Active magnetic bearings (AMBs) are an advanced solution for supporting a load without physical contact and can be found in industrial applications which require high precision, reliability, or are subject to extreme conditions. A magnetic bearing system comprising of a controller, current amplifiers, electromagnetic actuators, rotor assembly, and position sensors is a nonlinear dynamical system because the force generated by the actuator is square proportional with electrical current, and inverse square proportional with the distance to the target. These nonlinearities add complications to the controllability of AMB systems because control strategies either must accommodate for the changing system conditions or rely on linear assumptions to simplify the system to maintain small errors in the output. Until recently, nonlinear control methods have not been applied to AMB systems in an industrial setting due to their complexity. Industrial applications of magnetic bearings approach the problem of nonlinearity using an assumption of linearity in a small region about an operating point. In this method, a linear controller is designed at a specified current and displacement (referred to as the operating point) and the goal of operation is to minimize deviation from the operating point. As a result, the goal of controller design is disturbance rejection, which is commonly performed by modelling the AMB system and designing a controller to satisfy the requirement.

Industry focuses on modelling the dynamic response of AMB systems using finite element modelling (FEM) and nonparametric measurement of the rotor rigid and flexible dynamics. In contrast, academic literature has emphasized estimating stiffness coefficients and predicting the closed loop response of the AMB system to a disturbance using parametric model estimation from measurement data. Review of previous academic works reveal that linear modelling techniques have been applied in the AMB research community for the benefit of rendering a simple model of the system, given that perturbations to the system remain sufficiently small as to neglect the error of nonlinearity. The linear model is a useful tool in controller design which can predict the output of the system to a given input or disturbance. Van Schoor et al. [1] identified Linear Time-Invariant (LTI) models of single radially configured AMB with 2 degrees of freedom with the focus of minimizing the model error in predicting the incremental output. The LTI models were evaluated by comparing the output of the model prediction to that of a nonlinear simulation and studying the difference. The predictions of the linear and nonlinear models diverged as frequency of excitation increased. Khader et al. [2] identified linear models for separate system components using MATLAB model estimation tools. The authors utilized frequency response data and the Prediction Error Method (PEM) to estimate a linear model for the system based around an operating point. The rigid and flexible modes of the rotor were estimated separately using different mathematical models and combined to form the general rotor model. The plant model was constructed by combining the different components and validated against frequency response data by evaluating the root mean square error and power spectrum of the model residuals as compared to the measured frequency response. The measured frequency response showed strong agreement in the prediction of flexible modes. Further comparisons [3–5] have been conducted between LTI modelling methods and nonlinear simulations of AMB systems, as well as developing linear and nonlinear models of the closed loop AMB system. Sahinkaya et al. [6] performed another such study on the parametric identification of AMB systems, wherein they proposed a method to automate the design of an H-infinity AMB levitation controller through the frequency domain identification of the plant and application of the subspace method to estimate a parametric model. The authors of this study used a grey-box approach to identify the system parameters using through the constraint of a mathematical model of the rigid and flexible rotor dynamics. Sahinkaya et al. claimed this method was advantageous over PEM because less engineering experience was required to formulate the problem. This study aims to identify a parametric system model in a similar way; however, the PEM is used in a black-box context, without requiring formulation of the dynamic equations. This method will not constrain the context of the system identification, allowing for the parameter estimation method to freely include dynamics not characteristic of the rotor assembly, such as the stator structure. In all studies examined, the models were either estimated by measurement of the closed loop system response, or the models were initially built from or constrained to an expected mathematical model. However, none of the examined studies used measurement data without an initial model in a black-box method to identify all the dynamics of the system as measured by the sensor elements. In contrast to the goals of previous research, this paper aims primarily to identify a model to facilitate controller synthesis without knowledge of the equations of motion of the rotor-AMB-structure system. In common industry practice, the characteristics of the actuators, sensors, and amplifiers are known from calculation using finite element analysis or from direct measurement. A first step in commissioning of AMB machinery is the estimation of a system model to facilitate controller synthesis to meet stability and performance criteria.

