Providing a Basin of Attraction to a Target Region by Computation of Lyapunov-like Functions

In this paper, we present a method for computing a basin of attraction to a target region for non-linear ordinary differential equations. This basin of attraction is ensured by a Lyapunov-like polynomial function that we compute using an interval based branch-and-relax algorithm. This algorithm relaxes the necessary conditions on the coefficients of the Lyapunov-like function to a system of linear interval inequalities that can then be solved exactly, and iteratively reduces the relaxation error by recursively decomposing the state space into hyper-rectangles. Tests on an implementation are promising.

[1]  Tomás Vejchodský,et al.  Discrete maximum principle for higher-order finite elements in 1D , 2007, Math. Comput..

[2]  Editors , 1986, Brain Research Bulletin.

[3]  Christopher W. Brown.  QEPCAD B: a system for computing with semi-algebraic sets via cylindrical algebraic decomposition , 2004, SIGS.

[4]  K. Forsman,et al.  Construction of Lyapunov functions using Grobner bases , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[5]  A. Papachristodoulou,et al.  On the construction of Lyapunov functions using the sum of squares decomposition , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[6]  Enric Rodríguez-Carbonell,et al.  Automatic Generation of Polynomial Loop Invariants: Algebraic Foundations , 2004, ISSAC '04.

[7]  J. Worrell.  Decidable Theories , 2007 .

[8]  Christian Jansson,et al.  Rigorous Lower and Upper Bounds in Linear Programming , 2003, SIAM J. Optim..

[9]  Toby Walsh,et al.  Handbook of Constraint Programming , 2006, Handbook of Constraint Programming.

[10]  A. Tarski.  A Decision Method for Elementary Algebra and Geometry , 2023 .

[11]  Henny B. Sipma,et al.  Constructing invariants for hybrid systems , 2008, Formal Methods Syst. Des..

[12]  Anders Rantzer,et al.  Primal-dual tests for safety and reachability , 2005 .

[13]  Stefan Ratschan,et al.  Efficient solving of quantified inequality constraints over the real numbers , 2002, TOCL.

[14]  George J. Pappas,et al.  Hybrid Systems: Computation and Control , 2004, Lecture Notes in Computer Science.

[15]  Anders Rantzer,et al.  Primal-Dual Tests for Safety and Reachability , 2005, HSCC.

[16]  Frédéric Benhamou,et al.  Continuous and Interval Constraints , 2006, Handbook of Constraint Programming.

[17]  V. Lakshmikantham,et al.  Practical Stability Of Nonlinear Systems , 1990 .

[18]  Ali Jadbabaie,et al.  Safety Verification of Hybrid Systems Using Barrier Certificates , 2004, HSCC.

[19]  Markus Müller-Olm,et al.  Computing polynomial program invariants , 2004, Inf. Process. Lett..

[20]  Stefan Ratschan,et al.  Continuous First-Order Constraint Satisfaction , 2002, AISC.

[21]  Ashish Tiwari,et al.  Generating Polynomial Invariants for Hybrid Systems , 2005, HSCC.

[22]  Andreas Podelski,et al.  Model Checking of Hybrid Systems: From Reachability Towards Stability , 2006, HSCC.

[23]  P. Giesl.  Construction of Global Lyapunov Functions Using Radial Basis Functions , 2007 .

[24]  H. Burchardt,et al.  Estimating the region of attraction of ordinary differential equations by quantified constraint solving , 2007 .

[25]  B. F. Caviness,et al.  Quantifier Elimination and Cylindrical Algebraic Decomposition , 2004, Texts and Monographs in Symbolic Computation.

[26]  A. Fuller,et al.  Stability of Motion , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[27]  Franco Blanchini,et al.  Set invariance in control , 1999, Autom..

[28]  Thomas Sturm,et al.  REDLOG: computer algebra meets computer logic , 1997, SIGS.

[29]  Sigurdur Hafstein,et al.  A CONSTRUCTIVE CONVERSE LYAPUNOV THEOREM ON EXPONENTIAL STABILITY , 2004 .

[30]  J. Rohn,et al.  Linear interval inequalities , 1994 .

[31]  Mark R. Greenstreet,et al.  Hybrid Systems: Computation and Control , 2002, Lecture Notes in Computer Science.

[32]  D. N. Shields.  The behaviour of optimal Lyapunov functions , 1975 .

[33]  Stefan Ratschan,et al.  Quantified Constraints Under Perturbation , 2002, J. Symb. Comput..

[34]  Arnold Neumaier,et al.  Safe bounds in linear and mixed-integer linear programming , 2004, Math. Program..

[35]  A. Vicino,et al.  On the estimation of asymptotic stability regions: State of the art and new proposals , 1985 .

[36]  Pablo A. Parrilo,et al.  Semidefinite Programming Relaxations and Algebraic Optimization in Control , 2003, Eur. J. Control.