Overview This project applies the tagged-signal model to explain the semantics of piecewise continuous signals. Then it illustrates an operational way to realize that semantics via a few case studies. In the end, a simulation strategy for hybrid systems is proposed. November 18, 2004 Piecewise Continuous Signals in Hybrid Systems Haiyang Zheng Edward A. Lee Signals in hybrid systems are piecewise continuous. With the tagged-signal model, a piecewise continuous signal is a partial function f : T → R, where T is the tag set and T = R × N. R is the set of real values representing time and N is the set of natural numbers representing indexes. A discontinuity is the effect of a set of simultaneous discrete events. These events are totally ordered in that they have the same time values but different indexes. Therefore piecewise continuous signals are functional. Case 3: Sampling a piecewise continuous signal. Sampling Discontinuities When a sampler samples a discontinuity, the sampled result must be deterministic. We choose to sample the value of the signal just before the discontinuity happens. Therefore, the output signal from a sampler is discrete and the indexes of its tags are always 1. Operations on Piecewise Continuous Signals A desired piecewise continuous signal. An incorrect signal can't capture discontinuities. The signal generated from the left model. Case 1: Generating a piecewise continuous signal. If an event happens in the continuous phase of execution, the detector needs to refine step size and locate the exact time point where this event happens. During a discrete phase of execution, the detector does not try to refine the step size because it is 0.0 already. Instead, the detector just compares the values of two consecutive discrete events to detect level crossings. Generating Discontinuities Case 4: Handling transient states in hybrid systems. Simulink models introduce a nonzero delay in transient states. Transient states in Stateflow. Handling Simultaneous Events ― Transient States A transient state can be easily found in a network of interacting hybrid automata. Semantically, zero time is spent in transient states. An execution must be faithful to this semantics. Piecewise Continuous Signals Current and Future Work Simulating Hybrid Systems We propose an operational semantics which only evaluates signals at a discrete subset D of the tag set, where D is carefully chosen such that it contains tags of all discontinuities. Signals between any two consecutive tags in D can be evaluated with a numerical ODE solver. An execution of a hybrid system is an interleaving of discrete phases and continuous phases of execution at the tags in D, where discrete phases of execution only happen at discontinuities. A discrete phase of execution evaluates all the events with the same tag from unknown to be either having a value or absent. This process is formally a fixed-point iteration. A sequence of discrete phases of execution that happen at the same time is also a fixed-point iteration. The order of the phases of execution is defined by the tags of events they evaluate. The fixed point is a discrete phase of execution that evaluates the events of all signals to be absent. A continuous phase of execution evaluates the signals at a tag t from the signals at the immediately preceding tag t’ with an ODE solver, given that the system satisfies the Lipschitz condition in ( t’, t ). There are two ongoing efforts. 1. We are applying Banach fixed-point theorem using a new metric space that provides a theoretical foundation for the above operational semantics. 2. We are unifying the operational semantics of the Synchronous Reactive model of computation with that of discrete- event systems and hybrid systems to achieve heterogeneous models. Case 2: Detecting level crossings at discontinuities. Intuitively, a piecewise continuous signal is a composition of discrete signals and continuous signals shown in the above figure. One way to generate discontinuities is to use Modal Models, where transitions represent discrete events. See case 1. Note that a transition must take zero ( model ) time. Detecting Events at Discontinuities An event detector must be able to detect level crossings that happen at both continuous and discrete phases of execution. See case 2.