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Computation of local ISS Lyapunov functions with low gains via linear programming
Separable Lyapunov functions for monotone systems: Constructions and limitations
1. | Institut für Mathematik, Universität Würzburg, Campus Hubland Nord, Emil-Fischer-Str. 40, 97074 Würzburg, Germany |
2. | Department of Systems Design and Informatics, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka 820-8502, Japan |
3. | Automatic Control LTH, Lund University, Box 118, SE-221 00 Lund, Sweden |
4. | School of Mathematical & Physical Sciences, Faculty of Science & IT, The University of Newcastle (UON), University Drive, Callaghan NSW 2308, Australia |
References:
[1] |
D. Angeli and A. Astolfi, A tight small-gain theorem for not necessarily ISS systems,, Systems Control Lett., 56 (2007), 87.
doi: 10.1016/j.sysconle.2006.08.003. |
[2] |
D. Angeli, E. D. Sontag and Y. Wang, A characterization of integral input-to-state stability,, IEEE Trans. Autom. Control, 45 (2000), 1082.
doi: 10.1109/9.863594. |
[3] |
A. Bacciotti, F. Ceragioli and L. Mazzi, Differential inclusions and monotonicity conditions for nonsmooth Lyapunov functions,, Set-Valued Analysis, 8 (2000), 299.
doi: 10.1023/A:1008763931789. |
[4] |
A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, Academic Press, (1979).
|
[5] |
N. P. Bhatia and G. P. Szegö, Stability Theory of Dynamical Systems,, Die Grundlehren der mathematischen Wissenschaften, (1970).
|
[6] |
F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory,, Springer, (1998).
|
[7] |
G. Como, E. Lovisari and K. Savla, Throughput optimality and overload behavior of dynamical flow networks under monotone distributed routing,, IEEE Trans. Contr. Network Systems, 2 (2014), 57.
doi: 10.1109/TCNS.2014.2367361. |
[8] |
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. |
[9] |
P. De Leenheer, D. Angeli and E. D. Sontag, Monotone chemical reaction networks,, Journal of Mathematical Chemistry, 41 (2007), 295.
doi: 10.1007/s10910-006-9075-z. |
[10] |
D. Hinrichsen and A. J. Pritchard, Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness,, Springer, (2005).
doi: 10.1007/b137541. |
[11] |
M. W. Hirsch and H. Smith, Monotone dynamical systems,, in Handbook of Differential Equations: Ordinary Differential Equations, (2005), 239.
|
[12] |
M. W. Hirsch, Systems of differential equations which are competitive or cooperative. I. Limit sets,, SIAM J. Math. Anal., 13 (1982), 167.
doi: 10.1137/0513013. |
[13] |
M. W. Hirsch, Systems of differential equations that are competitive or cooperative. II. Convergence almost everywhere,, SIAM J. Math. Anal., 16 (1985), 423.
doi: 10.1137/0516030. |
[14] |
M. W. Hirsch, Systems of differential equations which are competitive or cooperative. III. Competing species,, Nonlinearity, 1 (1988), 51.
doi: 10.1088/0951-7715/1/1/003. |
[15] |
M. W. Hirsch, Systems of differential equations that are competitive or cooperative. V. Convergence in $3$-dimensional systems,, J. Diff. Eqns., 80 (1989), 94.
doi: 10.1016/0022-0396(89)90097-1. |
[16] |
M. W. Hirsch, Systems of differential equations that are competitive or cooperative. IV. Structural stability in three-dimensional systems,, SIAM J. Math. Anal., 21 (1990), 1225.
doi: 10.1137/0521067. |
[17] |
M. W. Hirsch, Systems of differential equations that are competitive or cooperative. VI. A local $C^r$ closing lemma for 3-dimensional systems,, Ergodic Theory Dynam. Systems, 11 (1991), 443.
doi: 10.1017/S014338570000626X. |
[18] |
M. W. Hirsch and H. L. Smith, Competitive and cooperative systems: Mini-review,, in Positive systems (Rome, (2003), 183.
doi: 10.1007/978-3-540-44928-7_25. |
[19] |
H. Ito, A Lyapunov approach to cascade interconnection of integral input-to-state stable systems,, IEEE Trans. Autom. Control, 55 (2010), 702.
doi: 10.1109/TAC.2009.2037457. |
[20] |
H. Ito and Z.-P. Jiang, Necessary and sufficient small gain conditions for integral input-to-state stable systems: A Lyapunov perspective,, IEEE Trans. Autom. Control, 54 (2009), 2389.
