We present a novel method to compute contraction metrics for general nonlinear dynamical systems with exponentially stable equilibria. Such a contraction metric delivers information on the long term behaviour of the system and is robust with respect to perturbations of the dynamics, even perturbations that shift the equilibrium. We prove that our method is always able to deliver a contraction metric in any compact subset of the basin of attraction of an exponentially stable equilibrium. Further, we demonstrate the applicability of the method by computing contraction metrics for two three-dimensional systems from the literature.
Citation: |
Figure 1. Example 3.1: the blue surface (left) is the boundary between the area where the verification condition (VP1) is satisfied and where it is not satisfied, while the green surface (right) describes the verification condition (VP2) suggested by the Numerical Integration-CPA method. Hence, $ P $ is a contraction metric within the intersection of the areas bounded by both surfaces
Figure 2. Example 3.1: The yellow dots indicate the failing points of the Lyapunov-like function while the red closed shape is a level set of it. The green surface is the boundary of the area where verification conditions (VP2) are satisfied; the area where the constraints (VP1) are satisfied is a superset of this set, hence both (VP1) and (VP2) are satisfied within the area bounded by the green surface. From the results we can conclude that there is exactly one equilibrium in the set bounded by the red surface, that it is exponentially stable, and that the red set is a subset of its basin of attraction
Figure 3. Example 3.2. Left: The magenta dots are the equilibria. The blue surface is the boundary between the area where the verification condition (VP1) is satisfied and where it is not satisfied. Middle: The green surface is the boundary between the area where the verification condition (VP2) is satisfied and where it is not satisfied; hence, $ P $ is a contraction metric within the intersection of the areas bounded by the blue and green sets. Clearly, in this example the blue set is far behind the green set and does not have any effect on limiting the area. Right: This plot shows a level set of a Lyapunov-like function in red and the failing points of that function in yellow. The set bounded by the red surface is positively invariant subset of the basin of attraction of an exponentially stable equilibrium in its interior
Figure 4. System (33): The green surface indicates the boundary of the area where (VP2) is satisfied. The yellow points indicate where the Lyapunov-like function fails to satisfy $ (V_P)'_+( {\bf x})<0 $, and the red set is a level set of $ V_P $. From left to right, the first plot is for the small perturbation $ \varepsilon = 0.03 $, the second plot shows the problem with the large perturbation $ \varepsilon = 0.1 $, and the last plot displays a new Lyapunov-like function computed for the large perturbation $ \varepsilon = 0.1 $. Hence, the set bounded by the red surface is a positively invariant subset of the basin of attraction of an exponentially stable equilibrium in the interior of the set for the perturbed system (33) with $ \varepsilon = 0.03 $ (left) and $ \varepsilon = 0.1 $ (right)
[1] | R. Agarwal, S. Hodis and D. O'Regan, 500 Examples and Problems of Applied Differential Equations, Springer, 2019. doi: 10.1007/978-3-030-26384-3. |
[2] | E. Akin, The General Topology of Dynamical Systems, American Mathematical Society, 1993. doi: 10.1090/gsm/001. |
[3] | S. Albertsson, P. Giesl, S. Gudmundsson and S. Hafstein, Simplicial complex with approximate rotational symmetry: A general class of simplicial complexes, J. Comp. Appl. Math., 363 (2020), 413-425. doi: 10.1016/j.cam.2019.06.019. |
[4] | Z. Aminzare and E. Sontag, Contraction methods for nonlinear systems: A brief introduction and some open problems, Proceedings of the 53rd IEEE Conference on Decision and Control, (2014), 3835-3847. |
[5] | J. Auslander, Generalized recurrence in dynamical systems, Contr. to Diff. Equ., 3 (1964), 65-74. |
[6] | E. M. Aylward, P. A. Parrilo and J.-J. Slotine, Stability and robustness analysis of nonlinear systems via contraction metrics and SOS programming, Automatica, 44 (2008), 2163-2170. |
[7] | P. Bernhard and S. Suhr, Lyapounov functions of closed cone fields: From Conley theory to time functions, Commun. Math. Phys., 359 (2018), 467-498. doi: 10.1007/s00220-018-3127-7. |
[8] | N. Bhatia and G. Szegő, Dynamical Systems: Stability Theory and Applications, Springer, Berlin. Lecture Notes in Mathematics 35, 1967. |
