Advanced Search
Article Contents
Article Contents

A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems

  • * Corresponding author: Morgan Jones

    * Corresponding author: Morgan Jones 

The authors of this manuscript were supported by NSF grant NSD CMMI-1931270.
We would like to thank the reviewers of this paper for their valuable comments and suggestions

Abstract Full Text(HTML) Figure(3) Related Papers Cited by
  • Many dynamical systems described by nonlinear ODEs are unstable. Their associated solutions do not converge towards an equilibrium point, but rather converge towards some invariant subset of the state space called an attractor set. For a given ODE, in general, the existence, shape and structure of the attractor sets of the ODE are unknown. Fortunately, the sublevel sets of Lyapunov functions can provide bounds on the attractor sets of ODEs. In this paper we propose a new Lyapunov characterization of attractor sets that is well suited to the problem of finding the minimal attractor set. We show our Lyapunov characterization is non-conservative even when restricted to Sum-of-Squares (SOS) Lyapunov functions. Given these results, we propose a SOS programming problem based on determinant maximization that yields an SOS Lyapunov function whose $ 1 $-sublevel set has minimal volume, is an attractor set itself, and provides an optimal outer approximation of the minimal attractor set of the ODE. Several numerical examples are presented including the Lorenz attractor and Van-der-Pol oscillator.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Graph showing an estimation of the Lorenz attractor (Example 1) given by the red transparent surface. This surface is the $ 1 $-sublevel set of a solution to the SOS Problem (68). The grayed shaded surfaces represent the projection of the our Lorenz attractor estimation on the xy, xz, and yz axes. The black line is an approximation of the attractor found by simulating a Lorenz trajectory using Matlab's $ \texttt{ODE45} $ function

    Figure 2.  Graph showing an estimation of the attractor (given by the red area) of the ODE (70) in Example 2. This red area is the $ 1 $-sublevel set of a solution to the SOS Problem (68). The two black lines are simulated solution maps of the ODE (71) using Matlab's $ \texttt{ODE45} $ function initialized outside of the limit cycle

    Figure 3.  Graph showing an estimation of the attractor (given by the red area) of the ODE (71) in Example 3. This red area is the $ 1 $-sublevel set of a solution to the SOS Problem (68). The four black lines are simulated solution maps initialized at $ [ \pm 1, \pm 1]^T $ of the ODE (71) using Matlab's $ \texttt{ODE45} $ function

