Article Contents
Article Contents

# Input-to-state stability of continuous-time systems via finite-time Lyapunov functions

• * Corresponding author: Huijuan Li

This work was partially supported by National Natural Science Foundation of China [NSFC11701533]

• In this paper, input-to-state stability (ISS) of continuous-time systems is analyzed via finite-time Lyapunov functions. ISS of a continuous-time system is first proved via finite-time robust Lyapunov functions for an introduced auxiliary system of the considered system. It is then obtained that the existence of a finite-time ISS Lyapunov function implies that the continuous-time system is ISS. The converse finite-time ISS Lyapunov theorem is proposed. Furthermore, we explore the properties of finite-time ISS Lyapunov functions for the continuous-time system on a bounded and compact set without a small neighborhood of the origin. The effectiveness of our results is illustrated by four examples.

Mathematics Subject Classification: Primary: 37B25, 93D25; Secondary: 34D20.

 Citation:

•  [1] D. Aeyels and J. Peuteman, A new asymptotic stability criterion for nonlinear time-variant differential equations, IEEE Transactions on Automatic Control, 43 (1998), 968-971.  doi: 10.1109/9.701102. [2] A. Browder, Mathematical Analysis. An introduction, Springer, 1996. doi: 10.1007/978-1-4612-0715-3. [3] S. Dashkovskiy, B. Rüffer and F. Wirth, A small-gain type stability criterion for large scale networks of ISS systems, Proc. of 44th IEEE Conference on Decision and Control and European Control Conference (ECC 2005), 2005, 5633–5638. [4] S. Dashkovskiy, B. Rüffer and F. Wirth, An ISS Lyapunov function for networks of ISS systems, in Proc. 17th Int. Symp. Math. Theory of Networks and Systems (MTNS 2006), Kyoto, Japan, July 24-28, 2006, 77–82. [5] S. Dashkovskiy, B. Rüffer and F. Wirth, An ISS small-gain theorem for general networks, Math. Control Signals Systems, 19 (2007), 93-122.  doi: 10.1007/s00498-007-0014-8. [6] S. Dashkovskiy, B. Rüffer and F. Wirth, Small gain theorems for large scale systems and construction of ISS Lyapunov functions, SIAM Journal on Control and Optimization, 48 (2010), 4089-4118.  doi: 10.1137/090746483. [7] A. Doban and M. Lazar, Computation of Lyapunov functions for nonlinear differential equations via a Yoshizawa-type construction, IFAC-PapersOnLine, 49 (2016), 29–34, 10th IFAC Symposium on Nonlinear Control Systems NOLCOS 2016. [8] R. Geiselhart, Advances in the Stability Analysis of Large-Scale Discrete-Time Systems, PhD thesis, Universität Würzburg, 2015. [9] R. Geiselhart and F. Wirth, Solving iterative functional equations for a class of piecewise linear -functions, Journal of Mathematical Analysis and Applications, 411 (2014), 652-664.  doi: 10.1016/j.jmaa.2013.10.016. [10] R. Geiselhart and F. Wirth, Relaxed ISS small-gain theorems for discrete-time systems, SIAM Journal on Control and Optimization, 54 (2016), 423-449.  doi: 10.1137/14097286X. [11] Z.-P. Jiang, I. M. Y. Mareels and Y. Wang, A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems, Automatica J. IFAC, 32 (1996), 1211-1215.  doi: 10.1016/0005-1098(96)00051-9. [12] I. Karafyllis, Can we prove stability by using a positive definite function with non sign-definite derivative?, IMA Journal of Mathematical Control and Information (2012), 29 (2012), 147-170.  doi: 10.1093/imamci/dnr035. [13] C. Kellett, A compendium of comparison function results, Math. Control Signals Systems, 26 (2014), 339-374.  doi: 10.1007/s00498-014-0128-8. [14] M. Lazar, A. I. Doban and N. Athanasopoulos, On stability analysis of discrete-time homogeneous dynamics, in System Theory, Control and Computing (ICSTCC), 2013 17th International Conference, 2013,297–305. doi: 10.1109/ICSTCC.2013.6688976. [15] H. Li and A. Liu, Computation of non-monotonic Lyapunov functions for continuous-time systems, Communications in Nonlinear Science and Numerical Simulation, 50 (2017), 35-50.  doi: 10.1016/j.cnsns.2017.02.017. [16] Y. Lin, E. D. Sontag and Y. Wang, A smooth converse Lyapunov theorem for robust stability, SIAM J. Control and Optimization, 34 (1996), 124–160. doi: 10.1137/S0363012993259981. [17] E. D. Sontag, Comments on integral variants of ISS, Systems Control Lett., 34 (1998), 93-100.  doi: 10.1016/S0167-6911(98)00003-6. [18] E. D. Sontag, Further facts about input to state stabilization, IEEE Trans. Automat. Control, 35 (1990), 473-476.  doi: 10.1109/9.52307. [19] E.D.Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition., Springer, 1998. doi: 10.1007/978-1-4612-0577-7. [20] E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control, 34 (1989), 435-443.  doi: 10.1109/9.28018. [21] E. D. Sontag, Some connections between stabilization and factorization, in Proc. of the 28th IEEE Conference on Decision and Control (CDC 1989), Vol. 1–3 (Tampa, FL, 1989), IEEE, New York, 1989,990–995. [22] E. D. Sontag and Y. Wang, New characterizations of input-to-state stability, IEEE Trans. Automat. Control, 41 (1996), 1283-1294.  doi: 10.1109/9.536498. [23] E. D. Sontag and Y. Wang, On characterizations of the input-to-state stability property, Systems Control Lett., 24 (1995), 351-359.  doi: 10.1016/0167-6911(94)00050-6.