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Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem
Mechanical Systems and Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem Cong Peng a,n, Yahong Fan b, Ziyuan Huang a, Bangcheng Han a, Jiancheng Fang a a b
School of Instrumentation Science and Opto-Electronics Engineering, Beihang University, Beijing, China Beijing Institute of Control Engineering, Beijing, China
a r t i c l e i n f o
abstract
Article history: Received 20 April 2015 Received in revised form 27 April 2016 Accepted 23 May 2016
This paper presents a novel synchronous micro-vibration suppression method on the basis of the small gain theorem to reduce the frequency-varying synchronous micro-vibration forces for a magnetically suspended flywheel (MSFW). The proposed synchronous micro-vibration suppression method not only eliminates the synchronous current fluctuations to force the rotor spinning around the inertia axis, but also considers the compensation caused by the displacement stiffness in the permanent-magnet (PM)-biased magnetic bearings. Moreover, the stability of the proposed control system is exactly analyzed by using small gain theorem. The effectiveness of the proposed micro-vibration suppression method is demonstrated via the direct measurement of the disturbance forces for a MSFW. The main merit of the proposed method is that it provides a simple and practical method in suppressing the frequency varying micro-vibration forces and preserving the nominal performance of the baseline control system. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Magnetic bearing Flywheel Synchronous micro-vibration Micro-vibration suppression Frequency varying Robust control
1. Introduction The micro-vibration mainly caused by rotary equipment onboard satellites can degrade the performance of the payload. With the increasing stability requirements for the satellite platform, suppressing the micro-vibrations onboard satellites becomes an important problem. These disturbances usually have high frequency with low amplitude, hence they cannot be controlled or suppressed by attitude and orbit control systems [1]. Generally, one of the largest micro-vibration sources onboard satellites is considered from the flywheels [2–4]. The flywheel [5–8], with varying the rotor's speed continually, exchanges the momentum with satellite to stabilize its attitude. According to the suspended bearing for the high speed rotor, flywheel can be divided into the mechanical bearing flywheel and the magnetically suspended flywheel (MSFW). Traditionally, minimizing the micro-vibrations of the mechanical bearing flywheel is to use vibration isolators [9–12]. Although the vibration isolators usually have good isolation performance for the micro-vibrations, they are not suitable for the ultra-precision observation tasks since the normal performance of the spacecraft would affected by the heavy and bulky isolators. MSFW provides a potential scheme for the higher vibration suppression performance [13–17]. It does not need isolators, the real-time vibration suppression algorithm can be directly applied in the magnetic bearing control system. Moreover, the vibration suppression algorithms in MSFW can be generalized more easily compared with the traditional vibration isolators designed for mechanical bearing flywheel. n
Correspondence author. E-mail address: [email protected] (C. Peng).
http://dx.doi.org/10.1016/j.ymssp.2016.05.033 0888-3270/& 2016 Elsevier Ltd. All rights reserved.
Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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The dominant source of the micro-vibration in MSFW is the rotor mass imbalance, which generates the synchronous vibration at the first harmonic of the rotational speed. Imbalance occurs if the inertial axis of the rotor is not coincident with its geometric axis. It is required to make the rotor in MSFW rotate around its inertial axis to suppress the vibration forces, which can be transmitted to the machine housing and is harmful for the stability of the satellite attitude. Specifically, since MSFW operates with time-varying rotational speed, imbalance compensation for MSFW should work for the frequency varying condition. Another important problem is the necessity for suppressing the residual synchronous vibration in the displacement stiffness since the MSFW is usually composed of electromagnets and permanent magnets (PM). The vibration force remains in the PMs should be reduced by the displacement stiffness compensation. Hence, the complete microvibration suppression for the MSFW demands for a robust method to reduce the synchronous component in both control current and displacement stiffness in the presence of time-varying frequency. There are lots of methods for compensating the mass imbalance in active magnetic bearing (AMB) system in the literature. Early imbalance compensation technique was based on the inserted notch filter in the control loop [18].The major drawback of such method is that the notch filter can reduce the stability margin of the closed-loop system so that they are only available for the constant speed. Then some state observer based control strategies were proposed to cancel the vibration signals by injecting a synchronous reference signal [19,20]. However, the observer-based control method can also worsen the stability of the closed-loop system by altering its complementary sensitivity function. Ahrens and Kučera [21] analyzed the unbalance dynamics behaviors of a MSFW rotor and proposed to compensate the rotor imbalance by the balancing data from the magnetic bearings. However, the balancing data can be significantly influenced by the housing dynamics. Furthermore, the proposed balancing method was not mentioned to be adaptive to the frequency varying harmonic. To extend the stability margin to deal with the frequency varying disturbance, Herzog [22] proposed a generalized notch filter, which employs an inverse sensitivity matrix for adaptively compensating the rotor imbalance disturbance with time-varying frequency. [23–26] developed adaptive control methods to achieve rotating about the inertia axis. The adaptive method identified the physical rotor imbalance parameters on-line, and then the identification parameters were used to update the feedback controller. The closed-loop stability was guaranteed by continuous parameters updating. In recent years, some modern control approaches were applied in the AMB system for imbalance compensation. An automatic learning control, based on time-domain iterative learning control and gain-scheduled control, was presented in [27] to eliminate unbalance effects. The variable learning cycle and variable learning gain were used in the learning process to achieve the control system stability over a wide range of operational speed. As for the displacement stiffness compensation, Li [28] proposed a double-loop feedforward control scheme to eliminate the fluctuating force from the negative stiffness. The double-loop needs successive switch to asymptotically achieve the compensation, which is complex. Tang [29] proposed to offer an additional control current to compensate the vibration remains in the permanent magnets. Although much effect has been paid to suppress the dominant vibration forces caused by mass imbalance, most of the methods in the publications are only available for a constant speed condition, which cannot ascertain the absolute stability for the time-varying operational speed. On the other hand, the existing methods for the adaptive disturbance suppression are usually complicated and require large memory space, which aggravates much computation load in digital processors. The application is limited in the space engineering. Moreover, the displacement stiffness compensation method mentioned in the previous literature was not proved to work in a wide speed range since the stability of the closed-loop system was not exactly analyzed and discussed. In this study, a practical approach is developed for completely suppressing the synchronous micro-vibration of a MSFW, including both the synchronous current fluctuations and the residual synchronous forces existed in the negative displacement stiffness. The work proposes a robust resonant controller on the basis of small gain theorem to eliminate the synchronous current fluctuations. The proposed resonant controller employs a compensation phase which is updated to the variable rotational speed. The compensation phase is used to guarantee the stability of the closed-loop system. The stability of the proposed control system is analyzed by means of small gain theorem, which is widely used in practical engineering [30–32]. Another advantage of the proposed control method is that it obtains a good tradeoff between the stability margin and the suppression precision. Finally, the micro-vibration forces are measured by the Kilster table, which is a standard direct forces/torques measurement equipment. Experimental results prove the great effectiveness of the proposed synchronous vibration suppression method for the MSFW. This paper is organized as follows. The configuration of the MSFW and the synchronous micro-vibration dynamics are analyzed in Section 2. In Section 3, a robust composite controller is proposed for the MSFW to suppress the frequency varying synchronous vibration forces and ascertain the stability of the baseline control system. Experimental results are presented in Section 4. Finally, conclusions are given in Section 5.
