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Abstract

Magnetic bearings have been used in flywheel energy storage systems to improve their performance because these kinds of bearings can provide non-contact supporting so that the friction between the rotor and the bearings is reduced significantly. However, the gyroscopic coupling, parameter coupling, and imbalance force affect the operating performance and stability of a magnetic suspended flywheel energy storage system with asymmetric rotor; therefore, the main purpose of this study is to propose a control method for achieving decoupling and stable operation of the aforementioned system. To this end, after deriving the mathematical model of a radial 4-degree-of-freedom rotor–bearing system, a novel cross-feedback-based modal decoupling controller is designed for vibration suppression caused by gyroscopic coupling, parameter coupling, and imbalance force. Better performance is obtained through comparing the decoupling performance, control performance, and disturbance rejection performance with a traditional decentralized proportional–integral–derivative controller and a cross-feedback controller via ADAMS–MATLAB co-simulation technology and experimental results.

Keywords

Flywheel energy storage system magnetic bearings vibration suppression modal decoupling control disturbance rejection

Introduction

Flywheel energy storage systems which are also called flywheel battery have been used more and more for energy harvesting and recycling in many situations, such as space satellites, renewable energy power generation systems, and electric vehicles, because of their cleanliness and high-density energy storage. Their capacity of energy storage almost depends on the rotor velocity; however, the traditional flywheel energy storage systems have significant friction between the rotor and the bearings. Hence, recently the applications of flywheel energy storage technologies are limited.

Magnetic bearings have been used in flywheel energy storage systems to improve their performance because these kinds of bearings can provide non-contact supporting. As the rotation is frictionless, there is no wear and no lubrication requirement, and very high rotation speeds are possible.¹ However, the gyroscopic coupling, parameter coupling, and imbalance force could cause rotor vibration so that the operating performance and stability of flywheel energy storage systems could be degraded. Hence, the main purpose of this study is to propose a control method for achieving decoupling and stable operation of the aforementioned system.

To this end, various control schemes have been presented. These methods can be classified into two types: decentralized control, which is a combination of several single-degree-of-freedom (DOF) controllers,^2–5 and centralized control, which is a multiple DOF controller.^6–10 Chen et al. presented a decentralized proportional–integral–derivative neural network (PIDNN) control scheme for regulating and stabilizing a 5-DOF active magnetic bearing (AMB). Good control performance and robustness were verified experimentally.² Zhang et al. proposed a controller with local damping enhancement and cross feedback to suppress the gyroscopic effect and bending vibration in a flywheel energy storage system. The effectiveness of the proposed method was demonstrated by experimental results.³ Ouyang et al. proposed a double-stage cross-feedback controller for suppressing the vibration caused by gyroscopic coupling effect and imbalance force in a wind power generator with magnetic bearings. Comparative simulations showed that the proposed method has steady accuracy and disturbance rejection performance.⁴ Zhang et al. presented a modal decoupling controller for a magnetic bearing–supported flywheel–rotor system. The results confirmed that the proposed controller can separately regulate each modal stiffness and damping through decoupling between conical and parallel modes, and obviously improve the dynamic behaviors and anti-interference performance.⁵ Peng et al. presented a robust controller with a disturbance observer for a hybrid magnetic bearing system used in a control moment gyro device. The controller gains were determined by a linear matrix inequality (LMI)–based approach. Better disturbance rejection and robust performance were obtained when compared with a traditional $H_{\infty}$ controller via simulations and experiments.⁶ Abouelsoud et al. presented a model reference adaptive controller for compensating the gyroscopic coupling in a vertical shaft magnetic bearing system. The stability of the whole system was confirmed by Lyapunov approach, and its effectiveness was verified by simulations.⁷ Balini et al. presented an unstable $H_{\infty}$ controller for an AMB spindle with flexible dynamics. The effectiveness of the proposed method was demonstrated by experimental results.⁸ Zheng et al. presented an optimal controller to compensate for gyroscopic coupling in an AMB system with a flexible rotor. The effectiveness of the proposed method was verified using the experimental results.⁹ Fang et al. presented a µ-synthesis control scheme for a magnetic bearing–supported double-gimbal control moment gyro device. Good vibration suppression and robust performance were obtained by the proposed method via simulations and experiments.¹⁰

Other methods have also been presented for AMB control including sliding mode control,^11–13 nonlinear smooth feedback control,¹⁴ adaptive control,¹⁵ and fuzzy logic control.¹⁶

However, only symmetry rotor was considered in most of the studies in the literature. Due to the limitation of the manufacturing process and the requirement of practical application, magnetic bearing systems are usually asymmetrical in size. In this situation, not only gyroscopic coupling but also parameter coupling could affect the stability and control performance significantly. Although centralized control methods such as sliding mode control could be applied to solve the aforementioned problem, decentralized control method such as cross-feedback controller or modal decoupling controller have a simple structure and these can be achieved only by adding cross channels based on decentralized controllers. Furthermore, it has good compensating effect on the gyroscopic coupling.³ However, the traditional cross-feedback control or modal decoupling control method shows worst decoupling and robust performance to deal with asymmetric rotor systems. Hence, we combine the advantages of these algorithms and propose a cross-feedback-based controller that can solve the problem mentioned in this study. To this end, after deriving the mathematical model of a radial 4-DOF rotor–bearing system, a novel cross-feedback-based modal decoupling controller is designed. Better performance is obtained through comparing the decoupling performance, control performance, and disturbance rejection performance with a traditional decentralized proportional–integral–derivative (PID) controller and a cross-feedback controller via ADAMS–MATLAB co-simulation technology and experimental results.

In summary, the major contributions of this study are as follows.

To the best of our knowledge, the proposed method is the first cross-feedback-based modal decoupling control method for magnetic suspended flywheel energy storage system.

The proposed controller has better decoupling performance and robustness with respect to parameter variation such as rotor velocity.

The designed controller system has clear physical concept than the existing centralized control method for magnetic suspended flywheel energy storage system.

Modeling of the rotor–bearing system

In this section, the mathematical model of a magnetic suspended flywheel energy storage rotor system is presented as shown in Figure 1. Two radial AMBs and two position sensors are implemented on both ends of the rotor, respectively. $o - xyz$ , $o_{l} - x_{l} y_{l}$ , $o_{r} - x_{r} y_{r}$ , $o_{sl} - x_{sl} y_{sl}$ , and $o_{sr} - x_{sr} y_{sr}$ denote the center of mass coordinate, left-side and right-side bearing coordinates, and left-side and right-side sensor coordinates, respectively.

Figure 1.

Schematic model of the rotor.

Using the assumptions mentioned in Ouyang et al.,⁴ the equations of translational and rotational motions for the rotor system are¹

m {\overset{\cdot\cdot}{x}}_{c} = F_{Lx} + F_{Rx} + F_{dLx} + F_{dRx}

(1)

m {\overset{\cdot\cdot}{y}}_{c} = F_{Ly} + F_{Ry} + F_{dLy} + F_{dRy}

(2)

J_{x} {\overset{\cdot\cdot}{θ}}_{x} - J_{z} ω {\overset{\cdot}{θ}}_{y} = a F_{Ly} - b F_{Ry} + d_{L} F_{dLy} - d_{R} F_{dRy}

(3)

J_{x} {\overset{\cdot\cdot}{θ}}_{y} + J_{z} ω {\overset{\cdot}{θ}}_{y} = a F_{Ly} - b F_{Ry} + d_{L} F_{dLx} - d_{R} F_{dRx}

(4)

where the parameters $x_{c}$ , $y_{c}$ , $θ_{x}$ , and $θ_{y}$ denote the center of mass of the rotor position and the pitch angles around the x- and y-axes in the generalized coordinate, respectively. The parameters $ω$ , m, $J_{x}$ , $J_{y}$ , and $J_{z}$ denote the angular velocity of the rotor, the rotor mass, the diametral moment of inertia in the x- and y-directions, and the polar moment of inertia, respectively. The parameters $F_{Lx}$ , $F_{Rx}$ , $F_{Ly}$ , and $F_{Ry}$ denote the radial bearing forces F acting on the rotor plane in the x- and y-directions, respectively. The parameters $F_{dLx}$ , $F_{dRx}$ , $F_{dLy}$ , and $F_{dRy}$ denote the load forces $F_{d}$ acting on the rotor plane in the x- and y-directions, respectively.

