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

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

Received  June 2014 Revised  March 2015 Published  August 2015

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.
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
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show all references

References:
[1]

Internat. J. Control, 34 (1981), 371-381. doi: 10.1080/00207178108922536.  Google Scholar

[2]

Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 33-56. doi: 10.3934/dcdsb.2012.17.33.  Google Scholar

[3]

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.  Google Scholar

[4]

in Proc. European Control Conference (ECC 2009), Budapest, Hungary, 2009, 91-96. Google Scholar

[5]

Springer-Verlag, Berlin, 1998.  Google Scholar

[6]

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.  Google Scholar

[7]

in Proc. 17th Int. Symp. Math. Theory of Networks and Systems (MTNS 2006), Kyoto, Japan, 2006, 77-82. Google Scholar

[8]

Math. Control Signals Systems, 19 (2007), 93-122. doi: 10.1007/s00498-007-0014-8.  Google Scholar

[9]

SIAM J. Control Optim., 48 (2010), 4089-4118. doi: 10.1137/090746483.  Google Scholar

[10]

Discrete Contin. Dyn. Syst., 32 (2012), 3539-3565. doi: 10.3934/dcds.2012.32.3539.  Google Scholar

[11]

in Proc. of the 52nd IEEE Conference on Decision and Control (CDC 2013), 2013, 1732-1737. Google Scholar

[12]

Discrete Contin. Dyn. Syst., 10 (2004), 657-678. doi: 10.3934/dcds.2004.10.657.  Google Scholar

[13]

Dyn. Syst., 20 (2005), 281-299. doi: 10.1080/14689360500164873.  Google Scholar

[14]

Texas State University-San Marcos, Department of Mathematics, San Marcos, TX, 2007. Available from: http://ejde.math.txstate.edu.  Google Scholar

[15]

IEEE Trans. Automat. Control, 50 (2005), 1681-1697. doi: 10.1109/TAC.2005.858691.  Google Scholar

[16]

Automatica J. IFAC, 32 (1996), 1211-1215. doi: 10.1016/0005-1098(96)00051-9.  Google Scholar

[17]

in Proc. 20th Int. Symp. Math. Theory of Networks and Systems (MTNS 2012), Melbourne, Australia, 2012, Paper No. 184, 8 pages. Google Scholar

[18]

Dyn. Syst., 17 (2002), 137-150. doi: 10.1080/0268111011011847.  Google Scholar

[19]

Circuits Systems Signal Process., 1 (1982), 171-202. doi: 10.1007/BF01600051.  Google Scholar

[20]

IEEE Trans. Automat. Control, 34 (1989), 435-443. doi: 10.1109/9.28018.  Google Scholar

[21]

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]

IEEE Trans. Automat. Control, 35 (1990), 473-476. doi: 10.1109/9.52307.  Google Scholar

[23]

Systems Control Lett., 24 (1995), 351-359. doi: 10.1016/0167-6911(94)00050-6.  Google Scholar

[24]

IEEE Trans. Automat. Control, 41 (1996), 1283-1294. doi: 10.1109/9.536498.  Google Scholar

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