A methodology for estimating the region of attraction for autonomous nonlinear systems is developed. The methodology is based on a proof that the region of attraction can be estimated accurately by the zero sublevel set of an implicit function which is the viscosity solution of a time-dependent Hamilton–Jacobi equation. The methodology starts with a given initial domain and yields a sequence of region of attraction estimates by tracking the evolution of the implicit function. The resulting sequence is contained in and converges to the exact region of attraction. While alternative iterative methods for estimating the region of attraction have been proposed, the methodology proposed in this paper can compute the region of attraction to achieve any desired accuracy in a dimensionally independent and efficient way. An implementation of the proposed methodology has been developed in the Matlab environment. The correctness and efficiency of the methodology are verified through a few examples.
AlbertoLFCChiangHD (2012) Characterization of stability region for general autonomous nonlinear dynamical systems. IEEE Transactions on Automatic Control57(6): 1564–1569.
2.
AnghelMMilanoFPapachristodoulouA (2013) Algorithmic construction of Lyapunov functions for power system stability analysis. IEEE Transactions on Circuits and Systems60(9): 2533–2546.
3.
AubinJPCellinaA (2012) Differential Inclusions: Set-Valued Maps and Viability Theory, volume 264. Springer Science & Business Media: Berlin, Germany.
4.
BardiMFalconeMSoraviaP (1999) Numerical Methods for Pursuit-Evasion Games via Viscosity Solutions. Boston: Birkhäuser, pp. 105–175.
5.
CamilliFGrüneLWirthF (2000) A Regularization of Zubov’s Equation for Robust Domains of Attraction. London: Springer, pp. 277–289.
6.
CamilliFGrüneLWirthF (2001) A generalization of Zubov’s method to perturbed systems. SIAM Journal on Control and Optimization40(2): 496–515.
7.
CamilliFGrüneLWirthF (2008) Control Lyapunov functions and Zubov’s method. SIAM Journal on Control and Optimization47(1): 301–326.
8.
ChakrabortyASeilerPBalasGJ (2011) Nonlinear region of attraction analysis for flight control verification and validation. Control Engineering Practice19(4): 335–345.
9.
ChesiG (2009) Estimating the domain of attraction for non-polynomial systems via LMI optimizations. Automatica45(6): 1536–1541.
10.
ChiangHDHirschMWuF (1988) Stability regions of nonlinear autonomous dynamical systems. IEEE Transactions on Automatic Control33(1): 16–27.
11.
ChiangHDThorpJ (1989) Stability regions of nonlinear dynamical systems: A constructive methodology. IEEE Transactions on Automatic Control34(12): 1229–1241.
12.
ChiangHDWangT (2016) On the number and types of unstable equilibria in nonlinear dynamical systems with uniformly-bounded stability regions. IEEE Transactions on Automatic Control61(2): 485–490.
13.
CrandallMGEvansLCLionsPL (1984) Some properties of viscosity solutions of Hamilton–Jacobi equations. Transactions of the American Mathematical Society282(2): 487–502.
14.
FujisakiYSakuwaR (2006) Estimation of asymptotic stability regions via homogeneous polynomial Lyapunov functions. International Journal of Control79(6): 617–623.
15.
GhoshRTomlinC (2004) Symbolic reachable set computation of piecewise affine hybrid automata and its application to biological modelling: Delta-notch protein signalling. Systems Biology1(1): 170–183.
16.
GieslPHafsteinS (2015) Review on computational methods for lyapunov functions. Discrete and Continuous Dynamical Systems - Series B20(8): 2291–2331.
17.
HahnW (1967) Stability of Motion. Heidelberg: Springer Berlin.
18.
JiangGSPengD (2000) Weighted eno schemes for Hamilton–Jacobi equations. SIAM Journal on Scientific Computing21(6): 2126–2143.
19.
KaynamaSOishiMMitchellIMet al. (2011) The continual reachability set and its computation using maximal reachability techniques. In: 50th IEEE conference on decision and control and European control conference, Orlando, USA, 12–15 December 2011, pp. 6110–6115. Piscataway, NJ: IEEE.
