Stream- and State-Based Semantics of Hierarchy in Block Diagrams

Abstract

Block diagrams are often used in embedded system design for modeling both plant and controller, typically with continuous and discrete modeling, respectively. Though easy to use, advanced users and implementers of these languages often run afoul of subtle semantic problems these seemingly simple languages can have. Based on the stream-and state-based approaches, this paper discusses how the specialized state-based semantics of continuous-time block diagrams can interoperate hierarchichally with discrete-time block diagrams. The languages presented may serve as a reference of sorts, helping to clarify some of the underlying choices in block diagram language design, and in the process shedding light on the differences between, and limitations of, existing block diagram languages.

Determinate composition of FMUs for co-simulation

“…Very relevant to this problem is the work reported by Denckla and Mosterman [7] and Mosterman et al [19], who present stream-and state-based functional interfaces for a Simulink-type of language. Also, Sander and Jantsch [25] present the ForSyDe modeling framework, which provides a set of libraries for capturing heterogeneous MoCs based on the functional programming language Haskell.…”

A type-based analysis of causality loops in hybrid systems modelers

“…As integration is performed globally, 8 The same issue exists in release R2014a. 9 In contrast, the application of a unit delay 1 z to a continuous-time signal is statically detected and results in a warning. mem(y) may cause strange behaviors as the previous value of a continuously changing signal x depends precisely on when the solver decides to stop!…”

“…Indeed, models are used not only for simulation, but also for test-case generation, formal verification and translation to embedded code. This explains the need for formal operational semantics for specifying their implementations and proving them correct [15,9]. The underlying mathematical model is the synchronous parallel composition of stream equations, Ordinary Differential Equations (ODEs), hierarchical automata, and imperative features.…”

“…9 If x is a signal, mem(x) is a piecewise constant signal which refers to the value of x at the previous integration step (or major step). If those steps are taken at increasing instants ti ∈ R, mem(x)(t0) = x0 where t0 = 0 and x0 is an explicitly defined initial value, mem(x)(ti) = x(ti−1) for i > 0 and mem(x)(ti + δ) = x(ti−1) for 0 ≤ δ < ti+1 − ti.…”

A modular formal semantics for Ptolemy

“…Perhaps the work most closely related to our paper is (Denckla and Mosterman, 2008). There, the authors present two types of semantics for a block-diagram language with hierarchy: a stream-based semantics where blocks are viewed as functions from streams to streams; and a state-based semantics where each block is represented by an initial state and a kind of 'step' function which, given the current input and state, returns the current output and an 'implicit output'.…”