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.

Modern complex systems, such as radiotherapy machines, require robust strategies for fault detection, diagnosis, and prognosis to ensure operational continuity and patient safety. While data-driven methods have gained traction, few studies address diagnostic and prognostic tasks using multimodal operational data under unsupervised or semi-supervised learning settings. This gap is particularly critical given the scarcity of labeled failure data in real-world environments. This work aims to design a unified approach for fault detection, diagnosis, and prognosis using multimodal data in the absence of complete labeling. To this end, autoencoders (AEs) are employed due to their suitability for unsupervised and self-supervised learning, flexibility in handling heterogeneous data, and ability to construct latent representations optimized for various downstream tasks. A specific implementation based on a Long Short-Term Memory β-Variational Autoencoder (LSTM-β-VAE) was developed to detect anomalies in machine logs. This framework is applied to TomoTherapy® systems - a highly complex and under-explored use case within the radiotherapy domain. Initial results demonstrate strong anomaly detection performance on both a public benchmark dataset (HDFS) and a proprietary dataset derived from real-world TomoTherapy® machine faults. Beyond methodology, the paper includes a concise literature review of multimodal learning and data-driven diagnosis and prognosis with a focus on AEs. Based on this review, key research directions are identified for the continuation of the thesis, especially the integration of explainable AI as a means to enhance diagnosis capabilities in the absence of labeled faults.

Structural analysis-based diagnosis allows for the extraction of a wealth of information and properties by studying a structural model that represents a physical system. This diagnosis approach is centered on structurally overdetermined sets, which enable the generation of residuals for fault detection and isolation. As the 'for Dummies' editorial collection, this article aims at taking on complex concepts and making them easy to understand. It aims to clarify and compare key concepts in structural analysis, focusing on Minimally Structurally Overdetermined (MSO) sets and Minimal Test Equation Supports (MTES). Additionally, we explain and illustrate the Dulmage-Mendelsohn decomposition, which helps identify structurally overdetermined parts of the system and plays a important role in the structural analysis process. Through detailed exploration and practical examples, we demonstrate the roles, applications, and interrelations of these sets, highlighting their respective strengths and limitations. The paper provides an overview of the algorithms used to identify and use these sets, including a theoretical and practical comparison of their computational efficiency and diagnostic capabilities.

Air data sensors provide essential measurements to ensure the availability of autopilot and to maintain aircraft performance, flight envelope protection and optimal aerodynamic surfaces control laws. The importance of these sensors imply the existence of embedded fault tolerance features, mainly represented by hardware redundancy. The latter is prone to fail in case of common fault of multiple sensors, especially if the faults are coherent and simultaneous. Increasing the robustness of fault detection and isolation (FDI) techniques for air data sensors to the aforementioned conditions is essential for the development of more autonomous aircraft, reducing crew workload and guaranteeing flight protections under adverse conditions. This paper reviews recent works on Air Data System (ADS) FDI, assessing proposed model, data and signal-driven approaches. We finally argue in favor of data-driven and hybrid approaches for the development of virtual sensors and semi-supervised anomaly detectors, offering an overview of ways forward.

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.

This paper addresses the capability of autonomous robots to achieve flexible goals in dynamic environments. In such a setting numerous challenges jeopardize the robustness of such systems. Thus, we propose a hierarchical diagnosis concept for layered control architectures, that can detect and deal with such challenges to maintain a consistent knowledge about the world and to allow reliable decision-making. Layered control systems use various knowledge representations and decision-making mechanisms teamed with specialized isolated fault-handling approaches. However, some issues can only be identified if the information from different layers is combined. Our approach addresses challenges like failing actions, uncertain observations, and unmodeled events by propagating observations and diagnoses results throughout the hierarchy. This enhances adaptability and dependability in various domains. In this paper, we present a prototype architecture following this approach.

This paper examines two prominent Fault Detection and Isolation methodologies: the Signature Matrix approach, traditionally used in hardware systems, and the Spectrum-based approach, applied in software fault localization. Despite their distinct operational domains, both methods share the objective of precisely identifying and isolating faults. This study aims to compare these approaches and to highlight their similarities in principle. Through a comparative analysis, we assess how the structured pattern recognition of the Signature Matrix method and the statistical analysis capabilities of the Spectrum-based approach can be synergized to enhance diagnostic processes of cyber-physical systems that are composed of both hardware and software components. The investigation is motivated by the prospect of developing a hybrid Fault Detection and Isolation strategy that incorporates the robust detection mechanisms of hardware diagnostics with the techniques used in software fault localization. The findings are intended to advance the theoretical framework of Fault Detection and Isolation systems and suggest practical implementations across varied technological platforms, thereby improving the reliability and efficiency of fault detection and isolation in both hardware and software contexts.

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.

