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Li, Huijuan; Baier, Robert; Grüne, Lars; Hafstein, Sigurdur F.; Wirth, Fabian R.

Computation of local ISS Lyapunov functions with low gains via linear programming. (English) Zbl 1366.37146

Discrete Contin. Dyn. Syst., Ser. B 20, No. 8, 2477-2495 (2015).
Summary: 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.

Cited in 6 Documents

MSC:

37M99 Approximation methods and numerical treatment of dynamical systems
93D09 Robust stability
93D30 Lyapunov and storage functions
90C05 Linear programming
93D25 Input-output approaches in control theory
90C90 Applications of mathematical programming

Keywords:

nonlinear system; local input-to-state stability; Lyapunov function; linear programming
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\textit{H. Li} et al., Discrete Contin. Dyn. Syst., Ser. B 20, No. 8, 2477--2495 (2015; Zbl 1366.37146)
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References:

[1] M. Abu Hassan, Numerical determination of domains of attraction for electrical power systems using the method of Zubov,, Internat. J. Control, 34, 371 (1981) · Zbl 0466.93021
[2] R. Baier, Linear programming based Lyapunov function computation for differential inclusions,, Discrete Contin. Dyn. Syst. Ser. B, 17, 33 (2012) · Zbl 1235.93222
[3] F. Camilli, A regularization of Zubov’s equation for robust domains of attraction,, in Nonlinear Control in the Year 2000, 277 (2000) · Zbl 1239.93048
[4] F. Camilli, Domains of attraction of interconnected systems: A Zubov method approach,, in Proc. European Control Conference (ECC 2009), 91 (2009)
[5] F. H. Clarke, <em>Nonsmooth Analysis and Control Theory</em>,, Springer-Verlag (1998) · Zbl 1047.49500
[6] S. Dashkovskiy, 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), 5633 (2005)
[7] S. Dashkovskiy, An ISS Lyapunov function for networks of ISS systems,, in Proc. 17th Int. Symp. Math. Theory of Networks and Systems (MTNS 2006), 77 (2006)
[8] S. Dashkovskiy, An ISS small-gain theorem for general networks,, Math. Control Signals Systems, 19, 93 (2007) · Zbl 1125.93062
[9] S. N. Dashkovskiy, Small gain theorems for large scale systems and construction of ISS Lyapunov functions,, SIAM J. Control Optim., 48, 4089 (2010) · Zbl 1202.93008
[10] P. Giesl, Existence of piecewise linear Lyapunov functions in arbitrary dimensions,, Discrete Contin. Dyn. Syst., 32, 3539 (2012) · Zbl 1266.37010
[11] L. Grüne, Numerical ISS controller design via a dynamic game approach,, in Proc. of the 52nd IEEE Conference on Decision and Control (CDC 2013), 1732 (2013)
[12] S. F. Hafstein, A constructive converse Lyapunov theorem on exponential stability,, Discrete Contin. Dyn. Syst., 10, 657 (2004) · Zbl 1070.93044
[13] S. F. Hafstein, A constructive converse Lyapunov theorem on asymptotic stability for nonlinear autonomous ordinary differential equations,, Dyn. Syst., 20, 281 (2005) · Zbl 1102.34038
[14] S. F. Hafstein, <em>An Algorithm for Constructing Lyapunov Functions</em>, vol. 8 of Electron. J. Differ. Equ. Monogr.,, Texas State University-San Marcos (2007) · Zbl 1131.37021
[15] S. Huang, A unified approach to controller design for achieving ISS and related properties,, IEEE Trans. Automat. Control, 50, 1681 (2005) · Zbl 1365.93173
[16] Z.-P. Jiang, A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems,, Automatica J. IFAC, 32, 1211 (1996) · Zbl 0857.93089
[17] H. Li, 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, 137 (2002) · Zbl 1013.34051
[19] A. N. Michel, Stability analysis of complex dynamical systems: Some computational methods,, Circuits Systems Signal Process., 1, 171 (1982) · Zbl 0493.93040
[20] E. D. Sontag, Smooth stabilization implies coprime factorization,, IEEE Trans. Automat. Control, 34, 435 (1989) · Zbl 0682.93045
[21] E. D. Sontag, Some connections between stabilization and factorization,, in Proc. of the 28th IEEE Conference on Decision and Control (CDC 1989), 1 (1989)
[22] E. D. Sontag, Further facts about input to state stabilization,, IEEE Trans. Automat. Control, 35, 473 (1990) · Zbl 0704.93056
[23] E. D. Sontag, On characterizations of the input-to-state stability property,, Systems Control Lett., 24, 351 (1995) · Zbl 0877.93121
[24] E. D. Sontag, New characterizations of input-to-state stability,, IEEE Trans. Automat. Control, 41, 1283 (1996) · Zbl 0862.93051
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.