On the complexity and dynamical properties of mixed logical dynamical systems via an automaton-based realization of discrete-time hybrid automaton

Modeling of hybrid systems using mixed logical dynamical (MLD) systems is an art. The MLD framework often introduces numerous constraints and auxiliary binary and continuous variables, which, in turn, increase the computational complexity of the optimization problems. This paper presents an automaton-based realization for discrete-time hybrid automaton (DHA) with both controlled and uncontrolled switching phenomena by which it is attempted to develop efficient translation techniques to MLD systems and reduce the total number of decision variables in the MLD model. Based on this DHA model, a modified version of MLD systems, which is called extended MLD (EMLD) is formally defined and represented. EMLD is derived based on the concept of forward evolution in which the one-step delay between the change of discrete states and continuous dynamics of the existing evolution in the conventional MLD systems is eliminated. The result is that the size of EMLD model in terms of the number of variables, which determines the complexity of the synthesis problems on MLD systems, is reduced. The dynamical properties such as the nonblocking, determinism, and well-posedness of the proposed DHA and its equivalent MLD models are investigated, and the necessary and sufficient conditions are derived. The effectiveness of the proposed ideas is shown by the numerical examples.