doi: 10.3934/dcdsb.2019192

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

* Corresponding author: Huijuan Li

Received  November 2018 Revised  March 2019 Published  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.

Citation: Huijuan Li, Junxia Wang. Input-to-state stability of continuous-time systems via finite-time Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 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. DashkovskiyB. 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. DashkovskiyB. 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. JiangI. 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. DashkovskiyB. 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. DashkovskiyB. 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. JiangI. 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

[1]

Peter Giesl. Construction of a finite-time Lyapunov function by meshless collocation. Discrete & 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 & Continuous Dynamical Systems - A, 2011, 31 (4) : 1023-1038. doi: 10.3934/dcds.2011.31.1023

[3]

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 & Continuous Dynamical Systems - B, 2015, 20 (8) : 2477-2495. doi: 10.3934/dcdsb.2015.20.2477

[4]

Andrii Mironchenko, Hiroshi Ito. Characterizations of integral input-to-state stability for bilinear systems in infinite dimensions. Mathematical Control & Related Fields, 2016, 6 (3) : 447-466. doi: 10.3934/mcrf.2016011

[5]

Łukasz Struski, Jacek Tabor. Expansivity implies existence of Hölder continuous Lyapunov function. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3575-3589. doi: 10.3934/dcdsb.2017180

[6]

Arno Berger. On finite-time hyperbolicity. Communications on Pure & Applied Analysis, 2011, 10 (3) : 963-981. doi: 10.3934/cpaa.2011.10.963

[7]

Fritz Colonius, Guilherme Mazanti. Decay rates for stabilization of linear continuous-time systems with random switching. Mathematical Control & Related Fields, 2019, 9 (1) : 39-58. doi: 10.3934/mcrf.2019002

[8]

Arno Berger, Doan Thai Son, Stefan Siegmund. Nonautonomous finite-time dynamics. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 463-492. doi: 10.3934/dcdsb.2008.9.463

[9]

Jianjun Paul Tian. Finite-time perturbations of dynamical systems and applications to tumor therapy. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 469-479. doi: 10.3934/dcdsb.2009.12.469

[10]

Grzegorz Karch, Kanako Suzuki, Jacek Zienkiewicz. Finite-time blowup of solutions to some activator-inhibitor systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4997-5010. doi: 10.3934/dcds.2016016

[11]

Joon Kwon, Panayotis Mertikopoulos. A continuous-time approach to online optimization. Journal of Dynamics & Games, 2017, 4 (2) : 125-148. doi: 10.3934/jdg.2017008

[12]

Hanqing Jin, Xun Yu Zhou. Continuous-time portfolio selection under ambiguity. Mathematical Control & Related Fields, 2015, 5 (3) : 475-488. doi: 10.3934/mcrf.2015.5.475

[13]

Peter Giesl. Construction of a global Lyapunov function using radial basis functions with a single operator. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 101-124. doi: 10.3934/dcdsb.2007.7.101

[14]

Andrei Korobeinikov, Philip K. Maini. A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence. Mathematical Biosciences & Engineering, 2004, 1 (1) : 57-60. doi: 10.3934/mbe.2004.1.57

[15]

Robert Baier, Lars Grüne, Sigurđur Freyr Hafstein. Linear programming based Lyapunov function computation for differential inclusions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 33-56. doi: 10.3934/dcdsb.2012.17.33

[16]

Hjörtur Björnsson, Sigurdur Hafstein, Peter Giesl, Enrico Scalas, Skuli Gudmundsson. Computation of the stochastic basin of attraction by rigorous construction of a Lyapunov function. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4247-4269. doi: 10.3934/dcdsb.2019080

[17]

Giovanni Colombo, Khai T. Nguyen. On the minimum time function around the origin. Mathematical Control & Related Fields, 2013, 3 (1) : 51-82. doi: 10.3934/mcrf.2013.3.51

[18]

Fatiha Alabau-Boussouira, Vincent Perrollaz, Lionel Rosier. Finite-time stabilization of a network of strings. Mathematical Control & Related Fields, 2015, 5 (4) : 721-742. doi: 10.3934/mcrf.2015.5.721

[19]

Hui Meng, Fei Lung Yuen, Tak Kuen Siu, Hailiang Yang. Optimal portfolio in a continuous-time self-exciting threshold model. Journal of Industrial & Management Optimization, 2013, 9 (2) : 487-504. doi: 10.3934/jimo.2013.9.487

[20]

Shui-Nee Chow, Xiaojing Ye, Hongyuan Zha, Haomin Zhou. Influence prediction for continuous-time information propagation on networks. Networks & Heterogeneous Media, 2018, 13 (4) : 567-583. doi: 10.3934/nhm.2018026

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (21)
  • HTML views (64)
  • Cited by (0)

Other articles
by authors

[Back to Top]