Journal of Modern Mathematics Frontier Volume 2 Issue 3, September 2013 www.sjmmf.org 103 A Process-theoretic Approach to Supervisory Control of Interactive Markov Chains J. Markovski Department of Mechanical Engineering, Eindhoven University of Technology Den Dolech 2, 5612MH, Eindhoven, Netherlands j.markovski@tue.nl Abstract We propose a process-theoretic approach to supervisory control of stochastic discrete-event systems with unrestricted nondeterminism. The model of choice is termed Interactive Markov Chains, a natural semantic model for stochastic variants of process calculi and Petri nets. We employ a stochastic extension of the behavioral preorder partial bisimulation to capture the notion of controllability and preserve correct stochastic behavior. The stochastic behavior is preserved up to lumping of Markovian delays. To synthesize a supervisor, we abstract from the stochastic behavior and show that the obtained supervisor is suitable for the original system as well. Keywords Supervisory Control Theory; Interactive Markov Chains; Partial Bisimulation Preorder; Controllability; Supervisor Synthesis Introduction Development costs for control software of high-tech systems are constantly increasing due to ever-rising complexity of the machines and demands for better quality, safety, and performance. Traditionally, the control requirements are formulated in informal documents and translated into control software, followed by code validation and testing. However, this iterative process becomes time-consuming due to frequent changes and ambiguity of the specification documents. This issue gave rise to supervisory control theory developed by Ramadge and Wonham (1987), where supervisory controllers that coordinate discrete- event system behaviour are synthesized automatically based on formal models of the hardware and the control requirements. The supervisory controller observes machine behavior by receiving signals from ongoing activities and sends feedback in terms of control signals about allowed activities. We work under the standard assumption that the supervisory controller reacts sufficiently fast on machine input. In this case this feedback loop can be modeled as a pair of synchronizing processes, cf. Cassandras and Lafortune (2004). We refer to the model of the machine as plant, which is restricted by synchronization with the model of the controller, known as a supervisor. Model-based Systems Engineering We structure the modelling process in a model-based systems engineering framework depicted in Fig. 1, which extends previous proposals of Schiffelers et al. (2009), Markovski et al. (2010), and Markovski (2011b). Following the model-based methodology, domain engineers initially specify the functionality of the desired controlled system. This leads to a design, developed by the domain and software engineers together. This design defines the modeling level of abstraction and control architecture and it results in informal specifications of the plant, the control, and the performance requirements. Next, the plant and control requirements are modeled in parallel. We synthesize a supervisor based on the abstracted version of the plant, which is coupled with the original variant of the plant to obtain the complete stochastic supervised behavior of the system. Note that the control requirements specify only desired safety functional properties of the system. The succeeding steps validate that the control is meaningful, i.e., desired functionalities of the controlled plant are preserved. This step involves stochastic verification of the supervised plant based on the model of the performance requirements, e.g. in the vein of Baier et al (2010), or validation by simulation, as proposed in Schiffelers et al. (2009). If validation fails, then the control requirements are remodeled, and sometimes a complete revision proves necessary. Finally, the control software is generated auto- matically based on the validated models, shifting the focus of software engineers from coding to modeling.