[9] | J. Björnsson, P. Giesl, S. Hafstein, C. Kellett and H. Li, Computation of Continuous and Piecewise Affine Lyapunov Functions by Numerical Approximations of the Massera Construction, Proceedings of the CDC, 53rd IEEE Conference on Decision and Control (Los Angeles (CA), USA), 2014. |
[10] | J. Björnsson, P. Giesl, S. Hafstein, C. Kellett and H. Li, Computation of Lyapunov functions for systems with multiple attractors, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 4019-4039. doi: 10.3934/dcds.2015.35.4019. |
[11] | G. Borg, A Condition for the Existence of Orbitally Stable Solutions of Dynamical Systems, Kungl. Tekn. Högsk. Handl. 153, 1960. |
[12] | G. Chesi, Domain of Attraction: Analysis and Control via SOS Programming, Springer, 2011. |
[13] | C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series no. 38, American Mathematical Society, 1978. |
[14] | B. P. Demidovič, On the dissipativity of a certain non-linear system of differential equations. Ⅰ, Vestnik Moskov. Univ. Ser. I Mat. Meh., 1961 (1961), 19-27. |
[15] | A. Doban, Stability Domains Computation and Stabilization of Nonlinear Systems: Implications for Biological Systems, PhD thesis: Eindhoven University of Technology, 2016. |
[16] | A. Doban and M. Lazar, Computation of lyapunov functions for nonlinear differential equations via a Yoshizawa-type construction, IFAC-PapersOnLine, 49 (2016), 29-34. |
[17] | F. Forni and R. Sepulchre, A differential lyapunov framework for contraction analysis, IEEE Transactions on Automatic Control, 59 (2014), 614-628. |
[18] | P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, Lecture Notes in Math. 1904, Springer, 2007. |
[19] | P. Giesl, Converse theorems on contraction metrics for an equilibrium, J. Math. Anal. Appl., 424 (2015), 1380-1403. doi: 10.1016/j.jmaa.2014.12.010. |
[20] | P. Giesl, Computation of a contraction metric for a periodic orbit using meshfree collocation, SIAM J. Appl. Dyn. Syst., 18 (2019), 1536-1564. doi: 10.1137/18M1220182. |
[21] | P. Giesl and S. Hafstein, Construction of a CPA contraction metric for periodic orbits using semidefinite optimization, Nonlinear Anal., 86 (2013), 114-134. |
[22] | P. Giesl and S. Hafstein, Revised CPA method to compute lyapunov functions for nonlinear systems, J. Math. Anal. Appl., 410 (2014), 292-306. doi: 10.1016/j.jmaa.2013.08.014. |
[23] | P. Giesl and S. Hafstein, Computation and verification of Lyapunov functions, SIAM J. Appl. Dyn. Syst., 14 (2015), 1663-1698. |
[24] | P. Giesl and S. Hafstein, Review of computational methods for Lyapunov functions, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2291-2331. |
[25] | P. Giesl and S. Hafstein, Uniformly regular triangulations for parameterizing Lyapunov functions, Proceedings of the 18th International Conference on Informatics in Control, Automation and Robotics (ICINCO), (2021), 549-557. |
[26] | P. Giesl, S. Hafstein and C. Kawan, Review on contraction analysis and computation of contraction metrics, J. Comput. Dyn, 10 (2023), 1-47. |
[27] | P. Giesl, S. Hafstein and I. Mehrabinezhad, Computation and verification of contraction metrics for exponentially stable equilibria, J. Comput. Appl. Math., 390 (2021), 113332, 21 pp. doi: 10.1016/j.cam.2020.113332. |
[28] | P. Giesl, S. Hafstein and I. Mehrabinezhad, Computation and verification of contraction metrics for periodic orbits, J. Math. Anal. Appl., 503 (2021), Paper No. 125309, 32 pp. |
[29] | P. Giesl, S. Hafstein and I. Mehrabinezhad, Computing contraction metrics for three-dimensional systems, IFAC PapersOnLine, 54 (2021), 297-303. |
[30] | P. Giesl, S. Hafstein and I. Mehrabinezhad, Contraction Metrics by Numerical Integration and Quadrature: Uniform Error estimate, Proceedings of the 20th International Conference on Informatics in Control, Automation and Robotics (ICINCO), 2023. |
[31] | P. Giesl, S. Hafstein and I. Mehrabinezhad, Positively Invariant Sets for ODEs and Numerical Integration, Proceedings of the 20th International Conference on Informatics in Control, Automation and Robotics (ICINCO), 2023. |
[32] | P. Giesl and H. Wendland, Construction of a contraction metric by meshless collocation, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3843-3863. doi: 10.3934/dcdsb.2018333. |
[33] | S. Hafstein, Implementation of simplicial complexes for CPA functions in C++11 using the armadillo linear algebra library, Proceedings of the 3rd International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH) (Reykjavik, Iceland), (2013), 49-57. |
[34] | S. Hafstein, C. Kellett and H. Li, Computing continuous and piecewise affine Lyapunov functions for nonlinear systems, Journal of Computational Dynamics, 2 (2015), 227-246. doi: 10.3934/jcd.2015004. |
[35] | S. Hafstein and S. Suhr, Smooth complete Lyapunov functions for ODEs, J. Math. Anal. Appl., 499 (2021), 125003, 15 pp. doi: 10.1016/j.jmaa.2021.125003. |