Based on the success of past academic studies on linear model estimation from measurement data, this study aims to adopt the approach of estimating a parametric model from measurement data to achieve the industry goal of realizing a satisfactory controller by identifying a model to simulate the rigid and flexible dynamics of the rotor and mechanical structure in the AMB system. The rigid and flexible rotor models can be analytically identified and combined to form a canonical state-space representation of the overall rotor dynamics, as demonstrated by Vuojolainen [7]. While the mathematical model of the system is necessary to characterize the dynamic behaviour of the system, it is not necessary to empirically identify a model to represent the dynamics of the system, and the constraint of an expected model form will omit eigenmodes contributed by the structural components. The method proposed in this paper aims to achieve a holistic identification of the system without prior constraint of the model form. Identification of the AMB system must be performed in the closed-loop condition because the negative position stiffness renders the AMB open-loop unstable [8]. This study also attempts to use the identified rotor model to construct a full system model to predict the closed-loop response which will be validated against measurements of the closed-loop system. The development of the parametric model of the full system will be a segmental model which more closely represents the successive transformation of signals through the real system as opposed to nonparametric (numerical) response data. This segmental model may be used to predict how the system changes when elements are updated or substituted. The dynamics of the mechanical structure will be modelled using a single set of measurements. This approach will focus on predicting the behaviour of the system as observed by the rotor position sensors and will not differentiate between different contributors to dynamic interactions.

This study differs from previous academic work by identifying the complete rotor and structure in a single series of measurements without constraining a canonical form and building a model of the closed loop system by combining the components of the system to simulate the output. This study builds on industry practice of building system models using direct plant measurement combinations by making use of academically demonstrated methods of model estimation to identify parametric models of the plant to achieve a more informative AMB system model which can be used in further applications than the current standard of iterative controller design based on engineering experience. This paper explores a shift towards a data-driven approach which utilizes measurements of the real system for model identification. Application of the measurement-based identification method across the entire operating range of the system renders a digital twin of the AMB system, which can be updated periodically to account for changes in the system over time. This shift towards the data-driven approach and the use of digital twins was explored in depth in the manufacturing space by Jeong [9]. In their paper, Jeong proposed that the data-driven digital twin demonstrated significant potential to revolutionize the manufacturing environment when combined with emergent AI technologies. AMB commissioning and long-term operation can benefit from this combination of technologies to improve predictive maintenance and system performance. Further application of the digital twin can be used to realize an algorithm for the controller synthesis to optimize stability margins or maximize AMB stiffness. Such an application of digital twin was successfully demonstrated by Sim and Lee [10], wherein the optimal conditions for minimizing machining time and production cost were derived using a genetic algorithm. Although the application space is different, the potential of a digital twin to solve a complex optimization problem is broadly applicable.

This paper is organized as follows: Section 2 introduces the demonstration AMB system to be studied and the excitation signal provided to perform the experiment. Section 3 details the structure of the system including the signals which will define the closed loop and the plant, and the mathematical form of the system model. Section 4 demonstrates the case study using the proposed framework. The conclusions and recommendations for future works are given in section 5.

This paper proposes a method to estimate a segmental parametric model of a levitated AMB system using measurement data, the components of which can be manipulated to validate design changes in the controller or other system hardware. The benefit of the proposed measurement-based model estimation method is to assist in intelligent controller design by facilitating development of representative dynamics models of the complete system architecture without analytical analysis of the system. The full range of dynamic interaction within a specified bandwidth is captured without any required knowledge of the physics in the system. The fitment of the estimated models to the measurement data proved that a representative model can be identified for the MIMO system and be used to predict the behaviour of a SISO controller with acceptable accuracy. The parametric models identified in this study predict the frequency response of the rotor with 97.9% accuracy or greater, and can provide further information about the AMB system than current industry standards by simulating the response to new inputs not provided in the experiment, while also having removed the influence of measurement noise in the simulated response The segmental models constructed in this study extend the advantage of industrial methods of identifying the rotor model by offering an accurate prediction of the impact of changing elements within the system. This method provides a comprehensive model of the system in the stationary condition, which is a critical stage in commissioning of rotating machinery supported by AMBs. The same method demonstrated in this study can be repeated while the rotor is rotating to identify the gyroscopic effects and iterate the controller design to ensure rigid and flexible rotor modes remain stable.