2. Micro-vibration dynamics modeling and analysis 2.1. Configuration of the MSFW The schematic diagram of the MSFW in this work is shown in Fig. 1. The rotor is supported by five degree-of-freedom (DOF) PM-biased hybrid magnetic bearings, which are composed of two 3-DOF axial magnetic bearings and one 2-DOF Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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Fig. 1. The schematic diagram of the MSFW.
radial magnetic bearing. The 3-DOF axial magnetic bearings are used to control the axial translational and the radial rotational motion of the rotor, while the radial magnetic bearing works for the radial translational motion. This structure is different from the conventional 5-DOF hybrid magnetic bearings with a pair of 2-DOF radial magnetic bearings and a pair of 1-DOF axial magnetic bearings. The pair of the 2-DOF radial magnetic bearings are used for both translational and rotational motion of the rotor in the radial direction. The major advantage of the novel structure for the magnetic bearings results in the short axial length and provides a better reliability and stability control. More details about the structure of the magnetic bearings can be referred to [33,34]. The rotor is driven by the brushless DC motor. An angular contact ball bearing is used as the touchdown bearing to prevent the sudden landing of the rotor in case of power loss and control failure. 2.2. Position sensor system The system is equipped with eddy current displacement sensors to measure the displacement of the rotor. The displacement information is then provided to the feedback control system of the magnetic bearings, as shown in Fig. 2. The eddy current displacement sensors have the merits of high sensibility, reliability, and wide operating temperature, which are significantly suitable for detecting the small air gap length of the magnetic bearings. The configuration of the displacement sensors are shown in Fig. 3. The radial translational displacements ( x, y ) of the rotor are obtained from the four radial probes x+, x−, y+ , and y− , which can be presented as
x = x + − x− y = y+ − y−
(1)
The radial rotational displacements ( α , β ) are calculated from the four axial probes z1, z2, z3 and z 4 , which are described as
α = ( z1 − z3 )/rs β = ( z2 − z 4 )/rs
(2)
where rs is the distance from the geometric center to the axial sensor. That is, the radial rotational motion of the rotor is control by the 3-DOF axial magnetic bearings in a differential mode.
Fig. 2. The coordinates of the MSFW.
Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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Fig. 3. The schematic of the sensor system.
2.3. Micro-vibration dynamics modeling Synchronous micro-vibration is mainly caused by the rotor mass imbalance. The rotor mass imbalance leads to the offset of the mass center from the geometric center and the misalignment of the principal axis from the geometric axis, as shown in Fig. 2. Since the imbalance effects appear in radial directions, only the dynamics of the rotor in the radial DOF are analyzed in this study. Generally, the misalignment relationship between the geometric center OG ( x, y, α , β ) and the mass center OI ( xi , yi , αi, βi ) can be presented as
⎧x ⎪ i ⎪ ⎪ yi ⎨ ⎪ αi ⎪ ⎪ ⎩ βi
= x + εs cos ( Ωt + φs ) = y + εs sin ( Ωt + φs ) = α + εd cos ( Ωt + φd ) = β + εd sin ( Ωt + φd )
(3)
where εs and εd are the eccentricity between OG and OI , φs and φd are the phase angle of the eccentricity in relation to the
( x, y ) and ( α , β ) directions, respectively. Ω is the rotational speed of the rotor. According to Newton's second law, the dynamic model of the MSFW is described as
⎧ mx¨ i = Fx ⎪ ⎪ my¨i = Fy ⎨ ⎪ Jx α¨ i + Jz Ωβi̇ = Px ⎪ J β¨ − J Ωα = P i̇ y z ⎩ y i
(4)
where m is the mass of the rotor, Jx and Jy are the moments of inertial in the radial directions, while Jz is the inertial of the rotor in the axial direction. Fx and Fy are the magnetic forces in the x and y directions, respectively. Px and Py are the magnetic torques in the α and β directions, respectively. Substituting (3) into (4), the equations of motion of the center of the rotor are derived as
⎧ mx¨ = Fx + Fdx ⎪ ⎪ my¨ = Fy + Fdy ⎨ ⎪ Jx α¨ + Jz Ωβ ̇ = Px + Pdx ⎪ J β¨ − J Ωα = P + P ̇ y dy z ⎩ y
(5)
with Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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Fig. 4. Block diagram of the synchronous micro-vibration control system.
⎧ 2 cos Ωt + φ ⎪ Fdx = mεs Ω ( s) ⎨ ⎪ 2 ⎩ Fdy = mεs Ω sin ( Ωt + φs )
(6)
⎧ P = ( J − J ) ε Ω2 cos ( Ωt + φ ) ⎪ dx x z d d ⎨ 2 sin Ωt + φ ⎪ P = J − J ε Ω ( d y z d) ⎩ dy
(7)
(
)
where Fdx and Fdy are the micro-vibration forces in the x and y directions, respectively. Pdx and Pdy are the micro-vibration torques around the radial directions. The micro-vibration forces are caused by the static imbalance, which leads to the offset of the mass center and the geometric center. On the other hand, the micro-vibration torques in radial directions are due to the dynamic imbalance, which is rising from the misalignment of the inertial axis and the geometric axis. The microvibration forces/torques synchronized with the rotational speed would transmit to the housing and further influence the stability of the satellites.
3. Frequency varying micro-vibration suppression The objective of the micro-vibration suppression method is to reduce the synchronous micro-vibration forces/torques as much as possible in the presence of time-varying frequency. In this work, an improved micro-vibration actively control scheme is proposed and the associated stability performance is analyzed by using the small gain theorem. Since the MSFW is dominantly governed by control current acting on the coils in magnetic bearings, the micro-vibration suppression control can contribute to the power saving and stability enhancement performance. The total control system is composed of two parts, the micro-vibration suppression algorithm and the baseline feedback controller for stabilizing the rotor. The micro-vibration suppression algorithm works as a patch for the baseline controller and it should not influence the original stability performance of the magnetic bearing control system. Fig. 4 illustrates the K block diagram of the proposed micro-vibration suppression controller. In the control block diagram, P ( s ) = 2 x is the ms −Ki controlled object, i.e., the magnetic bearing-rotor dynamics [36]. The overall control scheme is given as follows. (1) The baseline controller C ( s ) is designed on the basis of the nominal magnetic bearing system for stabilizing the rotor. The PID control method is used as the baseline controller in the MSFW for the simple and practical application. The excellent PID control performance for suspending and stabilizing the magnetic bearing-rotor system has been widely studied in [35]. See the references for the PID performance in greater details. (2) An adaptive resonant controller Gr ( s ) is proposed to plug into the baseline magnetic bearing control system to reduce the synchronous current fluctuations. Elimination of the synchronous current fluctuations means that the imbalance has no influence on the magnetic bearing actuator and makes the rotor to spin around its inertial axis. When the rotor rotates around its mass center under the closed-loop control, the eccentricity ε → 0, then the micro-vibration forces will vanish spontaneously. (3) The negative displacement stiffness compensation GD ( s ) is designed to compensate for the residual synchronous vibration forces caused by the negative displacement stiffness of the PM-biased magnetic bearing configuration. The details of the proposed micro-vibration suppression method are presented as follows. 3.1. Frequency varying synchronous current fluctuation suppression The objective of the resonant controller [37] is to eliminate the synchronous current fluctuations in the coil current to obtain the inertial axis rotation. Specifically, considering the MSFW operating with continues speed variation, the stability margin of the micro-vibration suppression algorithm should be significantly wide and update with the varied rotational speed. In general, for a fixed rotational speed or above certain critical speed, the reconstructed synchronous component through Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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the resonant controller is designed as [16,28]
cx ( t ) = γ ( a sin Ωt + b cos Ωt )
(8)
However, when the conventional resonant controller inserts into the baseline control system for the magnetic bearings, the sensitivity function of the original control system is changed. Hence, the stability performance deteriorates in the presence of the varied rotational speed. In order to make the resonant controller effective in a wide speed range, the reconstructed synchronous signal is designed as
cx ( t ) = γ ⎡⎣ a sin ( Ωt + ϕ) + b cos ( Ωt + ϕ) ⎤⎦
(9)
where ϕ is the compensation phase, which is used to guarantee the stability of the resonant controller according to the different rotational speed, γ is the gain of the resonant controller, which decides the convergence speed of the suppression. Hence, an improved resonant controller is proposed as
Gr ( s) =
ex ( s ) s cos ϕ + Ω sin ϕ =γ cx ( s ) s2 + Ω2
(10)
It is noted that the proposed resonant controller is affected by the varied rotational speed. The gain γ and the compensation phase ϕ is updated according to the rotational speed. The selection of the parameters are associated with the stability of the improved resonant controller. In order to evaluate the relationship between the compensation phase and the rotational speed, the sensitivity function of the baseline control system is defined as
Fig. 5. Movement of control system poles without and with compensation phase. (a2) and (b2) is the local enlarged plots of (a1) and (b1), respectively.
Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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S ( s) =
1 1 + C ( s) P ( s)
7
(11)
Hence, the characteristic equation for the overall system from ex to dx is derived as follows:
T ( s) = 1 + Gr ( s) + C ( s) P ( s)
(12)
Eq. (12) can be rewritten as follows by substituting (11) into (12):
T ( s ) = s2 + Ω2 + γ ( s cos ϕ + Ω sin ϕ) S ( s ) + G r ( s ) ( s2 + Ω2) S ( s )
(13)
The closed-loop poles can be presented as a function of γ by taking the derivative of s as
∂s ∂γ
= ( jΩ cos ϕ + Ω sin ϕ) S ( jΩ) s = jΩ
(14)
Hence, the phase angle of ∂s/∂γ is
∠
∂s = ∠S ( jΩ) + ϕ ∂γ
(
)
(15)
It is known that the system is stable when all the poles of the closed-loop transfer function are included in the unit circle [38]. In order to ascertain the stability of the overall system in the presence of the time-varying frequency, the condition holds as:
90° < ∠
∂s < 270° ∂γ
(16)
According to (15) and (16), the stability condition is presented as
−90° < ∠S ( jΩ) + ϕ < 90°
(17)
Eq. (17) indicates the relationship between the compensation phase and the rotational speed. It is noted that the compensation phase changes with different operational speed dynamically. The stability margin for the time-varying frequency can be ascertained by different compensation phases. Fig. 5 shows the pole–zero plot of the system with the proposed plug-in resonant controller. It can be seen from Fig. 5(a2) that with the conventional resonant controller, the system is unstable with not all the poles of the closed-loop transfer function are included in the unit circle. The unstable critical speed is 3600 rpm. That is, only when the wheel speed beyond 3600 rpm, the system is stable. Compared to Fig. 5(a2), the poles distribution is changed when the compensation phase φ = 120°, as shown in Fig. 5(b2). The system is stable when the wheel speed beyond 1800 rpm. That is, the stability region for different rotational speed is achieved by different compensation phase.
3.2. Residual synchronous micro-vibration forces compensation It is known the electromagnetic force can be approximately linearized by the Taylor series expansion as
Fx = Kx ( x + Θ) + Ki ix
(18)
where Kx is the displacement stiffness and Ki is the current stiffness. Θ is the synchronous component in the displacement. It is noted that the imbalance compensation by resonant controller only eliminates the synchronous component in the control current, there still exists the synchronous component in the displacement stiffness. Consequently, the complete synchronous vibration suppression also has to compensate the synchronous vibration rising from the negative displacement stiffness existed in the PM-biased magnetic bearings. The compensation for the negative displacement stiffness is to add the compensation control current as
i′ = −
Kx Θ (t ), Ki
(19)
Substituting (19) into (18), the magnetic force is derived as
Fx = Kx (x + Θ) + Ki (ix + i′) = Kx x + Ki ix.
(20)
As we can observed from (20), the synchronous component has been eliminated completely in the magnetic force, including both the synchronous component elimination in control current, and the compensation for the negative displacement stiffness. That is, the synchronous vibration force is minimized and there is almost no vibration force transmitted to the housing. Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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Fig. 6. M − Δ loop for stability analysis with small gain theorem.
3.3. Stability analysis of the proposed control system In this section, the stability of the proposed control system which composed of the baseline controller and the microvibration suppression scheme is analyzed. Moreover, the selection of the appropriate compensation phase ϕ is discussed to ensure the global asymptomatic stability of the closed-loop system in the presence of varied rotational speed. The small gain theorem is a powerful method to analyze the stability characteristics of the systems subject to disturbances and model uncertainties as shown in Fig. 6. In order to obtain the exact stability region of the proposed control system with the application of small gain theorem, the baseline control system should be inverted into the equivalent system as presented in Fig. 7. According to Fig. 7(b), the actual plant Pn ( s ) is composed of Gr ( s ) and P ( s ), i.e.,
Pn ( s ) = P ( s ) + G r ( s )
(21)
Then the model uncertainty Δ( s ) is calculated as
Δ( s ) = Pn ( s ) − P ( s ) = G r ( s )
(22)
That is, the micro-vibration suppression scheme can be regarded as the uncertainty for the controlled plant. On the basis of the small gain theorem, a sufficient condition for the closed-loop stability is given by
Δ ( jω) M ( jω)
∞
<1∀ω∈
(23)
where M ( jω) is the complementary sensitivity function, which is described as
M ( jω) =
C ( jω) ⎡⎣ P ( jω) + G r ( jω) ⎤⎦ 1 + C ( jω) ⎡⎣ P ( jω) + G r ( jω) ⎤⎦
(24)
Fig. 7. Block diagram of the proposed control system. (a) Block diagram of the closed-loop control system. (b) Equivalent block diagram for M − Δ loop transformation.
Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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Fig. 8. Frequency magnitude response using small gain theorem under the rotational speed 2000 rpm with different compensation phase ϕ = 120° (red line), ϕ = 90° (black line), ϕ = 60° (blue line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
It is noted that the magnitude of the uncertainty Δ ( jω) is affected by the compensation phase ϕ , By selecting appropriate compensation phase, the inequality (23) can be satisfied and the global asymptomatic stability is achieved for the closedloop system in the presence of varied rotational speed. Fig. 8 shows the magnitude Bode plot with the rotational speed of 2000 rpm, different magnitude response is plotted with different compensation phases ϕ = 60°, 90°, 120°. It can be observed that with ϕ = 120°, the magnitude of the uncertainty is above the magnitude of the complementary sensitivity function all over the frequencies. That is, Eq. (24) is satisfied and the stability condition is guaranteed. The similar conclusion can be drawn from Fig. 9, when the rotational speed is 4000 rpm under the compensation phase ϕ = 60°. In addition, Fig. 10 shows the frequency magnitude response with adaptive compensation phase under different rotational speed. From Fig. 10, it can be observed that with the adaptive compensation phase, the magnitude of the uncertainty can be adjusted above the magnitude of the complementary sensitivity, the stability is guaranteed under different rotational speed.
