Polynomial synthesis of supervisor for partially observed discrete-event systems by allowing nondeterminism in control

Abstract

We study the supervisory control of discrete-event systems (DESs) under partial observation using nondeterministic supervisors. We formally define a nondeterministic control policy and also a control & observation compatible nondeterministic state machine and prove their equivalence. The control action of a nondeterministic supervisor is chosen online, nondeterministically from among a set of choices determined offline. Also, the control action can be changed online nondeterministically (prior to any new observation) in accordance with choices determined offline. The online choices, once made, can be used to affect the set of control action choices in future. We show that when control is exercised using a nondeterministic supervisor, the specification language is required to satisfy a weaker notion of observability, which we define in terms of recognizability and achievability. Achievability serves as necessary and sufficient condition for the existence of a nondeterministic supervisor, and it is weaker than controllability and observability combined. When all events are controllable, achievability reduces to recognizability. We show that both existence, and synthesis of nondeterministic supervisors can be determined polynomially. (For deterministic supervisors, only existence can be determined polynomially.) Both achievability and recognizability are preserved under union, and also under intersection (when restricted over prefix-closed languages). Using the intersection closure property we derive a necessary and sufficient condition for the range control problem for the prefix-closed case. Unlike the deterministic supervisory setting where the complexity of existence is exponential, our existence condition is polynomially verifiable, and also a supervisor can be polynomially synthesized.

Maximal Synthesis for Hennessy-Milner Logic

“…Other approaches use a different formalism for the automaton before and after synthesis [21] or present an approach that selects control actions online [15]. A number of articles do not take into account the maximality or maximal permissiveness of the synthesized products, see for instance [14] and [15]. As a further difference between this paper and earlier work, we define synthesis upon non-deterministic transition systems, while other authors explicitly require the input LTS to be deterministic [12], [14], [16].…”

“…The field of supervisory control [23] is by far the main contributor to solutions in line with our approach [14], [15], [28], [21], [1], [3], even maturing into various algorithms (and realizations thereof) [22], [29]. In this paper we do not take into account a number of specific aspects of supervisory control theory, such as marker state reachability, controllability and deadlock avoidance [23].…”

“…Other papers use seemingly more extended logics [8] but omit some key elements such as a complete coverage of the ∨-operator. Other approaches use a different formalism for the automaton before and after synthesis [21] or present an approach that selects control actions online [15]. A number of articles do not take into account the maximality or maximal permissiveness of the synthesized products, see for instance [14] and [15].…”

Control of Nondeterministic Discrete-Event Systems for Bisimulation Equivalence

“…Moreover, when the existence conditions are satisfied, serves as a supervisor, i.e., the complexity of synthesizing a supervisor is linear in the size of the specification. It is interesting to note that when specification is deterministic but the plant is nondeterministic, the complexity of existence as well as synthesis is again polynomial [20]. In contrast, the situation seems to be different when both the plant and the specification are nondeterministic.…”

“…The notion of nondeterministic control was formalized in [20] and used for control under partial observation for language specification, and the notion of achievability (a property weaker than controllability and observability combined) was introduced. The problem of finding a nondeterministic supervisor so that it's parallel-composition with plant conforms to a deterministic specification (via language containment) is studied in [41].…”