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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 [41] . The models used are LTSs, say G, and the associated languages, say Im10 ${\#8466 (G)}$ , 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 [41] . 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 [41] . Namely, if S denotes the specification of the system and $ \upper_phi$ is a safety property that has to be ensured on S (i.e. $ \models$$ \upper_phi$ ), the problem consists in computing a supervisor Im11 $\#119966 $ , such that

Im12 $\mtable{...}$(1)

where Im13 $\#8741 $ is the classical parallel composition between two LTSs. Given S, some events of S are said to be uncontrollable ($ \upper_sigma$uc ), i.e. the occurrence of these events cannot be prevented by a supervisor, while the others are controllable ($ \upper_sigma$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.

Im14 $\mtable{...}$(2)

Namely, when acting on S, a supervisor is not allowed to disable uncontrollable events. Given a safety property $ \upper_phi$ , that can be modeled by an LTS Im15 $A_\#934 $ , there actually exists 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 [41] 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 Im15 $A_\#934 $ w.r.t. S and $ \upper_sigma$uc . In some situations, it may also be interesting to force the controlled system to be non-blocking (See [41] 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 Im15 $A_\#934 $ and to remove the states of the obtained LTS that may lead to states that violate the property by triggering only uncontrollable events.

Optimal Control. We are also interested in the Optimal Control Problem. The purpose of optimal control is to study the behavioral properties of a system in order to generate a supervisor that constrains the system to a desired behavior according to quantitative and qualitative requirements. In this spirit, we have been working on the optimal scheduling of a system through a set of multiple goals that the system had to visit one by one [37] . We have also extended the results of [44] to the case of partial observation in order to handle more realistic applications [38] .

Control of Structured Discrete Event System. In many applications and control problems, LTS are the starting point to model fragments of a large scale system, which usually consists of several composed and nested sub-systems. Knowing that the number of states of the global system grows exponentially with the number of parallel and nested sub-systems, we have been interested in designing algorithms that perform the controller synthesis phase by taking advantage of the structure of the plant without expanding the system. Given a concurrent system and a safety property, modeled as a language , also called specification that have to be ensured on this system, we have investigated in e.g. [2] the computation of the supremal controllable language contained in the expected language. To do so, we use a modular centralized approach and perform the control on some approximations of the plant derived from the behavior of each component. The behavior of these approximations is restricted so that they respect a new language property for discrete event systems called partial controllability condition that depends on the safety property. It is shown that, under some assumptions the intersection of these ``controlled approximations'' corresponds to the supremal controllable language contained in the specification with respect to the plant. This computation is performed without building the whole plant, hence avoiding the state space explosion induced by the concurrent nature of the plant.

Similarly, in order to take into account nested behaviors, some techniques based on model aggregation methods [46] , [32] have been proposed to deal with hierarchical control problems. Another direction has been proposed in [31] . Brave and Heimann in [31] introduced Hierarchical State Machines which constitute a simplified version of the Statecharts . Compared to the classical state machines, they add concurrency and hierarchy features. Some other works dealing with control and hierarchy can be found in [35] , [36] . This is the direction we have chosen in the VerTeCs Team [3] .


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