### 1 Introduction

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.

### 2 Experimental Setup

### 2.1 AMB Demonstration System

Figs. 1, 2, and 3 illustrate the AMB test setup used for this study. The system hardware and control architecture are provided by SKF Magnetic Mechatronics. Fig. 1 depicts the levitation control architecture of the experimental setup. Each AMB is comprised of 2 axes of control, while each control axis contains 2 opposing electromagnets. The entire AMB machine contains 4 control axes and 8 power amplifiers. while each axis of control has a dedicated controller and position sensor. The position sensors are comprised of hall effect sensors which target the rotor surface and are driven by a carrier signal provided by the controller. Changes in the distance from the rotor to the sensor result in a change in voltage across the sensor body, which is fed back to the controller and compared to a setpoint. The difference between the sensor signal and the setpoint defines the position error, which is conditioned by the controller to provide an input to the AMB power amplifier. In the identification experiment, a known disturbance is added to the input to the AMB power amplifier and the response at the sensor is measured. Each electromagnet with the AMBs provide a maximum force capacity of 265 N per axis at maximum current and a centered rotor, with a designed current capacity of 0–3A. With the rotor levitated, the nominal airgap between the rotor and the AMB pole face is 0.45 mm. The identification experiment is performed with the system operating, and the rotor in the levitated position.

Figs. 2 and 3 depict the rotor assembly which is supported by one AMB assembly on each end which applies force to the rotor in the radial direction. Each AMB is situated within a cartridge assembly, along with one sensor assembly, and one backup ball bearing. There is no axial AMB in this system, rather the rotor is constrained in the axial direction using a bellows coupling which connects the motor drive. The AMB system provided by SKF Magnetic Mechatronics and is an industry development test bench for research and development in control strategies. The system is designed to rotate at 200 Hz, which is higher than the first flexible mode of the rotor, classifying the system as supercritical. This machine was chosen for this study because it is designed to be representative of the dynamics observed in dynamically rich high-speed machinery such as turboexpanders and compressors and serves as a viable tool to evaluate the identification and control methods applied to industrial applications. The electrical connections in the assembly meet at a junction box connecting to the AMB control system by means of an integrated cable harness. A single interface panel connects the SKF MB4150g5 magnetic bearing controller to a STAC SPDuo 2-channel signal analyzer tool, which provides access to measure various signals within the control loop, as well as add externally generated signals to the signal path. The control coordinates shown in Fig. 3 illustrate the coordinate convention used by the control system to position the rotor. The control axes are described by a cartesian frame of reference, rotated from a pure vertical and horizontal reference by 45 degrees. Control of AMB1 is denoted V13 and W13 axes, while AMB2 is allocated V24 and W24 axes. Further detail on the control axes is provided in Fig. 3.

Fig. 4 further shows the convention for control axes and their labels. The combined controllability provided by the two radial AMBs form the V-plane and W-planes, which are mutually orthogonal, while each plane is comprised of a set of axes. In this convention, V13 and W13 are comparable to the commonly used

*x*and*y*coordinates in a cartesian frame of reference, where V1 corresponds to*x+*, V3 corresponds to*x*−, W1 corresponds to*y+*, and W3 corresponds to*y*−. In the case of AMB1, the*xy*-plane of a cartesian frame of reference is referred to as the VW13-plane. A separate but parallel plane is denoted for AMB2, referred to as the VW24 plane. The V13 and V24 axis are parallel to each other, as are the W13 and W24 axes. The numbers 1 and 2 represent the + direction while 2 and 4 represent the - direction. The Z axis is not magnetically controlled in this system but is rather separately constrained by a motor coupling. In other AMB machines, the Z axis is actively controlled by a separate AMB.Each radial AMB in the system is configured to independently control the position of the rotor using only the signal from its paired position sensor. This is referred to as single input single output (SISO) configuration. Each AMB acts on the rotor to maintain the position of the rotor within the geometric centre of the system, and controller design is linearized around this operating point. The AMBs respond to disturbances such as the force of rotation or an external load by modulating the applied force to return the sensor reading to the target position. In this system, the controller maintains levitation of the rotor, while the signal interface panel allows the user to add external signals and perform measurements at various points within the system during operation. Fig. 5 shows a simplified diagram of the rotor, including approximate locations of the AMB actuators and rotor position sensors. The placement of the AMBs affects the controllability of each resonant frequency of the rotor, while the placement of the position sensors affects the observability of each resonant frequency of the rotor. The relative position between the each AMB and each position sensor determines the magnitude and phase of the response observed at each sensor due to an excitation applied at each AMB. The V13 and V24 axes of control are identified explicitly in this study, while the W13 and W24 axes are not considered due to the symmetry of the system.