[36] | S. Hafstein and A. Valfells, Efficient computation of Lyapunov functions for nonlinear systems by integrating numerical solutions, Nonlinear Dynamics, 97 (2019), 1895-1910. |
[37] | W. Hahn, Stability of Motion, Springer, Berlin, 1967. |
[38] | P. Hartman, On stability in the large for systems of ordinary differential equations, Canadian J. Math., 13 (1961), 480-492. |
[39] | P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964. |
[40] | M. Hurley, Chain recurrence, semiflows, and gradients, J. Dyn. Diff. Equat., 7 (1995), 437-456. doi: 10.1007/BF02219371. |
[41] | M. Hurley, Lyapunov functions and attractors in arbitrary metric spaces, Proc. Amer. Math. Soc., 126 (1998), 245-256. doi: 10.1090/S0002-9939-98-04500-6. |
[42] | T. Johansen, Computation of Lyapunov functions for smooth, nonlinear systems using convex optimization, Automatica, 36 (2000), 1617-1626. |
[43] | P. Julian, J. Guivant and A. Desages, A parametrization of piecewise linear Lyapunov functions via linear programming, Int. J. Control, 72 (1999), 702-715. doi: 10.1080/002071799220876. |
[44] | N. N. Krasovski$\mathop {\rm{i}}\limits^{˘} $, Problems of the Theory of Stability of Motion, Mir, Moskow, 1959, English translation by Stanford University Press, 1963. |
[45] | D. Lewis, Differential equations referred to a variable metric, Amer. J. Math., 73 (1951), 48-58. doi: 10.2307/2372159. |
[46] | D. C. Lewis, Metric properties of differential equations, Amer. J. Math., 71 (1949), 294-312 (English). doi: 10.2307/2372245. |
[47] | W. Lohmiller and J.-J. Slotine, On Contraction Analysis for Non-linear Systems, Automatica, 34 (1998), 683-696. doi: 10.1016/S0005-1098(98)00019-3. |
[48] | A. M. Lyapunov, Problème général de la stabilité du mouvement, Ann. Fac. Sci. Toulouse, 9 (1907), 203-474. doi: 10.5802/afst.246. |
[49] | S. Marinósson, Lyapunov function construction for ordinary differential equations with linear programming, Dynamical Systems: An International Journal, 17 (2002), 137-150. doi: 10.1080/0268111011011847. |
[50] | B. McLaren and R. Peterson, Wolves, moose, and tree rings on isle royale, Science, 266 (1994), 1555-1558. |
[51] | P. Parrilo, Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimiziation, PhD thesis: California Institute of Technology Pasadena, California, 2000. |
[52] | M. Peet and A. Papachristodoulou, A converse sum of squares Lyapunov result with a degree bound, IEEE Transactions on Automatic Control, 57 (2012), 2281-2293. doi: 10.1109/TAC.2012.2190163. |
[53] | C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, 2^{nd} edition, Studies in Advanced Mathematics, CRC Press, 1999. |
[54] | J. Simpson-Porco and F. Bullo, Contraction theory on Riemannian manifolds, Systems Control Lett., 65 (2014), 74-80. doi: 10.1016/j.sysconle.2013.12.016. |
[55] | A. Vannelli and M. Vidyasagar, Maximal Lyapunov functions and domains of attraction for autonomous nonlinear systems, Automatica J. IFAC, 21 (1985), 69-80. doi: 10.1016/0005-1098(85)90099-8. |
[56] | T. Yoshizawa, Stability Theory by Liapunov's Second Method, Publications of the Mathematical Society of Japan, No. 9, The Mathematical Society of Japan, Tokyo, 1966. |
[57] | V. Zubov, Methods of A. M. Lyapunov and Their Application, Translation prepared under the auspices of the United States Atomic Energy Commission; edited by Leo F. Boron, P. Noordhoff Ltd, Groningen, 1964. |
Example 3.1: the blue surface (left) is the boundary between the area where the verification condition (VP1) is satisfied and where it is not satisfied, while the green surface (right) describes the verification condition (VP2) suggested by the Numerical Integration-CPA method. Hence,
Example 3.1: The yellow dots indicate the failing points of the Lyapunov-like function while the red closed shape is a level set of it. The green surface is the boundary of the area where verification conditions (VP2) are satisfied; the area where the constraints (VP1) are satisfied is a superset of this set, hence both (VP1) and (VP2) are satisfied within the area bounded by the green surface. From the results we can conclude that there is exactly one equilibrium in the set bounded by the red surface, that it is exponentially stable, and that the red set is a subset of its basin of attraction
Example 3.2. Left: The magenta dots are the equilibria. The blue surface is the boundary between the area where the verification condition (VP1) is satisfied and where it is not satisfied. Middle: The green surface is the boundary between the area where the verification condition (VP2) is satisfied and where it is not satisfied; hence,
System (33): The green surface indicates the boundary of the area where (VP2) is satisfied. The yellow points indicate where the Lyapunov-like function fails to satisfy