4. Experimental results 4.1. Experimental setup To evaluate the effectiveness of the proposed control method, experiments were carried out on the MSFW as shown in Fig. 11. The micro-vibration forces are tested by the Kistler table, which is a standard forces measurement equipment. The Kistler table can monitor the forces in 3-DOF. The interface table as shown in Fig. 11 is to isolate the noise and ensure the rigid contact of the MSFW with the Kistler table. During the test, the MSFW operates in its operational speed range from 0– 4000 rpm. For every 50 rpm, the micro-vibration data is collected by the data acquisition system and transferred to real-
Fig. 9. Frequency magnitude response using small gain theorem under the rotational speed 4000 rpm with different compensation phase ϕ = 60° (red line), ϕ = 90° (black line), ϕ = 120° (blue line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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Fig. 10. Frequency magnitude response with adaptive compensation phase under different rotational speed, Ω = 1000 rpm (pink line), Ω = 2000 rpm (green line), Ω = 3000 rpm (black line), Ω = 4000 rpm (blue line), Ω = 5000 rpm (red line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 11. The experimental set-up.
time multi-analyzer for further analysis. It is noted that the instability possibility aggravates especially in the middle speed range, as revealed in the bode plot of the sensitivity function shown in Fig. 12. For the high speed range, the slope of the sensitivity function converges to zero and the compensation phase almost unchanged with chosen as zero. The proposed controller model can be scaled up and be suitable for even higher rotational speed. In the experiment, the current stiffness for radial and axial magnetic bearings are 4.3 105 N/m and 1.3 105 N/m,
Fig. 12. Bode plot of the sensitivity function of the baseline control systems.
Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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Fig. 13. Current fluctuations without/with synchronous suppression. (a1) Ω = 1200 rpm , (a2) Ω = 2400 rpm , (a3)Ω = 3600 rpm .
respectively. The radial displacement stiffness is 230 N/A and the axial displacement stiffness is 128 N/A. The mass of the rotor is 4.08 kg. The moments of inertia are Jx ¼Jy ¼0.062 kg m2 and Jz ¼0.102 kg m2. The control parameters for the baseline PID controller are listed as kP = 0.288, kI = 4 , kD = 0.00085. In addition, the error between the simulated model and the experimental model is small, especially in the nominal operating point. 4.2. Synchronous current fluctuation suppression First, the proposed resonant controller is used to suppress the synchronous current fluctuations in the coils. Control current fluctuations with and without the proposed resonant controller are compared. A portion of the experimental results at the constant rotational speed of 1200 rpm, 2400 rpm, and 3600 rpm are shown in Figs. 13–15 with different compensation phases and the optimal convergence gain γ . Fig. 13 shows the control current fluctuations in time domain with the synchronous current suppression triggering at about 0.5 s. Fig. 14 presents the frequency spectrum of the recorded control current (a1)–(a3) without synchronous current suppression and (b1)–(b3) with the proposed synchronous current suppression at the rotational speed of 1200 rpm, 2400 rpm, 3600 rpm, respectively. Fig. 15 (a1)–(a3) shows the control current orbits of the radial translational magnetic bearings without synchronous current suppression, while Fig. 15 (b1–b3) presents the control current orbits of the radial translational magnetic bearings with synchronous current suppression at the rotational speed of 1200 rpm, 2400 rpm, and 3600 rpm, respectively. It can be observed from Fig. 13, the peak-peak amplitude of the current fluctuations reduces to less than 0.01 A, which is almost 20% of that without the synchronous current suppression. This is also demonstrated by the current frequency spectrum as shown in Fig. 14. The peaks at 20 Hz, 40 Hz, and 60 Hz in the frequency spectrum completely vanish when the suppression method on. Moreover, the control current orbit converges toward its center quickly after suppressing the synchronous current fluctuations, as shown in Fig. 15. The significant reduction in the coil current fluctuations in different constant rotational speed proves that the proposed resonant controller is effective to the frequency varying synchronous disturbance. 4.3. Micro-vibration forces suppression performance To comprehensively evaluate the performance of the proposed method in attenuating the synchronous micro-vibrations, the micro-vibration forces of the MSFW in the full operational speed range are measured by Kilster table, as illustrated in the experimental set-up.
Fig. 14. Current frequency spectrum without/with suppression. (a1)–(b1) Ω = 1200 rpm , (a2)–(b2) Ω = 2400 rpm , (a3)–(b3) Ω = 3600 rpm .
Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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Fig. 15. Current orbits without/with suppression. (a1)–(b1) Ω = 1200 rpm , (a2)–(b2) Ω = 2400 rpm , (a3)–(b3) Ω = 3600 rpm .
Fig. 16. Experimental results for synchronous micro-vibration forces without suppression.
Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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Fig. 17. Experimental results for synchronous micro-vibration forces with synchronous current fluctuation suppression.
The micro-vibration forces of the MSFW are presented by waterfall plots, which are widely used for evaluating the forces with both frequency and rotational speed. The comparative results are given in Figs. 16–18 for demonstrating the effectiveness of the proposed synchronous micro-vibration suppression method. Fig. 16 shows the waterfall plots of the microvibration forces without any micro-vibration suppression method, Fig. 17 shows the waterfall plots with only the synchronous current fluctuations elimination and Fig. 18 shows the waterfall plots with both synchronous current elimination and residual synchronous vibration compensation. From Fig. 16, it can be seen that there are diagonal ridges of micro-vibration forces. As these micro-vibration forces slide along the frequency axis with the increasing of the wheel speed, the frequency of these micro-vibrations are linearly depended on the wheel speed. They are the wheel harmonics. The synchronous harmonic is the largest of these diagonal ridges, which is highlighted with the red line. As we can observed from Fig. 16, the significant amplification of the synchronous micro-vibration forces illustrates it as the dominant vibration to the housing. Fig. 17 presents the results with eliminating the synchronous current fluctuations. From Fig. 17, it can be observed that the synchronous vibration force does not increase with the rotational speed, it remains at a relatively small value. Specifically, take Fx as an example, the maximum value of the synchronous vibration force without any suppression, as shown in Fig. 16(a), reaches almost 9.1 N. While when the synchronous current elimination brought into effect, as shown in Fig. 17(a), the maximum synchronous vibration force is less than 1.8 N at the maximum rotational speed 4000 rpm. About 80% reduction of the micro-vibration force proves that the proposed synchronous current reduction method is effective in suppressing the synchronous micro-vibration forces. Fig. 18 shows the results with not only eliminating the synchronous current fluctuations but also suppressing the residual synchronous disturbance in the displacement stiffness. Also take Fx as an example, the maximum value of the microvibration force Fx is 0.8 N at the rotational speed 4000 rpm with the complete suppression. Comparing Fig. 18 with Fig. 17, the complete suppression method has a smaller micro-vibration forces, which decrease to 33% of that with only synchronous current fluctuation suppression. The comparative results of the micro-vibration forces are concluded in Table 1. The micro-vibration forces in other DOFs have the similar performance with Fx. Moreover, the proposed suppression method is tested in continues speed run-up condition, and it achieves the good synchronous vibration forces suppression performance with the time-varying rotational speed. It indicates that, by the compensation phases adjustment for the proposed resonant controller when it plugged into the baseline control system under different rotational speed, the proposed method achieves a good robustness stability performance in the presence of frequency varying disturbances. Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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Fig. 18. Experimental results for synchronous micro-vibration forces with both synchronous current suppression and residual synchronous vibration forces compensation in displacement stiffness.
Table 1 Experimental results for micro-vibration forces comparison. Synchronous vibration force (maximum value)
Fx (N)
Fy (N)
Fz (N)
Without suppression With synchronous current suppression Complete suppression
9.1 1.8 0.6
8.8 1.2 0.5
4.1 0.6 0.2
Consequently, these experimental results demonstrate that the proposed method has significant effectiveness in rejecting the frequency varying synchronous micro-vibrations for the MSFW and retaining the good nominal performance of the baseline controller.
