Acta Polytechnica https://doi.org/10.14311/AP.2022.62.0479 Acta Polytechnica 62(4):479–487, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague ACTIVE DISTURBANCE REJECTION CONTROL-BASED ANTI-COUPLING METHOD FOR CONICAL MAGNETIC BEARINGS Danh Huy Nguyena, Minh Le Vua, Hieu Do Tronga, Danh Giang Nguyenb, Tung Lam Nguyena, ∗ a Hanoi University of Science and Technology, School of Electrical Engineering, 1 Dai Co Viet st, Hanoi 100000, Vietnam b National University of Civil Engineering, Faculty of Mechanical Engineering, 55 Giai Phong st, Hanoi 100000, Vietnam ∗ corresponding author: lam.nguyentung@hust.edu.vn Abstract. Conical-shape magnetic bearings are currently a potential candidate for various magnetic force-supported applications due to their unique geometric nature reducing the number of required active magnets. However, the bearing structure places control-engineering related problems in view of underactuated and coupling phenomena. The paper proposes an Adaptive Disturbance Rejection Control (ADRC) for solving the above-mentioned problem in the conical magnetic bearing. At first, virtual current controls are identified to decouple the electrical sub-system, then the active disturbance rejection control is employed to eliminate coupling effects owing to rotational motions. Comprehensive simulations are provided to illustrate the control ability. Keywords: Conical active magnetic bearings, over-actuated systems, ADRC, coupling mechanism, linearization. 1. Introduction Recently, active magnetic bearing (AMB) has been of increasing interest to the manufacturing industry due to its contactless, lubrication-free, no mechan- ical wear, and high-speed capability [1–3]. These characteristics enable them to be employed in a va- riety of applications, including artificial hearts [4], vacuum pumps [5], and flywheel energy storage sys- tems [6] and [7], etc. The motion resolution of the suspended object in translation or high-speed rotation is restricted solely by the actuators, sensors, and the servo system utilised due to the non-contact nature of a magnetic suspension. As a result, magnetic bearings can be utilised in almost any environment as long as the electromagnetic parts are suitably shielded, for example, in open air at temperatures ranging from 235 °C to 450 °C [8]. Many researchers, in particular, have endeavored to design a range of AMBs that are compact and simple-structured while still performing well. Because of the advantages of a cone-shaped active magnetic bearing (AMB) system, such as its simple structure, low heating, and high dependability, there is an increasing number of studies on it [9, 10]. The structure of a conical magnetic bearing is iden- tical to that of a regular radial magnetic bearing, with the exception that both the stator and rotor working surfaces are conical, allowing force to be ap- plied in both axial and radial directions [11, 12]. The conical form saves axial space, which can be used to install gears and other components for an added mechanical benefit. It also conserves energy for an op- timal load support. Conical electromagnetic bearings feature two coupled properties as compared to ordi- nary radial electromagnetic bearings: current-coupled and geometry-coupled effects, making dynamic mod- elling and control of these systems particularly diffi- cult. The current-coupled effect exists because the axial and radial control currents flow in the bearing coils simultaneously. Furthermore, the inclined angle of the magnet core causes a geometry-coupled effect. Coupled dynamic characteristics of the rotor conical magnetic bearing system became known due to the existence of the two coupled effects. So far, several researchers have discussed the modelling and control of cone-shaped AMBs [2, 13, 14]. Lee CW and Jeong HS presented a control method for conical magnetic bearings in [12], which allows the rotor to float in the air stably. They proposed a completely connected linearised dynamic model for the cone-shaped magnet coil that covers the relationships between the input voltage