IEEE Robotics & Automation Magazine - June 2015 - 92

process types as long as the routing decisions of these process
types are independent of the underlying resource-allocation
function and the imposed DAP; this requirement can be
observed by treating the process routing as uncontrollable by
the designed policy. Furthermore, extensive computational
experimentation with a series of pretty sizable RASs has also
demonstrated the scalability and the computational tractability
of the method. Figure 6 demonstrates the application of this
incremental synthesis method for the development of a set of
linear inequalities-or a linear classifier-and the corresponding monitor places implementing the maximally permissive
DAP for the example RAS of Figure 1. On the other hand, it is
also true that the application of this method to the computation
of the maximally permissive DAP for RASs with nonbinary
state spaces is practically limited by the previously discussed
potential inability to represent this DAP by a set of linear
inequalities on the RAS state.
We close our discussion on the existing results regarding the
structural characterizations of the RAS deadlock through the
concept of the PN siphon, and the implications of these characterizations for the analysis and control of the relevant RAS behavior, by briefly mentioning an additional line of research that
has sought to employ these structural characterizations to provide some explanation for the observed possibility of ensuring
sufficient control of all the potential deadlocks of a given RAS
by controlling explicitly only a limited subset of them. The results of this line can be traced in [32] and [48], while the work
in [32] has also applied these results in a control-synthesis process through the introduction of the concept of elementary siphons. (Obviously, a straightforward alternative explanation for
these observed dependencies among the RAS deadlocks is provided by the classification theory that is discussed in the "The
Optimal DAP as a Classifier of the RAS States" section, and, in
particular, the notion of covering the RAS unsafe states that is
effected by the classifying inequalities.)
Going Forward
It should be evident from the entire discussion of the previous
parts of this article that the RAS concept and its accompanying
theory constitute a well-established theme in the academic research community. This discussion also reveals the methodological richness and the analytical and computational
strength that characterize the existing developments in this
area. The presented results offer rigorous and structured solutions to some ubiquitous problems that must be addressed by
the engineering community as it tries to increase the automation levels for a broad spectrum of technological applications
with ever increasing operational scale and complexity. In particular, the "Modeling an RAS as a Finite State Algorithm: The
Optimal Nonblocking Supervisor and Its Complexity" and
"Dealing with the Computational Challenges of the Optimal
Nonblocking Supervisor" sections, on the past developments
regarding the RAS deadlock problem and its efficient resolution through the deployment of the maximally permissive
DAP or some good approximation to this policy, demonstrate
how automation-related research can benefit from the effective
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utilization of the existing formal methods for behavioral verification and synthesis. These two sections also show how the
RAS-related research has contributed to the broader DES theory by capitalizing upon the special structure and the more concrete insights that are offered by the target application domains.
The previous discussion also unveils a series of directions
in which the current RAS theory can be extended and
strengthened, and some important open challenges that must
be systematically addressed by the relevant research community to eventually materialize the control paradigm that is epitomized by Figure 3. Hence, when it comes to the behavioral
RAS theory, one can consider the further development of the
existing theory to address more complex classes and behaviors
of the RAS taxonomy presented in the "The RAS Modeling
Abstraction, the Corresponding Control Paradigmn and an
RAS Taxonomy" section, than the usually studied class of disjunctive/conjunctive RAS. More specifically, while the currently existing results can provide liveness characterization and
assessment for these broader RAS classes, there is a remaining
need for methodology that will support the expedient synthesis of the maximally permissive DAP for these RASs or some
pertinent approximations of this policy. One can also consider
the extension of the basic RAS behavioral control problem
addressed in this article by considering application environments that provide only partial observability of the underlying
RAS function or systems that, due to their scale and/or structure, might require more distributed supervision than the centralized control scheme of Figure 3. The reactive or proactive
accommodation of random capacity losses with respect to certain resources due to the occurrence of unexpected events is
another issue that has received only limited attention in the
current RAS literature. All these extensions can be formalized
through relevant results in the existing DES theory, but it is
also expected that, as in the case of the past developments, the
special and rich structure of the RAS concept will enable customized analyses and solutions for this set of problems as well.
The RAS behavioral control problem can also be extended by extending the set of specifications that it addresses beyond the issue of deadlock avoidance and the establishment
of nonblocking behavior. As a more concrete example of
such an extension, one can consider the enforcement of a
production ratio constraint for the manufacturing cell of Figure 1, stipulating that the two supported product types must
be produced in lockstep or that the difference of the cumulative productions for these two products should observe certain bounds at any point in time. Then, one can seek to
characterize and compute the maximally permissive policy
that ensures the liveness of the considered RAS while observing this additional constraint. The resulting supervisory control problem can be addressed using the notion of fairness in
the relevant DES theory, and a first set of results for this
problem is provided in [23]. Other behavioral constraints of
a similar flavor can arise, for instance, by the need to observe
certain formations or some patrolling procedures by a fleet of
mobile agents and by the enforcement of aging and other
priority schemes in the resource allocation taking place



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