On condition/event systems with discrete state realizations

This paper introduces condition/event (C/E) systems as a class of continuous-time discrete event dynamic systems (DEDS) with two types of discrete-valued input and output signals: condition signals and event signals. In applications such as discrete control, C/E systems provide an intuitive continuous-time modeling framework amenable to block diagram representation. In this paper we consider C/E systems with discrete state realizations, and study the relationship between continuous-time C/E systems and untimed models of their sequential inputoutput behavior called C/E languages. We show that C/E systems with discrete state realizations are necessarily time-change invariant (Theorem 3.t), which means the ensemble of admissible continuous-time input-output behaviors is completely characterized by the C/E language for the system (Theorem 4.1). It is also shown that deterministic C/E systems with discrete state realizations are necessarily discrete-time (clocked) systems (Corollary 3.1), and that finite discrete state realizations exist for a C/E system only if its related C/E language has a finite state generator (Theorem 4.2). Finally, we develop equivalent discrete-state realizations for C/E systems resulting from cascade and feedback interconnections. The paper concludes with a discussion of several directions for future research. 1. I n t r o d u c t i o n It is often observed that there are two kinds of mathemat ica l models for discrete event dynamic systems (DEDS) : logical o r un t imed models that character ize only the o rder in which events occur, and t imed models which include the t imes at which events occur. Unt imed models are useful for analyzing qualitative properties such as liveness and sequencing conditions. T imed models are used for quantitative pe r fo rmance analysis and verif icat ion o f real t ime propert ies . In this paper we introduce a class o f cont inuoust ime D E D S cal led condition/event systems (C/E systems) ~ and consider propert ies o f C / E systems for which the internal system dynamics can be mode led in terms of a discrete state space. We show that the ensemble of admissible input-output behaviors for a C /E system with a continuoust ime discrete state real izat ion can be comple te ly character ized by an unt imed mode l of the system's sequential behavior called the C/E language. We also develop discrete state realizations for cascade and feedback interconnect ions o f C / E systems. *Please direct correspondence concerning this paper to B.H. Krogh at the above address. 210 R.S. SREENIVAS AND B.H. KROGH There are two classes of input and output signals for a C/E system: condition signals, which are piecewise constant signals taking on values in a finite set of conditions; and event signals, which are null except at discrete points in time at which they assume a value in a finite set of events. In block diagrams for C/E systems, signal flow lines carrying event signals are distinguished from signal flow lines carrying condition signals by the symbol I B ' ~ ' the standard symbol for interrupt signals in real-time control-flow diagrams ennett 1988] (see Figure 1). When the internal dynamics of C/E systems are modeled by a discrete state realization (defined in Section 3), state transitions can be inhibited or enabled by the input condition signal and "forced" by the input event signal. The value of the output condition signal is a function of the system state and input condition signal, while the ouptut events are a function of state transitions and input events. The state trajectory is a condition signal. The primary motivation for introducing C/E systems is to model interconnected DEDS with continuous-time input and output signals. In applications such as discrete control, condition and event signals characterize the flow of information between subsystems. Discrete sensor values, for example, are piecewise constant condition signals, whereas interrupts, clocks, and event triggers are naturally modeled as event signals. Condition and event signals are also useful for representing the causal interactions between physical components of a discrete control system. This is illustrated by the portion of a conveyor system shown in Figure 2, which transports van bodies through an automated painting facility? In this system, van bodies arrive on skids and are held in place on the conveyor by a stop until they are ready to be painted (Figure 2a). The stop is then lowered and the van advances to grounding bars (Figure 2b). To push the skid onto the grounding bars, the stop is raised and the pusher is extended (Figure 2c). Another pusher is used to move the van off the grounding bars (not shown). To model the conveyor system in Figure 2, it is most convenient to model the pusher and stop separately from the motion of the van body skid on the conveyor, and then combine the models to represent the complete system. Figure 3 illustrates a C/E block diagram of these coupled physical systems. The stop and pusher motions can be controlled independently of the van body skid position, as indicated by the control input signal which triggers transitions in the state of the stop/pusher system. The motion of the van body skid is affected by the stop and pusher in two distinct ways, as indicated by the condition and event signals