-
Previous Article
On steady state of some Lotka-Volterra competition-diffusion-advection model
- DCDS-B Home
- This Issue
-
Next Article
Derivation of cable equation by multiscale analysis for a model of myelinated axons
Input-to-state stability of continuous-time systems via finite-time Lyapunov functions
1. | School of Mathematics and Physics, China University of Geosciences (Wuhan), 430074, Wuhan, China |
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.
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. |
[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. |
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. |
[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. |
[1] |
Peter Giesl. Construction of a finite-time Lyapunov function by meshless collocation. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2387-2412. doi: 10.3934/dcdsb.2012.17.2387 |
[2] |
Khalid Addi, Samir Adly, Hassan Saoud. Finite-time Lyapunov stability analysis of evolution variational inequalities. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1023-1038. doi: 10.3934/dcds.2011.31.1023 |
[3] |
Sanjeeva Balasuriya. Uncertainty in finite-time Lyapunov exponent computations. Journal of Computational Dynamics, 2020, 7 (2) : 313-337. doi: 10.3934/jcd.2020013 |
[4] |
Hiroshi Ito. Input-to-state stability and Lyapunov functions with explicit domains for SIR model of infectious diseases. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5171-5196. doi: 10.3934/dcdsb.2020338 |
[5] |
Huijuan Li, Robert Baier, Lars Grüne, Sigurdur F. Hafstein, Fabian R. Wirth. Computation of local ISS Lyapunov functions with low gains via linear programming. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2477-2495. doi: 10.3934/dcdsb.2015.20.2477 |
[6] |
Andrii Mironchenko, Hiroshi Ito. Characterizations of integral input-to-state stability for bilinear systems in infinite dimensions. Mathematical Control and Related Fields, 2016, 6 (3) : 447-466. doi: 10.3934/mcrf.2016011 |
[7] |
Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 821-836. doi: 10.3934/dcdsb.2021066 |
[8] |
Łukasz Struski, Jacek Tabor. Expansivity implies existence of Hölder continuous Lyapunov function. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3575-3589. doi: 10.3934/dcdsb.2017180 |
[9] |
Arno Berger. On finite-time hyperbolicity. Communications on Pure and Applied Analysis, 2011, 10 (3) : 963-981. doi: 10.3934/cpaa.2011.10.963 |
[10] |
Fritz Colonius, Guilherme Mazanti. Decay rates for stabilization of linear continuous-time systems with random switching. Mathematical Control and Related Fields, 2019, 9 (1) : 39-58. doi: 10.3934/mcrf.2019002 |
[11] |
Arno Berger, Doan Thai Son, Stefan Siegmund. Nonautonomous finite-time dynamics. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 463-492. doi: 10.3934/dcdsb.2008.9.463 |
[12] |
Jianjun Paul Tian. Finite-time perturbations of dynamical systems and applications to tumor therapy. Discrete and Continuous Dynamical Systems - B, 2009, 12 (2) : 469-479. doi: 10.3934/dcdsb.2009.12.469 |
[13] |
Grzegorz Karch, Kanako Suzuki, Jacek Zienkiewicz. Finite-time blowup of solutions to some activator-inhibitor systems. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4997-5010. doi: 10.3934/dcds.2016016 |
[14] |
Tianhu Yu, Jinde Cao, Chuangxia Huang. Finite-time cluster synchronization of coupled dynamical systems with impulsive effects. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3595-3620. doi: 10.3934/dcdsb.2020248 |
[15] |
Joon Kwon, Panayotis Mertikopoulos. A continuous-time approach to online optimization. Journal of Dynamics and Games, 2017, 4 (2) : 125-148. doi: 10.3934/jdg.2017008 |
[16] |
Hanqing Jin, Xun Yu Zhou. Continuous-time portfolio selection under ambiguity. Mathematical Control and Related Fields, 2015, 5 (3) : 475-488. doi: 10.3934/mcrf.2015.5.475 |
[17] |
Ran Dong, Xuerong Mao. Asymptotic stabilization of continuous-time periodic stochastic systems by feedback control based on periodic discrete-time observations. Mathematical Control and Related Fields, 2020, 10 (4) : 715-734. doi: 10.3934/mcrf.2020017 |
[18] |
Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331 |
[19] |
Qian Zhang, Huaicheng Yan, Jun Cheng, Xisheng Zhan, Kaibo Shi. Fault detection filtering for continuous-time singular systems under a dynamic event-triggered mechanism. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022023 |
[20] |
Ruofeng Rao, Shouming Zhong. Input-to-state stability and no-inputs stabilization of delayed feedback chaotic financial system involved in open and closed economy. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1375-1393. doi: 10.3934/dcdss.2020280 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]