The above equations can be represented as follows

\begin{matrix} M {\overset{\cdot\cdot}{q}}_{c} + G {\overset{\cdot}{q}}_{c} = T_{L} F + T_{d} F_{d} + P_{unb} \\ y = T_{s} q_{c} \end{matrix}

(5)

\begin{matrix} M = [\begin{matrix} m & 0 & 0 & 0 \\ 0 & m & 0 & 0 \\ 0 & 0 & J_{x} & 0 \\ 0 & 0 & 0 & J_{y} \end{matrix}], G = [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & J_{z} ω \\ 0 & 0 & - J_{z} ω & 0 \end{matrix}], \\ T_{L} = [\begin{matrix} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & a & - b \\ a & - b & 0 & 0 \end{matrix}] \\ T_{s} = [\begin{matrix} 1 & 0 & 0 & c \\ 1 & 0 & 0 & - d \\ 0 & 1 & c & 0 \\ 0 & 1 & - d & 0 \end{matrix}], T_{d} = [\begin{matrix} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & g & - h \\ g & - h & 0 & 0 \end{matrix}], \\ q_{c} = [\begin{matrix} x \\ y \\ - θ_{x} \\ θ_{y} \end{matrix}], F = [\begin{matrix} F_{Lx} \\ F_{Rx} \\ F_{Ly} \\ F_{Ry} \end{matrix}] \\ F_{d} = [\begin{matrix} F_{dLx} \\ F_{dRx} \\ F_{dLy} \\ F_{dRy} \end{matrix}], P_{unb} = [\begin{matrix} me ω^{2} \cos (ω t + θ_{1}) \\ me ω^{2} \sin (ω t + θ_{1}) \\ (J_{x} - J_{z}) ε ω^{2} \cos (ω t + θ_{2}) \\ (J_{y} - J_{z}) ε ω^{2} \sin (ω t + θ_{2}) \end{matrix}] \end{matrix}

The parameters e, $ε$ , $θ_{1}$ , and $θ_{2}$ are the eccentricity, the inclination of the spindle, the static unbalance angular position, and the dynamic unbalance angular position, respectively.

On the other hand, the linearized radial bearing forces F at equilibrium point can be represented as follows⁴

F = K_{i} i - K_{s} q_{L}

(6)

\begin{matrix} K_{i} = [\begin{matrix} k_{iLx} & 0 & 0 & 0 \\ 0 & k_{iRx} & 0 & 0 \\ 0 & 0 & k_{iLy} & 0 \\ 0 & 0 & 0 & k_{iRy} \end{matrix}], \\ K_{s} = [\begin{matrix} k_{sLx} & 0 & 0 & 0 \\ 0 & k_{sRx} & 0 & 0 \\ 0 & 0 & k_{sLy} & 0 \\ 0 & 0 & 0 & k_{sRy} \end{matrix}], i = [\begin{matrix} i_{Lx} \\ i_{Rx} \\ i_{Ly} \\ i_{Ry} \end{matrix}], q_{L} = [\begin{matrix} x_{l} \\ x_{r} \\ y_{l} \\ y_{r} \end{matrix}] \end{matrix}

where $K_{i}$ , $K_{s}$ , i, and $q_{L}$ denote the current stiffness matrix, the position stiffness matrix, the controlled current, and the rotor position variables within the magnetic bearings.

The dynamic model of the rotor–bearing system can be represented as follows by combining equations (5) and (6)

M {\overset{\cdot\cdot}{q}}_{c} + G {\overset{\cdot}{q}}_{c} + T_{L} K_{s} T_{L}^{T} q_{c} = T_{L} K_{i} i + U_{d}

(7)

where $U_{d} = T_{d} F_{d} + P_{unb}$ , which can be considered as the disturbance of the system. The block diagram of the 4-DOF magnetic bearings is shown in Figure 2. It is noted that the mathematical model of the rotor system is only used for controller design.

Figure 2.

Four-degree-of-freedom magnetic bearing system.

Cross-feedback-based modal decoupling controller design

As mentioned previously, the vibration in the asymmetric rotor system was caused by not only the gyroscopic coupling but also by the modal coupling due to the asymmetric system parameters. Meanwhile, the mass unbalance and external loads are also the major reasons causing rotor vibration. Many methods have been proposed to solve the aforementioned problem. However, the traditional decentralized control and centralized control showed worst control performance, especially the decoupling performance. Hence, a cross-feedback-based modal decoupling controller is proposed for suppressing the vibration of the magnetic suspended flywheel energy storage system.

The following PID controller is applied to each DOF of the rotor system

\begin{matrix} G_{t} (s) = P_{t} (s) + I_{t} (s) + D_{t} (s) \\ P_{t} (s) = diag {k_{p 1}, k_{p 2}, k_{p 3}, k_{p 4}} \\ I_{t} (s) = diag {k_{I 1} \times \frac{1}{T_{i 1} s}, k_{I 2} \times \frac{1}{T_{i 2} s}, k_{I 3} \times \frac{1}{T_{i 3} s}, k_{I 4} \times \frac{1}{T_{i 4} s}} \\ D_{t} (s) = diag {k_{D 1} \times \frac{τ_{1} s + 1}{T_{f 1} s + 1}, k_{D 2} \times \frac{τ_{2} s + 1}{T_{f 2} s + 1}, k_{D 3} \times \frac{τ_{3} s + 1}{T_{f 3} s + 1}, k_{D 4} \times \frac{τ_{4} s + 1}{T_{f 4} s + 1}} \end{matrix}

(8)

where $k_{Pn}$ , $k_{In}$ , $k_{Dn}$ , $T_{in}$ , $T_{fn}$ , and $τ_{n} (n = 1, 2, 3, 4)$ are the proportional coefficients, the integral coefficients, the differential coefficients, the inertia loop coefficients, the small inertia loop coefficients, and the natural differential coefficients, respectively.

Subsisting equation (8) into equation (7) and using the Laplace transform, we have

\begin{matrix} M s^{2} q_{c} (s) + Gs q_{c} (s) = - T_{L} K_{s} T_{L}^{T} q_{c} (s) \\ + T_{L} K_{i} G_{t} (s) T_{L}^{T} q_{c} (s) + U_{d} (s) \end{matrix}

(9)

T_{L} K_{s} T_{L}^{T} = [\begin{matrix} k_{sLx} + k_{sRx} & 0 & 0 & a k_{sLx} - b k_{sRx} \\ 0 & k_{sLy} + k_{sRy} & a k_{sLy} - b k_{sRy} & 0 \\ 0 & a k_{sLy} - b k_{sRy} & a^{2} k_{sLy} + b^{2} k_{sRy} & 0 \\ a k_{sLx} - b k_{sRx} & 0 & 0 & a^{2} k_{sLx} + b^{2} k_{sRx} \end{matrix}]

(10)

T_{L} K_{i} P_{t} (s) T_{L}^{T} = [\begin{matrix} k_{iLx} k_{P 1} + k_{iRx} k_{P 2} & 0 & 0 & a k_{iLx} k_{P 1} - b k_{iRx} k_{P 2} \\ 0 & k_{iLy} k_{P 3} + k_{iRy} k_{P 4} & a k_{iLy} k_{P 3} - b k_{iRy} k_{P 4} & 0 \\ 0 & a k_{iLy} k_{P 3} - b k_{iRy} k_{P 4} & a^{2} k_{iLy} k_{P 3} + b^{2} k_{iRy} k_{P 4} & 0 \\ a k_{iLx} k_{P 1} - b k_{iRx} k_{P 2} & 0 & 0 & a^{2} k_{iLx} k_{P 1} + b^{2} k_{iRx} k_{P 2} \end{matrix}]

(11)