To perform the identification experiment, an excitation signal is provided to the amplifier input while the rotor is levitated using a decentralized pole placement controller, and the response is measured using a signal analyzer. Frequency domain is the primary choice when dealing with vibration analysis of mechanical systems [11].

### 2.2 Input Signals

The identification experiment is conducted by providing a swept sine voltage waveform to the signal injection point. The swept sine was chosen because of the high SNR and configurability of the signal amplitude, experiment duration, and frequency range. The amplitude of the excitation signal is designed to preserve a small disturbance to the rotor position during the experiment and minimize the nonlinear changes in the system while maintaining a strong SNR. 0.3 V is chosen based on measurements of rotor position, and linearly decreasing to 0.1 V at the amplifier bandwidth of 2,000 Hz. The signal decreases in amplitude with increasing frequency to minimize the nonlinear changes in the system. The same excitation signal is used for each identification experiment.

### 3 Measurement-based Dynamic Modelling

### 3.1 AMB System Architecture

AMB systems may be operated in a SISO condition. This simplification works well for feedback control of the AMB system in many cases. However, if an accurate system model is required, the interaction between channels through the dynamics of the mechanical components must be considered. The position of the rotor at sensor 1 results from a combined contribution of the actions of both AMB 1 and AMB 2, and the same holds true for sensor 2 with different amounts. The relative impact of each AMB on the sensor reading is determined by the rigid and flexible dynamics of the mechanical system, comprised of the rotor and housing structure. Stable levitation of the rotor assembly is maintained by a series of independent controllers, one for each axis of control. Each controller modulates the input signal to the AMB power amplifier to match the position sensor output signal with the position setpoint request. The response of the identified system is scaled by each position sensor, compared with the position setpoint, and the difference (or error) is returned to the controller.

Fig. 7 shows a simplified representation of the system layout for two control axes. This representation demonstrates a SISO configuration for the control loop, separating the feedback of each sensor signal to its respective controller. The contribution from the rotor to each output must be considered, maintaining a MIMO plant configuration within the system. Each controller receives a position reference signal (

*p**) which is compared to the output of the position sensor. The controller (*_{i}*K**) is situated in the forward loop while the plant is comprised of the bearing amplifier and actuator (*_{i}*B**), the rigid and flexible rotor dynamics (*_{i}*A**), and the sensor conditioning (*_{ij}*H**). The subscript 1 or 2 denotes the relevant control axis. Subscripts in the*_{i}*A**terms indicate interaction between the two control axes. The sensor output is fed directly back and subtracted from the position setpoint reference. The experiments in this study are performed by adding a signal to the amplifier command (excitation point*_{ij}*f**) and measuring the response from the sensor (*_{i}*y**) to the AMB amplifier input signal (*_{i}*v**). This test measures the influence of the amplifier, bearing, rotor assembly, and sensor while excluding the controller. The excitation is provided to one AMB and frequency response data is measured at each sensor (*_{i}*y*_{i}*, y**) and amplifier input of the opposite AMB (*_{j}*v**). The experiment is repeated for the opposite AMB, yielding six measurements in total. The closed loop empirical transfer function estimate (ETFE) for each axis is measured by recording the response at the amplifier input (*_{j}*u**) to an excitation (*_{i}*f**). Frequency domain identification is chosen in this experiment for the ability to enhance signal to noise ratio (SNR) through averaging of measurement data. All system signals are processed in raw voltage and converted to engineering units by the control system when communicating with the user interface. This consideration allows for the system to be represented using unity feedback, which enables the controller designer to quantify closed loop stability with simple metrics for the open loop response. Fig. 8 provides an illustration of Eqs. (1) and (2), which provide rotor position as seen by sensors 1 and 2 due to inputs to bearing amplifier 1 and 2. In the case of this system, both AMB cartridges are the same design which simplifies both bearing and sensor influence to be the same. The symmetry between the two AMB cartridges in this system simplifies the bearing and sensor representation such that*_{i}*B*_{1}=*B*_{2}=*B*, and*H*_{1}=*H*_{2}=*H*.The series of identification measurements provide the system of equations needed to solve for the 4 rotor dynamic elements. The signals which contained an excitation have been noted with ‘