doi: 10.1109/TAC.2009.2028980. |
[21] |
H. Ito, State-dependent scaling problems and stability of interconnected iISS and ISS systems,, IEEE Trans. Autom. Control, 51 (2006), 1626.
doi: 10.1109/TAC.2006.882930. |
[22] |
H. Ito, S. Dashkovskiy and F. Wirth, Capability and limitation of max- and sum-type construction of Lyapunov functions for networks of iISS systems,, Automatica J. IFAC, 48 (2012), 1197.
doi: 10.1016/j.automatica.2012.03.018. |
[23] |
H. Ito, Z.-P. Jiang, S. Dashkovskiy and B. S. Rüffer, Robust stability of networks of iISS systems: Construction of sum-type Lyapunov functions,, IEEE Trans. Autom. Control, 58 (2013), 1192.
doi: 10.1109/TAC.2012.2231552. |
[24] |
H. Ito, B. S. Rüffer and A. Rantzer, Max- and sum-separable Lyapunov functions for monotone systems and their level sets,, in Proc. 53rd IEEE Conf. Decis. Control, (2014), 2371.
doi: 10.1109/CDC.2014.7039750. |
[25] |
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. |
[26] |
H. K. Khalil, Nonlinear systems,, 3rd edition, (2002). Google Scholar |
[27] |
V. Lakshmikantham and S. Leela, Differential and Integral Inequalities: Theory and Applications, Vol. I: Ordinary Differential Equations,, Academic Press, (1969).
|
[28] |
M. Margaliot and T. Tuller, Stability analysis of the ribosome flow model,, IEEE/ACM Trans. Comp. Biology & Bioinformatics, 9 (2012), 1545.
doi: 10.1109/TCBB.2012.88. |
[29] |
A. Rantzer, Distributed control of positive systems,, in Proc. 50th IEEE Conf. Decis. Control and Europ. Contr. Conf., (2011), 6608.
doi: 10.1109/CDC.2011.6161293. |
[30] |
A. Rantzer, Optimizing positively dominated systems,, in Proc. 51st IEEE Conf. Decis. Control, (2012), 272.
doi: 10.1109/CDC.2012.6426312. |
[31] |
A. Rantzer, B. S. Rüffer and G. Dirr, Separable Lyapunov functions for monotone systems,, in Proc. 52nd IEEE Conf. Decis. Control, (2013), 4590.
doi: 10.1109/CDC.2013.6760604. |
[32] |
B. S. Rüffer, Monotone inequalities, dynamical systems, and paths in the positive orthant of Euclidean $n$-space,, Positivity, 14 (2010), 257.
doi: 10.1007/s11117-009-0016-5. |
[33] |
B. S. Rüffer, Small-gain conditions and the comparison principle,, IEEE Trans. Autom. Control, 55 (2010), 1732.
doi: 10.1109/TAC.2010.2048053. |
[34] |
B. S. Rüffer, P. M. Dower and H. Ito, Computational comparison principles for large-scale system stability analysis,, in Proc. of the 10th SICE Annual Conference on Control Systems, (2010). Google Scholar |
[35] |
B. S. Rüffer, H. Ito and P. M. Dower, Computing asymptotic gains of large-scale interconnections,, in Proc. 49th IEEE Conf. Decis. Control, (2010), 7413. Google Scholar |
[36] |
B. S. Rüffer, C. M. Kellett and S. R. Weller, Connection between cooperative positive systems and integral input-to-state stability of large-scale systems,, Automatica J. IFAC, 46 (2010), 1019.
doi: 10.1016/j.automatica.2010.03.012. |
[37] |
H. L. Smith, Monotone Dynamical Systems,, Mathematical Surveys and Monographs, (1995).
|
[38] |
E. D. Sontag, Input to state stability,, in The Control Systems Handbook: Control System Advanced Methods (ed. W. S. Levine), (2010), 1. Google Scholar |
[39] |
E. D. Sontag, Smooth stabilization implies coprime factorization,, IEEE Trans. Autom. Control, 34 (1989), 435.
doi: 10.1109/9.28018. |
[40] |
E. D. Sontag, Comments on integral variants of ISS,, Systems Control Lett., 34 (1998), 93.
doi: 10.1016/S0167-6911(98)00003-6. |
[41] |
E. D. Sontag, Monotone and near-monotone biochemical networks,, Systems and Synthetic Biology, 1 (2007), 59.
doi: 10.1007/s11693-007-9005-9. |
[42] |
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. |
[43] |
J. A. Yorke, Extending Liapunov's second method to non-Lipschitz Liapunov functions,, Bull. Amer. Math. Soc., 74 (1968), 322.