T_{L} K_{i} I_{t} (s) T_{L}^{T} = [\begin{matrix} k_{iLx} \frac{k_{I 1}}{T_{i 1} s} + k_{iRx} \frac{k_{I 2}}{T_{i 2} s} & 0 & 0 & a k_{iLx} \frac{k_{I 1}}{T_{i 1} s} - b k_{iRx} \frac{k_{I 2}}{T_{i 2} s} \\ 0 & k_{iLy} \frac{k_{I 3}}{T_{i 3} s} + k_{iRy} \frac{k_{I 4}}{T_{i 4} s} & a k_{iLy} \frac{k_{I 3}}{T_{i 3} s} - b k_{iRy} \frac{k_{I 4}}{T_{i 4} s} & 0 \\ 0 & a k_{iLy} \frac{k_{I 3}}{T_{i 3} s} - b k_{iRy} \frac{k_{I 4}}{T_{i 4} s} & a^{2} k_{iLy} \frac{k_{I 3}}{T_{i 3} s} + b^{2} k_{iRy} \frac{k_{I 4}}{T_{i 4} s} & 0 \\ a k_{iLx} \frac{k_{I 1}}{T_{i 1} s} - b k_{iRx} \frac{k_{I 2}}{T_{i 2} s} & 0 & 0 & a^{2} k_{iLx} \frac{k_{I 1}}{T_{i 1} s} + b^{2} k_{iRx} \frac{k_{I 2}}{T_{i 2} s} \end{matrix}]

(12)

\begin{matrix} T_{L} K_{i} D_{t} (s) T_{L}^{T} = [\begin{matrix} k_{iLx} k_{D 1} + k_{iRx} k_{D 2} & 0 & 0 & a k_{iLx} k_{D 1} - b k_{iRx} k_{D 2} \\ 0 & k_{iLy} k_{D 3} + k_{iRy} k_{D 4} & a k_{iLy} k_{D 3} - b k_{iRy} k_{D 4} & 0 \\ 0 & a k_{iLy} k_{D 3} - b k_{iRy} k_{D 4} & a^{2} k_{iLy} k_{D 3} + b^{2} k_{iRy} k_{D 4} & 0 \\ a k_{iLx} k_{D 1} - b k_{iRx} k_{D 2} & 0 & 0 & a^{2} k_{iLx} k_{D 1} + b^{2} k_{iRx} k_{D 2} \end{matrix}] \\ + [\begin{matrix} k_{iLx} k_{D 1} τ_{1} s + k_{iRx} k_{D 2} τ_{2} s & 0 & 0 & a k_{iLx} k_{D 1} τ_{1} s - b k_{iRx} k_{D 2} τ_{2} s \\ 0 & k_{iLy} k_{D 3} τ_{3} s + k_{iRy} k_{D 4} τ_{4} s & a k_{iLy} k_{D 3} τ_{3} s - b k_{iRy} k_{D 4} τ_{4} s & 0 \\ 0 & a k_{iLy} k_{D 3} τ_{3} s - b k_{iRy} k_{D 4} τ_{4} s & a^{2} k_{iLy} k_{D 3} τ_{3} s + b^{2} k_{iRy} k_{D 4} τ_{4} s & 0 \\ a k_{iLx} k_{D 1} τ_{1} s - b k_{iRx} k_{D 2} τ_{2} s & 0 & 0 & a^{2} k_{iLx} k_{D 1} τ_{1} s + b^{2} k_{iRx} k_{D 2} τ_{2} s \end{matrix}] \end{matrix}

(13)

It is obvious that the equations of motion of the above system are coupled in each DOF by the gyroscopic matrix G and other matrixes.

Firstly, a cross-feedback controller is designed to compensate for the gyroscopic coupling effect. Because the natural differential coefficients of the differentiators are different, the following differential loops are designed in parallel with the differentiators in the PID controller to compensate for the coupling effect

D_{com} = \frac{β_{n} s + 1}{T_{f β n} s + 1}, n = 1, 2, 3, 4

(14)

where $β_{n}$ and $T_{f β n}$ are the differential coefficients and the small inertia loop coefficients. Then, the transfer function matrix of the velocity cross channel is presented as follows

k_{φ} = [\begin{matrix} 0 & 0 & k_{φ 3} (2 + τ_{3} s + β_{3} s) & - k_{φ 3} (2 + τ_{4} s + β_{4} s) \\ 0 & 0 & k_{φ 4} (2 + τ_{3} s + β_{3} s) & - k_{φ 4} (2 + τ_{4} s + β_{4} s) \\ k_{φ 1} (2 + τ_{1} s + β_{1} s) & - k_{φ 1} (2 + τ_{2} s + β_{2} s) & 0 & 0 \\ k_{φ 2} (2 + τ_{1} s + β_{1} s) & - k_{φ 2} (2 + τ_{2} s + β_{2} s) & 0 & 0 \end{matrix}]

(15)

If the conditions $τ_{1} - τ_{2} = β_{2} - β_{1}$ and $τ_{3} - τ_{4} = β_{4} - β_{3}$ are satisfied, the cross-feedback coefficients are

k_{φ 1} = \frac{J_{z} ω}{(a (τ_{1} + β_{1}) + b (τ_{2} + β_{2})) (a + b) k_{iLy}}

(16)

k_{φ 2} = \frac{- J_{z} ω}{(a (τ_{1} + β_{1}) + b (τ_{2} + β_{2})) (a + b) k_{iRy}}

(17)

k_{φ 3} = \frac{- J_{z} ω}{(a (τ_{3} + β_{3}) + b (τ_{4} + β_{4})) (a + b) k_{iLx}}

(18)

k_{φ 4} = \frac{J_{z} ω}{(a (τ_{3} + β_{3}) + b (τ_{4} + β_{4})) (a + b) k_{iRx}}

(19)

Because the position terms are included in equation (15), the transfer function matrix of the position cross channel is presented as follows

k_{φ P} = [\begin{matrix} 0 & 0 & k_{φ P 3} & - k_{φ P 3} \\ 0 & 0 & k_{φ P 4} & - k_{φ P 4} \\ k_{φ P 1} & - k_{φ P 1} & 0 & 0 \\ k_{φ P 2} & - k_{φ P 2} & 0 & 0 \end{matrix}]

(20)

Substituting equation (20) into equation (9) to compensate for the coupling term, we have

k_{φ P 1} = \frac{- 2 (a k_{φ 1} k_{iLy} - b k_{φ 2} k_{iRy})}{(a + b) k_{iLy}}

(21)

k_{φ P 2} = \frac{2 (a k_{φ 1} k_{iLy} - b k_{φ 2} k_{iRy})}{(a + b) k_{iRy}}

(22)

k_{φ P 3} = \frac{- 2 (a k_{φ 3} k_{iLx} - b k_{φ 4} k_{iRx})}{(a + b) k_{iLx}}

(23)

k_{φ P 4} = \frac{2 (a k_{φ 3} k_{iLx} - b k_{φ 4} k_{iRx})}{(a + b) k_{iRx}}

(24)

Next, a controller is designed to eliminate the anti-diagonal terms in equations (10)–(13). The following compensator is substituted into equation (9) to compensate for the integral coupling term in equation (12)

k_{ci} = [\begin{matrix} 0 & \frac{k_{ciA}}{T_{i 2} s} & 0 & 0 \\ \frac{k_{ciB}}{T_{i 1} s} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{k_{ciC}}{T_{i 4} s} \\ 0 & 0 & \frac{k_{ciD}}{T_{i 3} s} & 0 \end{matrix}]

(25)

where $k_{ciA}$ , $k_{ciB}$ , $k_{ciC}$ , and $k_{ciD}$ are the coefficients of each integral decoupling channel. Then, we have

k_{ciA} = \frac{- (a^{2} + ab) k_{iLx} \frac{k_{I 1}}{T_{i 1}} + (b^{2} + ab) k_{iRx} \frac{k_{I 2}}{T_{i 2}}}{(a^{2} - b^{2}) k_{iLx}} T_{i 2}

(26)

k_{ciB} = \frac{- (a^{2} + ab) k_{iLx} \frac{k_{I 1}}{T_{i 1}} + (b^{2} + ab) k_{iRx} \frac{k_{I 2}}{T_{i 2}}}{(a^{2} - b^{2}) k_{iRx}} T_{i 1}

