Hybrid systems combine both continuous and discrete behavior. These systems serve as models in many fields, including control systems, robotics, and industrial processes. However, due to their complexity, finding an accurate model is a challenge. This paper presents a holistic approach to learning models of hybrid systems using symbolic regression. Our method leverages symbolic regression to automatically discover accurate and interpretable mathematical models in the form of hybrid systems from observed data. An advantage of our algorithm is that it detects transitions between different behavioral modes of a system based on the inherent dynamics. From learned expressions for the dynamical behavior of a system, we form a hybrid system by combining the learned expressions with a decision tree determining the current behavioral mode from data. This hybrid decision tree serves regression, prediction, and further related tasks. Our results demonstrate that symbolic regression can effectively identify the underlying dynamics of a real hybrid system and predict output signals on new input data with high accuracy. • Hybrid systems are powerful models combining continuous and discrete dynamics. • A data-driven identification automates model generation. • Often, dynamic modes are identified from signal similarities. • Different initial conditions lead to dissimilar signals even for same dynamics. • Dynamics-based identification is a more effective approach.

Identification of hybrid systems requires learning models that capture both discrete transitions and continuous dynamics from observational data. Traditional approaches follow a stepwise process, separating trace segmentation and mode-specific regression, which often leads to inconsistencies due to unmodeled interdependencies. In this paper, we propose a new iterative learning paradigm that jointly optimizes segmentation and flow function identification. The method incrementally constructs a hybrid model by evaluating and expanding candidate flow functions over observed traces, introducing new modes only when existing ones fail to explain the data. The approach is modular and agnostic to the choice of the regression technique, allowing the identification of hybrid systems with varying levels of complexity. Empirical results on benchmark examples demonstrate that the proposed method produces more compact models compared to traditional techniques, while supporting flexible integration of different regression methods. By favoring fewer, more generalizable modes, the resulting models are not only likely to reduce complexity but also simplify diagnostic reasoning, improve fault isolation, and enhance robustness by avoiding overfitting to spurious mode changes.

Hybrid systems, which combine both continuous and discrete behavior, are used in many fields, including robotics, biological systems, and control systems. However, due to their complexity, finding an accurate model is a challenge. This paper discusses the usage of symbolic regression to learn hybrid systems from data and specifically analyses learning parameters for a recent algorithm. Symbolic regression is a powerful tool that can automatically discover accurate and interpretable mathematical models in the form of symbolic expressions. Models generated by symbolic regression are a valuable tool for system identification and diagnosis, e.g., to predict future system behavior or detect anomalies. A major opportunity of our approach is the ability to detect transitions between different continuous behaviors of a system directly based on the dynamics. From a diagnosis perspective, this can advantageously be used to detect the system entering fault modes and identify their models. This paper presents a parameter study for a symbolic regression based identification algorithm.

Symbolic regression has shown potential in the identification of physical systems. Hybrid systems, which combine both continuous and discrete behavior, are a relevant extension of purely physical systems, used in many fields, including robotics, biological systems, and control systems. However, due to their complexity, finding an accurate model is a challenge. This paper presents a novel approach to learning models of hybrid systems using symbolic regression. Our method leverages the power of genetic programming to automatically discover accurate and interpretable mathematical models in the form of hybrid systems from observed data. Symbolic regression detects transitions between different continuous behavior of a system directly based on the dynamics, instead of pure distances of observed trajectories. Furthermore, models generated by symbolic regression can be used to predict future system behavior, detect anomalies, and identify the underlying dynamics of the system while providing a human-readable representation. Our results demonstrate that symbolic regression can effectively identify the underlying dynamics of a real system represented in a hybrid model, providing a valuable tool for system identification and diagnosis.

In the domain of cyber-physical systems, there is an increasing relevance of data-driven approaches for the learning of hybrid system dynamics. In particular, accurate models have been successfully abstracted from continuous (real-valued) traces and applied for various goals. However, industrial applications involving online modeling or rapid prototyping have two additional requirements: 1) runtime efficiency and 2) the interpretability of the approach and results.

