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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.
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.
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, (2000), 277.
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), (2009), 91. Google Scholar |
[5] |
F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory,, Springer-Verlag, (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.
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), (2006), 77. Google Scholar |
[8] |
S. Dashkovskiy, B. Rüffer and F. Wirth, An ISS small-gain theorem for general networks,, Math. Control Signals Systems, 19 (2007), 93.
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.
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.
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. Google Scholar |
[12] |
S. F. Hafstein, A constructive converse Lyapunov theorem on exponential stability,, Discrete Contin. Dyn. Syst., 10 (2004), 657.
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.
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, (2007).
|
[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.
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.
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), (2012). Google Scholar |
[18] |
S. F. Marinósson, Lyapunov function construction for ordinary differential equations with linear programming,, Dyn. Syst., 17 (2002), 137.
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.
doi: 10.1007/BF01600051. |
[20] |
E. D. Sontag, Smooth stabilization implies coprime factorization,, IEEE Trans. Automat. Control, 34 (1989), 435.
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), (1989), 1.
|
[22] |
E. D. Sontag, Further facts about input to state stabilization,, IEEE Trans. Automat. Control, 35 (1990), 473.
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.
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.
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.
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.
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, (2000), 277.
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), (2009), 91. Google Scholar |
[5] |
F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory,, Springer-Verlag, (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.
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), (2006), 77. Google Scholar |
[8] |
S. Dashkovskiy, B. Rüffer and F. Wirth, An ISS small-gain theorem for general networks,, Math. Control Signals Systems, 19 (2007), 93.
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.
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.
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. Google Scholar |
[12] |
S. F. Hafstein, A constructive converse Lyapunov theorem on exponential stability,, Discrete Contin. Dyn. Syst., 10 (2004), 657.
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.
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, (2007).
|
[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.
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.
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), (2012). Google Scholar |
[18] |
S. F. Marinósson, Lyapunov function construction for ordinary differential equations with linear programming,, Dyn. Syst., 17 (2002), 137.
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.
doi: 10.1007/BF01600051. |
[20] |
E. D. Sontag, Smooth stabilization implies coprime factorization,, IEEE Trans. Automat. Control, 34 (1989), 435.
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), (1989), 1.
|
[22] |
E. D. Sontag, Further facts about input to state stabilization,, IEEE Trans. Automat. Control, 35 (1990), 473.
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.
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.
doi: 10.1109/9.536498. |
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