Feedback is an essential composition operator in many classes of reactive and other systems. This paper studies feedback in the context of compositional theories with refinement. Such theories allow to reason about systems on a component-by-component basis, and to characterize substitutability as a refinement relation. Although compositional theories of feedback do exist, they are limited either to deterministic systems (functions) or input-receptive systems (total relations). In this work we propose a compositional theory of feedback which applies to non-deterministic and non-input-receptive systems (e.g., partial relations). To achieve this, we use the semantic frameworks of predicate and property transformers, and relations with fail and unknown values. We show how to define instantaneous feedback for stateless systems and feedback with unit delay for stateful systems. Both operations preserve the refinement relation, and both can be applied to non-deterministic and non-input-receptive systems.
In @cite_22 a model for synchronous nondeterministic systems is used for studying preservation of security properties by composition of systems, including delayed feedback. However this model does not treat non-input receptive systems.
@cite_22
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We give a denotational framework (a "meta model") within which certain properties of models of computation can be compared. It describes concurrent processes in general terms as sets of possible behaviors. A process is determinate if, given the constraints imposed by the inputs, there are exactly one or exactly zero behaviors. Compositions of processes are processes with behaviors in the intersection of the behaviors of the component processes. The interaction between processes is through signals, which are collections of events. Each event is a value-tag pair, where the tags can come from a partially ordered or totally ordered set. Timed models are where the set of tags is totally ordered. Synchronous events share the same tag, and synchronous signals contain events with the same set of tags. Synchronous processes have only synchronous signals as behaviors. Strict causality (in timed tag systems) and continuity (in untimed tag systems) ensure determinacy under certain technical conditions. The framework is used to compare certain essential features of various models of computation, including Kahn process networks, dataflow, sequential processes, concurrent sequential processes with rendezvous, Petri nets, and discrete-event systems. This work puts forward a novel technique to generate finite abstractions of autonomous and nonautonomous Max-Plus-Linear (MPL) models, a class of discrete-event systems used to characterize the dynamics of the timing related to successive events that synchronize autonomously. Nonautonomous versions of MPL models embed within their dynamics nondeterminism, namely a signal choice that is usually regarded as an exogenous control or schedule. In this paper, abstractions of MPL models are characterized as finite-state Labeled Transition Systems (LTS). LTS are obtained first by partitioning the state space (and, for the nonautonomous model, by covering the input space) of the MPL model and by associating states of the LTS to the introduced partitions, then by defining relations among the states of the LTS based on dynamical transitions between the corresponding partitions of the MPL state space, and finally by labeling the LTS edges according to the one-step timing properties of the events of the original MPL model. In order to establish formal equivalences, the finite abstractions are proven to either simulate or to bisimulate the original MPL model. This approach enables the study of general properties of the original MPL model by verifying (via model checking) equivalent logical specifications over the finite LTS abstraction. The computational aspects related to the abstraction procedure are thoroughly discussed and its performance is tested on a numerical benchmark. In this paper, we address an output feedback stabilization problem for a class of linear impulsive systems that accommodate arbitrarily-spaced impulse times and possibly singular state transition matrices. By combining recent results for state feedback stabilization and state estimation, we show that a separation property holds and formulate an output feedback compensation scheme in which the feedback loop is closed between a discrete-time measurement and a continuoustime control input. Rather than directly adopting an observerbased structure involving the time-varying gains associated with the separate stabilization and estimation problems, we construct a purely discrete-time compensator followed by a memoryless generalized hold device that achieves closed-loop exponential stability. The paper investigates the synchronization of a network of identical linear state-space models under a possibly time-varying and directed interconnection structure. The main result is the construction of a dynamic output feedback coupling that achieves synchronization if the decoupled systems have no exponentially unstable mode and if the communication graph is uniformly connected. The result can be interpreted as a generalization of classical consensus algorithms. Stronger conditions are shown to be sufficient-but to some extent, also necessary-to ensure synchronization with the diffusive static output coupling often considered in the literature.