An improved suboptimal solution for supervisory control and observation problem

In some discrete event dynamical systems (DEDS) only partial observations of the system behavior are available. The supervisory control and observation problem (SCOP) is of particular importance in these applications. In the DEDS literature, a necessary and sufficient condition for the existence of a solution for SCOP has been developed. Given a desired closed loop language K, SCOP has a solution if and only if it is both observable and controllable. However, the set of all observable and controllable sublanguages of K, denoted by DD(K), is not closed under the set union operation. Thus an optimal solution does not exist. An adopted method to deal with this problem is to consider the set of all L_m(G)-closed sublanguages of K whose prefix closures are normal and controllable, denoted by DD´(K). DD´(K) is a proper subset of DD(K) and is closed under the set union operation. Therefore DD´(K) contains a unique supremal element, sup DD´(K). We introduce a new property for nonclosed languages similar to the normality condition. We use this property to construct an improved suboptimal solution for SCOP