(27)

k_{ciC} = \frac{- (a^{2} + ab) k_{iLy} \frac{k_{I 3}}{T_{i 3}} + (b^{2} + ab) k_{iRy} \frac{k_{I 4}}{T_{i 4}}}{(a^{2} - b^{2}) k_{iLy}} T_{i 4}

(28)

k_{ciD} = \frac{- (a^{2} + ab) k_{iLy} \frac{k_{I 3}}{T_{i 3}} + (b^{2} + ab) k_{iRy} \frac{k_{I 4}}{T_{i 4}}}{(a^{2} - b^{2}) k_{iRy}} T_{i 3}

(29)

Then, the following compensator is substituted into equation (9) to compensate for the differential coupling term in equation (13)

k_{c} = [\begin{matrix} 0 & k_{cA} (1 + τ_{2} s) & 0 & 0 \\ k_{cB} (1 + τ_{1} s) & 0 & 0 & 0 \\ 0 & 0 & 0 & k_{cC} (1 + τ_{4} s) \\ 0 & 0 & k_{cD} (1 + τ_{3} s) & 0 \end{matrix}]

(30)

where $k_{cA}$ , $k_{cB}$ , $k_{cC}$ and $k_{cD}$ are the coefficients of each differential decoupling channel. Then, we have

k_{cA} = \frac{- (a^{2} + ab) k_{iLx} k_{D 1} τ_{1} + (b^{2} + ab) k_{iRx} k_{D 2} τ_{2}}{(a^{2} - b^{2}) k_{iLx} τ_{2}}

(31)

k_{cB} = \frac{- (a^{2} + ab) k_{iLx} k_{D 1} τ_{1} + (b^{2} + ab) k_{iRx} k_{D 2} τ_{2}}{(a^{2} - b^{2}) k_{iRx} τ_{1}}

(32)

k_{cC} = \frac{- (a^{2} + ab) k_{iLy} k_{D 3} τ_{3} + (b^{2} + ab) k_{iRy} k_{D 4} τ_{4}}{(a^{2} - b^{2}) k_{iLy} τ_{4}}

(33)

k_{cD} = \frac{- (a^{2} + ab) k_{iLy} k_{D 3} τ_{3} + (b^{2} + ab) k_{iRy} k_{D 4} τ_{4}}{(a^{2} - b^{2}) k_{iRy} τ_{3}}

(34)

On the other hand, the anti-diagonal terms and proportional terms including those in equations (10), (11), (13), and (30) may affect the control performance of the system; therefore, they should be compensated. The following compensator is designed

k_{cs} = [\begin{matrix} 0 & k_{csA} & 0 & 0 \\ k_{csB} & 0 & 0 & 0 \\ 0 & 0 & 0 & k_{csC} \\ 0 & 0 & k_{csD} & 0 \end{matrix}]

(35)

where $k_{csA}$ , $k_{csB}$ , $k_{csC}$ , and $k_{csD}$ are the coefficients of each proportional decoupling channel. Then, we have

k_{csA} = \frac{X_{1} - {ak}_{iLx} k_{D 1} + {bk}_{iRx} k_{D 2}}{(a - b) k_{iLx}} - k_{cA}

(36)

k_{csB} = \frac{X_{1} - {ak}_{iLx} k_{D 1} + {bk}_{iRx} k_{D 2}}{(a - b) k_{iRx}} - k_{cB}

(37)

k_{csC} = \frac{X_{2} - {ak}_{iLy} k_{D 3} + {bk}_{iRy} k_{D 4}}{(a - b) k_{iLy}} - k_{cC}

(38)

\begin{matrix} k_{csD} = \frac{X_{2} - {ak}_{iLy} k_{D 3} + {bk}_{iRy} k_{D 4}}{(a - b) k_{iRy}} - k_{cD} \\ X_{1} = a k_{sLx} - b k_{sRx} - a k_{iLx} k_{P 1} + b k_{iRx} k_{P 2} \\ X_{2} = a k_{sLy} - b k_{sRy} - a k_{iLy} k_{P 3} + b k_{iRy} k_{P 4} \end{matrix}

(39)

It is noted that additional diagonal terms are introduced into equation (9), and the effect on the whole system caused by these terms should be considered. Generally, the selection of PID controller coefficients is made based on the design stiffness and damping of the AMBs. If the transfer function matrices of the system include the aforementioned terms, the controller coefficients of each channel may change. Indeed, the whole system becomes unstable.

First, the following compensator is designed to compensate for the additional diagonal terms caused by equation (25)

k_{in} = [\begin{matrix} c_{i 1} \frac{1}{T_{i 1} s} & c_{i 2} \frac{1}{T_{i 2} s} & 0 & 0 \\ c_{i 3} \frac{1}{T_{i 1} s} & c_{i 4} \frac{1}{T_{i 2} s} & 0 & 0 \\ 0 & 0 & c_{i 5} \frac{1}{T_{i 3} s} & c_{i 6} \frac{1}{T_{i 4} s} \\ 0 & 0 & c_{i 7} \frac{1}{T_{i 3} s} & c_{i 8} \frac{1}{T_{i 4} s} \end{matrix}]

(40)

where $c_{in}, n = 1, 2, 3, 4, 5, 6, 7, 8$ , are the coefficients of the back-end channel of integrators, and we have

\begin{matrix} c_{i 1} = \frac{(ab - b^{2}) (k_{iLx} k_{ciA} \frac{T_{i 1}}{T_{i 2}} + k_{iRx} k_{ciB})}{k_{iLx} {(a + b)}^{2}}, \\ c_{i 2} = - 2 \frac{ab (k_{iLx} k_{ciA} + k_{iRx} k_{ciB} \frac{T_{i 2}}{T_{i 1}})}{k_{iLx} {(a + b)}^{2}} \end{matrix}

(41)

\begin{matrix} c_{i 3} = - 2 \frac{ab (k_{iLx} k_{ciA} \frac{T_{i 1}}{T_{i 2}} + k_{iRx} k_{ciB})}{k_{iRx} {(a + b)}^{2}}, \\ c_{i 4} = \frac{(ab - a^{2}) (k_{iLx} k_{ciA} + k_{iRx} k_{ciB} \frac{T_{i 2}}{T_{i 1}})}{k_{iRx} {(a + b)}^{2}} \end{matrix}

(42)

\begin{matrix} c_{i 5} = \frac{(ab - b^{2}) (k_{iLy} k_{ciC} \frac{T_{i 3}}{T_{i 4}} + k_{iRy} k_{ciD})}{k_{iLy} {(a + b)}^{2}}, \\ c_{i 6} = - 2 \frac{ab (k_{iLy} k_{ciC} + k_{iRy} k_{ciD} \frac{T_{i 4}}{T_{i 3}})}{k_{iLy} {(a + b)}^{2}} \end{matrix}

(43)

\begin{matrix} c_{i 7} = - 2 \frac{ab (k_{iLy} k_{ciC} \frac{T_{i 3}}{T_{i 4}} + k_{iRy} k_{ciD})}{k_{iRy} {(a + b)}^{2}}, \\ c_{i 8} = \frac{(ab - a^{2}) (k_{iLy} k_{ciC} + k_{iRy} k_{ciD} \frac{T_{i 4}}{T_{i 3}})}{k_{iRy} {(a + b)}^{2}} \end{matrix}

(44)

Then, the following compensator is designed to compensate for the additional diagonal terms caused by equation (30)

\begin{matrix} k_{vn} = [\begin{matrix} c_{v 1} (1 + τ_{1} s) & c_{v 2} (1 + τ_{2} s) & 0 & 0 \\ c_{v 3} (1 + τ_{1} s) & c_{v 4} (1 + τ_{2} s) & 0 & 0 \\ 0 & 0 & c_{v 5} (1 + τ_{3} s) & c_{v 6} (1 + τ_{4} s) \\ 0 & 0 & c_{v 7} (1 + τ_{3} s) & c_{v 8} (1 + τ_{4} s) \end{matrix}] \\ + [\begin{matrix} c_{v 9} (1 + β_{1} s) & c_{v 10} (1 + β_{2} s) & 0 & 0 \\ c_{v 11} (1 + β_{1} s) & c_{v 12} (1 + β_{2} s) & 0 & 0 \\ 0 & 0 & c_{v 13} (1 + β_{3} s) & c_{v 14} (1 + β_{4} s) \\ 0 & 0 & c_{v 15} (1 + β_{3} s) & c_{v 16} (1 + β_{4} s) \end{matrix}] \end{matrix}

