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## Section: Scientific Foundations

### Controller Synthesis

The Supervisory Control Problem is concerned with ensuring (not only checking) that a computer-operated system works correctly. More precisely, given a specification model and a required property, the problem is to control the specification's behavior, by coupling it to a supervisor, such that the controlled specification satisfies the property  [45] . The models used are LTSs and the associated languages, which make a distinction between controllable and non-controllable actions and between observable and non-observable actions. Typically, the controlled system is constrained by the supervisor, which acts on the system's controllable actions and forces it to behave as specified by the property. The control synthesis problem can be seen as a constructive verification problem: building a supervisor that prevents the system from violating a property. Several kinds of properties can be ensured such as reachability, invariance (i.e. safety), attractivity, etc. Techniques adapted from model checking are then used to compute the supervisor w.r.t. the objectives. Optimality must be taken into account as one often wants to obtain a supervisor that constrains the system as few as possible.

The Supervisory Control Theory overview. Supervisory control theory deals with control of Discrete Event Systems. In this theory, the behavior of the system S is assumed not to be fully satisfactory. Hence, it has to be reduced by means of a feedback control (named Supervisor or Controller) in order to achieve a given set of requirements  [45] . Namely, if S denotes the specification of the system and is a safety property that has to be ensured on S (i.e. ), the problem consists in computing a supervisor , such that

 (1)

where is the classical parallel composition between two LTSs. Given S , some events of S are said to be uncontrollable (uc ), i.e. the occurrence of these events cannot be prevented by a supervisor, while the others are controllable (c ). It means that all the supervisors satisfying (1 ) are not good candidates. In fact, the behavior of the controlled system must respect an additional condition that happens to be similar to the ioco conformance relation that we previously defined in  3.3 . This condition is called the controllability condition and is defined as follows.

 (2)

Namely, when acting on S , a supervisor is not allowed to disable uncontrollable events. Given a safety property , that can be modeled by an LTS , there actually exist many different supervisors satisyfing both (1 ) and (2 ). Among all the valid supervisors, we are interested in computing the supremal one, ie the one that restricts the system as few as possible. It has been shown in  [45] that such a supervisor always exists and is unique. It gives access to a behavior of the controlled system that is called the supremal controllable sub-language of w.r.t. S and uc . In some situations, it may also be interesting to force the controlled system to be non-blocking (See  [45] for details).

The underlying techniques are similar to the ones used for Automatic Test Generation. It consists in computing a product between the specification and and to remove the states of the obtained LTS that may lead to states that violate the property by triggering only uncontrollable events.

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