Comparative Evaluation and Improvement of Computational Approaches to Reachability Analysis of Linear Hybrid Systems

This thesis addresses the problem of reachability analysis with the focus on linear hybrid systems. Hybrid systems are a mixture of continuous and discrete behaviors. The Hybrid automaton consisting of a graph, in which the locations describe the continuous and the transitions the discrete behavior, represents the best formal model for such kind of systems. It provides a formalism integrating differential equations and logic expressions in a same framework, thus opening new horizons in research and development of new methods and novel algorithms. Despite recent progress made in this field in the last years, actual verification methods and available tools have exhibited their shortcomings.

We started this work with the assessment of some verification tools using a suite of benchmarks conceived specially for this task. The benchmarks possess particular characteristics for testing of efficiency, applicability, scalability, capability and performances of these tools.

We offer a theoretical overview of existing methods for computing an overapproximation of reachable sets for linear time invariant hybrid systems. This covers approaches for overapproximating reachable sets of the continuous dynamics with and without invariants as well as methods for solving the problem of guard intersection at transitions. We furthermore propose new overapproximation techniques for treating the continuous part as well as the discrete part of the hybrid automaton. We suggest scalable, modular implementations of these diverse methods allowing thereby possible combinations between them first using support functions and then with zonotopes. The implementations include different approaches for handling invariants, guards and transitions for the above-mentioned set representations. Two toolboxes are the results of this implementation effort. Both tools integrate the methods described above. They offer a GUI and allow for a user-configurable reachability analysis. We use both tools to carry out a performance comparison of different methods. We note thereby that there is a correlation between these performances and the complexity of the tested example. However, we note during this survey using the proposed benchmark suite that the difference in the performance with regards to the tightness of the over-approximation and the computation time is not so crucial for low dimensional systems.

We propose a networked platoon of vehicles to demonstrate different context where reachability analysis can be useful. We first perform a reachability analysis to determine unsafe gaps between the vehicles which are controlled using LMIformalism. Reachability analysis can be helpful for control design. The choice between controllers on the basis of reachability results has led to controller ensuring the best compromise between safe and small gaps when applying H2 or H∞ control design techniques. Reachability can also be used to determine time-critical conditions. As demonstration, we opt for a platoon approaching an intersection.