Application-Based Discussion of Verified Simulations of Interval Enclosure Techniques

Dynamical systems are often subject to uncertainties, whether it be parameter uncertainties or the interpretation of state dependencies in quasi-linear state-space representations as some kind of time-varying uncertain parameters. The first group of uncertainties arises from mathematical model simplifications, manufacturing tolerances, and imperfect measurements. Here, uncertainties can be represented in different forms like probability distributions in the stochastic case or interval representations in a bounded error framework. On the one hand, there exist numerous techniques to handle stochastic uncertainty, for example Monte-Carlo methods, but those do not allow for the computation of worst-case bounds of the sets of reachable states. On the other hand, approaches based on interval analysis are capable of the latter aspect. This paper deals with those methods from the perspective of an application scenario in the form of a guaranteed robust stabilization of an inverted pendulum by using constant controller gains. Here, we assume an interval representation for the bounded influence of state dependencies in the system matrices by a polytopic uncertainty model. When dealing with interval uncertainty, the rigorous computation of guaranteed state enclosures is a difficult task. Due to conservatism and/or the wrapping effect, overestimation is a common problem. This paper discusses different approaches to perform a verified reachability analysis by means of interval enclosures to be interfaced with the controller parameterization and gives guidelines on which available technique to use in a real-life robust control task.