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CHOPtrey: Contextual online polynomial extrapolation for enhanced multi-core co-simulation of complex systems
Abstract and Figures
The growing complexity of Cyber-Physical Systems (CPS), together with increasingly available par-allelism provided by multi-core chips, fosters the parallelization of simulation. Simulation speed-ups are expected from co-simulation and parallelization based on model splitting into weak-coupled sub-models, as for instance in the framework of Functional Mockup Interface (FMI). However, slackened synchronization between sub-models and their associated solvers running in parallel introduces integration errors, which must be kept inside acceptable bounds. CHOPtrey denotes a forecasting framework enhancing the performance of complex system co-simulation, with a trivalent articulation. First, we consider the framework of a Computationally Hasty Online Prediction system (CHOPred). It allows to improve the trade-off between integration speed-ups, needing large communication steps, and simulation precision, needing frequent updates for model inputs. Second, smoothed adaptive forward prediction improves co-simulation accuracy. It is obtained by past-weighted extrapolation based on Causal Hopping Oblivious Polynomials (CHOPoly). And third, signal behavior is segmented to handle the discontinuities of the exchanged signals: the segmentation is performed in a Contextual & Hierarchical Ontology of Patterns (CHOPatt). Implementation strategies and simulation results demonstrate the framework ability to adaptively relax data communication constraints beyond synchronization points which sensibly accelerate simulation. The CHOPtrey framework extends the range of applications of standard Lagrange-type methods, often deemed unstable. The embedding of predictions in lag-dependent smoothing and discontinuity handling demonstrates its practical efficiency.
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... 3.2). Temporary switching to zero-order hold in a signal reconstruction context when the signal becomes hard to predict is seen in [12]. This is called a macro-discontinuity, and can be detected on a single output independently of the other outputs. ...
... in CLS mode (see (12)), and where q ∶= p k, As in (22), for the rest of Sect. 3.1.3, ...
... strategy is to take into account all the (q + 1) points that have been used in the determination of the chosen order, and the most recent point as well. We thus have (q + 2) points to adjust a polynomial of degree at most q: an extrapolation process cannot be made, but the "best fitting" polynomial can be found for the CLS criterion (12). Please note that removing the constraint Ω CLS q−2 (t [1] ) =z [1] in (12) corresponds to the relaxation technique on the past referred to as "method 1" in [23] in the particular case of q = 0. ...
This paper introduces the FORNITS non-iterative co-simulation algorithm in which F stands for the 3 flexible aspects of the method: flexible polynomial order representation of coupling variables, flexible time-stepper applying variable co-simulation step size rules on subsystems allowing it, and flexible scheduler orchestrating the meeting times among the subsystems and capable of asynchronousness when subsystems’ constraints require it. The motivation of the FORNITS method is to accept any kind of co-simulation model as far as they represent circuits (0D models, such as ODE or DAE), including any kind of subsystem (open circuits), regardless on their available capabilities. Indeed, one of the major problems in industry is that the subsystems usually have constraints or lack of advanced capabilities making it impossible to implement most of the advanced co-simulation algorithms on them. The method makes it possible to preserve the dynamics of the coupling constraints when necessary as well as to avoid breaking smoothness at communication times, and also to adapt the co-simulation step size in a way that is robust both to zero-crossing variables (contrary to classical relative error-based criteria) and to jumps. Five test cases are presented to illustrate the robustness of the FORNITS method as well as its higher accuracy than the non-iterative Jacobi coupling algorithm (the most commonly used method in industry) for a smaller number of co-simulation steps.
... One way to achieve this is to switch to the naïve ZOH algorithm whenever a discontinuity has occurred within the last − 1 time step, while is the number of past signal values the current coupling algorithm uses. The idea of switching the coupling algorithm based on specific signal properties is also pursued in offline co-simulations to reduce reconstruction errors [53]. Figure 6 shows an example of how discontinuity detection with coupling algorithm switching works. ...