5. Conclusion In this paper, an adaptive and robust control method for frequency varying synchronous micro-vibration suppression is proposed for a MSFW. In order to completely minimize the synchronous micro-vibration forces in the presence of the timevarying rotational speed, an improved resonant controller with phase compensation is proposed to eliminate the synchronous current fluctuations and the compensation for the synchronous vibration forces remains in negative displacement stiffness is considered. The stability of the closed-loop system is provided by using the small gain theorem. Finally, the performance and effectiveness of the proposed synchronous micro-vibration suppression method are verified by the direct vibration forces testing experiments for the MSFW via a standard forces measurement equipment. Through the experimental results, the synchronous micro-vibration forces with the proposed suppression method can be decreased to about 6.5% of that without any suppression. In addition, experimental results are tested in the full operational speed range, which demonstrates the robustness of the proposed method when it plugged into the baseline control system. The future work will focus on extending the proposed resonant controller to the multi-frequency micro-vibration suppression. Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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Acknowledgment The work was supported by the National Natural Science Foundation of China under Grant 61403392, the Fundamental Research Funds for the Central Universities under Grant YWF-16-BJ-J-24, and the National Natural Science Foundation of China under Grant 61573032.
References [1] W. Zhou, D. Li, Experimental research on a vibration isolation platform for momentum wheel assembly, J. Sound Vib. 332 (5) (2013) 1157–1171. [2] V. Nagabhushan, N.G. Fitz-Coy, On-orbit jitter control in momentum actuators using a three-flywheel system, Acta Astronaut. 95 (2014) 61–81. [3] H.U. Oh, S. Taniwaki, N. Kinjyo, K. Izawa, Flywheel vibration isolation test using a variable-damping isolator, Smart Mater. Struct. 15 (2) (2006) 365–373. [4] Q. Luo, D. Li, W. Zhou, Studies on vibration isolation for a multiple flywheel system in variable configurations, J. Vib. Control 21 (2015) 105–123. [5] C. Peng, J. Fang, X. Xu, Mismatched disturbance rejection control for voltage-controlled active magnetic bearing via state-space disturbance observe, IEEE Trans. Power Electron. 30 (5) (2015) 2753–2762. [6] D. Gerada, A. Mebarki, N.L. Brown, C. Gerada, A. Cavagino, A. Boglietti, High-speed electrical machines: technologies, trends, and developments, IEEE Trans. Ind. Electron. 61 (6) (2014) 2946–2959. [7] Y. Ren, D. Su, J. Fang, Whirling modes stability criterion for a magnetically suspended flywheel rotor with significant gyroscopic effects and bending modes, IEEE Trans. Ind. Electron. 28 (12) (2013) 5890–5901. [8] C. Peng, J. Fang, Composite anti-disturbance controller for magnetically suspended control moment gyro subject to mismatched disturbances, Nonlinear Dyn. 79 (2015) 1563–1573. [9] L.L. Sun, C.H. Hansen, C. Doolan, Evaluation of the performance of a passive-active vibration isolation systems, Mech. Syst. Signal Process. 50–51 (2015) 480–497. [10] W. Wu, X. Chen, Y. Shan, Analysis and experiment of a vibration isolator using a novel magnetic spring with negative stiffness, J. Sound Vib. 333 (13) (2014) 2958–2970. [11] W. Sun, H. Gao, O. Kaynak, Vibration isolation for active suspensions with performance constraints and actuator saturation, IEEE/ASME Trans. Mechatron. 20 (2) (2015) 675–683. [12] C. Liu, X. Jing, S. Daley, F. Li, Recent advances in micro-vibration isolation, Mech. Syst. Signal Process. 56–57 (2015) 55–80. [13] G. Schweitzer, E.H. Maslen, Magnetic Bearings: Theory, Design, and Application to Rotating Machinery, Springer, Berlin, 2009. [14] K. Jiang, C. Zhu, L. Chen, X. Qian, Multi-DOF rotor model based measurement of stiffness and damping for active magnetic bearing using multifrequency excitation, Mech. Syst. Signal Process. 60–61 (2015) 358–374. [15] K. Jiang, C. Zhu, Multi-frequency periodic vibration suppressing in active magnetic bearing-rotor systems via response matching in frequency domain, Mech. Syst. Signal Process. 25 (2011) 1417–1429. [16] Q. Chen, G. Liu, S. Zheng, Suppression of imbalance vibration for AMBs controlled driveline system using double-loop structure, J. Sound Vib. 337 (2015) 1–13. [17] J. Fang, S. Zheng, B. Han, AMB vibration control for structural resonance of double-gimbal control moment gyro with high-speed magnetically suspended rotor, IEEE/ASME Trans. Mechatron. 18 (1) (2013) 32–43. [18] C.R. Knospe, S.M. Tamer, S.J. Fedigan, Robustness of adaptive rotor vibration control to structured uncertainty, ASME J. Dyn. Syst. Meas. Control 119 (1997) 243–250. [19] F. Matsumura, M. Fujita, K. Okawa, Modeling and control of magnetic bearing systems achieving a rotation around the axis of inertia, in: Proc. of the 2nd Int. Symp. Magnetic Bearings, Tokyo, Japan, 1990, pp. 273–280. [20] T. Mizuno, T. Higuchi, Design of magnetic bearing controllers based on disturbance estimation, in: Proc. of the 2nd Int. Symp. Magnetic Bearings, Tokyo, Japan, Jul. 12–14, 1990, pp. 281–288. [21] M. Ahrens, L. Kučera, R. Larsonneur, Performance of a magnetically suspended flywheel energy storage device, IEEE Trans. Control Syst. Technol. 4 (5) (1996) 494–502. [22] R. Herzog, P. Buhler, C. Gahler, R. Larsonneur, Unbalance compensation using generalized notchfilters in the multivariable feedback of magnetic bearings, IEEE Trans. Control Syst. Technol. 4 (5) (1996) 580–586. [23] J.D. Setiewan, R. Mukherjee, E.H. Maslen, Synchronous sensor runout and unbalance compensation in active magnetic bearings using bias current excitation, ASME J. Dyn. Syst. Meas. Control 124 (1) (2002) 14–24. [24] B. Shafai, S. Beale, P. LaRocca, E. Cusson, Magnetic bearing control systems and adaptive forced balancing, IEEE Control Syst. Mag. 14 (2) (1994) 4–13. [25] C. Bi, D. Z. Wu, Q. Jiang, Z. J. Liu, Optimize control current in magnetic bearings using automatic learning control, in: Proc. of the IEEE Int. Conf. Mechatronics, Istanbul, Turkey, Jun. 3–5, 2004. [26] J. Shi, R. Zmood, L. Qin, Synchronous disturbance attenuation in magnetic bearing systems using adaptive compensation signals, Control Eng. Pract. 12 (3) (2004) 283–290. [27] C. Bi, D. Wu, Q. Jiang, Z. Liu, Automatic learning control for unbalance compensation in active magnetic bearings, IEEE Trans. Magn. 41 (7) (2005) 2270–2280. [28] L. Li, T. Shinshi, C. Iijima, X. Zhang, A. Shimokohbe, Compensation of rotor imbalance for precision rotation of a planar magnetic bearing rotor, Precis. Eng. 27 (2003) 140–150. [29] J. Tang, B. Liu, J. Fang, S.S. Ge, Suppression of vibration caused by residual unbalance of rotor for magnetically suspended flywheel, J. Vib. Control 19 (13) (2012) 1962–1979. [30] X.M. Sun, D. Wu, G. Liu, W. Wang, Input-to-state stability for networked predictive control with random delays in both feedback and forward channels, IEEE Trans. Ind. Electron. 61 (7) (2014) 3519–3526. [31] G. Escobar, M. Hernandez-Gomez, P.R. Martinez, M.F. Martinez-Montejano, A repetitive-based controller for a power factor precompensator, in: IEEE Transactions on Circuits and Systems I, Regular Paper, Vol. 54(9), 2007, pp. 1968–1976. [32] J.H. Su, B.C. Hsu, Application of small-gain theorem in the dead-time compensation of voltage-source-inverter drives, IEEE Trans. Ind. Electron. 52 (5) (2005) 1456–1458. [33] J. Fang, J. Sun, H. Liu, J. Tang, A novel 3-DOF axial hybrid magnetic bearing, IEEE Trans. Magn. 46 (12) (2010) 4034–4045. [34] J. Sun, J. Fang, A novel structure of permanent-magnet-biased radial hybrid magnetic bearing, J. Magn. Magn. Mater. 323 (2) (2011) 202–208. [35] J. Fang, Y. Ren, Y. Fan, Nutation and precession stability criterion of magnetically suspended rigid rotors with gyroscopic effects based on positive and negative frequency characteristics, IEEE Trans. Ind. Electron. 61 (4) (2014) 2003–2014. [36] C. Peng, J. Fang, P. Cui, Dynamics modeling and measurement of the microvibrations for a magnetically suspended flywheel, IEEE Trans. Instrum. Meas. 64 (12) (2015) 3239–3252. [37] Y. Yang, K. Zhou, M. Cheng, B. Zhang, Phase compensation resonant controller for PWM converters, IEEE Trans. Ind. Inform. 9 (2) (2013) 957–964. [38] K. Ogata, Discrete-Time Control Systems, 2nd edition, Prentice Hall, 1994.
