Verified simulation of control systems with interval parameters using an exponential state enclosure technique

A large number of control systems is characterized by uncertainties which result from manufacturing tolerances, imperfect measurement, and simplifications of mathematical models during the design of both controllers and observers. Different numerical techniques were developed in recent years to characterize the influence of the above-mentioned types of uncertainty. In this field, stochastic simulation techniques, exploiting for example Monte-Carlo methods, are widely used in engineering applications. However, these techniques do not allow for a computation of guaranteed worst-case bounds of the sets of reachable states as soon as the uncertain variables are given by a set-valued description. In this case, the use of interval arithmetic is a promising alternative. Although interval arithmetic provides the possibility to determine verified enclosures of those domains in the state-space that contain all reachable states, the problems of overestimation and large computing time have led to the fact that interval methods are still not widely used in engineering. To overcome these problems, a novel simulation approach is presented in this paper, which allows for a computation of tight interval bounds for systems of uncertain linear ordinary differential equations with both aperiodic and oscillatory behavior. Simulation results for a closed-loop controller of a flexible high-bay rack feeder system conclude this paper.