Lagrangian Reachabililty

We introduce LRT, a new Lagrangian-based ReachTube computation algorithm that conservatively approximates the set of reachable states of a nonlinear dynamical system. LRT makes use of the Cauchy-Green stretching factor (SF), which is derived from an overapproximation of the gradient of the solution-flows. The SF measures the discrepancy between two states propagated by the system solution from two initial states lying in a well-defined region, thereby allowing LRT to compute a reachtube with a ball-overestimate in a metric where the computed enclosure is as tight as possible. To evaluate its performance, we implemented a prototype of LRT in C++/Matlab, and ran it on a set of well-established benchmarks. Our results show that LRT compares very favorably with respect to the CAPD and Flow* tools.

[1]  G. Dahlquist.  Stability and error bounds in the numerical integration of ordinary differential equations , 1961 .

[2]  D. Schaeffer,et al.  A two-current model for the dynamics of cardiac membrane , 2003, Bulletin of mathematical biology.

[3]  Antonis Papachristodoulou,et al.  A Decomposition Technique for Nonlinear Dynamical System Analysis , 2012, IEEE Transactions on Automatic Control.

[4]  Alexandre Donzé,et al.  Breach, A Toolbox for Verification and Parameter Synthesis of Hybrid Systems , 2010, CAV.

[5]  Johan Löfberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004 .

[6]  D. Wilczak,et al.  C-Lohner algorithm , 2011 .

[7]  John N. Maidens,et al.  Reachability Analysis of Nonlinear Systems Using Matrix Measures , 2015, IEEE Transactions on Automatic Control.

[8]  James Kapinski,et al.  Locally optimal reach set over-approximation for nonlinear systems , 2016, 2016 International Conference on Embedded Software (EMSOFT).

[9]  J. Rohn.  Bounds on Eigenvalues of Interval Matrices , 1998 .

[10]  Martin Berz,et al.  Rigorous integration of flows and ODEs using taylor models , 2009, SNC '09.

[11]  P. Zgliczynski,et al.  The existence of simple choreographies for the N-body problem—a computer-assisted proof , 2003, math/0304404.

[12]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[13]  Xin Chen,et al.  Flow*: An Analyzer for Non-linear Hybrid Systems , 2013, CAV.

[14]  Piotr Zgliczynski,et al.  C1 Lohner Algorithm , 2002, Found. Comput. Math..

[15]  N. Nedialkov,et al.  Interval Tools for ODEs and DAEs , 2006, 12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006).

[16]  Hong Ling,et al.  Polı , 2011 .

[17]  橋口 公一,et al.  Introduction to finite strain theory for continuum elasto-plasticity , 2013 .

[18]  Oded Maler,et al.  Systematic Simulation Using Sensitivity Analysis , 2007, HSCC.

[19]  Jean-Jacques E. Slotine,et al.  On Contraction Analysis for Non-linear Systems , 1998, Autom..

[20]  Z. Galias,et al.  Computer assisted proof of chaos in the Lorenz equations , 1998 .

[21]  Daniel Wilczak,et al.  Heteroclinic Connections Between Periodic Orbits in Planar Restricted Circular Three Body Problem. Part II , 2006 .

[22]  David Angeli,et al.  A characterization of integral input-to-state stability , 2000, IEEE Trans. Autom. Control..

[23]  Xin Chen,et al.  Taylor Model Flowpipe Construction for Non-linear Hybrid Systems , 2012, 2012 IEEE 33rd Real-Time Systems Symposium.

[24]  Chuchu Fan,et al.  Bounded Verification with On-the-Fly Discrepancy Computation , 2015, ATVA.

[25]  Martin Berz,et al.  COSY INFINITY Version 9 , 2006 .

[26]  Martin Berz,et al.  Verified Integration of ODEs and Flows Using Differential Algebraic Methods on High-Order Taylor Models , 1998, Reliab. Comput..