... One way to achieve this is to switch to the naïve ZOH algorithm whenever a discontinuity has occurred within the last m − 1 time step, while m is the number of past signal values the current coupling algorithm uses. The idea of switching the coupling algorithm based on specific signal properties is also pursued in offline co-simulations to reduce reconstruction errors [53]. Figure 6 shows an example of how discontinuity detection with coupling algorithm switching works. ...
Currently, innovations in mechatronic products often occur at the system level, requiring consideration of component interactions throughout the entire development process. In the earlier phases of development, this is accomplished by coupling virtual prototypes such as simulation models. As the development progresses and real prototypes of certain system components become available, real-virtual prototypes (RVPs) are established with the help of network communication. However, network effects—all of which can be interpreted as latencies in simplified terms—distort the system behavior of RVPs. To reduce these distortions, we propose a coupling method for RVPs that compensates for latencies. We present an easily applicable approach by introducing a generic coupling algorithm based on error space extrapolation. Furthermore, we enable online learning by transforming coupling algorithms into feedforward neural networks. Additionally, we conduct a frequency domain analysis to assess the impact of coupling faults and algorithms on the system behavior of RVPs and derive a method for optimally designing coupling algorithms. To demonstrate the effectiveness of the coupling method, we apply it to a hybrid vehicle that is productively used as an RVP in the industry. We show that the optimally designed and trained coupling algorithm significantly improves the credibility of the RVP.
... Accuracy can be further improved by analyzing the current trend of each variable and adopting the extrapolation method accordingly. Both synchronous approaches [14,15] and asynchronous algorithms using explicit error estimation [16] exists. Éguillon et al. [10] later presented F 3 ORNITS, a co-simulation algorithm providing flexible extrapolation, timestepping, and scheduling. ...
Combining performance and numerical stability is a key issue in co-simulation. The Transmission Line Modeling method uses physically motivated communication delays to ensure numerical stability for stiff connections. However, using a fixed communication delay may limit performance for some models. This paper proposes Steady-State Identification for enabling variable communication delays. Three algorithms for online Steady-State Identification are evaluated in three different co-simulation models. All algorithms are able to identify steady state and can thereby determine when communication delays can be allowed to increase without compromising accuracy and stability. The results show a reduction in number of the solver derivative evaluations by roughly 40–60% depending on the model. The proposed method additionally supports connections with asymmetric communication delays, which allows each sub-model to independently control the delay of its input variables. Models supporting delay-size control can thereby be connected to those that do not, so that the step length of each individual sub-model is maximized. Controlling the delay-size in sub-models also makes the method independent of the master co-simulation algorithm.
... However, the selection of an appropriate extrapolation technique is often problem-dependent, and finding the strategy that delivers the best possible results can become a challenging task [11,12]. The extrapolation method can also be adjusted as the simulation progresses, based on previously experienced system behaviour [13]. When additional information about the subsystems is available, other solutions to approximate the values of the coupling variables can be proposed, for instance if the subsystems provide the partial derivatives of their output with respect to their state and input [14], or if the dynamics of some subsystems are known in detail, so that model-based coupling techniques can be used [15][16][17]. ...
Co-simulation is an effective and versatile way to determine the forward-dynamics behaviour of complex engineering applications. In co-simulation setups, the overall system dynamics is split into several subsystems that evolve in time separately. This makes it possible to use modelling and integration methods that can be tailored to the specific nature and behaviour of each of them. Co-simulation subsystems coordinate their execution by means of information exchanges through a discrete-time interface. In some cases, this limited exchange of data can cause accuracy and stability issues in the simulation process, especially when explicit coupling schemes are employed. Correction algorithms are then required to ensure the accuracy of the obtained results. This paper provides insight into the structure of explicit co-simulation problems, revealing the effect of input extrapolation at the discrete-time interface between subsystems. The resulting system equations are formulated in terms of control theory expressions, which can be then used to develop compensation solutions to correct the perturbations introduced at the co-simulation interface. The compensator architecture is chosen to ensure the eigenstructure assignability condition, which has been ad-hoc developed in this paper. These aspects are illustrated here in the explicit co-simulation of linear mechanical systems.