and output current. The connected controller uses a linear quadratic regulator with integral action to stabilise the AMB system, while the decoupled controller is used to stabilise the five single DOF sys- tems. Abdelfatah M. Mohamed et al. [11] proposed the Q-parameterization method for designing system stabilisation in terms of two free parameters. The proposed technique is validated using a digital simula- tion. As a result, plant parameters such as transient and forced response are good, and stiffness character- istics are obtained with small oscillation. Recently, in [15], E. E. Ovsyannikova and A. M. Gus’kov cre- 479 https://doi.org/10.14311/AP.2022.62.0479 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en Huy, Minh, Hieu et al. Acta Polytechnica ated a mathematical model of a rigid rotor suspended in a blood flow and supported by conical active mag- netic bearings. They used the proportional-integral differential (PID) control, which takes into account the influence of hydrodynamic moments, which affect the rotor from the side of the blood flow, as well as external influences on the person. The experimental findings are reported, with a rotor speed range of 5000 to 12000 rpm and a placement error of less than 0.2 mm. Nguyen et al. introduced a control approach considering input and output constraints in the mag- netic bearing system in [16] and [17]. The control restricts the rotor displacement in a certain range according to the system structure. In [18], modelling of a conical AMB structure for a complete support of 5-dof rotor system was reported by A. Katyayn and P. K. Agarwal, who improved the system’s per- formance by creating the Interval type-2 fuzzy logic controller (IT2FLC) with an uncertain bound algo- rithm. This controller reduces the need for a precise system modeling while also allowing for the handling of parameter uncertainty. The simulation results show that the proposed controller outperforms the type-1 fuzzy logic controller in terms of transient responses. In this paper, we examine the concept of conical magnetic bearings for both the radial and axial dis- placement control. The governing equations charac- terising the relationship between magnetic forces, air gaps, gyroscopic force, and control currents are used to build the nonlinear model of the conical magnetic bearing. The main contribution of the paper is that rotational motions are treated as disturbances and are handled by the Active Disturbance Rejection Control (ADRC) [19, 20] to stabilise the cone-shaped AMB system. ADRC was developed as an option that com- bines the easy applicability of conventional PID-type control methods with the strength of modern model- based approaches. The core of ADRC is an extended observer that treats actual disturbances and modelling uncertainty together, using only a very coarse process model to create a control loop. Because of the ex- cellent abilities of ADRC, the paper also tackles the unwanted dynamics due to rotational motions, which are normally neglected in other related works. The ignorance might lead to system degradation due to a high operating speed resulting in strong coupling effects. The effectiveness of the proposed control struc- ture for stabilising the rotor position and rejecting coupling-phenomenon-induced disturbance is numeri- cally evaluated through comprehensive scenarios. 2. Dynamic modelling of conical magnetic bearings Consider the simplified model of a conical magnetic bearing system as shown in Fig. 1. It is assumed that the rotor is rigid and its centre of mass and geometric centre are coincide. Furthermore, the assumptions of non-saturated circuit and negligible flux linkage between magnetic coils are made. Rm and β are the Figure 1. Model of a cone-shaped active magnetic bearing system. effective radius and inclined angle of the magnetic core, b1 and b2 are the distances between the two rad- ical magnetic bearing and the centre of gravity of the rotor; Fj (j =1 to 8) are the magnetic forces produced by the stator and exerted on the rotor; (x, y, z) and (θx, θy , θz ) are the displacement and angular coordi- nates defined with respect to the centre of mass. The cone-shaped active magnetic bearing system can be modelled as follows: mẍ = (F1 + F2 + F5 + F6) sin β − (F3 + F4 + F7 + F8) sin β − mg. mÿ = (F1 − F2 + F3 − F4)cosβ. mz̈ = (F5 − F6 + F7 − F8)cosβ. Jdθ̈y = [(F6 − F5)b1 + (F7 − F8)b2]cosβ + (F5 − F6 + F8 − F7)Rm sin β + J θ̇xθ̇z . Jdθ̈z = [(F1 − F2)b1 + (F4 − F3)b2]cosβ + (F2 − F1 + F3 − F4)Rm sin β + J θ̇xθ̇y (1) where J is the moment of inertia of the rotor about the axis of rotation. The mass and moment of inertia of the rotor are m and Jd, respectively. We also consider the effect of the x-axis rotation on the other two axes. Here, the first three equations in Eq. (1) are the kinematics of the rotor’s transverse motion, while the last two equations represent the rotor’s rotational dy- namics. In addition, in the two rotational kinematics equations, there is an additional component of the feedback force. Suppose that when the rotor rotates rapidly if a force is applied to the y-axis (z-axis) that is sufficiently large to deflect the rotor from the axis of motion by a small angle, the rotor itself will also react back to a torque of the corresponding magnitude equal to J θ̇xθ̇z . Similarly, the component of gyro force 480 vol. 62 no. 4/2022 Active disturbance rejection control-based . . . Figure 2. Simplified model of the cone-shaped AMB system. along the z-axis is computed. In order to linearise, the dynamic equation (1), small motions of the rotor are considered. Fig. 2 shows the change of the air gap of the cone-shaped magnet, which is written as: gy1,2 = go − x sin β ± (y + b1θz ) cos β gy3,4 = go + x sin β ± (y − b2θz ) cos β gz1,2 = go − x sin β ± (z + b1θy ) cos β gz3,4 = go + x sin β ± (z + b2θy ) cos β (2) where go is the steady-state nominal air gap. The magnetic force can be written regrading to he actual air gap and the current as: F1,2 = µoApN 2(Io1 + iy1,2 ) 2 4gy1,2 2 F3,4 = µoApN 2(Io2 + iy3,4 ) 2 4gy3,4 2 F5,6 = µoApN 2(Io1 + iz1,2 ) 2 4gz1,2 2 F7,8 = µoApN 2(Io2 + iz3,4 ) 2 4gz3,4 2 (3) where µo(= 4π × 10−7H/m) is the permeability of free space; Ap = A/cosβ, A is the cross-sectional area, N is the number of coil turns; iqj (j = 1, 4&q = y, z) is the control current of each magnet; Io1 and Io2 are the bias currents in the upper and lower bearing. Assume that the current change and the displacement of the rotor are small relative to the bias current Io and the nominal air gap. Apply Eq. (2) to Eq. (3) and use the Taylor expansion series to obtain the magnetic force, which is linearised as: F1,2 = Fo1 + Ki1 iy1,2 + Kq1 x sin β ± Kq1 (y + b1θz ) cos β F3,4 = Fo2 + Ki2 iy3,4 + Kq2 x sin β ± Kq2 (y − b2θz ) cos β F5,6 = Fo1 + Ki1 iz1,2 + Kq1 x sin β ± Kq1 (z − b1θy ) cos β F7,8 = Fo2 + Ki2 iz3,4 + Kq2 x sin β ± Kq2 (z + b2θy ) cos β (4) where Foj = µoApN 2Ioj 2 4go 2 , j = 1, 2 are the steady-state magnetic forces and kqj = 2Foj go , kij = 2Foj Ioj , j = 1, 2 are the position and current stiffnesses, respectively. From the combination of Eqs. (1) and (4), the linear differential equation showing the kinematics of the 5 degrees of freedom conical-AMB drive system can be rewritten as: Mbq̈b + Kbqb = Kibmim + Gq̇b (5) where qb = { x, y, z, θy , θz } T im = { iy1 , iy2 , iy3 , iy4 , iz1 , iz2 , iz3 , iz4 } T Kb=   −Kxx 0 0 0 0 0 −Kyy 0 0 −Kyθz 0 0 −Kzz −Kzθy 0 0 0 −K θy z −Kθy θy 0 0 −Kθz y 0 −Kθz θz 0   G =   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 J . θx 0 0 0 J . θx 0   Kibm=   Ki1 Sβ Ki1 Cβ 0 0 Ki1 σ Ki1 Sβ −Ki1 Cβ 0 0 −Ki1 σ −Ki2 Sβ Ki2 Cβ 0 0 Ki2 γ −Ki2 Sβ −Ki2 Cβ 0 0 −Ki2 γ Ki1 Sβ 0 Ki1 Cβ Ki1 α 0 Ki1 Sβ 0 −Ki1 Cβ −Ki1 α 0 −Ki2 Sβ 0 Ki2 Cβ Ki2 γ 0 −Ki2 Sβ 0 −Ki2 Cβ −Ki2 γ 0   Mb =   m 0 0 0 0 0 m 0 0 0 0 0 m 0 0 0 0 0 Jd 0 0 0 0 0 J d   Kxx = 4 (Kq1 + Kq2 ) S 2 β Kyy = Kzz = 2C2β (Kq1 + Kq2 ) Kyθz = Kzθy = 2C 2 β (Kq1 b1 + Kq2 b2) 481 Huy, Minh, Hieu et al. Acta Polytechnica Kθy z = 2C 2β (Kq1 b1 + Kq2 b2) + S(2β)Rm (Kq1 − Kq2 ) Kθz y = 2C 2β (Kq1 b1 − Kq2 b2) − S(2β)Rm (Kq1 − Kq2 ) Kθy θy = 2C 2β ( Kq1 b1 2 − Kq2 b2 2) + S(2β)Rm (Kq1 b1 + Kq2 b2) Kθz θz = 2C 2β ( Kq1 b1 2 + Kq2 b2 2) − S(2β)Rm (Kq1 b1 + Kq2 b2) α = b1Cβ + RmSβ; σ = b1Cβ − RmSβ γ = b2Cβ − RmS β Here, qb is the displacement vector defined in the mass centre coordinates; im is the control current vec- tor and Mb , Kb and Kibm are the mass, position stiffness, and current stiffness matrices, respectively. As can be observed, the system’s equation is compli- cated and coupled because the components outside the main diagonal of the matrices, Kb , Kibm and G are non-zero. Due to this characteristic, conventional linear control rules cannot be applied directly to each motion direction. As a result, the Active Disturbance Rejection Control (ADRC) algorithm is employed to handle the coupling effects by taking these effects as system disturbances. 3. Control system design The conical AMB system is naturally unstable, a closed-loop control is required to stabilise the ro- tor position. The control current of the system can be calculated through the control structure “different driving mode”, which is shown in Fig. 3. Figure 3. Conceptual control loop of the cone-shaped active magnetic bearings. The main principle of the aforementioned structure: where controlling the position of the rotor according to the Y − axis and Z − axis, the magnet pairs are in the poles that are opposite each other. For example, iy1 and iy2 magnets, as well as, iy3 and iy4 , iz1 and iz2 , iz3 and iz4 are similarly controlled by this structure. Here, the magnet in each pair is controlled by the sum of the bias current and control current, and the other with the difference of the bias current and control current. This means that when the rotor is displaced from its equilibrium position, the “different driving mode” controls the pairs of magnets, whereas when the rotor is in its equilibrium position, only the bias current is present on each pair of the magnets. When the rotor deviates from the equilibrium position, the current through the pairs of magnets is written as in the following equation:   iy1 iy2 iy3 iy4 iz1 iz2 iz3 iz4   =   Io1 Io1 Io2 Io2 Io1 Io1 Io2 Io2   +   1 0 0 0 −1 −1 0 0 0 −1 0 1 0 0 1 0 −1 0 0 1 0 0 1 0 −1 0 0 0 −1 −1 0 0 0 1 1 0 0 0 −1 1     Iyt Iyd Izt Izd Ix   (6) where Io = [Io1 , Io1 , Io2 , Io2 , Io1 ,Io1 , Io2 , Io2 ] T is the bias current. At steady-state, consider Io = 0, ir = [Iyt, Iyd, Izt, Izd, Ix] T is the x, y, and z axes’ virtual control current. Ix is the virtual control current of x-axes. The virtual control current in the upper half of the y and z axes is (Iyt, Izt) , whereas the virtual control current in the bottom half is (Iyd, Izd). H =   1 −1 0 0 0 0 0 0 0 0 1 −1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 −1 1 −1 −1 −1 1 1 −1 −1 1 1   T In this case, Eq. (5) can be rewritten as: Mbq̈b − Gq̇b + Kbqb = KibmHir (7) Since the control is performed in the bearing coor- dinates, rewriting the equations of motion in bear- ing coordinates utilising the relationship between the mass-centre coordinates (x, y, z, θy , θz ) and the bear- ing coordinates (x, y1, y2, z1, z2), given by: qb = { x, y, z, θy , θz } T and qse = { x, y1, y2, z1, z2} T qse = Tqb with T is the coordinate transfer matrix T =   1 0 0 0 0 0 1 0 0 b1 0 1 0 0 −b2 0 0 1 −b1 0 0 0 1 b2 0   Eq. (1) shows that the inter-channel effect occurs at Kb and KibmH because the major non-diagonal components are not zero. The Kb and KibmH are invertible. The following control structure is used to eliminate the interstitial component: ir = (KibmH)−1(v + KbT−1qse) (8) 482 vol. 62 no. 4/2022 Active disturbance rejection control-based . . . Figure 4. Control loop structure with active distur- bance rejection control (ADRC). Eq. (7) can be rewritten as: Mbq̈b − Gq̇b = v (9) where v is the new control signal’s vector. The intersti- tial component has been removed in the control chan- nels (x, y, z) leaving just the interstitial component in the control channel (θy , θz ) owing to the gyroscope force. The original model of the magnetic bearing is a com- plex, multivariable nonlinear system, through the pro- cess of linearity and decoupling, we