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Cong Peng received the B.S. degree from Southeast University, Nanjing, China, in 2010. She is currently working toward the Ph.D. degree in the School of Instrumentation Science and Opto-electronics Engineering, Beihang University, Beijing, China. She is currently a Research Member with Fundamental Science on Novel Inertial Instrument & Navigation System Technology Laboratory, Beihang University. Her research interests include robust control of the magnetic bearing system and the active vibration control of the magnetically suspended actuators.
Yahong Fan was born in Xi’an, China, in 1973. He received the B.S. degree in electronic engineering from Shaanxi Institute of Technology (now Shaanxi University of Technology), Hanzhong, China, in 1996 and the Ph.D. degree in precision instrument and machinery from Beihang University, Beijing, China, in 2011. He is currently a Researcher with Beijing Institute of Control Engineering, China Academy of Space Technology, Beijing. His research interests include spacecraft attitude dynamics and control, active vibration control, and novel inertial actuators like the magnetically suspended flywheel and control moment gyroscope. Dr. Fan received the first-class National Invention Award of China as the fourth inventor in 2007.
Ziyuan Huang received the M.S. degree in mechanical engineering from Xi’an University of Technology, Xi’an, China, in 2012. He is currently working toward the Ph.D. degree in instrument science and technology at the School of Instrumentation Science and Optoelectronics Engineering, Beihang University, Beijing, China. His research interests include high-speed permanent-magnet electrical machine structural design, rotor dynamics and strength analysis, motor losses, and thermal analysis.
Bangcheng Han was born in February 1974. He received the M.S. degree from Jilin University, Changchun, China, in 2001, and the Ph.D. degree from Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun, China, in 2004. In 2004, he was a Postdoctoral Research Fellow in the School of Instrumentation Science and Optoelectronics Engineering, Beihang University, Beijing, China. In 2006, he joined Beihang University, where he is currently a Professor in the School of Instrumentation Science and Optoelectronics Engineering. He has over 20 journal and conference publications. His research interests include mechatronics, magnetic suspension technology, and attitude control actuator of spacecraft.
Jiancheng Fang received the B.S. degree from Shandong University of Technology, Jinan, China, in 1983, the M.S. degree from Xi’an Jiaotong University, Xi’an, China, in 1988, and the Ph.D. degree from Southeast University, Nanjing, China, in 1996. His current research mainly focuses on the magnetic bearing technology and novel inertial instrument and equipment technology.Prof. Fang has the special appointment professorship with the title of “Cheung Kong Scholar,” which has been jointly established by the Ministry of Education of China and the Li Ka Shing Foundation. He is in the first group of Principal Scientists of the National Laboratory for Aeronautics and Astronautics of China. He received the first-class National Invention Award of China as the first inventor, and the second-class National Science and Technology Progress Award of China as the first contributor in 2007.
Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem Cong Peng a,n, Yahong Fan b, Ziyuan Huang a, Bangcheng Han a, Jiancheng Fang a a b
School of Instrumentation Science and Opto-Electronics Engineering, Beihang University, Beijing, China Beijing Institute of Control Engineering, Beijing, China
a r t i c l e i n f o
abstract
Article history: Received 20 April 2015 Received in revised form 27 April 2016 Accepted 23 May 2016
This paper presents a novel synchronous micro-vibration suppression method on the basis of the small gain theorem to reduce the frequency-varying synchronous micro-vibration forces for a magnetically suspended flywheel (MSFW). The proposed synchronous micro-vibration suppression method not only eliminates the synchronous current fluctuations to force the rotor spinning around the inertia axis, but also considers the compensation caused by the displacement stiffness in the permanent-magnet (PM)-biased magnetic bearings. Moreover, the stability of the proposed control system is exactly analyzed by using small gain theorem. The effectiveness of the proposed micro-vibration suppression method is demonstrated via the direct measurement of the disturbance forces for a MSFW. The main merit of the proposed method is that it provides a simple and practical method in suppressing the frequency varying micro-vibration forces and preserving the nominal performance of the baseline control system. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Magnetic bearing Flywheel Synchronous micro-vibration Micro-vibration suppression Frequency varying Robust control
1. Introduction The micro-vibration mainly caused by rotary equipment onboard satellites can degrade the performance of the payload. With the increasing stability requirements for the satellite platform, suppressing the micro-vibrations onboard satellites becomes an important problem. These disturbances usually have high frequency with low amplitude, hence they cannot be controlled or suppressed by attitude and orbit control systems [1]. Generally, one of the largest micro-vibration sources onboard satellites is considered from the flywheels [2–4]. The flywheel [5–8], with varying the rotor's speed continually, exchanges the momentum with satellite to stabilize its attitude. According to the suspended bearing for the high speed rotor, flywheel can be divided into the mechanical bearing flywheel and the magnetically suspended flywheel (MSFW). Traditionally, minimizing the micro-vibrations of the mechanical bearing flywheel is to use vibration isolators [9–12]. Although the vibration isolators usually have good isolation performance for the micro-vibrations, they are not suitable for the ultra-precision observation tasks since the normal performance of the spacecraft would affected by the heavy and bulky isolators. MSFW provides a potential scheme for the higher vibration suppression performance [13–17]. It does not need isolators, the real-time vibration suppression algorithm can be directly applied in the magnetic bearing control system. Moreover, the vibration suppression algorithms in MSFW can be generalized more easily compared with the traditional vibration isolators designed for mechanical bearing flywheel. n
Correspondence author. E-mail address: [email protected] (C. Peng).
http://dx.doi.org/10.1016/j.ymssp.2016.05.033 0888-3270/& 2016 Elsevier Ltd. All rights reserved.
Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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The dominant source of the micro-vibration in MSFW is the rotor mass imbalance, which generates the synchronous vibration at the first harmonic of the rotational speed. Imbalance occurs if the inertial axis of the rotor is not coincident with its geometric axis. It is required to make the rotor in MSFW rotate around its inertial axis to suppress the vibration forces, which can be transmitted to the machine housing and is harmful for the stability of the satellite attitude. Specifically, since MSFW operates with time-varying rotational speed, imbalance compensation for MSFW should work for the frequency varying condition. Another important problem is the necessity for suppressing the residual synchronous vibration in the displacement stiffness since the MSFW is usually composed of electromagnets and permanent magnets (PM). The vibration force remains in the PMs should be reduced by the displacement stiffness compensation. Hence, the complete microvibration suppression for the MSFW demands for a robust method to reduce the synchronous component in both control current and displacement stiffness in the presence of time-varying frequency. There are lots of methods for compensating the mass imbalance in active magnetic bearing (AMB) system in the literature. Early imbalance compensation technique was based on the inserted notch filter in the control loop [18].The major drawback of such method is that the notch filter can reduce the stability margin of the closed-loop system so that they are only available for the constant speed. Then some state observer based control strategies were proposed to cancel the vibration signals by injecting a synchronous reference signal [19,20]. However, the observer-based control method can also worsen the stability of the closed-loop system by altering its complementary sensitivity function. Ahrens and Kučera [21] analyzed the unbalance dynamics behaviors of a MSFW rotor and proposed to compensate the rotor imbalance by the balancing data from the magnetic bearings. However, the balancing data can be significantly influenced by the housing dynamics. Furthermore, the proposed balancing method was not mentioned to be adaptive to the frequency varying harmonic. To extend the stability margin to deal with the frequency varying disturbance, Herzog [22] proposed a generalized notch filter, which employs an inverse sensitivity matrix for adaptively compensating the rotor imbalance disturbance with time-varying frequency. [23–26] developed adaptive control methods to achieve rotating about the inertia axis. The adaptive method identified the physical rotor imbalance parameters on-line, and then the identification parameters were used to update the feedback controller. The closed-loop stability was guaranteed by continuous parameters updating. In recent years, some modern control approaches were applied in the AMB system for imbalance compensation. An automatic learning control, based on time-domain iterative learning control and gain-scheduled control, was presented in [27] to eliminate unbalance effects. The variable learning cycle and variable learning gain were used in the learning process to achieve the control system stability over a wide range of operational speed. As for the displacement stiffness compensation, Li [28] proposed a double-loop feedforward control scheme to eliminate the fluctuating force from the negative stiffness. The double-loop needs successive switch to asymptotically achieve the compensation, which is complex. Tang [29] proposed to offer an additional control current to compensate the vibration remains in the permanent magnets. Although much effect has been paid to suppress the dominant vibration forces caused by mass imbalance, most of the methods in the publications are only available for a constant speed condition, which cannot ascertain the absolute stability for the time-varying operational speed. On the other hand, the existing methods for the adaptive disturbance suppression are usually complicated and require large memory space, which aggravates much computation load in digital processors. The application is limited in the space engineering. Moreover, the displacement stiffness compensation method mentioned in the previous literature was not proved to work in a wide speed range since the stability of the closed-loop system was not exactly analyzed and discussed. In this study, a practical approach is developed for completely suppressing the synchronous micro-vibration of a MSFW, including both the synchronous current fluctuations and the residual synchronous forces existed in the negative displacement stiffness. The work proposes a robust resonant controller on the basis of small gain theorem to eliminate the synchronous current fluctuations. The proposed resonant controller employs a compensation phase which is updated to the variable rotational speed. The compensation phase is used to guarantee the stability of the closed-loop system. The stability of the proposed control system is analyzed by means of small gain theorem, which is widely used in practical engineering [30–32]. Another advantage of the proposed control method is that it obtains a good tradeoff between the stability margin and the suppression precision. Finally, the micro-vibration forces are measured by the Kilster table, which is a standard direct forces/torques measurement equipment. Experimental results prove the great effectiveness of the proposed synchronous vibration suppression method for the MSFW. This paper is organized as follows. The configuration of the MSFW and the synchronous micro-vibration dynamics are analyzed in Section 2. In Section 3, a robust composite controller is proposed for the MSFW to suppress the frequency varying synchronous vibration forces and ascertain the stability of the baseline control system. Experimental results are presented in Section 4. Finally, conclusions are given in Section 5.
2. Micro-vibration dynamics modeling and analysis 2.1. Configuration of the MSFW The schematic diagram of the MSFW in this work is shown in Fig. 1. The rotor is supported by five degree-of-freedom (DOF) PM-biased hybrid magnetic bearings, which are composed of two 3-DOF axial magnetic bearings and one 2-DOF Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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Fig. 1. The schematic diagram of the MSFW.
radial magnetic bearing. The 3-DOF axial magnetic bearings are used to control the axial translational and the radial rotational motion of the rotor, while the radial magnetic bearing works for the radial translational motion. This structure is different from the conventional 5-DOF hybrid magnetic bearings with a pair of 2-DOF radial magnetic bearings and a pair of 1-DOF axial magnetic bearings. The pair of the 2-DOF radial magnetic bearings are used for both translational and rotational motion of the rotor in the radial direction. The major advantage of the novel structure for the magnetic bearings results in the short axial length and provides a better reliability and stability control. More details about the structure of the magnetic bearings can be referred to [33,34]. The rotor is driven by the brushless DC motor. An angular contact ball bearing is used as the touchdown bearing to prevent the sudden landing of the rotor in case of power loss and control failure. 2.2. Position sensor system The system is equipped with eddy current displacement sensors to measure the displacement of the rotor. The displacement information is then provided to the feedback control system of the magnetic bearings, as shown in Fig. 2. The eddy current displacement sensors have the merits of high sensibility, reliability, and wide operating temperature, which are significantly suitable for detecting the small air gap length of the magnetic bearings. The configuration of the displacement sensors are shown in Fig. 3. The radial translational displacements ( x, y ) of the rotor are obtained from the four radial probes x+, x−, y+ , and y− , which can be presented as
x = x + − x− y = y+ − y−
(1)
The radial rotational displacements ( α , β ) are calculated from the four axial probes z1, z2, z3 and z 4 , which are described as
α = ( z1 − z3 )/rs β = ( z2 − z 4 )/rs
(2)
where rs is the distance from the geometric center to the axial sensor. That is, the radial rotational motion of the rotor is control by the 3-DOF axial magnetic bearings in a differential mode.
Fig. 2. The coordinates of the MSFW.
Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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Fig. 3. The schematic of the sensor system.
2.3. Micro-vibration dynamics modeling Synchronous micro-vibration is mainly caused by the rotor mass imbalance. The rotor mass imbalance leads to the offset of the mass center from the geometric center and the misalignment of the principal axis from the geometric axis, as shown in Fig. 2. Since the imbalance effects appear in radial directions, only the dynamics of the rotor in the radial DOF are analyzed in this study. Generally, the misalignment relationship between the geometric center OG ( x, y, α , β ) and the mass center OI ( xi , yi , αi, βi ) can be presented as
⎧x ⎪ i ⎪ ⎪ yi ⎨ ⎪ αi ⎪ ⎪ ⎩ βi
= x + εs cos ( Ωt + φs ) = y + εs sin ( Ωt + φs ) = α + εd cos ( Ωt + φd ) = β + εd sin ( Ωt + φd )
(3)
where εs and εd are the eccentricity between OG and OI , φs and φd are the phase angle of the eccentricity in relation to the
( x, y ) and ( α , β ) directions, respectively. Ω is the rotational speed of the rotor. According to Newton's second law, the dynamic model of the MSFW is described as
⎧ mx¨ i = Fx ⎪ ⎪ my¨i = Fy ⎨ ⎪ Jx α¨ i + Jz Ωβi̇ = Px ⎪ J β¨ − J Ωα = P i̇ y z ⎩ y i
(4)
where m is the mass of the rotor, Jx and Jy are the moments of inertial in the radial directions, while Jz is the inertial of the rotor in the axial direction. Fx and Fy are the magnetic forces in the x and y directions, respectively. Px and Py are the magnetic torques in the α and β directions, respectively. Substituting (3) into (4), the equations of motion of the center of the rotor are derived as
⎧ mx¨ = Fx + Fdx ⎪ ⎪ my¨ = Fy + Fdy ⎨ ⎪ Jx α¨ + Jz Ωβ ̇ = Px + Pdx ⎪ J β¨ − J Ωα = P + P ̇ y dy z ⎩ y
(5)
with Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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Fig. 4. Block diagram of the synchronous micro-vibration control system.