... To approximate the coupling variables between two macro-time points, Lagrange polynomials are used here. Other approximation approaches -acceleration based techniques [34,35], interface-model-based approaches [47], context-based approximation polynomials [7], methods applying a least square fit [42], etc. -are not considered in the current manuscript. ...
Considering co-simulation and solver coupling approaches, the coupling variables have to be approximated within a macro-time step (communication-time step), e.g., by using extrapolation/interpolation polynomials. Usually, the approximation order is assumed to be fixed. The efficiency and accuracy of a co-simulation may, however, be increased by using a variable approximation order. Therefore, a technique to control the integration order is required. Here, an order control algorithm for co-simulation and solver coupling methods is presented. The order controller is incorporated into the control algorithm for the macro-step size so that co-simulations with variable integration order and variable macro-step size can be carried out. Different numerical examples are presented, which illustrate the applicability and benefit of the proposed order control strategy. This contribution mainly focuses on mechanical systems. The presented techniques may, however, also be applied to nonmechanical dynamical systems.
... This allows internal solvers to take longer integration steps. The technique is used in CHOPTrey, a forecasting framework for hybrid dynamic co-simulation (11). However, this technique relies on slacked synchronization, which will always introduce some numerical errors and cannot provide guaranteed numerical stability for stiff connections. ...
Modeling and simulation are important tools for efficient product development. There is a growing need for collaboration, interdisciplinary simulation, and re-usability of simulation models. This usually requires simulation tools to be coupled together for co-simulation. However, the usefulness of co-simulation is often limited by poor performance and numerical instability. Achieving stability is especially hard for stiff mechanical couplings. A suitable method is to use transmission line modeling (TLM), which separates submodels using physically motivated time delays. The most established standard for tool coupling today is the Functional Mock-up Interface (FMI). Two example models in one dimension and three dimensions are used to demonstrate how the next version of FMI for co-simulation can be used in conjunction with TLM. The stability properties of TLM are also proven by numerical analysis. Results show that numerical stability can be ensured without compromising on performance. With the current FMI standard, this requires tailor-made models and custom solutions for the interpolation of input variables. Without using custom solutions, variables must be exchanged using sampled communication and extrapolation. In this case, stability properties can be improved by reducing communication step size. However, it is shown that stability cannot be achieved even when using unacceptably small communication steps. This motivates the need for the next version of FMI to include an intermediate update mode, where variables can be interchanged in between communication points. It is suggested that the FMI standard should be extended with optional callback functions for providing intermediate output variables and requesting intermediate input variables.
... Using Hermite polynomials [7] or acceleration-based extrapolation [8] can even extend the stability region communication interval sizes. Adaptive extrapolation schemes even determine the polynomial degree at run-time [9]. Based on the coupling type, the iterative [10] and non-iterative [11] co-simulation solutions can handle larger time-steps avoiding instabilities. ...
Modeling and simulation of mechatronic systems require a multi-disciplinary approach as several physical domains interact with each other. Due to this heterogeneity, each domain can benefit from a dedicated time integration strategy as well as modeling formalism to control the error in simulation and reduce computational overhead. Current multi-rate adaptive time-stepping approaches disregard the system dynamics knowledge obtained in previous simulations and adapt the time steps at run-time as a black-box model. Although current multi-rate adaptive time-stepping approaches do not exploit the time scale of the physics of each domain, this information can be exploited in co-simulations by having a predefined multi-rate time integration approach in design space exploration and parameter identification applications, which involve repeated simulations with similar dynamics. In this study, an offline adaptive co-simulation technique is introduced. The proposed method adapts the communication and time integration time steps before the simulation using the estimated error data from past co-simulation runs. The offline adaptive co-simulations performed faster against the run-time adaptive co-simulations for comparable accuracy levels in 1D and multi-level mechatronic systems with changing loading and initial conditions.