have the linear form of the system shown in Eq. (9) with 5 inputs and 5 outputs. The full form of Eq. (9) is shown as follows:   m .. x = v1 m .. y = v2 m .. z = v3 Jd .. θy +J . θx . θz = v4 Jd .. θz +J . θx . θy = v5 (10) Remark: It is noted that the first three equations of Eq. (10) characterise the transnational motions and can be readily stabilized using v1, v2, and v3. The last two equation indicate coupling mechanism related to the rotational motion of the rotor that is normally ignored. In practice, the magnetic bearing is normally employed to operate in high speed range, hence rotational motion effects can not be neglected. In this section, the ADRC controller will be used to remove the remaining coupling components as well as to stabilise the control object. An ADRC controller is used for each input and output pair (x, v1), (y, v2), (z, v3), (θy , v4), (θz , v5). In ADRC de- sign, f (t) is unknown and is considered as a “general- ized disturbance”, and b0 is the available information concerning the model. The according structure of the control loop with ADRC is presented in Fig. 4. The fundamental idea of ADRC is to implement an extended state observer (ESO) that provides an esti- mate, f̂ (t) , such that we can compensate the impact of f (t) on our system. The equation for the extended state observer is given as:  ˙̂x1 (t)˙̂x2 (t) ˙̂x3 (t)   =   0 1 00 0 1 0 0 0     x̂1 (t)x̂2 (t) x̂3 (t)   +   0b0 0   u (t) +   l1l2 l3   (y (t) − x̂1 (t)) =   −l1 1 0−l2 0 1 −l3 0 0   ︸ ︷︷ ︸ A−LC   x̂1 (t)x̂2 (t) x̂3 (t)   +   0b0 0   ︸ ︷︷ ︸ B u (t) +   l1l2 l3   y (t) (11) where x̂1(t) = ŷ(t); x̂2(t) = ˙̂y(t); x̂3(t) = f̂ (t). Re- moving the unknown components is done through the following control law: ÿ(t) = (f (t) − f̂ (t)) + u0(t) ≈ u0(t) ≈ KP .((r(t) − y(t)) − KD .ẏ(t)) (12) where r is the setpoint. In order to work properly, observer parameters, l1, l2, l3, in Eq. (11) still have to be determined. According to [20], the ADRC’s parameters can be chosen to tune the closed-loop to a critically damped behaviour and a desired 2% set- tling time Tsettle. The tuning procedure is summarised as follows: Kp = ( sCL )2 , KD = −2.sCL l1 = −3.sESO, l2 = 3. ( sESO )2 , l3 = ( sESO )3 (13) with sCL = − 6 Tsettle being the negative-real double closed-loop pole. sESO ≈ (3...10).sCL is the observer pole. Using the ADRC controller to calculate the variable x, y and z are calculated similarly: ẍ = ( 1 m .d(t) + ∆b.u(t))︸ ︷︷ ︸ f (t) +b01v1 = f (t) + b01.v1(t) v1(t) = KP 1.((r(t) − x̂(t)) − KD1. . x̂(t)) (14) For equations containing the two variables (θy and θz ), which have an interleaved component between the two equations. Because the interleaved component is unknown, the extended observer can be used to estimate and analyse it, using the ADRC controller with variable θy , as follows: θ̈y = ( 1 J d(t) + ∆b.v4(t) + J θxθz) + b04v4 = f (t) + b04.v4(t) b04 = 1 J v4(t) = KP 4.((r(t) − θ̂y (t)) − KD4. ˙̂ θy (t)) (15) 483 Huy, Minh, Hieu et al. Acta Polytechnica Name Symbol b01 = b02 = b03 1/m b04 = b05 1/J Tsettle 0.1 (s) sCL -60 KP i (i = 1, . . . , 5) 3600 KDi (i = 1, . . . , 5) 120 sESO -420 l1i (i = 1, . . . , 5) 1260 l2i (i = 1, . . . , 5) 529200 l3i (i = 1, . . . , 5) 74088000 Table 1. Controller parameters. 4. NUMERICAL SIMULATIONS In this section, we consider two scenarios to evalu- ate the effectiveness of using the ADRC controller in the case of variable speed rotation and rotor load disturbance. Bearing design parameters Value Radial air gap g0 0.5 mm Cross-sectional area A 18*10 mm Inclined angle β 10o Magnetic coils N 300 turns Resistance R 2 Ω Inductance of wire L0 20 mH Rotor mass M 1.86 Kg Moment of inertia Jd 0.00647 kgm2 Moment of inertia Jp 0.00121 kgm2 Bias current I01, I02 1.6 A,1 A Bearing span b1, b2 81.7 mm,71.6 mm Table 2. System parameters. 