The problem of maximal safe set determination has been solved for general hybrid systems with an elegant procedure that however is not guaranteed to terminate in a finite number of steps and is computationally involved. We propose a procedure for a restricted class of hybrid systems, called switching dynamical systems, where transitions between different configurations are determined by external uncontrollable events. This procedure can be generalized to cover general hybrid systems. However, while demonstrably more efficient than the one available in the literature, the procedure still suffers from computational complexity stemming from the computation of maximal controlled invariant sets of general dynamical systems. The procedure is made computationally more appealing by linearizing and discretizing the dynamical systems since procedures for the computation of maximal controlled invariant sets for discrete-time linear systems are well-known in the literature. Even in this case, the procedure for the determination of the maximal controlled invariant set may not converge in a finite number of steps.
We propose an inner approximation procedure that together with the
classical outer approximation procedure yields tight bounds for an error
due to the truncation
of the procedure after a finite number of steps. The theory is applied to
idle-speed regulation in
engine control.
Reading material
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