Interval methods for robust gain scheduling controllers

A novel interval-based approach for a gain-scheduled controller synthesis is presented in this paper which aims at stabilizing continuous-time dynamic systems of finite dimension in a guaranteed manner over predefined finitely long time horizons. The fundamental aim of the design is the temporal adaptation of the feedback gain in combination with a reduction of the interval widths which characterize outer enclosures of the states that are reachable in the worst case at a certain time instant while ensuring asymptotic stability of the closed-loop dynamics. For that purpose, feedback gains are computed first for an initial state enclosure. Second, they are tested for validity in such a way that the parameterization of the controller shall be applicable over the whole time discretization step. In case of the verification failing, the gain is modified after determining a bounding box for those states that can be reached over the considered prediction window. That means, an offline calculation of controller gains is possible so that prespecified performance indicators with respect to the closed-loop dynamics are guaranteed to be satisfied. The robust and/or optimal control problem is solved efficiently using linear matrix inequality (LMI) techniques, while methods from interval analysis are furthermore employed during the underlying reachability analysis. Additionally, efficient approaches are presented which reduce overestimation due to the unavoidable wrapping effect. Hence, the proposed design method aims, simultaneously, at the reduction of overestimation and a guaranteed stability verification. To conclude this paper, the resulting control strategy is verified numerically for an inverted pendulum as a prototypical benchmark application.