doi: 10.1090/S0002-9904-1968-11940-8. |
[44] |
T. Yoshizawa, Stability Theory by Liapunov's Second Method,, Publications of the Mathematical Society of Japan, (1966). Google Scholar |
show all references
References:
[1] |
D. Angeli and A. Astolfi, A tight small-gain theorem for not necessarily ISS systems,, Systems Control Lett., 56 (2007), 87.
doi: 10.1016/j.sysconle.2006.08.003. |
[2] |
D. Angeli, E. D. Sontag and Y. Wang, A characterization of integral input-to-state stability,, IEEE Trans. Autom. Control, 45 (2000), 1082.
doi: 10.1109/9.863594. |
[3] |
A. Bacciotti, F. Ceragioli and L. Mazzi, Differential inclusions and monotonicity conditions for nonsmooth Lyapunov functions,, Set-Valued Analysis, 8 (2000), 299.
doi: 10.1023/A:1008763931789. |
[4] |
A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, Academic Press, (1979).
|
[5] |
N. P. Bhatia and G. P. Szegö, Stability Theory of Dynamical Systems,, Die Grundlehren der mathematischen Wissenschaften, (1970).
|
[6] |
F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory,, Springer, (1998).
|
[7] |
G. Como, E. Lovisari and K. Savla, Throughput optimality and overload behavior of dynamical flow networks under monotone distributed routing,, IEEE Trans. Contr. Network Systems, 2 (2014), 57.
doi: 10.1109/TCNS.2014.2367361. |
[8] |
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. |
[9] |
P. De Leenheer, D. Angeli and E. D. Sontag, Monotone chemical reaction networks,, Journal of Mathematical Chemistry, 41 (2007), 295.
doi: 10.1007/s10910-006-9075-z. |
[10] |
D. Hinrichsen and A. J. Pritchard, Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness,, Springer, (2005).
doi: 10.1007/b137541. |
[11] |
M. W. Hirsch and H. Smith, Monotone dynamical systems,, in Handbook of Differential Equations: Ordinary Differential Equations, (2005), 239.
|
[12] |
M. W. Hirsch, Systems of differential equations which are competitive or cooperative. I. Limit sets,, SIAM J. Math. Anal., 13 (1982), 167.
doi: 10.1137/0513013. |
[13] |
M. W. Hirsch, Systems of differential equations that are competitive or cooperative. II. Convergence almost everywhere,, SIAM J. Math. Anal., 16 (1985), 423.
doi: 10.1137/0516030. |
[14] |
M. W. Hirsch, Systems of differential equations which are competitive or cooperative. III. Competing species,, Nonlinearity, 1 (1988), 51.
doi: 10.1088/0951-7715/1/1/003. |
[15] |
M. W. Hirsch, Systems of differential equations that are competitive or cooperative. V. Convergence in $3$-dimensional systems,, J. Diff. Eqns., 80 (1989), 94.
doi: 10.1016/0022-0396(89)90097-1. |
[16] |
M. W. Hirsch, Systems of differential equations that are competitive or cooperative. IV. Structural stability in three-dimensional systems,, SIAM J. Math. Anal., 21 (1990), 1225.
doi: 10.1137/0521067. |
[17] |
M. W. Hirsch, Systems of differential equations that are competitive or cooperative. VI. A local $C^r$ closing lemma for 3-dimensional systems,, Ergodic Theory Dynam. Systems, 11 (1991), 443.
doi: 10.1017/S014338570000626X. |
[18] |
M. W. Hirsch and H. L. Smith, Competitive and cooperative systems: Mini-review,, in Positive systems (Rome, (2003), 183.
doi: 10.1007/978-3-540-44928-7_25. |
[19] |
H. Ito, A Lyapunov approach to cascade interconnection of integral input-to-state stable systems,, IEEE Trans. Autom. Control, 55 (2010), 702.
doi: 10.1109/TAC.2009.2037457. |
[20] |
H. Ito and Z.-P. Jiang, Necessary and sufficient small gain conditions for integral input-to-state stable systems: A Lyapunov perspective,, IEEE Trans. Autom. Control, 54 (2009), 2389.
doi: 10.1109/TAC.2009.2028980. |
[21] |
H. Ito, State-dependent scaling problems and stability of interconnected iISS and ISS systems,, IEEE Trans. Autom. Control, 51 (2006), 1626.