*e*’ in the subtext to differentiate the same channels in different conditions.Rearranging to isolate the

*A**elements:*_{ij}##### (4)

The remaining step in the identification process is to perform the operations in Eq. (4) using the scalar frequency response data for each entry.

### 3.2 Parametric Model Estimation

The Prediction Error Modelling (PEM) method supports black box identification of a given system, where no physical insight into the system dynamics is required [12]. State-space models offer a clean way to handle MIMO systems. The general form of a continuous time state-space model from an input

*u*(*t*) to output*y*(*t*) is given by Eq. (5).This AMB system model will be treated as the output-error type, meaning the noise conditioning (

*K*) and feedthrough (*D*) are zero.*A*represents the system state matrix, which conditions the system state*x*(*t*),*B*is the input conditioning matrix, and*C*is the output conditioning matrix. Stochastic noise in the system is represented by*e*(*t*). The states of the system have not been modelled analytically, which means a different method is required to estimate the number of system states which the model should capture. In this approach, the frequency response data is plotted, and the number of resonant frequencies is totaled within the measurement range. This approach works well for estimating system models using frequency response data when there is no defined cutoff in the contribution of energy from each incremental system state as measured by the Hankel Singular Value. In this system, each degree of influence is estimated to be well represented by 50 system states.### 4 Experimental Results

### 4.1 Identified System Model

The full system model was obtained and compared to the measurement data. The units of the measurement data are in Volts (V) for all inputs and outputs, rendering a dimensionless model. Fig. 9 depicts the comparison between the calculated rotor dynamic model ([ 99.2 97.9 98.7 99.6 ] % , defined by the normalized root mean square error (NRMSE) between the simulated response and the measured output data [10].

*A**), and the measured response. Each of the four frequency response functions represent the dynamic interaction from each AMB output (*_{ij}*B**) to each sensor input (*_{i}*H**), forming the 2 × 2 MIMO system. The fit between the calculated and the measured frequency response for each input-to-output combination displayed in Fig. 9 was measured to be*_{i}The identified model for the rotor influence can be used to estimate the closed loop system response using the known influence of the controllers, bearings, and sensors as shown in Fig. 8. The combination of each element to build a model of the closed loop system provides a method to design a new controller. The controller is designed using proportional-integral (PI) control to maximize open loop gain at low frequencies (DC), while additional pole placement methods are used to stabilize the undamped poles of the rotor by implementing additional phase or gain margin where required. In many AMB applications, the goal of the controller is to maximize open loop gain in the DC region where levitation is achieved, while rejecting disturbances at higher frequencies either by adding damping or minimizing the magnitude of the open loop response. The identified rotor model provides a means to design a PI and pole placement controller to achieve satisfactory levitation control of the system. The resulting models in Figs. 10 and 11 represent the two independently controlled (SISO) axes, which is compared with the directly measured closed loop response. These models were calculated using the standard industry practice of combining the identified rotor dynamic models with the known controller and the calculated response of the bearings and sensors. The controller, bearings, and sensors are defined from SKF Magnetic Mechatronics design calculations, the accuracy of which is verified using the fit between the calculated and measured closed loop frequency response. The fit between the simulated and the measured closed loop frequency response is 72.4 and 85.3% for the V13 control axis and V24 control axis respectively, and the frequency response prediction of the model shows strong agreement with the measured response of the system.

### 5 Conclusions

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.