This research proposes a method for detecting subtle faults named snail trails for their visual similarity with the trail of a snail in photovoltaic modules. Snail trails do not significantly reduce panel performance but they are the main cause of serious panel deterioration such as microcracks and delamination and can go so far as to set the panel on fire. To detect these faults, this research uses an ensemble learning framework, named ensemble learning for diagnosis, which combines several complementary learning algorithms, namely Support Vector Machines, K-Nearest Neighbors, and Decision Trees. A set of features is obtained by extracting the time–frequency characteristics and statistics from the photovoltaic current signal of the photovoltaic panel. This is followed by a feature selection and dimensionality reduction step that delivers the input to the learning algorithms. The approach presented in this study is experimentally validated, independently for the 4 seasons of the year, with data from a real photovoltaic string of 16 panels. The results demonstrate that the proposed approach can efficiently classify healthy panels and panels with snail trails efficiently. Interestingly, the method only requires the electrical current signal, measured on panels with data acquisition systems that are standard in the photovoltaic industry. The genericity of the approach makes it a good candidate for detecting other photovoltaic faults and for solving diagnosis problems in other domains.

Health monitoring of current complex systems significantly impacts the total cost of the system. Centralized fault diagnosis architectures are sometimes prohibitive for large-scale interconnected systems, such as distribution systems, telecommunication networks, water distribution networks, or fluid power systems. Confidentiality constraints are also an issue. This article presents a decentralized fault diagnosis method that only requires the knowledge of local models and limited knowledge of their neighboring subsystems. The method, implemented in the decentralized diagnoser design ($D^{3}$) algorithm, is based on structural analysis and can advantageously be applied to high-dimensional systems, linear or nonlinear. Using the concept of isolation on request, a hierarchy is built according to diagnostic objectives. The resulting diagnoser is based on analytical redundancy relations (ARRs) generated along the hierarchy. Their number is optimized via binary integer linear programming (BILP) while still guaranteeing maximal diagnosability at each level.$D^{3}$proves of lower time complexity than its centralized equivalent. It is successfully applied to a nonlinear combined cycle gas-turbine power plant.

State tracking, i.e. estimating the state over time, is always an important problem in autonomous dynamic systems. Run-time requirements advocate for incremental estimation and memory limitations lead us to consider an estimation strategy that retains only one state out of the set of candidate estimates at each time step. This avoids the ambiguity of a high number of candidate estimates and allows the decision system to be fed with a clear input. However, this strategy may lead to dead-ends in the continuation of the execution. In this paper, we show that single-state trackability can be expressed in terms of the simulation relation between automata. This allows us to provide a complexity bound and a way to build estimators endowed with this property and, moreover, customizable along some correctness criteria. Our implementation relies on the Sat Modulo Theory solver MonoSAT and experiments show that our encoding scales up and applies to real world scenarios.

Complex technical systems usually show a dynamic behavior that is often conveniently represented with a discrete event model. Such a behavior is the result of dynamic components which interact with each other. Due to the complexity of technical systems faults are not totally avoidable. In order to deal with such faults diagnosing the system at run-time is of great interest. To perform such a diagnosis it is common to use fault models. Such models are in practice often hard to obtain. To address this problem we show a diagnosis approach for discrete event systems which uses the model of the nominal behavior only. In order to perform this diagnosis we adopt the well known idea of consistency based diagnosis.

In this paper, we present a dynamic clustering algorithm that efficiently deals with data streams and achieves several important properties which are not generally found together in the same algorithm. The dynamic clustering algorithm operates online in two different time-scale stages, a fast distance-based stage that generates micro-clusters and a density-based stage that groups the micro-clusters according to their density and generates the final clusters. The algorithm achieves novelty detection and concept drift thanks to a forgetting function that allows micro-clusters and final clusters to appear, drift, merge, split or disappear. This algorithm has been designed to be able to detect complex patterns even in multi-density distributions and making no assumption of cluster convexity. The performance of the dynamic clustering algorithm is assessed theoretically through complexity analysis and empirically through a set of experiments.

This paper presents the theoretical keystone for a decentralization of model-based diagnosis by proving the equivalence between decentralized and centralized residual generation. The proof is based on structural analysis and graph-theoretical concepts. The second contribution of this paper is the design of a decentralized fault-focused residual generation scheme advantageously implementing a strategy of fault isolation on request. Algorithms are tested on the attitude determination and control system of a low Earth orbit satellite.

This paper proposes a method to track the system mode and diagnose a hybrid system without building an entire diagnoser off-line. The method is supported by a hybrid automaton (HA) model that represents the hybrid system continuous and discrete behavioral dynamics. This model is built on request through parallel composition of the component HA models. Diagnosis is performed by interpreting the events and measurements issued by the physical system directly on the HA model. This interpretation allows us to construct the useful parts of the diagnoser developing only the branches that are required to explain the occurrence of incoming events. The resulting diagnoser adapts to the system operational life and is much less demanding in terms of memory storage than the entire diagnoser. In addition to this feature, the proposed framework subsumes previous works in that it copes with both structural and nonstructural faults. The method is validated by the application to a case study based on the sewer network of the city of Barcelona.

