Computation of local ISS Lyapunov functions with low gains via linear programming

Previous Article
Grid refinement in the construction of Lyapunov functions using radial basis functions
DCDS-B Home
This Issue
Next Article
Separable Lyapunov functions for monotone systems: Constructions and limitations

2015, 20(8): 2477-2495. doi: 10.3934/dcdsb.2015.20.2477

Computation of local ISS Lyapunov functions with low gains via linear programming

Huijuan Li ^1, , Robert Baier ^2, , Lars Grüne ^2, , Sigurdur F. Hafstein ^3, and Fabian R. Wirth ^4,

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

Received June 2014 Revised March 2015 Published August 2015

Abstract
Related Papers

In this paper, we present a numerical algorithm for computing ISS Lyapunov functions for continuous-time systems which are input-to-state stable (ISS) on compact subsets of the state space. The algorithm relies on a linear programming problem and computes a continuous piecewise affine ISS Lyapunov function on a simplicial grid covering the given compact set excluding a small neighborhood of the origin. The objective of the linear programming problem is to minimize the gain. We show that for every ISS system with a locally Lipschitz right-hand side our algorithm is in principle able to deliver an ISS Lyapunov function. For $C^2$ right-hand sides a more efficient algorithm is proposed.

Keywords: Nonlinear systems, linear programming., local ISS Lyapunov function, local input-to-state stability, robust Lyapunov function.

Mathematics Subject Classification: Primary: 37B25, 93D09, 93D30; Secondary: 34D20, 90C0.

Citation: 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

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.
[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.
[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.
[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).
[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.
[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.
[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.
[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).
[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.

[1]	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
[2]	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
[3]	Sigurdur Hafstein, Skuli Gudmundsson, Peter Giesl, Enrico Scalas. Lyapunov function computation for autonomous linear stochastic differential equations using sum-of-squares programming. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 939-956. doi: 10.3934/dcdsb.2018049
[4]	Jóhann Björnsson, Peter Giesl, Sigurdur F. Hafstein, Christopher M. Kellett. Computation of Lyapunov functions for systems with multiple local attractors. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4019-4039. doi: 10.3934/dcds.2015.35.4019
[5]	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
[6]	Martin Gugat, Günter Leugering, Ke Wang. Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-lyapunov function. Mathematical Control & Related Fields, 2017, 7 (3) : 419-448. doi: 10.3934/mcrf.2017015
[7]	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
[8]	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
[9]	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
[10]	Yongjian Yang, Zhiyou Wu, Fusheng Bai. A filled function method for constrained nonlinear integer programming. Journal of Industrial & Management Optimization, 2008, 4 (2) : 353-362. doi: 10.3934/jimo.2008.4.353
[11]	Paul L. Salceanu. Robust uniform persistence in discrete and continuous dynamical systems using Lyapunov exponents. Mathematical Biosciences & Engineering, 2011, 8 (3) : 807-825. doi: 10.3934/mbe.2011.8.807
[12]	Jifeng Chu, Jinzhi Lei, Meirong Zhang. Lyapunov stability for conservative systems with lower degrees of freedom. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 423-443. doi: 10.3934/dcdsb.2011.16.423
[13]	Dmitry Jakobson and Iosif Polterovich. Lower bounds for the spectral function and for the remainder in local Weyl's law on manifolds. Electronic Research Announcements, 2005, 11: 71-77.
[14]	Hans Rullgård, Eric Todd Quinto. Local Sobolev estimates of a function by means of its Radon transform. Inverse Problems & Imaging, 2010, 4 (4) : 721-734. doi: 10.3934/ipi.2010.4.721
[15]	Qinghua Ma, Zuoliang Xu, Liping Wang. Recovery of the local volatility function using regularization and a gradient projection method. Journal of Industrial & Management Optimization, 2015, 11 (2) : 421-437. doi: 10.3934/jimo.2015.11.421
[16]	Zhiqing Meng, Qiying Hu, Chuangyin Dang. A penalty function algorithm with objective parameters for nonlinear mathematical programming. Journal of Industrial & Management Optimization, 2009, 5 (3) : 585-601. doi: 10.3934/jimo.2009.5.585
[17]	Zhiyou Wu, Fusheng Bai, Guoquan Li, Yongjian Yang. A new auxiliary function method for systems of nonlinear equations. Journal of Industrial & Management Optimization, 2015, 11 (2) : 345-364. doi: 10.3934/jimo.2015.11.345
[18]	Markus Dick, Martin Gugat, Günter Leugering. A strict $H^1$-Lyapunov function and feedback stabilization for the isothermal Euler equations with friction. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 225-244. doi: 10.3934/naco.2011.1.225
[19]	Wacław Marzantowicz, Justyna Signerska. Firing map of an almost periodic input function. Conference Publications, 2011, 2011 (Special) : 1032-1041. doi: 10.3934/proc.2011.2011.1032
[20]	Rui Qian, Rong Hu, Ya-Ping Fang. Local smooth representation of solution sets in parametric linear fractional programming problems. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 45-52. doi: 10.3934/naco.2019004

2017 Impact Factor: 0.972

American Institute of Mathematical Sciences