The integration of physics-based and data-driven techniques in hybrid modeling offers a promising solution for dynamic system diagnostics. Physics-based and data-driven fault diagnostics approaches have their respective limitations. Data-driven methods rely heavily on historical data and may struggle to adapt to changes in the system, whereas model-based techniques rely on mathematical representations of the system, which may not be accurate enough for complex systems due to partially known physics and unavoidable simplifying assumptions. By combining these approaches into a hybrid framework it is possible to achieve an effective diagnostics solution.

Our team could develop a hybrid approach that combines physics-based models with data-driven models to provide a more comprehensive understanding of the system, resulting in improved diagnostic performance. This hybrid approach incorporates computational components in both forward and backward problems. The forward problem aims to develop a physics-based model of the dynamic system, which is analyzed using numerical or analytical methods to simulate both normal and abnormal conditions. The goal is to leverage the generalizability of the physics-based model to simulate the system's response with different fault conditions and varying degrees of severity, particularly in scenarios that are difficult to observe in collected data.

The forward problem leverages the capability of physics-based models to offer a generic mathematical representation of a dynamic system. This feature is crucial in application domains where collecting comprehensive data covering a wide range of dynamic behaviors is difficult. One such example is the challenge of unbalanced datasets in the diagnostics of engineering systems where the amount of faulty data is limited. In such cases, the physics-based model can generate additional synthetic data for faulty conditions, enhancing the information content of observations and leading to improved diagnostics.

Moreover, the analysis of the physics-based model in the forward problem uncovers the underlying causal relationships between the components of the system, providing more detailed information about the characteristics of faults and their impact on system performance. This information is beneficial to the performance of diagnostics. The goal is to extract information from the phase space of the dynamic system to achieve this objective.

Phase space encompasses all possible solutions of a dynamic system. It is a mathematical or graphical representation of the state of a system that takes into account all the variables and their interactions that explain the system's behavior over time. In phase space, each point represents a unique state of the system, and the trajectory of the system over time can be represented as a path through this space. Phase space also provides insight into the variability, stability, and predictability of the system, as well as its correlation with different health conditions. This information is crucial for understanding the behavior of dynamic systems and enriching the observations made on the dynamic system. There are many techniques aimed at quantifying the information contained within this space for applications such as regime of response detection, bifurcation analysis, and parameter estimation.

In the context of diagnostics, the information contained in this space, which is correlated with the health condition of the system, is extracted to be used for diagnostics in the inverse problem. When a machine learning model is employed, this extracted information is transformed into features that serve as inputs for the model. This is highly essential for the performance of a machine learning model, as it is significantly dependent on the information content of the extracted features from the input signal. The ideal extracted features should be robust, mathematically defined, computationally efficient, interpretable, correlated to the target domain, and generalizable. The physics-based model of a dynamic system can be used for such feature extraction. The generic, compact, and interpretable abstraction of a dynamic system obtained through physics-based modeling techniques can provide a foundation for feature extraction that meets these criteria.

The entire process, comprising both forward and inverse problems, can be considered a hybrid approach that leverages the strengths of both physics-based and data-driven models. The physics-based model acts as a prior to the data-driven model, providing physics-informed features that can guide the learning process and increase the accuracy of diagnostics. The combination of these two models results in a comprehensive solution that takes advantage of the strengths of both models and can be applied to a wide range of applications in the diagnostics of engineering systems [references].

The below figure illustrates the main computational components of both the forward and inverse problems. The phase spaces are generated by using a nonlinear pendulum, which is a prototype nonlinear dynamic setup capable of showing a wide range of response regimes with different representations in phase space. As discussed, the physics-informed feature extraction is a transferable component between the problems.

Computational components of forward and inverse problems

The discussed approach has been applied to diagnostics of a wide range of engineering systems, particularly crack diagnostics in a Jeffcott rotor. The Jeffcott rotor consists of a massless elastic shaft and a rigid disk.

The below figure illustrates the components of the forward and inverse problems modified for the cracked rotor case. In the forward problem, a physics-based model is developed based on dynamical and structural modeling concepts, resulting in the derivation of equations of motion. These equations allow for simulating the system dynamics in both healthy and faulty conditions. The physics-based model, as discussed in the previous section, enables physics-informed feature extraction. Extended phase space topology is used for this goal which is noted by EPST in the figure. This feature extraction process is the shared component between the forward and backward problems. In the backward problem, which is dedicated to diagnostics, the extracted features are input into a machine learning model to determine the health condition of the system and the fault severity (crack depth) in case of fault detection.

Computational components of forward and backward problems applied to crack diagnostics in a rotor

The diagnostic algorithm's predictions for unseen test samples show a high correlation coefficient, with values of 9.97e-1 and 9.89-1 for frequency ratios of = 1/2 and 1/3, respectively. These results demonstrate the high accuracy and generalization capability of the diagnostics algorithm in fault detection and quantifying its severity. The results provide strong evidence for the effectiveness of integrating physics-based modeling and machine learning techniques in a hybrid approach for achieving accurate and reliable diagnostics. By combining the strengths of both physics-based modeling and machine learning, a hybrid approach can effectively capture the complex dynamics of a system, accurately detect faults, and quantify fault severity.

1. Amirhassan Abbasi, Foad Nazari, C Nataraj, On Modeling of Vibration and Crack Growth in a Rotor for Prognostics.