(45)

where $c_{vn}, n = 1, 2, 3, 4, 5, 6, 7, 8$ , and $c_{vn}, n = 9, 10, 11, 12, 13, 14, 15, 16$ , are the coefficients of the back-end channel of differentiators in the PID controller and the coefficients of the back-end channel of the added differential loops. If the conditions $τ_{1} = β_{2}$ , $τ_{2} = β_{1}$ , $τ_{3} = β_{4}$ , and $τ_{4} = β_{3}$ are satisfied, then we have

c_{v 1} = \frac{(ab - b^{2}) k_{iRx} k_{cB}}{k_{iLx} {(a + b)}^{2}}, c_{v 2} = - 2 \frac{ab k_{iLx} k_{cA}}{k_{iLx} {(a + b)}^{2}}

(46)

c_{v 3} = - 2 \frac{ab k_{iRx} k_{cB}}{k_{iRx} {(a + b)}^{2}}, c_{v 4} = \frac{(ab - a^{2}) k_{iLx} k_{cA}}{k_{iRx} {(a + b)}^{2}}

(47)

c_{v 5} = \frac{(ab - b^{2}) k_{iRy} k_{cD}}{k_{iLy} {(a + b)}^{2}}, c_{v 6} = - 2 \frac{ab k_{iLy} k_{cC}}{k_{iLy} {(a + b)}^{2}}

(48)

c_{v 7} = - 2 \frac{ab k_{iRy} k_{cD}}{k_{iRy} {(a + b)}^{2}}, c_{v 8} = \frac{(ab - a^{2}) k_{iLy} k_{cC}}{k_{iRy} {(a + b)}^{2}}

(49)

c_{v 9} = \frac{(ab - b^{2}) k_{iLx} k_{cA}}{k_{iLx} {(a + b)}^{2}}, c_{v 10} = - 2 \frac{ab k_{iRx} k_{cB}}{k_{iLx} {(a + b)}^{2}}

(50)

c_{v 11} = - 2 \frac{ab k_{iLx} k_{cA}}{k_{iRx} {(a + b)}^{2}}, c_{v 12} = \frac{(ab - a^{2}) k_{iRx} k_{cB}}{k_{iRx} {(a + b)}^{2}}

(51)

c_{v 13} = \frac{(ab - b^{2}) k_{iLy} k_{cC}}{k_{iLy} {(a + b)}^{2}}, c_{v 14} = - 2 \frac{ab k_{iRy} k_{cD}}{k_{iLy} {(a + b)}^{2}}

(52)

c_{v 15} = - 2 \frac{ab k_{iLy} k_{cC}}{k_{iRy} {(a + b)}^{2}}, c_{v 16} = \frac{(ab - a^{2}) k_{iRy} k_{cD}}{k_{iRy} {(a + b)}^{2}}

(53)

In the same manner, the following compensator is designed to compensate for the additional diagonal terms caused by equation (35)

k_{sn} = [\begin{matrix} c_{s 1} & c_{s 2} & 0 & 0 \\ c_{s 3} & c_{s 4} & 0 & 0 \\ 0 & 0 & c_{s 5} & c_{s 6} \\ 0 & 0 & c_{s 7} & c_{s 8} \end{matrix}]

(54)

where $c_{sn}, n = 1, 2, 3, 4, 5, 6, 7, 8$ , are the coefficients of the front-end channel of proportional loops in the PID controller, and we have

\begin{matrix} c_{s 1} = \frac{(ab - b^{2}) (k_{csA} k_{iLx} + k_{csB} k_{iRx})}{{(a + b)}^{2} k_{iLx}}, \\ c_{s 2} = \frac{- 2 ab (k_{csA} k_{iLx} + k_{csB} k_{iRx})}{{(a + b)}^{2} k_{iLx}} \end{matrix}

(55)

\begin{matrix} c_{s 3} = \frac{- 2 ab (k_{csA} k_{iLx} + k_{csB} k_{iRx})}{{(a + b)}^{2} k_{iRx}}, \\ c_{s 4} = \frac{(ab - a^{2}) (k_{csA} k_{iLx} + k_{csB} k_{iRx})}{{(a + b)}^{2} k_{iRx}} \end{matrix}

(56)

\begin{matrix} c_{s 5} = \frac{(ab - b^{2}) (k_{csC} k_{iLy} + k_{csD} k_{iRy})}{{(a + b)}^{2} k_{iLy}}, \\ c_{s 6} = \frac{- 2 ab (k_{csC} k_{iLy} + k_{csD} k_{iRy})}{{(a + b)}^{2} k_{iLy}} \end{matrix}

(57)

\begin{matrix} c_{s 7} = \frac{- 2 ab (k_{csC} k_{iLy} + k_{csD} k_{iRy})}{{(a + b)}^{2} k_{iRy}}, \\ c_{s 8} = \frac{(ab - a^{2}) (k_{csC} k_{iLy} + k_{csD} k_{iRy})}{{(a + b)}^{2} k_{iRy}} \end{matrix}

(58)

Hence, the proposed control system shown in Figure 3 can be represented as follows

\begin{matrix} M s^{2} q_{c} (s) = - K_{s}^{'} q_{c} (s) + [P_{t}^{'} + D_{t}^{'} + I_{t}^{'}] q_{c} (s) + U_{d} (s) \\ K_{s}^{'} = [\begin{matrix} k_{sLx} + k_{sRx} & 0 & 0 & 0 \\ 0 & k_{sLy} + k_{sRy} & 0 & 0 \\ 0 & 0 & a^{2} k_{sLy} + b^{2} k_{sRy} & 0 \\ 0 & 0 & 0 & a^{2} k_{sLx} + b^{2} k_{sRx} \end{matrix}] \end{matrix}

(59)

\begin{matrix} P_{t}^{'} = [\begin{matrix} k_{ilx} k_{P 1} + k_{iRx} k_{P 2} & 0 & 0 & 0 \\ 0 & k_{iLy} k_{P 3} + k_{iRy} k_{P 4} & 0 & 0 \\ 0 & 0 & a^{2} k_{iLy} k_{P 3} + b^{2} k_{iRy} k_{P 4} & 0 \\ 0 & 0 & 0 & a^{2} k_{iLx} k_{P 1} + b^{2} k_{iRx} k_{P 2} \end{matrix}] \\ I_{t}^{'} = [\begin{matrix} k_{iLx} \frac{k_{I 1}}{T_{i 1} s} + k_{iRx} \frac{k_{I 2}}{T_{i 2} s} & 0 & 0 & 0 \\ 0 & k_{iLy} \frac{k_{I 3}}{T_{i 3} s} + k_{iRy} \frac{k_{I 4}}{T_{i 4} s} & 0 & 0 \\ 0 & 0 & a^{2} k_{iLy} \frac{k_{I 3}}{T_{i 3} s} + b^{2} k_{iRy} \frac{k_{I 4}}{T_{i 4} s} & 0 \\ 0 & 0 & 0 & a^{2} k_{iLx} \frac{k_{I 1}}{T_{i 1} s} + b^{2} k_{iRx} \frac{k_{I 2}}{T_{i 2} s} \end{matrix}] \\ D_{t}^{'} = [\begin{matrix} k_{iLx} k_{D 1} + k_{iRx} k_{D 2} & 0 & 0 & 0 \\ 0 & k_{iLy} k_{D 3} + k_{iRy} k_{D 4} & 0 & 0 \\ 0 & 0 & a^{2} k_{iLy} k_{D 3} + b^{2} k_{iRy} k_{D 4} & 0 \\ 0 & 0 & 0 & a^{2} k_{iLx} k_{D 1} + b^{2} k_{iRx} k_{D 2} \end{matrix}] \\ + [\begin{matrix} k_{iLx} k_{D 1} τ_{1} s + k_{iRx} k_{D 2} τ_{2} s & 0 & 0 & 0 \\ 0 & k_{iLy} k_{D 3} τ_{3} s + k_{iRy} k_{D 4} τ_{4} s & 0 & 0 \\ 0 & 0 & a^{2} k_{iLy} k_{D 3} τ_{3} s + b^{2} k_{iRy} k_{D 4} τ_{4} s & 0 \\ 0 & 0 & 0 & a^{2} k_{iLx} k_{D 1} τ_{1} s + b^{2} k_{iRx} k_{D 2} τ_{2} s \end{matrix}] \end{matrix}

Figure 3.

