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Computation of local ISS Lyapunov functions with low gains via linear programming
1. | School of Mathematics and Physics, Chinese University of Geosciences (Wuhan), 430074, Wuhan, China |
2. | Lehrstuhl für Angewandte Mathematik, Universität Bayreuth, 95440 Bayreuth, Germany, Germany |
3. | School of Science and Engineering, Reykjavik University, Menntavegi 1, Reykjavik, IS-101 |
4. | Fakultät für Informatik und Mathematik, Universität Passau, 94030 Passau, Germany |
References:
[1] |
M. Abu Hassan and C. Storey, Numerical determination of domains of attraction for electrical power systems using the method of Zubov, Internat. J. Control, 34 (1981), 371-381.
doi: 10.1080/00207178108922536. |
[2] |
R. Baier, L. Grüne and S. F. Hafstein, Linear programming based Lyapunov function computation for differential inclusions, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 33-56.
doi: 10.3934/dcdsb.2012.17.33. |
[3] |
F. Camilli, L. Grüne and F. Wirth, A regularization of Zubov's equation for robust domains of attraction, in Nonlinear Control in the Year 2000, Volume 1 (eds. A. Isidori, F. Lamnabhi-Lagarrigue and W. Respondek), Lecture Notes in Control and Inform. Sci., 258, NCN, Springer-Verlag, London, 2000, 277-289.
doi: 10.1007/BFb0110220. |
[4] |
F. Camilli, L. Grüne and F. Wirth, Domains of attraction of interconnected systems: A Zubov method approach, in Proc. European Control Conference (ECC 2009), Budapest, Hungary, 2009, 91-96. |
[5] |
F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag, Berlin, 1998. |
[6] |
S. Dashkovskiy, B. Rüffer and F. Wirth, A small-gain type stability criterion for large scale networks of ISS systems, in Proc. of 44th IEEE Conference on Decision and Control and European Control Conference (ECC 2005), 2005, 5633-5638.
doi: 10.1109/CDC.2005.1583060. |
[7] |
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, 2006, 77-82. |
[8] |
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. |
[9] |
S. N. Dashkovskiy, B. S. Rüffer and F. R. Wirth, Small gain theorems for large scale systems and construction of ISS Lyapunov functions, SIAM J. Control Optim., 48 (2010), 4089-4118.
doi: 10.1137/090746483. |
[10] |
P. Giesl and S. Hafstein, Existence of piecewise linear Lyapunov functions in arbitrary dimensions, Discrete Contin. Dyn. Syst., 32 (2012), 3539-3565.
doi: 10.3934/dcds.2012.32.3539. |
[11] |
L. Grüne and M. Sigurani, Numerical ISS controller design via a dynamic game approach, in Proc. of the 52nd IEEE Conference on Decision and Control (CDC 2013), 2013, 1732-1737. |
[12] |
S. F. Hafstein, A constructive converse Lyapunov theorem on exponential stability, Discrete Contin. Dyn. Syst., 10 (2004), 657-678.
doi: 10.3934/dcds.2004.10.657. |
[13] |
S. F. Hafstein, A constructive converse Lyapunov theorem on asymptotic stability for nonlinear autonomous ordinary differential equations, Dyn. Syst., 20 (2005), 281-299.
doi: 10.1080/14689360500164873. |
[14] |
S. F. Hafstein, An Algorithm for Constructing Lyapunov Functions, vol. 8 of Electron. J. Differ. Equ. Monogr., Texas State University-San Marcos, Department of Mathematics, San Marcos, TX, 2007. Available from: http://ejde.math.txstate.edu. |
[15] |
S. Huang, M. R. James, D. Nešić and P. M. Dower, A unified approach to controller design for achieving ISS and related properties, IEEE Trans. Automat. Control, 50 (2005), 1681-1697.
doi: 10.1109/TAC.2005.858691. |
[16] |
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. |
[17] |
H. Li and F. Wirth, Zubov's method for interconnected systems - a dissipative formulation, in Proc. 20th Int. Symp. Math. Theory of Networks and Systems (MTNS 2012), Melbourne, Australia, 2012, Paper No. 184, 8 pages. |
[18] |
S. F. Marinósson, Lyapunov function construction for ordinary differential equations with linear programming, Dyn. Syst., 17 (2002), 137-150.
doi: 10.1080/0268111011011847. |
[19] |
A. N. Michel, N. R. Sarabudla and R. K. Miller, Stability analysis of complex dynamical systems: Some computational methods, Circuits Systems Signal Process., 1 (1982), 171-202.
