Approximation Based Safety and Stability

With the advent of computers to control various physical processes, there has emerged a new class of systems which contain tight interactions between the “discrete” digital world and the “continuous” physical world. These systems which exhibit mixed discrete-continuous behaviors, called hybrid systems, are present everywhere in automotives, aeronautics, medical devices, robotics and are often deployed in safe-critical applications. Hence, reliability of the performance of these systems is a very important issue, and this thesis concerns automatic verification of their models to improve the confidence in these systems. Automatic verification of hybrid systems is an extremely challenging task, owing to the many undecidability results in the literature. Automated verification has been seen to be feasible for only simple classes of systems. These observations have emphasized the need for simplifying the complexity of the system before applying traditional verification techniques. This thesis explores the feasibility of such an approach for verifying complex systems. In this thesis, we take the approach of verifying a complex system by approximating it to a “simpler” system. We consider two important classes of properties that are required of hybrid systems in practice, namely, safety and stability. Intuitively, safety properties are used to express the fact that something bad does not happen during the execution of the system; and stability is used to capture the notion that small perturbations in the initial conditions or inputs to the system, do not result in drastic changes in the behaviors of the system. For safety verification, we present two techniques for approximation, an error based technique which allows one to compute as precise an approximation as desirable in terms of a quantified error between the original and the approximate system, and a property based technique which takes into account the property being analysed in constructing and refining the approximate