The proposed controller.

Simulation results

To demonstrate the effectiveness of the proposed controller, we carried out several simulations via ADAMS–MATLAB co-simulation technology for improving the simulation accuracy in this section.

ADAMS model for the rotor–bearing system

ADAMS is the most widely used dynamic simulation software, and SOLIDWORKS has a powerful three-dimensional (3D) modeling function as well as a good connection with ADAMS. In this article, we built the 3D model of the flywheel–rotor–bearing system via SOLIDWORKS which is shown in Figure 4, and this is the major part of the real system shown in Figure 19. The corresponding material information for each part of the model is shown in Table 1.

Figure 4.

Three-dimensional model of the flywheel–rotor–bearing system.

Table 1.

Material coefficients of the rotor–bearing system.

Component	Density $(kg / m^{3})$	Young’s modulus $(N / m^{2})$	Poisson’s ratio
Magnetic bearing-rotor	7870	1.20e+011	0.291
Motor rotor	7870	1.20e+011	0.291
Spindle	7830	2.09e+011	0.300
Flywheel	2800	7.30e+010	0.330

After defining the material information, the rotor system has mass information, center of mass, and moment of inertia which can be calculated through ADAMS software, and we established the coordinate system in the center of mass. In order to make the rotor suspend based on our requirements, we also need to add motion constraints for each part of the rotor. Considering physical applications, we added fixed constraint between the spindle and other parts such as bearing-rotors, motor rotor, and flywheel. We also add rotational constraints to the rotor itself. For defining the constraints of the rotor conveniently, we marked four points, which are MARKER51–MARKER54, respectively. Eight variables including four input variables and four output variables, which are shown in Table 2, were set up. Then, the ADAMS model of the rotor–bearing system is shown in Figure 5.

Table 2.

Definition of state variables.

Input variable	Output variable
$i_{Lx}$	$x_{l}$ DX(MARKER53,MARKER51)
$i_{Rx}$	$x_{r}$ DX(MARKER54,MARKER52)
$i_{Ly}$	$y_{l}$ DY(MARKER53,MARKER51)
$i_{Ry}$	$y_{r}$ DX(MARKER54,MARKER52)

Figure 5.

ADAMS model of the flywheel–rotor–bearing system.

Comparative controller and simulation conditions

To demonstrate the effectiveness of the proposed controller, comparative simulations were conducted by a traditional decentralized PID controller and a cross-feedback controller which are classical decentralized control strategies. The PID controller for each DOF magnetic bearing was designed as follows¹⁷

u = k_{P} e + k_{D} \overset{\cdot}{e} + k_{I} \int edt

(60)

where u, e, $k_{P}$ , $k_{D}$ , and $k_{I}$ are the control input, difference between the real position of the rotor and the desired one, proportional coefficient, differential coefficient, and integral coefficient, respectively.

On the other hand, the cross-feedback controller was designed as shown in Figure 6 which is based on the theory reported in Markus et al.¹⁸ The cross coefficient $k_{c}$ was determined as follows

k_{c} = κ \frac{- J_{z} ω}{{(a + b)}^{2} k_{i}}

(61)

The coefficient $κ$ was set to 0.75 when the best performance was obtained. In addition, the basic parameters $k_{P}$ , $k_{D}$ , and $k_{I}$ of the three controllers were assigned the same values as follows¹⁷

k_{P} = \frac{- 2 k_{s}}{k_{i}}

(62)

k_{D} = \frac{\sqrt{- m k_{s}}}{2 k_{i}}

(63)

k_{I} = 10 k_{P}

(64)

and the rest of the parameters of the proposed controller are shown in Table 3.

Figure 6.

Structure of the cross-feedback controller.

Table 3.

Parameters of the proposed controller.

Parameter	Value
$τ_{1}$	32
$τ_{2}$	40
$τ_{3}$	35
$τ_{4}$	35
$β_{1}$	40
$β_{2}$	32
$β_{3}$	43
$β_{4}$	35
$T_{i 1}$	31
$T_{i 2}$	35
$T_{i 3}$	37
$T_{i 4}$	33
$T_{f 1}$	$5.0 \times 10^{- 4}$
$T_{f 2}$	$5.6 \times 10^{- 4}$
$T_{f 3}$	$6.0 \times 10^{- 4}$
$T_{f 4}$	$6.2 \times 10^{- 4}$
$T_{f β 1}$	$2.4 \times 10^{- 4}$
$T_{f β 2}$	$1.8 \times 10^{- 4}$
$T_{f β 3}$	$3.3 \times 10^{- 4}$
$T_{f β 4}$	$3.1 \times 10^{- 4}$
e	$5.0 \times 10^{- 4}$
$ε$	$5.0 \times 10^{- 4}$

The initial position for $q_{c}$ is $[0.1 mm, - 0.1 mm, 0^{\circ}, 0^{\circ}]$ , and the desired position is $[0 mm, 0 mm, 0^{\circ}, 0^{\circ}]$ . The rest of the simulation conditions including the control period T, the magnitude characteristic $M_{r}$ , and the sampling time $t_{s}$ were set at $T = 10 s$ , $M_{r} = 0.0001 m$ , and $t_{s} = 0.005 s$ . The parameters of the rotor–bearing system are shown in Table 4.

Table 4.

Parameters of the rotor–bearing system.

Parameter	Value
m	$6.56 kg$
$J_{x}$	$0.083 kg m^{2}$
$J_{y}$	$0.083 kg m^{2}$
$J_{z}$	$0.006 kg m^{2}$
$k_{sLx}$	$- 1.00 \times 10^{5} N / m$
$k_{sRx}$	$- 4.00 \times 10^{5} N / m$
$k_{sLy}$	$- 1.25 \times 10^{5} N / m$
$k_{sRy}$	$- 3.60 \times 10^{5} N / m$
$k_{iLx}$	$5 N / A$
$k_{iRx}$	$10 N / A$
$k_{iLy}$	$8 N / A$
$k_{iRy}$	$12 N / A$
a	$0.117 m$
b	$0.069 m$
c	$0.154 m$
d	$0.107 m$
g	$0.217 m$
h	$0.139 m$

Note: The ‘N/A’s are the unit of the current stiffness k_iLx, k_iRx, k_iLy and k_iRy, which are the same meanings as others in this table.

Control performance

The control performance results for position x, position y, angular position $θ_{x}$ , and angular position $θ_{y}$ under the condition $ω = 25, 000 rpm$ obtained by the aforementioned controllers are shown in Figure 7. In this case, the imbalance force and other disturbance of the system were ignored. From the results shown in Figure 7(a) and Table 5, the overshot obtained by the proposed controller is 93.75% larger than that obtained by the PID controller and almost the same as that obtained by the cross-feedback controller. In Figure 7(b), the overshot obtained by the proposed controller is 28.85% and 21.82% larger than that obtained by the PID controller and the cross-feedback controller, respectively. In Figure 7(c), the overshot obtained by the proposed controller is 50% and 5.88% larger than that obtained by the PID controller and the cross-feedback controller, respectively. However, in Figure 7(d), the overshot obtained by the proposed controller is 50% and 72.73% smaller than that obtained by the PID controller and the cross-feedback controller, respectively; These results were obtained because the proposed controller provides larger control current in each direction of the bearings except the y-direction of the right-side bearing shown in Figure 8. In addition, almost the same steady-state errors are obtained by the three controllers.