In a co-simulation setup, the entire system is decomposed into a collection of individual subsystems that are interfaced together, with each subsystem being modelled and integrated separately according to its own requirements. To maintain the inter-connectivity and consolidation of the primary system, these subsystems must communicate with each other through the interface and transfer certain information at the end points of a defined time interval termed macro time step. Inside the macro time step, the evolution of the interface variables has to be approximated as information about them will only be available again at the end of the step. In real-time simulations, the size of the macro time step and the accuracy of the approximated interface variables are critical factors; if the interface variables are approximated accurately, the size of the macro time step can be kept large enough to provide interactive rates without loss of accuracy and stability. This work focuses on systems where unilateral contact interactions are important and proposes reduced interface model concepts for such non-smooth systems. The use of the proposed reduced interface model (RIM) is demonstrated in co-simulation to provide model-based approximation of the interface variables. The advantages of the proposed method are demonstrated through two representative case studies.
Sugarcane is one of the main crops in the world due to its economic value promoted by the sale of its derivatives, such as bioethanol and sugar. In order to achieve greater economic performance and productivity in the sugarcane field, several digital image processing studies have been conducted on sugarcane field images. However, mapping and measuring gaps in the planting rows are still being performed manually on-site to determine whether to replant the entire area or only the gaps. High cost of time and manpower is required to perform the manual measurement. Based on that, the aim of this study is to present a novel method to detect crop rows and measure gaps in crop fields. Our method is also able to deal with curved crop rows, which is a real problem and substantially limits numerous solutions in practical applications. The proposed method is evaluated using a mosaic of real scene image that was prepared with the support of a small remotely piloted aircraft. Experimental tests showed a low relative error of approximately
1.65
% compared to manual mapping in the planting regions, even for regions with gaps in the curved crop rows. It means that our proposal can identify and measure crop rows accurately, which enables automated inspections with high-precision measurements.
Simulation of complex systems requires the division to subsystems modelling each subsystem separately, and then interfacing and coupling them. The individual subsystems are often modelled in separate environments using different formulations and integrations. One way to couple subsystems is through co-simulation. For interactive, real-time simulation, non-iterative coupling is the main possibility. Such non-iterative co-simulation is very prone to instability problems that relate to the communication and information exchange between subsystems. This work focuses on non-smooth mechanical subsystem subjected to unilateral contacts and introduces an efficient way to interface it with the rest of the system using a reduced interface model. The proposed interface model accounts for contact attachments/detachments during the macro time step which can lead to better functionality and more accurate results. The proposed method and model is illustrated using an example of interfacing mechanical and hydraulic subsystems. The obtained results confirm that the proposed interface model enhances simulation accuracy, functionality and stability.
Here, we study the flow of energy between coupled simulators in a co-simulation environment using the concept of power bonds. We introduce energy residuals which are a direct expression of the coupling errors and, hence, the accuracy of co-simulation results. We propose a novel energy-conservation-based co-simulation method (ECCO) for adaptive macro step size control to improve accuracy and efficiency. In contrast to most other co-simulation algorithms, this method is non-iterative and only requires knowledge of the current coupling data. Consequently, it allows for significant speed-ups and the protection of sensitive information contained within simulator models. A quarter car model with linear and nonlinear damping serves as a co-simulation benchmark and verifies the capabilities of the energy residual concept: reductions in the errors of up to 93% are achieved at no additional computational cost.
Power system electromagnetic transient (EMT) simulation has been used to study the electromagnetic behavior of power system components. It generally comprises detailed models of the study area and an equivalent circuit which represents an external part of the study area. However, a detailed description of an external system that includes transmission or distribution system models is required to study the interaction among power system components because the number of high power converter based devices in a power grid have been increasing. Since detailed models of the system components are necessary to simulate a series of events such as cascading faults the computational burden of power system simulation has increased. Therefore a more effective and practical framework has been sought to handle this computational challenge. This paper proposes a co-simulation framework including a delay compensation algorithm to compensate the time delayed signals due to network segmentation and a fast and flexible simulation environment composed of non-real time power system EMT simulation on a general purpose computer with a multi core central processing unit (CPU), which is currently very popular owing to its performance. The proposed methods are applied to an AC/DC power system model.