  • [1] E. Ahbe, Region of Attraction Analysis of Uncertain Equilibrium Points and Limit Cycles - with Application to Airborne Wind Energy Systems, PhD thesis, ETH Zurich, Zurich, 2020.
    [2] A. Ahmadi, G. Hall, A. Makadia and V. Sindhwani, Geometry of 3d environments and sum of squares polynomials, 2017.
    [3] A. A. Ahmadi and B. E. Khadir, A globally asymptotically stable polynomial vector field with rational coefficients and no local polynomial Lyapunov function, Systems Control Lett., 121 (2018), 50-53.  doi: 10.1016/j.sysconle.2018.07.013.
    [4] J. Anderson and A. Papachristodoulou, Advances in computational Lyapunov analysis using sum-of-squares programming, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2361-2381.  doi: 10.3934/dcdsb.2015.20.2361.
    [5] J. AwrejcewiczD. BilichenkoA. K. CheibN. Losyeva and V. Puzyrov, Estimating the region of attraction based on a polynomial Lyapunov function, Appl. Math. Model., 90 (2021), 1143-1152.  doi: 10.1016/j.apm.2020.10.010.
    [6] D. P. Bertsekas, Dynamic Programming and Optimal Control, Athena scientific Belmont, MA, 2005.
    [7] N. P. Bhatia and G. P. Szegö, Dynamical Systems: Stability Theory and Applications, Lecture Notes in Mathematics, No. 35 Springer-Verlag, Berlin-New York 1967.
    [8] T. CunisJ.-P. Condomines and L. Burlion, Sum-of-Squares flight control synthesis for deep-stall recovery, J. Guidance, Control, and Dynamics, 43 (2020), 1498-1511. 
    [9] K. M CuomoA. V. Oppenheim and S. H. Strogatz, Synchronization of lorenz-based chaotic circuits with applications to communications, IEEE Transactions on Circuits and Systems Ⅱ: Analog and Digital Signal Processing, 40 (1993), 626-633. 
    [10] F. DabbeneD. Henrion and C. Lagoa, Simple approximations of semialgebraic sets and their application to control, Automatica J. IFAC, 78 (2017), 110-118.  doi: 10.1016/j.automatica.2016.11.021.
    [11] W. GaoL. YanM. Saeedi and H. S. Nik, Ultimate bound estimation set and chaos synchronization for a financial risk system, Math. Comput. Simulation, 154 (2018), 19-33.  doi: 10.1016/j.matcom.2018.06.006.
    [12] D. Goluskin, Bounding extrema over global attractors using polynomial optimisation, Nonlinearity, 33 (2020), 4878-4899.  doi: 10.1088/1361-6544/ab8f7b.
    [13] M. HirschS. Smale and  R. DevaneyDifferential Equations, Dynamical Systems and An Introduction To Choas, 2$^{nd}$ edition, Elsevier/Academic Press, Amsterdam, 2004. 
    [14] M. Jones and M. M. Peet, Polynomial approximation of value functions and nonlinear controller design with performance bounds, preprint, 2020, arXiv: 2010.06828.
    [15] M. Jones and M. M. Peet, Converse Lyapunov functions and converging inner approximations to maximal regions of attraction of nonlinear systems, preprint, 2021, arXiv: 2103.12825.
    [16] M. Jones and M. M. Peet, Using sos and sublevel set volume minimization for estimation of forward reachable sets, preprint, 2019, arXiv: 1901.11174.
    [17] M. Jones and M. M Peet, Using sos for optimal semialgebraic representation of sets: Finding minimal representations of limit cycles, chaotic attractors and unions, preprint, 2018, arXiv: 1809.10308.
    [18] H. Khalil, Nonlinear Systems, Macmillan Publishing Company, New York, 1992.
    [19] M. V. LakshmiG. FantuzziJ. D. Fernández-CaballeroY. Hwang and S. I. Chernyshenko, Finding extremal periodic orbits with polynomial optimization, with application to a nine-mode model of shear flow, SIAM J. Appl. Dyn. Syst., 19 (2020), 763-787.  doi: 10.1137/19M1267647.
    [20] J. B. Lasserre, Volume of sublevel sets of homogeneous polynomials, SIAM J. Appl. Algebra Geom., 3 (2019), 372-389.  doi: 10.1137/18M1222478.
    [21] E. Lee, Fundemental limits of cyber physical systems modeling, ACM Trans Cyber Phys, 2016.
    [22] D. LiJ. LuX. Wu and G. Chen, Estimating the bounds for the lorenz family of chaotic systems, Chaos Solitons Fractals, 23 (2005), 529-534.  doi: 10.1016/j.chaos.2004.05.021.
    [23] Y. LinE. D. Sontag and Y. Wang, A smooth converse lyapunov theorem for robust stability, SIAM J. Control Optim., 34 (1996), 124-160.  doi: 10.1137/S0363012993259981.
    [24] J. Lofberg, Yalmip: A toolbox for modeling and optimization in matlab, In 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No. 04CH37508), IEEE, (2004), 284-289.
    [25] A. Magnani, S. Lall and S. Boyd, Tractable fitting with convex polynomials via sum of squares, CDC, 2005.
    [26] M. M. Peet, Exponentially stable nonlinear systems have polynomial lyapunov functions on bounded regions, IEEE Trans. Automat. Control, 54 (2009), 979-987.  doi: 10.1109/TAC.2009.2017116.
    [27] M. M. Peet and A. Papachristodoulou, A converse sum-of-squares lyapunov result: An existence proof based on the picard iteration, In 49th IEEE Conference on Decision and Control (CDC), (2010), 5949-5954.
    [28] S. PrajnaA. Papachristodoulou and P. A. Parrilo, Introducing sostools: A general purpose sum of squares programming solver, Proceedings of the 41st IEEE Conference on Decision and Control, 1 (2002), 741-746. 
    [29] M. Putinar, Positive polynomials on compact semialgebriac sets, Math. J., 42 (1993), 969-984.  doi: 10.1512/iumj.1993.42.42045.
    [30] C. Schlosser and M. Korda, Converging outer approximations to global attractors using semidefinite programming, Automatica J. IFAC, 134 (2020), Paper No. 109900, 9 pp. doi: 10.1016/j.automatica.2021.109900.
    [31] A. R. Teel and L. Praly, A smooth lyapunov function from a class-kl estimate involving two positive semidefinite functions, ESAIM Control Optim. Calc. Var., 5 (2000), 313-367.  doi: 10.1051/cocv:2000113.
    [32] R. TütuncüK. Toh and M. Todd, Solving semidefinite-quadratic-linear programs using sdpt3, Math. Program., 95 (2003), 189-217.  doi: 10.1007/s10107-002-0347-5.
    [33] G. Valmorbida and J. Anderson, Region of attraction estimation using invariant sets and rational Lyapunov functions, Automatica, 75 (2017), 37-45.  doi: 10.1016/j.automatica.2016.09.003.
    [34] T.-C. WangS. Lall and M. West, Polynomial level-set method for attractor estimation, J. Franklin Inst., 349 (2012), 2783-2798.  doi: 10.1016/j.jfranklin.2012.08.014.
    [35] P. Yu and X. Liao, Globally attractive and positive invariant set of the lorenz system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 757-764.  doi: 10.1142/S0218127406015143.
    [36] Y. ZhaoW. ZhangH. Su and J. Yang, Observer-based synchronization of chaotic systems satisfying incremental quadratic constraints and its application in secure communication, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 50 (2018), 5221-5232. 
    [37] X. ZhengZ. SheJ. Lu and M. Li, Computing multiple Lyapunov-like functions for inner estimates of domains of attraction of switched hybrid systems, Internat. J. Robust Nonlinear Control, 28 (2018), 5191-5212.  doi: 10.1002/rnc.4280.
  • 加载中



Article Metrics

HTML views(167) PDF downloads(151) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint