Project : s4
Section: New Results
Classification and resolution of control problems through the quantified mu-calculus
The theory of control synthesis introduced by Ramadge and Wohnam is a generic method which can be described as follows: given a program and some expected behavior, known as control objective, the goal is to produce, by automated methods, a device (e.g., another program) with two main properties. First, this device must fulfill some constraints (e.g., it should belong to some particular class of programs), and second, it should be able to control the original program (e.g., by synchronous composition) in order to achieve the required behavior.
We have developed a logical formalism as a general formal language for the specification of control problems. The proposed framework extends the Mu-Calculus, a extremely expressive modal logic with fix-points operators, introduced by Dexter Kozen: we allow for quantifications over atomic propositions, yielding to a second order logic. We have established that checking for the existence of a solution to the control problem is equivalent to perform verification of formulas. Verification of formulas is often called Model-Checking. However, the is logic undecidable, as the decentralized control problems under partial observation can be expressed therein. We have explored various fragments of the logic. The fragments reveal to be expressive enough to specify interesting control problems, but small enough to remain decidable. An accurate study of the complexity of the satisfiability and model-checking problems, in these logical fragments, has been carried out.
In , we consider the fragment corresponding to the setting where the moves of the systems to be controlled are fully observable. These are the so called control problems under total observation. The logical setting offers a uniform way to describe, as parameters, the kind of system (closed or open), the control objective, the type of interaction between the controller and the system, optimality criteria (fairness, maximally permissive), etc. To our knowledge, none of the former approaches can capture such a wide range of concepts. Moreover, we have established that model-checking this fragment is decidable and that the synthesis of controllers can be obtained on the basis of the underlying model checking procedure.
In , the case of control requirements for systems under partial observation is studied. We have focused on a fragment expressive enough to specify the unobservable sets of events of (decentralized) controllers, and to allow for the joint unobservability and controllability of an event. We have identified the set of formulas representable by infinite tree automata. Technically, the automata constructions are borrowed from the work of André Arnold et al. . For formulas expressing control requirements, any model of the associated automaton provides an adequate controller. For example, given any Mu-Calculus definable control objective, a maximal permissive controller in some class of controllers under partial observation can be specified by a formula and synthesized in time 3EXP in the size of the formula.
This logical framework brings a new vision of the field, and makes discrete event system control theory much clearer. In particular, it provides a rigorous classification of control problems. Our logical framework is also expected to be relevant to problems related to control theory, such as diagnosis or test generation. This will be the objective of this continuing research work.