doi: 10.1007/BF01600051. |
[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, Further facts about input to state stabilization, IEEE Trans. Automat. Control, 35 (1990), 473-476.
doi: 10.1109/9.52307. |
[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. |
[24] |
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. |
show all references
References:
[1] |
M. Abu Hassan and C. Storey, Numerical determination of domains of attraction for electrical power systems using the method of Zubov, Internat. J. Control, 34 (1981), 371-381.
doi: 10.1080/00207178108922536. |
[2] |
R. Baier, L. Grüne and S. F. Hafstein, Linear programming based Lyapunov function computation for differential inclusions, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 33-56.
doi: 10.3934/dcdsb.2012.17.33. |
[3] |
F. Camilli, L. Grüne and F. Wirth, A regularization of Zubov's equation for robust domains of attraction, in Nonlinear Control in the Year 2000, Volume 1 (eds. A. Isidori, F. Lamnabhi-Lagarrigue and W. Respondek), Lecture Notes in Control and Inform. Sci., 258, NCN, Springer-Verlag, London, 2000, 277-289.
doi: 10.1007/BFb0110220. |
[4] |
F. Camilli, L. Grüne and F. Wirth, Domains of attraction of interconnected systems: A Zubov method approach, in Proc. European Control Conference (ECC 2009), Budapest, Hungary, 2009, 91-96. |
[5] |
F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag, Berlin, 1998. |
[6] |
S. Dashkovskiy, B. Rüffer and F. Wirth, A small-gain type stability criterion for large scale networks of ISS systems, in Proc. of 44th IEEE Conference on Decision and Control and European Control Conference (ECC 2005), 2005, 5633-5638.
doi: 10.1109/CDC.2005.1583060. |
[7] |
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, 2006, 77-82. |
[8] |
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. |
[9] |
S. N. Dashkovskiy, B. S. Rüffer and F. R. Wirth, Small gain theorems for large scale systems and construction of ISS Lyapunov functions, SIAM J. Control Optim., 48 (2010), 4089-4118.
doi: 10.1137/090746483. |
[10] |
P. Giesl and S. Hafstein, Existence of piecewise linear Lyapunov functions in arbitrary dimensions, Discrete Contin. Dyn. Syst., 32 (2012), 3539-3565.
doi: 10.3934/dcds.2012.32.3539. |
[11] |
L. Grüne and M. Sigurani, Numerical ISS controller design via a dynamic game approach, in Proc. of the 52nd IEEE Conference on Decision and Control (CDC 2013), 2013, 1732-1737. |
[12] |
S. F. Hafstein, A constructive converse Lyapunov theorem on exponential stability, Discrete Contin. Dyn. Syst., 10 (2004), 657-678.
doi: 10.3934/dcds.2004.10.657. |
[13] |
S. F. Hafstein, A constructive converse Lyapunov theorem on asymptotic stability for nonlinear autonomous ordinary differential equations, Dyn. Syst., 20 (2005), 281-299.
doi: 10.1080/14689360500164873. |
[14] |
S. F. Hafstein, An Algorithm for Constructing Lyapunov Functions, vol. 8 of Electron. J. Differ. Equ. Monogr., Texas State University-San Marcos, Department of Mathematics, San Marcos, TX, 2007. Available from: http://ejde.math.txstate.edu. |
[15] |
S. Huang, M. R. James, D. Nešić and P. M. Dower, A unified approach to controller design for achieving ISS and related properties, IEEE Trans. Automat. Control, 50 (2005), 1681-1697.
doi: 10.1109/TAC.2005.858691. |
[16] |
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. |
[17] |
H. Li and F. Wirth, Zubov's method for interconnected systems - a dissipative formulation, in Proc. 20th Int. Symp. Math. Theory of Networks and Systems (MTNS 2012), Melbourne, Australia, 2012, Paper No. 184, 8 pages. |
[18] |
S. F. Marinósson, Lyapunov function construction for ordinary differential equations with linear programming, Dyn. Syst., 17 (2002), 137-150.
doi: 10.1080/0268111011011847. |
[19] |
A. N. Michel, N. R. Sarabudla and R. K. Miller, Stability analysis of complex dynamical systems: Some computational methods, Circuits Systems Signal Process., 1 (1982), 171-202.
doi: 10.1007/BF01600051. |
[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, Further facts about input to state stabilization, IEEE Trans. Automat. Control, 35 (1990), 473-476.
doi: 10.1109/9.52307. |
[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. |
[24] |
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. |
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