A Statistical Approach for Computing Reachability of Non-linear and Stochastic Dynamical Systems

We present a novel approach to compute reachable sets of dynamical systems with uncertain initial conditions or parameters, leveraging state-of-the-art statistical techniques. From a small set of samples of the true reachable function of the system, expressed as a function of initial conditions or parameters, we emulate such function using a Bayesian method based on Gaussian Processes. Uncertainty in the reconstruction is reflected in confidence bounds which, when combined with template polyhedra ad optimised, allow us to bound the reachable set with a given statistical confidence. We show how this method works straightforwardly also to do reachability computations for uncertain stochastic models.

[1]  Carl E. Rasmussen,et al.  Gaussian processes for machine learning , 2005, Adaptive computation and machine learning.

[2]  Pravin Varaiya,et al.  On Ellipsoidal Techniques for Reachability Analysis. Part II: Internal Approximations Box-valued Constraints , 2002, Optim. Methods Softw..

[3]  Ezio Bartocci,et al.  On the Robustness of Temporal Properties for Stochastic Models , 2013, HSB.

[4]  Tarik Nahhal,et al.  Coverage-guided test generation for continuous and hybrid systems , 2009, Formal Methods Syst. Des..

[5]  Luca Bortolussi,et al.  Smoothed model checking for uncertain Continuous-Time Markov Chains , 2014, Inf. Comput..

[6]  Emilio Frazzoli,et al.  Incremental Search Methods for Reachability Analysis of Continuous and Hybrid Systems , 2004, HSCC.

[7]  Guido Sanguinetti,et al.  Learning and Designing Stochastic Processes from Logical Constraints , 2013, QEST.

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

[9]  Wpmh Maurice Heemels,et al.  Survey of modeling, analysis, and control of hybrid systems , 2009 .

[10]  Diego Latella,et al.  Continuous approximation of collective system behaviour: A tutorial , 2013, Perform. Evaluation.

[11]  Calin Belta,et al.  Controlling a Class of Nonlinear Systems on Rectangles , 2006, IEEE Transactions on Automatic Control.

[12]  Pravin Varaiya,et al.  On Ellipsoidal Techniques for Reachability Analysis. Part I: External Approximations , 2002, Optim. Methods Softw..

[13]  Thao Dang,et al.  Template-Based Unbounded Time Verification of Affine Hybrid Automata , 2011, APLAS.

[14]  Neil D. Lawrence,et al.  Modelling transcriptional regulation using Gaussian Processes , 2006, NIPS.

[15]  J. Douglas Faires,et al.  Numerical Analysis , 1981 .

[16]  Manuela L. Bujorianu,et al.  A Statistical Inference Method for the Stochastic Reachability Analysis , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[17]  Nasser M. Nasrabadi,et al.  Pattern Recognition and Machine Learning , 2006, Technometrics.

[18]  K. Worden,et al.  Bayesian sensitivity analysis of bifurcating nonlinear models , 2013 .

[19]  Antoine Girard,et al.  Efficient Computation of Reachable Sets of Linear Time-Invariant Systems with Inputs , 2006, HSCC.

[20]  Oded Maler,et al.  Computing reachable states for nonlinear biological models , 2009, Theor. Comput. Sci..

[21]  A. O'Hagan,et al.  Probabilistic sensitivity analysis of complex models: a Bayesian approach , 2004 .

[22]  Mahesh Viswanathan,et al.  Verification of annotated models from executions , 2013, 2013 Proceedings of the International Conference on Embedded Software (EMSOFT).

[23]  Andreas Krause,et al.  Information-Theoretic Regret Bounds for Gaussian Process Optimization in the Bandit Setting , 2009, IEEE Transactions on Information Theory.

[24]  H. Andersson,et al.  Stochastic Epidemic Models and Their Statistical Analysis , 2000 .

[25]  Thao Dang,et al.  Falsifying Oscillation Properties of Parametric Biological Models , 2013, HSB.

[26]  B. Krogh,et al.  Computing polyhedral approximations to flow pipes for dynamic systems , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[27]  Antoine Girard,et al.  Verification Using Simulation , 2006, HSCC.

[28]  Antoine Girard,et al.  Reachability Analysis of Polynomial Systems Using Linear Programming Relaxations , 2012, ATVA.

[29]  Antoine Girard,et al.  SpaceEx: Scalable Verification of Hybrid Systems , 2011, CAV.

[30]  Boxin Tang.  Orthogonal Array-Based Latin Hypercubes , 1993 .