The Design of dynamical observers for hybrid systems: Theory and Application to an Automotive Control Problem

A design methodology is presented for dynamical observers of hybrid systems with linear continuous-time dynamics that reconstruct the complete state (discrete location and continuous state) from the knowledge of the inputs and outputs of a hybrid plant. Given a current-location observable living hybrid system with minimum dwell-time, we prove that exponential ultimate boundedness for an hybrid observer can always be achieved. We also prove that the observer correctly identifies (apart from an initial finite number of transitions) the sequence of hybrid system locations even when the complementary outputs are generated with some delay with respect to the corresponding transitions in the plant.We then present the application of the theory to the problem of on–line identification of the actual engaged gear for a car, an important contribution. The relevance of this problem is related to engine control strategies achieving high performance and efficient emissions control which depend critically on the knowledge of the engaged gear. The performance of the observer was tested with (real and not simulated) experimental data obtained in Magneti Marelli Powertrain using an Opel Astra equipped with a Diesel engine and a robotized gearbox SeleSpeed.