Accurately detecting state events is critical for hybrid dynamic systems. However, it is difficult to locate the exact point of a state event in systems with singularities or systems in which event functions are of multiple roots. Overcoming these difficult situations depends on accurately predicting event functions and a reliable root-finding method. A carefully constructed extrapolation polynomial, which is derived from a variable-step multi-step integration method, is applied to predict values of event functions. Due to not using state values of the next time point, the polynomial can be employed to determine the integration step size by checking potential events in all situations. The extended interval Newton's method is employed to find all roots in a certain time interval, which is determined by the existing integration method. The procedure of the root non-existence test and root finding are combined into one. Simulation results show that the algorithm is effective under several different critical situations. By the algorithm, model singularities are found on time and simulation failures are avoided. The event detecting algorithm can be combined with the existing integration method.
From the Publisher:
For this inexpensive paperback edition of a groundbreaking classic, the author has extensively rearranged, rewritten and enlarged the material. Book is unique in its emphasis on the frequency approach and its use in the solution of problems. Contents include: Fundamentals and Algorithms; Polynomial Approximation Classical Theory; Fourier ApproximationModern Therory; Exponential Approximation.
Complex multi-disciplinary models in system dynamics are typically composed of subsystems. This modular structure of the model reflects the modular structure of complex engineering systems. In industrial applications, the individual subsystems are often modeled separately in different mono-disciplinary simulation tools. The Functional Mock-Up Interface (FMI) provides an interface standard for coupling physical models from different domains and addresses problems like export and import of model components in industrial simulation tools (FMI for Model Exchange) and the standardization of co-simulation interfaces in nonlinear system dynamics (FMI for Co-Simulation), see [8]. In November 2011, the third b -version of FMI for Model Exchange and Co-Simulation v2.0 was released [13] that supports advanced numerical techniques in co-simulation like higher order extrapolation and interpolation of subsystem inputs, step size control including step rejection and Jacobian based linearly implicit stabilization techniques. Well known industrial simulation tools for applied dynamics support Version 1.0 of this standard and plan to support the forthcoming Version 2.0 in the near future, see the “Tools” tab of website [8] for up-to-date information. The renewed interest in algorithmic and numerical aspects of co-simulation inspired some new investigations on error estimation and stabilization techniques in FMI for Model Exchange and Co-Simulation v2.0 compatible co-simulation environments. The present paper extends recent results from [3] on reliable error estimation and communication step size control in the framework of FMI for Model Exchange and Co-Simulation v2.0. Based on a strict mathematical analysis, we study the asymptotic behaviour of the local error and two error estimates that may be used to adapt the communication step size automatically to the changing solution behaviour during time integration. These theoretical results are illustrated by numerical tests for a (linear) quarter car model and provide a basis for future investigations with more complex cou
Coupling multiphysical systems by means of a co-simulation, the data between the subsystems are interchanged at a discrete macro time grid, also denoted as communication time grid. Between the communication points the coupling variables are approximated so that the numerical solvers in the subsystems can calculate the differential equations. Classical approximation techniques based on extrapolation entail a discontinuity in the equations in each macro time step which can slow down the numerical time integration. In the paper at hand a C0-continuous technique for approximating the coupling variables is reconsidered and the numerical stability and the local error of the method are compared to the classical Lagrange approximation approach. Further, the method is enhanced to a C1-continuous (continuous and differentiable) approximation technique. Both methods are investigated in combination with different numerical coupling approaches (sequential Gauss-Seidel scheme, parallel Jacobi scheme, force/displacement coupling, displacement/displacement coupling) which are commonly applied for co-simulation in technical applications. It is shown that the C1-continuous approximation technique yields a similar numerical stability and a similar local error as the Lagrange approach which results in a comparable or even better overall performance (taking into account the advantage of continuity at the numerical calculation of the subsystem differential equations). Applying the C0-continuous approach, a similar numerical stability is obtained. However, the order of the local error is significantly lower than for the C1-continuous method and the Lagrange approach.
This paper proposes a model based coupling technique for interconnected systems. It helps to overcome problems arising whenever the interconnections have a non-negligible influence on the overall system behavior. The main idea of the method is to use prediction schemes which compensate for performance degradation due to coupling imperfections. Exemplarily the so-called co-simulation scenario is selected to demonstrate the principles of the presented approach and its effectiveness by means of a complex real-time application.