KhodadadiLSamadiBKhaloozadehH (2014) Estimation of region of attraction for polynomial nonlinear systems: A numerical method. ISA Transactions53(1): 25–32.
22.
LygerosJ (2004) On reachability and minimum cost optimal control. Automatica40(6): 917–927.
23.
MatthewsMLWilliamsC (2012) Region of attraction estimation of biological continuous boolean models. In: 2012 IEEE international conference on systems, man, and cybernetics, Seoul, South Korea, 14–17 October 2012, pp. 1700–1705. Piscataway, NJ: IEEE.
24.
MitchellIM (2007) Comparing Forward and Backward Reachability as Tools for Safety Analysis. Heidelberg: Springer Berlin, pp. 428–443.
25.
MitchellIM (2008) The flexible, extensible and efficient toolbox of level set methods. Journal of Scientific Computing35(2–3): 300–329.
26.
MitchellIMBayenATomlinC (2005) A time-dependent Hamilton–Jacobi formulation of reachable sets for continuous dynamic games. IEEE Transactions on Automatic Control50(7): 947–957.
27.
NajafiEBabuškaRLopesGAD (2016) A fast sampling method for estimating the domain of attraction. Nonlinear Dynamics86(2): 823–834.
28.
OsherSFedkiwR (2003) Level Set Methods and Dynamic Implicit Surfaces. New York: Springer.
29.
OsherSFedkiwRP (2001) Level set methods: An overview and some recent results. Journal of Computational Physics169(2): 463–502.
30.
PalenciaCSanz–SernaJM (1984) An extension of the lax-richtmyer theory. Numerische Mathematik44(2): 279–283.
31.
PanditaRChakrabortyASeilerPet al. (2009) Reachability and region of attraction analysis applied to GTM dynamic flight envelope assessment. In: AIAA guidance, navigation, and control conference, Chicago, Illinois, 10–13 August 2009, pp. 1–20. Reston, VA: American Institute of Aeronautics and Astronautics.
32.
SethianJA (1999) Level Set Methods and Fast Marching Methods. New York: Cambridge University Press.
33.
SethianJAVladimirskyA (2003) Ordered upwind methods for static Hamilton–Jacobi equations: Theory and algorithms. SIAM Journal on Numerical Analysis41(1): 325–363.
34.
TanWPackardA (2008) Stability region analysis using polynomial and composite polynomial Lyapunov functions and sum-of-squares programming. IEEE Transactions on Automatic Control53(2): 565–571.
35.
TopcuUPackardASeilerP (2008) Local stability analysis using simulations and sum-of-squares programming. Automatica44(10): 2669–2675.
36.
TsaiYHRChengLTOsherSet al. (2003) Fast sweeping algorithms for a class of Hamilton–Jacobi equations. SIAM Journal on Numerical Analysis41(2): 673–694.
37.
ValmorbidaGAndersonJ (2017) Region of attraction estimation using invariant sets and rational Lyapunov functions. Automatica75: 37–45.
38.
VannelliAVidyasagarM (1985) Maximal Lyapunov functions and domains of attraction for autonomous nonlinear systems. Automatica21(1): 69–80.
39.
WangTChiangHD (2016) On the number of unstable equilibrium points on spatially-periodic stability boundary. IEEE Transactions on Automatic Control61(9): 2553–2558.
40.
WuMYanGLinZet al. (2009) Characterization of backward reachable set and positive invariant set in polytopes. In: 2009 American control conference, St. Louis, USA, 10–12 June 2009, pp. 4351–4356. Piscataway, NJ: IEEE.
ZaborszkyJHuangGZhengBet al. (1988a) On the phase portrait of a class of large nonlinear dynamic systems such as the power system. IEEE Transactions on Automatic Control33(1): 4–15.
43.
ZaborszkyJHuangGZhengBet al. (1988b) A counterexample of a theorem by Tsolas et al. and an independent result by Zaborszky et al. IEEE Transactions on Automatic Control33(3): 316–317.
44.
ZubovVI (1964) Methods of A.M. Lyapunov and Their Application. Groningen: Noordhoff.