doi: 10.1109/TAC.2006.882930. |
[22] |
H. Ito, S. Dashkovskiy and F. Wirth, Capability and limitation of max- and sum-type construction of Lyapunov functions for networks of iISS systems,, Automatica J. IFAC, 48 (2012), 1197.
doi: 10.1016/j.automatica.2012.03.018. |
[23] |
H. Ito, Z.-P. Jiang, S. Dashkovskiy and B. S. Rüffer, Robust stability of networks of iISS systems: Construction of sum-type Lyapunov functions,, IEEE Trans. Autom. Control, 58 (2013), 1192.
doi: 10.1109/TAC.2012.2231552. |
[24] |
H. Ito, B. S. Rüffer and A. Rantzer, Max- and sum-separable Lyapunov functions for monotone systems and their level sets,, in Proc. 53rd IEEE Conf. Decis. Control, (2014), 2371.
doi: 10.1109/CDC.2014.7039750. |
[25] |
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. |
[26] |
H. K. Khalil, Nonlinear systems,, 3rd edition, (2002). Google Scholar |
[27] |
V. Lakshmikantham and S. Leela, Differential and Integral Inequalities: Theory and Applications, Vol. I: Ordinary Differential Equations,, Academic Press, (1969).
|
[28] |
M. Margaliot and T. Tuller, Stability analysis of the ribosome flow model,, IEEE/ACM Trans. Comp. Biology & Bioinformatics, 9 (2012), 1545.
doi: 10.1109/TCBB.2012.88. |
[29] |
A. Rantzer, Distributed control of positive systems,, in Proc. 50th IEEE Conf. Decis. Control and Europ. Contr. Conf., (2011), 6608.
doi: 10.1109/CDC.2011.6161293. |
[30] |
A. Rantzer, Optimizing positively dominated systems,, in Proc. 51st IEEE Conf. Decis. Control, (2012), 272.
doi: 10.1109/CDC.2012.6426312. |
[31] |
A. Rantzer, B. S. Rüffer and G. Dirr, Separable Lyapunov functions for monotone systems,, in Proc. 52nd IEEE Conf. Decis. Control, (2013), 4590.
doi: 10.1109/CDC.2013.6760604. |
[32] |
B. S. Rüffer, Monotone inequalities, dynamical systems, and paths in the positive orthant of Euclidean $n$-space,, Positivity, 14 (2010), 257.
doi: 10.1007/s11117-009-0016-5. |
[33] |
B. S. Rüffer, Small-gain conditions and the comparison principle,, IEEE Trans. Autom. Control, 55 (2010), 1732.
doi: 10.1109/TAC.2010.2048053. |
[34] |
B. S. Rüffer, P. M. Dower and H. Ito, Computational comparison principles for large-scale system stability analysis,, in Proc. of the 10th SICE Annual Conference on Control Systems, (2010). Google Scholar |
[35] |
B. S. Rüffer, H. Ito and P. M. Dower, Computing asymptotic gains of large-scale interconnections,, in Proc. 49th IEEE Conf. Decis. Control, (2010), 7413. Google Scholar |
[36] |
B. S. Rüffer, C. M. Kellett and S. R. Weller, Connection between cooperative positive systems and integral input-to-state stability of large-scale systems,, Automatica J. IFAC, 46 (2010), 1019.
doi: 10.1016/j.automatica.2010.03.012. |
[37] |
H. L. Smith, Monotone Dynamical Systems,, Mathematical Surveys and Monographs, (1995).
|
[38] |
E. D. Sontag, Input to state stability,, in The Control Systems Handbook: Control System Advanced Methods (ed. W. S. Levine), (2010), 1. Google Scholar |
[39] |
E. D. Sontag, Smooth stabilization implies coprime factorization,, IEEE Trans. Autom. Control, 34 (1989), 435.
doi: 10.1109/9.28018. |
[40] |
E. D. Sontag, Comments on integral variants of ISS,, Systems Control Lett., 34 (1998), 93.
doi: 10.1016/S0167-6911(98)00003-6. |
[41] |
E. D. Sontag, Monotone and near-monotone biochemical networks,, Systems and Synthetic Biology, 1 (2007), 59.
doi: 10.1007/s11693-007-9005-9. |
[42] |
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. |
[43] |
J. A. Yorke, Extending Liapunov's second method to non-Lipschitz Liapunov functions,, Bull. Amer. Math. Soc., 74 (1968), 322.
doi: 10.1090/S0002-9904-1968-11940-8. |
[44] |
T. Yoshizawa, Stability Theory by Liapunov's Second Method,, Publications of the Mathematical Society of Japan, (1966). Google Scholar |
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