Figure 7.

Response trajectory of center of mass of the rotor (ω = 25,000 rpm) (control performance): (a) position x, (b) position y, (c) angular position θ_x, and (d) angular position θ_y.

Table 5.

Comparison of control performance.

Maximum amplitude of overshot	PID control				Cross-feedback control				Modal decoupling control
Maximum amplitude of overshot	x	y	$θ_{x}$	$θ_{y}$	x	y	$θ_{x}$	$θ_{y}$	x	y	$θ_{x}$	$θ_{y}$
$ω = 20, 000 rpm$	1.7 × 10⁻⁵	5.2 × 10⁻⁵	9.7 × 10⁻⁵	2.9 × 10⁻⁴	1.6 × 10⁻⁵	5.5 × 10⁻⁵	1.8 × 10⁻⁴	2.9 × 10⁻⁴	3.1 × 10⁻⁵	6.7 × 10⁻⁵	1.9 × 10⁻⁴	1.6 × 10⁻⁴
$ω = 25, 000 rpm$	1.6 × 10⁻⁵	5.2 × 10⁻⁵	1.2 × 10⁻⁴	3.0 × 10⁻⁴	1.6 × 10⁻⁵	5.5 × 10⁻⁵	1.7 × 10⁻⁴	2.8 × 10⁻⁴	3.1 × 10⁻⁵	6.7 × 10⁻⁵	1.8 × 10⁻⁴	1.5 × 10⁻⁴
$ω = 30, 000 rpm$	1.6 × 10⁻⁵	5.3 × 10⁻⁵	1.4 × 10⁻⁴	3.0 × 10⁻⁴	1.6 × 10⁻⁵	5.6 × 10⁻⁵	1.3 × 10⁻⁴	2.7 × 10⁻⁴	3.1 × 10⁻⁵	6.7 × 10⁻⁵	1.7 × 10⁻⁴	1.5 × 10⁻⁴

PID: proportional–integral–derivative.

Figure 8.

Control current (ω = 25,000 rpm) (control performance): (a) control current of the left-side bearing in the x-direction, (b) control current of the right-side bearing in the y-direction, (c) control current of the left-side bearing in the y-direction, and (d) control current of the right-side bearing in the y-direction.

The control performance results under the conditions $ω = 20, 000 rpm$ and $ω = 30, 000 rpm$ obtained by the aforementioned controllers are shown in Figures 9 and 10. Almost the same results were obtained which confirms that the three controllers have certain robustness with respect to rotor velocity changes in ideal situations.

Figure 9.

Response trajectory of center of mass of the rotor (ω = 20,000 rpm) (control performance): (a) position x, (b) position y, (c) angular position θ_x, and (d) angular position θ_y.

Figure 10.

Response trajectory of center of mass of the rotor (ω = 30,000 rpm) (control performance): (a) position x, (b) position y, (c) angular position θ_x, and (d) angular position θ_y.

Decoupling performance

In order to ensure that the magnetic suspension rotor will not impact the stator in the process of movement, the maximum force whose size is $40 N$ for the x-direction was applied at the center of mass of the rotor, the force and torque for the y-direction were set to 0, and the decoupling performance results for position x, position y, angular position $θ_{x}$ , and angular position $θ_{y}$ under the condition $ω = 25, 000 rpm$ obtained by the aforementioned controllers are shown in Figure 11. From the results shown in Figure 11(a) and Table 6, the overshot obtained by the proposed controller is 2.22% smaller than that obtained by the PID controller and almost the same as that obtained by the cross-feedback controller. In Figure 11(b), the overshot obtained by the proposed controller is 36% and 65.22% smaller than that obtained by the PID controller and the cross-feedback controller, respectively. In Figure 11(d), the overshot obtained by the proposed controller is 51.72% and 39.13% smaller than that obtained by the PID controller and the cross-feedback controller, respectively. However, in Figure 11(c), the overshot obtained by the proposed controller is 3.49% larger than that obtained by the PID controller and is 44.38% smaller than that obtained by the cross-feedback controller. These results were obtained because the control current obtained by the proposed controller is larger than that obtained by the PID controller and cross-feedback controller which are shown in Figure 12. In addition, the steady-state errors obtained by the PID controller and the cross-feedback controller are larger than that obtained by the proposed controller. It is confirmed that the proposed controller shows better decoupling performance than the PID controller and the cross-feedback controller.

Figure 11.

Response trajectory of center of mass of the rotor (ω = 25,000 rpm) (decoupling performance): (a) position x, (b) position y, (c) angular position θ_x, and (d) angular position θ_y.

Table 6.

Comparison of decoupling performance.

Maximum amplitude of overshot	PID control				Cross-feedback control				Modal decoupling control
Maximum amplitude of overshot	x	y	$θ_{x}$	$θ_{y}$	x	y	$θ_{x}$	$θ_{y}$	x	y	$θ_{x}$	$θ_{y}$
$ω = 20, 000 rpm$	4.5 × 10⁻⁵	2.3 × 10⁻⁶	7.1 × 10⁻⁵	3.0 × 10⁻⁴	4.5 × 10⁻⁵	4.1 × 10⁻⁶	8.0 × 10⁻⁵	2.6 × 10⁻⁴	4.4 × 10⁻⁵	1.4 × 10⁻⁶	1.4 × 10⁻⁴	1.5 × 10⁻⁴
$ω = 25, 000 rpm$	4.5 × 10⁻⁵	2.5 × 10⁻⁶	8.6 × 10⁻⁵	2.9 × 10⁻⁴	4.5 × 10⁻⁵	4.6 × 10⁻⁶	1.6 × 10⁻⁴	2.3 × 10⁻⁴	4.4 × 10⁻⁵	1.6 × 10⁻⁶	8.9 × 10⁻⁵	1.4 × 10⁻⁴
$ω = 30, 000 rpm$	4.5 × 10⁻⁵	2.3 × 10⁻⁶	7.1 × 10⁻⁵	3.0 × 10⁻⁴	4.5 × 10⁻⁵	4.1 × 10⁻⁶	8.0 × 10⁻⁵	2.6 × 10⁻⁴	4.4 × 10⁻⁵	1.4 × 10⁻⁶	1.4 × 10⁻⁴	1.5 × 10⁻⁴

PID: proportional–integral–derivative.

Figure 12.

Control current (ω = 25,000 rpm) (decoupling performance): (a) control current of the left-side bearing in the x-direction, (b) control current of the right-side bearing in the y-direction, (c) control current of the left-side bearing in the y-direction, and (d) control current of the right-side bearing in the y-direction.

The decoupling performance results under the conditions $ω = 20, 000 rpm$ and $ω = 30, 000 rpm$ obtained by the aforementioned controllers are shown in Figures 13 and 14. Almost the same results, especially the steady-state error, are obtained by the proposed controller as those obtained under $ω = 25, 000 rpm$ ; however, the steady-state errors obtained by the PID controller and the cross-feedback controller are larger than those shown in Figure 11. It is confirmed that the proposed controller has better robustness with respect to rotor velocity changes.

Figure 13.

Response trajectory of center of mass of the rotor (ω = 20,000 rpm) (decoupling performance): (a) position x, (b) position y, (c) angular position θ_x, and (d) angular position θ_y.

Figure 14.

Response trajectory of center of mass of the rotor (ω = 30,000 rpm) (decoupling performance): (a) position x, (b) position y, (c) angular position θ_x, and (d) angular position θ_y.

