The problem with time in mixed continuous/discrete time modelling

Abstract

The design of cyber-physical systems requires the use of mixed continuous time and discrete time models. Current modelling tools have problems with time transformations (such as a time delay) or multi-rate systems.We will present a novel approach that implements signals as functions of time, directly corresponding to their mathematical representation. This enables an exact implementation of time transformations and as an additional advantage enables local control over time. A representation of components and signals in both domains is provided, together with composition operators to allow the specification of signal flow diagrams.

UniTi: Unified Composition and Time for Multi-domain Model-based Design

“…All the researched tools and languages [2,8,30,33] use ordinary differential equation (ODE) solvers for simulating the CT domain; an equation system is set up and a global time step is applied for numerical approximation, thereby implementing CT signals as a sequence of values. Time transformations such as a time delay therefore buffer values and interpolate between available values introducing inaccuracies caused by the modelling tool [22,24,25].…”

“…More importantly, we have found no tool that distinguishes different notions of time. Different notions of time are necessary to provide an exact implementation of time transformations, such as a time delay, while retaining efficiency [22,24,25]. Current tools extract a set of equations from the model and use a solver on these equations at a global time step for simulation.…”

Multi-domain transformational design flow for embedded systems

“…All these tools implement the CT domain by solving a set of differential equations using a global time step, thereby implementing CT signals as a sequence of values. Time transformations such as a time delay therefore buffer values and interpolate between available values introducing inaccuracies caused by the modelling tool [7]. Our approach applies exact time transformations without the need for a solver.…”

“…Technically, this is achieved by the framework by embedding a DF component in a DT component and a DT component in a CT component such that for simulation purposes the CT domain is the unifying domain. Because we have local control over time in a CT component, this gives the advantage that simulation can be done without loosing efficiency [7].…”

“…As an example to illustrate the necessity to integrate the various domains we mention an ADSL modem, which uses adaptive transmission based on cable conditions. Current design tools implement the interaction between the above mentioned domains by discretising a global simulation time representing signals as a sequence of values, which prevents exact transformations with respect to time such as variable time delays (see [7]). Thus, based on these tools it is not possible to exactly verify the correct operation of the coding, modulation and error-correction without first developing a hardware prototype, because we can not differentiate between the error caused by the modeling tool and an error caused by an incorrect specification or implementation.…”