Verified Interval Enclosure Techniques for Robust Gain Scheduling Controllers
In real-life applications, dynamic systems are often subject to uncertainty due to model simplications, measurement inaccuracy or approximation errors which can be mapped to specific parameters. Uncertainty in dynamic systems can come either in stochastic forms or as interval representations, when they are considered as bounded as it will be done in this paper. The main idea, here, is to find a joint approach for an interval-based gain scheduling controller while simultaneously reducing overestimation by enclosing state intervals with the least amount of conservativity. The robust and/ or optimal control design is realized using linear matrix inequalities (LMIs) to find an efficient solution and aims at a guaranteed stabilization of the system dynamics over a predefined time horizon. Since the resulting system is assumed to be asymptotically stable, a temporal reduction of the widths of intervals representing worst-case bounds of the system states at a specific point of time should occur. However, for commonly used approaches in the computation of interval enclosures those interval widths seemingly blow up due to the wrapping effect in many cases. To avoid this, we provide two interval enclosure techniques --- an exploitation of cooperativity and an exponential approach --- and discuss their applicability taking into account two real-life applications, a high-bay rack feeder and an inverse pendulum.