American Institute of Mathematical Sciences

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March  2020, 25(3): 841-857. doi: 10.3934/dcdsb.2019192

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

Huijuan Li , and  Junxia Wang

1. 

School of Mathematics and Physics, China University of Geosciences (Wuhan), 430074, Wuhan, China

* Corresponding author: Huijuan Li

Received  November 2018 Revised  March 2019 Published  March 2020 Early access  September 2019

Fund Project: 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.

Keywords: Continuous-time systems, input-to-state stability, ISS Lyapunov function, finite-time ISS Lyapunov function.
Mathematics Subject Classification: Primary: 37B25, 93D25; Secondary: 34D20.
Citation: Huijuan Li, Junxia Wang. Input-to-state stability of continuous-time systems via finite-time Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 841-857. doi: 10.3934/dcdsb.2019192
References:
[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.  Google Scholar

[2]

A. Browder, Mathematical Analysis. An introduction, Springer, 1996. doi: 10.1007/978-1-4612-0715-3.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[8]

R. Geiselhart, Advances in the Stability Analysis of Large-Scale Discrete-Time Systems, PhD thesis, Universität Würzburg, 2015. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

C. Kellett, A compendium of comparison function results, Math. Control Signals Systems, 26 (2014), 339-374.  doi: 10.1007/s00498-014-0128-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

E. D. Sontag, Further facts about input to state stabilization, IEEE Trans. Automat. Control, 35 (1990), 473-476.  doi: 10.1109/9.52307.  Google Scholar

[19]

E.D.Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition., Springer, 1998. doi: 10.1007/978-1-4612-0577-7.  Google Scholar

[20]

E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control, 34 (1989), 435-443.  doi: 10.1109/9.28018.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[2]

A. Browder, Mathematical Analysis. An introduction, Springer, 1996. doi: 10.1007/978-1-4612-0715-3.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[8]

R. Geiselhart, Advances in the Stability Analysis of Large-Scale Discrete-Time Systems, PhD thesis, Universität Würzburg, 2015. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

C. Kellett, A compendium of comparison function results, Math. Control Signals Systems, 26 (2014), 339-374.  doi: 10.1007/s00498-014-0128-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

E. D. Sontag, Further facts about input to state stabilization, IEEE Trans. Automat. Control, 35 (1990), 473-476.  doi: 10.1109/9.52307.  Google Scholar

[19]

E.D.Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition., Springer, 1998. doi: 10.1007/978-1-4612-0577-7.  Google Scholar

[20]

E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control, 34 (1989), 435-443.  doi: 10.1109/9.28018.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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