Diagnosability of hybrid automata with measurement uncertainty

We propose a diagnosability notion that depends on two parameters denoted as (¿_d, ¿_m) for the general class of transition systems where the observable output is given by discrete symbols and by the delay between the generation of two symbols. The first parameter specifies that if a trajectory (state execution) has visited the faulty set, this can be detected only using the external trajectory (observable output) within a delay upper-bounded by ¿_d. The second parameter specifies the available precision ¿_m when measuring time delays of the external trajectory. Given an approximate abstraction T₁ of a system T₂ with precision ¿, we derive a relation between diagnosability properties of T₁ and T₂ with respect to the parameters ¿_d, ¿_m, ¿. We apply our results to an electromagnetic valve system for cam-less engines.