Testing of black-box systems is a difficult task, because no prior knowledge on the system is given that can be used for design and evaluation of tests. Learning a model of a black-box system from observations enables model-based testing (MBT). We take a recent approach using decision tree learning to create a model of a black-box system and discuss the usage of such a decision tree model for test generation. In this scope, we define a test coverage metric for decision tree models. Furthermore, we identify different modes of testing and explain that a decision tree model especially facilitates model-based testing for black-box systems with limited controllability of inputs and the inability to reset the system to a specific state. A case study on a discrete system illustrates our MBT approach.

Modeling Cyber-Physical Systems (CPS) requires knowledge from various domains, including computer science, electrical and mechanical engineering, and control theory. In addition, a solid understanding of the application domain, e.g., intralogistics, maritime technology, or grid control technology is required to ensure relevant and accurate models. In order to reduce the knowledge required for modeling CPS, we envision a framework for Automatic Generation of models for CPS (AGenC) supporting design and operation. Typical tasks in design and operation are summarized by the terms prediction, monitoring, and testing. Thus, the proposed framework employs learning techniques to generate models of CPS that predict the system’s performance in various scenarios, monitor the system in real-time to detect anomalies or failures, and automatically generate test cases. This research has the potential to significantly reduce the time and effort required for designing, testing, and maintaining CPS, making them more reliable and efficient.

Abstract models of embedded systems are useful for various tasks, ranging from diagnosis, through testing to monitoring at run-time. However, deriving a model for an unknown system is difficult. Generic learners like decision trees can identify specific properties of systems and have been applied successfully, e.g., for anomaly detection and test case identification. We consider Decision Tree Learning (DTL) to derive a new type of model from given observations with bounded history for systems that have a Mealy machine representation. We prove theoretical limitations and evaluate the practical characteristics in an experimental validation.

Cyber-Physical Systems (CPS) are often black-box systems, i.e., knowledge of the inner workings or a system model is not available. Nevertheless, models of CPS are needed for various tasks, ranging from verification, over testing to monitoring at runtime. For these tasks, finite and discrete models facilitating understandability, compactness, and efficiency are often desirable. Deriving a discrete model of a continuous-valued CPS is difficult. A simple abstraction is achieved with a time and value discretization through sampling and discretization intervals. We consider observing the system with bounded history and apply decision tree learning on discretized observations to generate a model of the system. The model supports the identification of system characteristics and predicts a valid next output based on the bounded history. We prove an upper bound on the error size for the prediction of an output. Experimental results give practical insight and present a comparison to automata learning.

Absolute position accuracy is the key performance criterion of an Indoor Localization System (ILS). Since ILS are heterogeneous and complex cyber-physical systems, the localization accuracy depends on various influences from the environment, system configuration, and the application processes. To determine the position accuracy of a system in a reproducible, comparable, and realistic manner, these factors must be taken into account. We propose a strategy for analyzing the influences on the position accuracy of ILS using decision trees in combination with application-related or technology-related categorization. The proposed strategy is validated using empirical data from 120 experiments. The accuracy of an Ultra-Wideband and a LiDAR-based ILS was determined under different application-driven influencing factors, considering the application of autonomous mobile robots in warehouses. Finally, the opportunities and limitations of analyzing decision trees to compare system performance, find a suitable system, optimize the environment or system configuration, and understand the relevance of different influencing factors are presented.

In this paper we examine two case studies in which we learn finite state machines from models of CPS using automata learning. We explore how well automata learning is suited as an approach for learning CPS. What challenges and problems exist when trying to learn a model of a CPS using automata learning. Automata learning can reliably learn finite state machines of systems like embedded systems or software systems. CPS pose different challenges, like continuous components, for which different levels of abstractions and considerations have to be used, so the resulting finite state machines are useful representations of the systems. Through the small, yet insightful case studies we show examples of how automata learning can be applied to CPS and what information the resulting automata can represent.

The design of energy- and area-efficient systems is important for modern technology. One approach to increase these efficiencies is approximate computing. During the last years, efficient approximations for combinational hardware components, e.g., adders or multipliers, have been proposed.We focus on quality metrics for the evaluation of approximations in sequential circuits with streaming in- and outputs. We propose the usage of sequence distance metrics for analysis of the sequential behavior after approximation and compare their performance to other metrics like mean errors and accumulated errors. We present case studies on some exemplary circuits. The experimental results show that our sequential metrics provide additional information to common mean errors and for stochastic applications yield the best guidance in selecting approximate sequential circuits.