A Novel Thick Ellipsoid Approach for Verified Outer and Inner State Enclosures of Discrete-Time Dynamic Systems

Abstract Simulating dynamic systems with bounded uncertainty in initial conditions and selected parameters is a common task for the reliability analysis of closed-loop control structures as well as for a simulation-based parameter identification on the basis of uncertain measurements. However, dealing with bounded uncertainty is not a trivial task. On the one hand, the naive application of interval analysis often leads to excessively large bounds which may yield state enclosures that are by far too pessimistic to be useful in practice. On the other hand, the use of grid-based or probabilistic and Monte-Carlo like simulation approaches suffers from the disadvantage that they do not provide a guarantee of the correctness of the obtained solutions. Using insufficiently many or badly chosen samples may lead to the phenomenon that critical system states are not detected so that the computed results underestimate the range of reachable states. Therefore, a novel ellipsoidal state enclosure technique is presented which does not require the online solution of linear matrix inequalities (LMIs). It uses the newly introduced representation of state domains by means of thick ellipsoids. These domains simultaneously represent inner and outer enclosures of the reachable states and directly provide a measure for the tightness of the obtained results, regardless of whether the system is linear or nonlinear or whether it is stable or not.