Disturbance rejection performance

In practical applications, external disturbance such as imbalance force always affects the system stability and control performance. Hence, the designed controller should satisfy the disturbance rejection requirement. In this section, we conducted several simulations considering the imbalance force terms in equation (1). The disturbance rejection performance results for position x, position y, angular position $θ_{x}$ , and angular position $θ_{y}$ under the condition $ω = 25, 000 rpm$ obtained by the aforementioned controllers are shown in Figure 15. From the results shown in Figure 15(a) and Table 7, the overshot obtained by the proposed controller is 48.15% and 37.93% larger than that obtained by the PID controller and the cross-feedback controller, respectively. In Figure 11(b), the overshot obtained by the proposed controller is 40.58% and 34.72% larger than that obtained by the PID controller and the cross-feedback controller, respectively. These results were obtained because the control current obtained by the proposed controller is larger than that obtained by the PID controller and cross-feedback controller which are shown in Figure 16. However, in Figure 15(c), the overshot obtained by the proposed controller is the same as that obtained by the PID controller and is 26.09% smaller than that obtained by the cross-feedback controller. It is shown that the proposed controller shows better disturbance rejection performance than the cross-feedback controller. In Figure 15(d), the overshot obtained by the proposed controller is 4% smaller than that obtained by the PID controller and is 20% larger than that obtained by the cross-feedback controller.

Figure 15.

Response trajectory of center of mass of the rotor (ω = 25,000 rpm) (disturbance rejection performance): (a) position x, (b) position y, (c) angular position θ_x, and (d) angular position θ_y.

Table 7.

Comparison of disturbance rejection performance.

Maximum amplitude of overshot	PID control				Cross-feedback control				Modal decoupling control
Maximum amplitude of overshot	x	y	$θ_{x}$	$θ_{y}$	x	y	$θ_{x}$	$θ_{y}$	x	y	$θ_{x}$	$θ_{y}$
$ω = 20, 000 rpm$	2.6 × 10⁻⁵	6.9 × 10⁻⁵	1.5 × 10⁻⁴	2.9 × 10⁻⁴	2.7 × 10⁻⁵	7.1 × 10⁻⁵	2.1 × 10⁻⁴	2.7 × 10⁻⁴	3.8 × 10⁻⁵	9.4 × 10⁻⁵	1.8 × 10⁻⁴	2.4 × 10⁻⁴
$ω = 25, 000 rpm$	2.7 × 10⁻⁵	6.9 × 10⁻⁵	1.7 × 10⁻⁴	2.5 × 10⁻⁴	2.9 × 10⁻⁵	7.2 × 10⁻⁵	2.3 × 10⁻⁴	2.2 × 10⁻⁴	4.0 × 10⁻⁵	9.7 × 10⁻⁵	1.7 × 10⁻⁵	2.4 × 10⁻⁴
$ω = 30, 000 rpm$	2.8 × 10⁻⁵	7.9 × 10⁻⁵	2.1 × 10⁻⁴	2.5 × 10⁻⁴	3.0 × 10⁻⁵	8.1 × 10⁻⁵	2.5 × 10⁻⁴	1.9 × 10⁻⁴	4.2 × 10⁻⁵	10.0 × 10⁻⁵	1.9 × 10⁻⁴	2.2 × 10⁻⁴

PID: proportional–integral–derivative.

Figure 16.

Control current (ω = 25,000 rpm) (disturbance rejection performance): (a) control current of the left-side bearing in the x-direction, (b) control current of the right-side bearing in the y-direction, (c) control current of the left-side bearing in the y-direction, and (d) control current of the right-side bearing in the y-direction.

The disturbance rejection performance results under the conditions $ω = 20, 000 rpm$ and $ω = 30, 000 rpm$ obtained by the aforementioned controllers are shown in Figures 17 and 18, respectively. Almost the same results are obtained by the three controllers under $ω = 25, 000 rpm$ .

Figure 17.

Response trajectory of center of mass of the rotor (ω= 20,000 rpm) (disturbance rejection performance): (a) position x, (b) position y, (c) angular position θ_x, and (d) angular position θ_y.

Figure 18.

Response trajectory of center of mass of the rotor (ω= 30,000 rpm) (disturbance rejection performance): (a) position x, (b) position y, (c) angular position θ_x, and (d) angular position θ_y.

Experimental verifications

To demonstrate the effectiveness of the proposed controller, we also carried out several experimental verifications with the same parameters and conditions as used in the simulations.

Experimental system

The experimental system shown in Figure 19 includes many parts such as magnetic bearings, motor, flywheel, and dynamometer. The control system shown in Figure 20 for the flywheel system includes a TMS320F28335 digital signal processor (DSP), power amplifiers, control coils, eddy current position sensors, and signal condition circuits.

Figure 19.

Experimental system.

Figure 20.

Control system.

Experimental results

Because the imbalance force of the actual magnetic suspended flywheel storage system always exists, we just need to analyze the control performance and decoupling performance of this system in experiments.

The control performance results for position x, position y, angular position $θ_{x}$ , and angular position $θ_{y}$ under the conditions $ω = 25, 000 rpm$ , $ω = 20, 000 rpm$ , and $ω = 30, 000 rpm$ obtained by the aforementioned controllers are shown in Figures 21 –23, respectively. Almost the same results were obtained when compared with the co-simulations which also confirms that the three controllers have certain robustness within the rated speed. However, the imbalance force cannot be ignored in the actual experiment; there is a certain vibration in response trajectory. It is shown that the three controllers have certain robustness.

Figure 21.

Response trajectory of center of mass of the rotor (ω = 25,000 rpm) (control performance): (a) position x, (b) position y, (c) angular position θ_x, and (d) angular position θ_y.

Figure 22.

Response trajectory of center of mass of the rotor (ω = 20,000 rpm) (control performance): (a) position x, (b) position y, (c) angular position θ_x, and (d) angular position θ_y.

Figure 23.

Response trajectory of center of mass of the rotor (ω = 30,000 rpm) (control performance): (a) position x, (b) position y, (c) angular position θ_x, and (d) angular position θ_y.

The decoupling performance results for position x, position y, angular position $θ_{x}$ , and angular position $θ_{y}$ under the conditions $ω = 25, 000 rpm$ , $ω = 20, 000 rpm$ , and $ω = 30, 000 rpm$ obtained by the aforementioned controllers are shown in Figures 24 –26, respectively. Position x and position y obtained by the three controllers were the same, but the overshot of angular position $θ_{x}$ and angular position $θ_{y}$ obtained by the modal decoupling controller are smaller than those obtained by the PID controller and the cross-feedback controller. Especially, the response trajectory of angular position $θ_{y}$ obtained by the modal decoupling controller is more stable than that obtained by the other two controllers. It is shown that the proposed controller shows better decoupling performance than the PID controller and the cross-feedback controller.

Figure 24.

Response trajectory of center of mass of the rotor (ω = 25,000 rpm) (decoupling performance): (a) position x, (b) position y, (c) angular position θ_x, and (d) angular position θ_y.

Figure 25.

Response trajectory of center of mass of the rotor (ω = 20,000 rpm) (decoupling performance): (a) position x, (b) position y, (c) angular position θ_x, and (d) angular position θ_y.

Figure 26.

Response trajectory of center of mass of the rotor (ω = 30,000 rpm) (decoupling performance): (a) position x, (b) position y, (c) angular position θ_x, and (d) angular position θ_y.

Conclusion

In order to suppress the vibration caused by gyroscopic coupling, parameter coupling, and imbalance force in a magnetic suspended flywheel energy storage system with an asymmetric rotor, a novel cross-feedback-based modal decoupling controller was designed after deriving the mathematical model of a radial 4-DOF rotor–bearing system. Better performance was obtained through comparing the decoupling performance, control performance, and disturbance rejection performance with a traditional decentralized PID controller and a cross-feedback controller via ADAMS–MATLAB co-simulation technology. The experimental results also verify the effectiveness of the proposed controller.

Footnotes

Handling Editor: Crinela Pislaru

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported in part by the National Natural Science Foundation of China under Grant No. 61703202 and the Key Research and Development Project of Jiangsu Province under Grant No. BE2017164.

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Vibration suppression for flywheel energy storage system using modal decoupling control

Abstract

Keywords

Introduction

Modeling of the rotor–bearing system

Cross-feedback-based modal decoupling controller design

Simulation results

ADAMS model for the rotor–bearing system

Comparative controller and simulation conditions

Control performance

Decoupling performance

Disturbance rejection performance

Experimental verifications

Experimental system

Experimental results

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References