Traffic Characterization of Aperiodic Control Systems

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In networked systems, particularly over wireless or shared channels, the scarcity of communication resources makes the application of traditional control strategies with periodic sampling problematic. Alternative approaches with aperiodic sampling, such as: event triggered control and self triggered control, have been recently proposed to reduce network usage. Nonetheless, their features have not been exploited to propose less conservative hardware designs. Because hybrid nature of embedded systems along with aperiodic control executions lead to complicated infinite-state systems. In other words, deriving formal approaches for behavioral analysis of these systems is cumbersome. In this study, modeling the traffic generated by aperiodic control systems is of interest. By doing so, one can use the resulting models to guarantee correct functioning of networked control systems, for example in appropriate dimensioning of the underlying communication system or conducting schedulability analysis. We propose a technique to remove temporal and spatial dependencies to derive a timed automaton (or a quotient system) inspired by approaches in the literature. Based on the proposed construction technique, the timed automaton captures the behavior of the controlled system. The derivation of transitions in the timed automaton are based on reachability analysis of each region in the state space using the following facts: the state space is abstracted to a finite number of convex regions and the convexity of a set is preserved under a linear map. Two cases are considered and for each, LMI-based conditions are derived. First, a self triggered strategy, that is state dependent and based on Lyapunov Razumikhin stability conditions, is considered. The self triggered technique is modified to a more suitable formulation inspired by an approach in stability analysis of switched systems. We show that it suffices to only derive the sampling function for half of the state space and then generalize it to the whole state space with an appropriate mapping. Then, reachability analysis is used to derive the automaton. Second, it is shown that for a class of event triggered control systems, the triggering mechanism can be formulated as a quadratic function of states at triggering instants. Then, one can abstract away temporal and spatial dependencies in the triggering mechanism to derive a quotient system. For each region of the state abstraction, the lower and upper bounds of inter-execution intervals are derived. Then, a reachability analysis of dynamical systems is applied to outer approximate the reachable sets over the derived time intervals. Using the results from reachability analysis, the timed automaton is constructed.