHJ. Razumikhin‐type stability theorems for discrete delay systems. Automatica. 2007;43(7):1219‐1225. · Zbl 1123.93065 |
| [26] | HillDJ, MoylanPL. General instability results for interconnected systems. SIAM J Control and Opti. 1983;21(2):256‐279. · Zbl 0505.93052 |
| [27] | SteinG. Respect the unstable. IEEE Contr Syst Mag. 2003;23:12‐25. |
| [28] | QiuL. Quantify the unstable. In: Procd. the 19th International Symposium on Mathematical Theory of Networks and Systems. Budapest, Hungary; 2010:981‐986. |
| [29] | LiuB, HillDJ. Stability via hybrid‐event‐time Lyapunov function and impulsive stabilization for discrete‐time delayed switched systems. SIAM J Control and Optim. 2014;52(2):1338‐1365. · Zbl 1292.93103 |
| [30] | YangT. Impulsive Control Theory. Berlin: Springer‐Verlag, 2001. · Zbl 0996.93003 |
| [31] | GuanZH, HillDJ, ShenX. On hybrid impulsive and switching systems and application to nonlinear control. IEEE Trans Autom Control. 2005;50(7):1058‐1062. · Zbl 1365.93347 |
| [32] | ZhangY, SunJT. Stability of impulsive linear differential equations with time delays. IEEE Trans on Circuits Syst‐II. 2005;52:701‐705. |
| [33] | LiX, SongS. Impulsive control for existence, uniqueness and global stability of periodic solutions of recurrent neural networks with discrete and continuously distributed delays. IEEE Trans Neural Netw. 2013;24:868‐877. |
| [34] | BapatRB, RaghavanTES. Nonnegative Matrices and Applications. Cambridge: Cambridge University Press; 1997. · Zbl 0879.15015 |
| [35] | LiaoXX, YuP. Absolute Stability of Nonlinear Control Systems, 2nd edn. New York: Springer; 2008. · Zbl 1151.93002 |
| [36] | RughWJ. Linear System Theory, 2nd edn. Upper Saddle River, NJ: Prentice‐Hall; 1996. · Zbl 0892.93002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