Hybrid systems serve as a powerful modeling paradigm for representing complex continuous controlled systems that exhibit discrete switches in their dynamics. The system and the models of the system are nondeterministic due to operation in uncertain environment. Bayesian belief update approaches to stochastic hybrid system state estimation face a blow up in the number of state estimates. Therefore, most popular techniques try to maintain an approximation of the true belief state by either sampling or maintaining a limited number of trajectories. These limitations can be avoided by using bounded intervals to represent the state uncertainty. This alternative leads to splitting the continuous state space into a finite set of possibly overlapping geometrical regions that together with the system modes form configurations of the hybrid system. As a consequence, the true system state can be captured by a finite number of hybrid configurations. A set of dedicated algorithms that can efficiently compute these configurations is detailed. Results are presented on two systems of the hybrid system literature.

In this paper we propose a hybrid system modeling framework aimed at analyzing diagnosability. In this framework, the hybrid system is seen as the composition of an underlying discrete event and an underlying continuous systems. Diagnosability of these two underlying systems are fully analyzed and new results are provided for the underlying continuous system (called the multimode system). Based on these results, a hybrid language that contains ‘natural’ discrete events and discrete events capturing the continuous dynamics, is defined. On the basis of this language the diagnosability definition of hybrid systems is provided. With respect to this definition, we prove that the diagnosability of the underlying continuous or the discrete event system is only a sufficient condition. Diagnosability of hybrid systems must be decided by coupling both discrete event and continuous informations. Finally, the necessary and sufficient condition of hybrid diagnosability is given.

Many techniques in model-based diagnosis and other research fields find the hitting sets of a group of sets. Existing techniques apply to sets of finite elements only. This paper addresses the computation of the hitting sets of a group of sets whose elements are convex or non-convex, bounded or unbounded continuous regions. We assume the conflict sets are known and we present a novel procedure, the Continuous Hitting Set algorithm (CHS) for transforming conflict sets of continuous elements into minimal hitting sets.

Real-life complex systems are often required to offer high reliability and quality of service and must be provided with self-management abilities, even in faulty situations. They are expected to be self-aware of their current state and survive autonomously the occurrence of faults, still managing to provide the desired functionality. In other words, such systems must be self-healing [2]. Designing self-healing systems requires to be able to evaluate the joint degree of self-awareness and reactiveness. In the artificial intelligence community, these two properties are better known as diagnosability [3, 1], i.e. the capability of a system to exhibit different observables for different anticipated faulty situations, and repairability, i.e. the ability of a system and its repair actions to cope with any unexpected situation. Checking separately diagnosability and repairability leads to a conservative assessement of self-healability. In this paper, we show that neither standard diagnosability nor repairability of every anticipated fault are necessary to achieve self-healability. Our main contribution consists of defining self-healability as a joint property bridging diagnosability and repairability, which requires a new definition of diagnosability that allows diagnosable subsets of faults to overlap, as opposed to the standard definitions which rely on a partition.

It is commonly accepted that the requirements for maintenance and diagnosis should be considered at the earliest stages of design. For this reason, methods for analyzing the diagnosability of a system and determining which sensors are needed to achieve the desired degree of diagnosability are highly valued. This paper clarifies the different diagnosability properties of a system and proposes a model-based method for: 1) assessing the level of discriminability of a system, i.e., given a set of sensors, the number of faults that can be discriminated, and its degree of diagnosability, i.e., the discriminability level related to the total number of anticipated faults; and 2) characterizing and determining the minimal additional sensors that guarantee a specified degree of diagnosability. The method takes advantage of the concept of component-supported analytical redundancy relation, which considers recent results crossing over the fault detection and isolation and diagnosis communities. It uses a model of the system to analyze in an exhaustive manner the analytical redundancies associated with the availability of sensors and performs from that a full diagnosability assessment. The method is applied to an industrial smart actuator that was used as a benchmark in the Development and Application of Methods for Actuator Diagnosis in Industrial Control Systems European project.

Two distinct and parallel research communities have been working along the lines of the model-based diagnosis approach: the fault detection and isolation (FDI) community and the diagnostic (DX) community that have evolved in the fields of automatic control and artificial intelligence, respectively. This paper clarifies and links the concepts and assumptions that underlie the FDI analytical redundancy approach and the DX consistency-based logical approach. A formal framework is proposed in order to compare the two approaches and the theoretical proof of their equivalence together with the necessary and sufficient conditions is provided.