Numerical Verification of Affine Systems with up to a Billion Dimensions

Affine systems reachability is the basis of many verification methods. With further computation, methods exist to reason about richer models with inputs, nonlinear differential equations, and hybrid dynamics. As such, the scalability of affine systems verification is a prerequisite to scalable analysis for more complex systems. In this paper, we improve the scalability of affine systems verification, in terms of the number of dimensions (variables) in the system. The reachable states of affine systems can be written in terms of the matrix exponential, and safety checking can be performed at specific time steps with linear programming. Unfortunately, for large systems with many state variables, this direct approach requires an intractable amount of memory while using an intractable amount of computation time. We overcome these challenges by combining several methods that leverage common problem structure. Memory is reduced by exploiting initial states that are not full-dimensional and safety properties (outputs) over a few linear projections of the state variables. Computation time is saved by using numerical simulations to compute only projections of the matrix exponential relevant for the verification problem. Since large systems often have sparse dynamics, we use Krylov-subspace simulation approaches based on the Arnoldi or Lanczos iterations. Our method produces accurate counter-examples when properties are violated and, in the extreme case with sufficient problem structure, can analyze a system with one billion real-valued state variables

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

An hybrid system approach to nonlinear optimal control problems

We consider a nonlinear ordinary differential equation and want to control its behavior so that it reaches a target by minimizing a cost function. Our approach is to use hybrid systems to solve this problem: the complex dynamic is replaced by piecewise affine approximations which allow an analytical resolution. The sequence of affine models then forms a sequence of states of a hybrid automaton. Given a sequence of states, we introduce an hybrid approximation of the nonlinear controllable domain and propose a new algorithm computing a controllable, piecewise convex approximation. The same way the nonlinear optimal control problem is replaced by an hybrid piecewise affine one. Stating a hybrid maximum principle suitable to our hybrid model, we deduce the global structure of the hybrid optimal control steering the system to the target

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

Randomized hybrid linear modeling by local best-fit flats

The hybrid linear modeling problem is to identify a set of d-dimensional affine sets in a D-dimensional Euclidean space. It arises, for example, in object tracking and structure from motion. The hybrid linear model can be considered as the second simplest (behind linear) manifold model of data. In this paper we will present a very simple geometric method for hybrid linear modeling based on selecting a set of local best fit flats that minimize a global l1 error measure. The size of the local neighborhoods is determined automatically by the Jones' l2 beta numbers; it is proven under certain geometric conditions that good local neighborhoods exist and are found by our method. We also demonstrate how to use this algorithm for fast determination of the number of affine subspaces. We give extensive experimental evidence demonstrating the state of the art accuracy and speed of the algorithm on synthetic and real hybrid linear data.Comment: To appear in the proceedings of CVPR 201

Global Trajectory Optimisation : Can We Prune the Solution Space When Considering Deep Space Manoeuvres? [Final Report]

This document contains a report on the work done under the ESA/Ariadna study 06/4101 on the global optimization of space trajectories with multiple gravity assist (GA) and deep space manoeuvres (DSM). The study was performed by a joint team of scientists from the University of Reading and the University of Glasgow