The following technical report is available from http://aib.informatik.rwth-aachen.de:
Reachability Analysis of Non-Linear Hybrid Systems Using Taylor Models Xin Chen AIB 2015-09
With the ubiquitous use of computers in controlling physical systems, it requires to have a new formalism that could model both continuous flows and discrete jumps. Hybrid systems are introduced to this purpose. A hybrid system, which is modeled by a hybrid automaton in the thesis, is equipped with finitely many discrete modes and continuous real-valued variables. A state of it is then represented by a mode along with a valuation of the variables. Given that the system is in a mode l, the variable values are changed continuously according to the Ordinary Differential Equation (ODE) associated to l, or discretely by a jump starting from l. The thesis focuses on the techniques to compute all reachable states over a bounded time horizon and finitely many jumps for a hybrid system with non-linear dynamics. The results of that can then be used in safety verification of the system.
Although a great amount of work has been devoted to the reachability analysis of hybrid systems with linear dynamics, there are few effective approaches proposed for the non-linear case which is very often in applications. The difficulty is twofold. Firstly, it is not easy to find an over-approximation with acceptable accuracy for a set of the solutions of a non-linear ODE. Secondly, to detect and compute the reachable states under a jump requires solving non-linear real arithmetic problems which is also difficult in general. In the thesis, we present our approaches to deal with the above difficulties. For the first one, we present the use of Taylor models as the over-approximate representations for non-linear ODE solutions. Our work can be viewed as a variant of the Taylor model method proposed by Berz et al., such that we are able to efficiently deal with some examples with more than $10$ variables. Besides, we also extend the work of Lin and Stadtherr to handle the ODEs with bounded time-varying parameters. For the second difficulty, we present two techniques: (a) domain contraction and (b) range over-approximation to compute an enclosure for the reachable set from which a jump is enabled. They can be seen as Satisfiability Modulo Theories (SMT) solving algorithms which are specialized for the reachability analysis of hybrid systems. In order to reduce the computational cost, we also propose different heuristics for aggregating Taylor models. Besides the above contributions, we describe a method to fast generate Taylor model over-approximations for linear ODE solutions. Its performance is demonstrated via a comparison with the tool SpaceEx.
To make our methods accessible by other people, we implement them in a tool named Flow*. To examine the effectiveness, we thoroughly compare it with some related tools which are popularly used, according to their functionalities, over a set of non-trivial benchmarks that are collected by us from the areas of mechanics, biology, electronic engineering and medicine. From the experimental results, the advantage of Flow* over the other tools becomes more apparent when the scale of the system grows. On the other hand, it also shows that Flow* can be applied to analyzing realistic systems.