Assessment of Quality of Hybrid Simulation Using Reachability Analysis

Hybrid simulation is an experimentally based method for investigating the response of a structure to dynamic excitation using a hybrid model. A hybrid model is an assemblage of one or more physical and one or more numerical, consistently scaled, substructures. The equation of motion of a hybrid model under dynamic excitation is solved during a hybrid simulation test. Results from hybrid simulation and shaking table tests have been shown to be comparable as long as propagation of experimental errors is successfully mitigated during the hybrid simulation. This paper addresses the issue of quality of hybrid simulation using a control-theory concept of reachability. Reachability analysis of a dynamic system deterministically predicts the sets of states the systems can attain under uncertain dynamic excitation starting from uncertain initial conditions evaluated using uncertain measurements. A method to compute the outside (worst-case) ellipsoidal approximation of these reachable sets is presented. Then, this method is used to predict an ellipsoidal tube of the envelope of states that can be reached starting from a state currently attained by the hybrid model during a hybrid simulation. A technique to evaluate the quality of hybrid simulation based on the size of the ellipsoidal tube approximation is developed and corresponding criteria for measuring the quality of hybrid simulation based on the how tightly the ellipsoidal tube envelopes the actual trajectory of the hybrid model in its state space are suggested.