4.1. Simulation scenario 1: We design an ADRC controller with a rotor rotation speed of 3000 rpm. The initial values of the rotor’s centre of mass position are: x0 = 0.25.10−3; y0 = 0.2.10−3; z0 = 0.125.10−3; θy = 0.1.10−3; θz = 0.2.10−3. Select the coefficients of the ADRC as fol- lows sCL = − 60.1 , s ESO = 7sCL, KP = (sCL)2, KD = −2sCL, l1 = −3sESO; l2 = 3 ( sESO )2 , l3 = −(sESO)3. The position of the centre of mass and the deflection angle of the rotor return to the equilibrium position after a time interval of 0.1 seconds and there is no overshoot in Fig. 5 and Fig. 6. From Fig. 7, initially, when the rotor position deviates from the equilibrium position, a control current is generated to bring the rotor back to the equilibrium position. After the rotor is in the equilibrium position, the control current is zero so that the bias currents I01 and I02 keep the rotor in this equilibrium state. The impact force of the magnet is shown in Fig. 8 as having a significant value Figure 5. Response to the position of the x, y, z axes. Figure 6. The position of the axis angle θy , θz . Figure 7. Control current response. at first to bring the rotor to equilibrium, but once the rotor returns to equilibrium, the force is kept stable at the values F01 and F02. From the above results, it can be concluded that the controller is designed to completely satisfy the requirements. 484 vol. 62 no. 4/2022 Active disturbance rejection control-based . . . Figure 8. Impact force of electromagnets. Figure 9. Velocity deviation of x, y, z axes according to observer. Figure 10. Velocity deviation of θy , θz axes according to observer. Based on Fig. 9 and Fig. 10, the observer satisfied the requirements, and the estimated velocity values were near to the real velocity value after 0.1 s. 4.2. Simulation scenario 2: The rotor speed will be changed to 12000 rpm to eval- uate the controllability of the controller when the rotor is in the high-speed region, the initial value of the rotor’s centre of mass is: x0 = 0.25.10−3; y0 = 0.2.10−3; z0 = 0.125.10−3; θy = 0.1.10−3; θz = 0.2.10−3. The simulation results on the x, y, and z axes are identical to the first simulation scenario, as shown in Fig. 12, where the angular position responses of the axes θy , θz have an undershoot and the response time has been increased to 0.2 seconds. Only the θy and θz axes are impacted when the rotor rotates at high speeds, but it soon returns to equilibrium. The suggested controller takes into account the rotor speed factor and demonstrates its capacity to function well in the high-speed region. Figure 11. Response to the position of the x, y, z axes. Figure 12. The position of the axis angle θy , θz . 485 Huy, Minh, Hieu et al. Acta Polytechnica Figure 13. Control current response. Figure 14. Impact force of electromagnets. 5. Conclusions In the paper, we consider the cone-shaped magnetic bearing, which is characterised as a class of under- actuated and strongly coupled systems. Based on control current distribution, the coupling mechanism in electrical sub-system is solved. Subsequently, an active disturbance control is adopted to tackle the rotational-motion-induced disturbance acting on the system. The simulations are carried out proving that the proposed control can effectively bring the the rotor to equilibrium. 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Electronics 2(3):246–279, 2013. https://doi.org/10.3390/electronics2030246. 487 https://doi.org/10.1109/TMECH.2005.859830 https://doi.org/10.1109/TMAG.2013.2263284 https://doi.org/10.30880/ijie.2021.13.05.026 https://doi.org/10.1109/TMAG.2013.2295060 https://doi.org/10.1109/CDC.1989.70176 https://doi.org/10.1016/0967-0661(96)00149-9 https://doi.org/10.1177/0954406213517488 https://doi.org/10.1023/A:1024137007918 https://doi.org/10.3103/S1052618820010100 https://doi.org/10.11591/ijece.v8i5.pp3666-3677 https://doi.org/10.11591/ijpeds.v11.i4.pp2154-2163 https://doi.org/10.11591/ijpeds.v11.i4.pp2154-2163 https://doi.org/10.1109/ICPCES.2017.8117616 https://doi.org/10.1109/TIE.2008.2011621 https://doi.org/10.3390/electronics2030246 Acta Polytechnica 62(4):479–487, 2022 1 Introduction 2 Dynamic modelling of conical magnetic bearings 3 Control system design 4 NUMERICAL SIMULATIONS 4.1 Simulation scenario 1: 4.2 Simulation scenario 2: 5 Conclusions Acknowledgements References