Hybrid Reachability Analysis for Kuramoto-Lanchester Model

Cyber-physical systems are ubiquitous nowadays and play a significant role in people's daily life. These systems include, e.g., autonomous vehicles and aerospace systems. Since human lives rely on the performance of these systems, it is of utmost importance to ensure their reliability. However, their complexity makes analysis particularly challenging and computationally expensive. Thus, it is crucial to develop tools to efficiently analyze cyber-physical systems and their safety properties. Cyber-physical systems are often modeled by hybrid automata, i.e. finite-state machines augmented with ordinary differential equations. In the thesis, we investigate reachability analysis methods for hybrid automata. In particular, we extend JuliaReach, a framework for fast prototyping set-based reachability analysis algorithms, to support verification of hybrid automata. For this purpose, we add to JuliaReach concrete and lazy discrete post operators. Lazy operations are particularly efficient in flowpipe based reachability analysis with long sequences of computations. The implemented algorithms are interchangeable and support all three reachability scenarios available in JuliaReach for the purely continuous setting: techniques to analyze linear systems using support functions and zonotopes as well as Taylor model based analysis for nonlinear systems. In order to evaluate our methods, we apply them to the Kuramoto-Lanchester model. This model exhibits highly nonlinear dynamics and can be easily scaled, and thus is well-suited to assess performance of reachability analysis methods for hybrid automata

Reachability analysis of linear hybrid systems via block decomposition

Reachability analysis aims at identifying states reachable by a system within a given time horizon. This task is known to be computationally expensive for linear hybrid systems. Reachability analysis works by iteratively applying continuous and discrete post operators to compute states reachable according to continuous and discrete dynamics, respectively. In this paper, we enhance both of these operators and make sure that most of the involved computations are performed in low-dimensional state space. In particular, we improve the continuous-post operator by performing computations in high-dimensional state space only for time intervals relevant for the subsequent application of the discrete-post operator. Furthermore, the new discrete-post operator performs low-dimensional computations by leveraging the structure of the guard and assignment of a considered transition. We illustrate the potential of our approach on a number of challenging benchmarks.Comment: Accepted at EMSOFT 202

Algorithmic Verification of Continuous and Hybrid Systems

We provide a tutorial introduction to reachability computation, a class of computational techniques that exports verification technology toward continuous and hybrid systems. For open under-determined systems, this technique can sometimes replace an infinite number of simulations.Comment: In Proceedings INFINITY 2013, arXiv:1402.661

Reach Set Approximation through Decomposition with Low-dimensional Sets and High-dimensional Matrices

Approximating the set of reachable states of a dynamical system is an algorithmic yet mathematically rigorous way to reason about its safety. Although progress has been made in the development of efficient algorithms for affine dynamical systems, available algorithms still lack scalability to ensure their wide adoption in the industrial setting. While modern linear algebra packages are efficient for matrices with tens of thousands of dimensions, set-based image computations are limited to a few hundred. We propose to decompose reach set computations such that set operations are performed in low dimensions, while matrix operations like exponentiation are carried out in the full dimension. Our method is applicable both in dense- and discrete-time settings. For a set of standard benchmarks, it shows a speed-up of up to two orders of magnitude compared to the respective state-of-the art tools, with only modest losses in accuracy. For the dense-time case, we show an experiment with more than 10.000 variables, roughly two orders of magnitude higher than possible with previous approaches

Forward Stochastic Reachability Analysis for Uncontrolled Linear Systems using Fourier Transforms

We propose a scalable method for forward stochastic reachability analysis for uncontrolled linear systems with affine disturbance. Our method uses Fourier transforms to efficiently compute the forward stochastic reach probability measure (density) and the forward stochastic reach set. This method is applicable to systems with bounded or unbounded disturbance sets. We also examine the convexity properties of the forward stochastic reach set and its probability density. Motivated by the problem of a robot attempting to capture a stochastically moving, non-adversarial target, we demonstrate our method on two simple examples. Where traditional approaches provide approximations, our method provides exact analytical expressions for the densities and probability of capture.Comment: V3: HSCC 2017 (camera-ready copy), DOI updated, minor changes | V2: Review comments included | V1: 10 pages, 12 figure