⎧ 2 cos Ωt + φ ⎪ Fdx = mεs Ω ( s) ⎨ ⎪ 2 ⎩ Fdy = mεs Ω sin ( Ωt + φs )
(6)
⎧ P = ( J − J ) ε Ω2 cos ( Ωt + φ ) ⎪ dx x z d d ⎨ 2 sin Ωt + φ ⎪ P = J − J ε Ω ( d y z d) ⎩ dy
(7)
(
)
where Fdx and Fdy are the micro-vibration forces in the x and y directions, respectively. Pdx and Pdy are the micro-vibration torques around the radial directions. The micro-vibration forces are caused by the static imbalance, which leads to the offset of the mass center and the geometric center. On the other hand, the micro-vibration torques in radial directions are due to the dynamic imbalance, which is rising from the misalignment of the inertial axis and the geometric axis. The microvibration forces/torques synchronized with the rotational speed would transmit to the housing and further influence the stability of the satellites.
3. Frequency varying micro-vibration suppression The objective of the micro-vibration suppression method is to reduce the synchronous micro-vibration forces/torques as much as possible in the presence of time-varying frequency. In this work, an improved micro-vibration actively control scheme is proposed and the associated stability performance is analyzed by using the small gain theorem. Since the MSFW is dominantly governed by control current acting on the coils in magnetic bearings, the micro-vibration suppression control can contribute to the power saving and stability enhancement performance. The total control system is composed of two parts, the micro-vibration suppression algorithm and the baseline feedback controller for stabilizing the rotor. The micro-vibration suppression algorithm works as a patch for the baseline controller and it should not influence the original stability performance of the magnetic bearing control system. Fig. 4 illustrates the K block diagram of the proposed micro-vibration suppression controller. In the control block diagram, P ( s ) = 2 x is the ms −Ki controlled object, i.e., the magnetic bearing-rotor dynamics [36]. The overall control scheme is given as follows. (1) The baseline controller C ( s ) is designed on the basis of the nominal magnetic bearing system for stabilizing the rotor. The PID control method is used as the baseline controller in the MSFW for the simple and practical application. The excellent PID control performance for suspending and stabilizing the magnetic bearing-rotor system has been widely studied in [35]. See the references for the PID performance in greater details. (2) An adaptive resonant controller Gr ( s ) is proposed to plug into the baseline magnetic bearing control system to reduce the synchronous current fluctuations. Elimination of the synchronous current fluctuations means that the imbalance has no influence on the magnetic bearing actuator and makes the rotor to spin around its inertial axis. When the rotor rotates around its mass center under the closed-loop control, the eccentricity ε → 0, then the micro-vibration forces will vanish spontaneously. (3) The negative displacement stiffness compensation GD ( s ) is designed to compensate for the residual synchronous vibration forces caused by the negative displacement stiffness of the PM-biased magnetic bearing configuration. The details of the proposed micro-vibration suppression method are presented as follows. 3.1. Frequency varying synchronous current fluctuation suppression The objective of the resonant controller [37] is to eliminate the synchronous current fluctuations in the coil current to obtain the inertial axis rotation. Specifically, considering the MSFW operating with continues speed variation, the stability margin of the micro-vibration suppression algorithm should be significantly wide and update with the varied rotational speed. In general, for a fixed rotational speed or above certain critical speed, the reconstructed synchronous component through Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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the resonant controller is designed as [16,28]
cx ( t ) = γ ( a sin Ωt + b cos Ωt )
(8)
However, when the conventional resonant controller inserts into the baseline control system for the magnetic bearings, the sensitivity function of the original control system is changed. Hence, the stability performance deteriorates in the presence of the varied rotational speed. In order to make the resonant controller effective in a wide speed range, the reconstructed synchronous signal is designed as
cx ( t ) = γ ⎡⎣ a sin ( Ωt + ϕ) + b cos ( Ωt + ϕ) ⎤⎦
(9)
where ϕ is the compensation phase, which is used to guarantee the stability of the resonant controller according to the different rotational speed, γ is the gain of the resonant controller, which decides the convergence speed of the suppression. Hence, an improved resonant controller is proposed as
Gr ( s) =
ex ( s ) s cos ϕ + Ω sin ϕ =γ cx ( s ) s2 + Ω2
(10)
It is noted that the proposed resonant controller is affected by the varied rotational speed. The gain γ and the compensation phase ϕ is updated according to the rotational speed. The selection of the parameters are associated with the stability of the improved resonant controller. In order to evaluate the relationship between the compensation phase and the rotational speed, the sensitivity function of the baseline control system is defined as
Fig. 5. Movement of control system poles without and with compensation phase. (a2) and (b2) is the local enlarged plots of (a1) and (b1), respectively.
Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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S ( s) =
1 1 + C ( s) P ( s)
7
(11)
Hence, the characteristic equation for the overall system from ex to dx is derived as follows:
T ( s) = 1 + Gr ( s) + C ( s) P ( s)
(12)
Eq. (12) can be rewritten as follows by substituting (11) into (12):
T ( s ) = s2 + Ω2 + γ ( s cos ϕ + Ω sin ϕ) S ( s ) + G r ( s ) ( s2 + Ω2) S ( s )
(13)
The closed-loop poles can be presented as a function of γ by taking the derivative of s as
∂s ∂γ
= ( jΩ cos ϕ + Ω sin ϕ) S ( jΩ) s = jΩ
(14)
Hence, the phase angle of ∂s/∂γ is
∠
∂s = ∠S ( jΩ) + ϕ ∂γ
(
)
(15)
It is known that the system is stable when all the poles of the closed-loop transfer function are included in the unit circle [38]. In order to ascertain the stability of the overall system in the presence of the time-varying frequency, the condition holds as:
90° < ∠
∂s < 270° ∂γ
(16)
According to (15) and (16), the stability condition is presented as
−90° < ∠S ( jΩ) + ϕ < 90°
(17)
Eq. (17) indicates the relationship between the compensation phase and the rotational speed. It is noted that the compensation phase changes with different operational speed dynamically. The stability margin for the time-varying frequency can be ascertained by different compensation phases. Fig. 5 shows the pole–zero plot of the system with the proposed plug-in resonant controller. It can be seen from Fig. 5(a2) that with the conventional resonant controller, the system is unstable with not all the poles of the closed-loop transfer function are included in the unit circle. The unstable critical speed is 3600 rpm. That is, only when the wheel speed beyond 3600 rpm, the system is stable. Compared to Fig. 5(a2), the poles distribution is changed when the compensation phase φ = 120°, as shown in Fig. 5(b2). The system is stable when the wheel speed beyond 1800 rpm. That is, the stability region for different rotational speed is achieved by different compensation phase.
3.2. Residual synchronous micro-vibration forces compensation It is known the electromagnetic force can be approximately linearized by the Taylor series expansion as
Fx = Kx ( x + Θ) + Ki ix
(18)
where Kx is the displacement stiffness and Ki is the current stiffness. Θ is the synchronous component in the displacement. It is noted that the imbalance compensation by resonant controller only eliminates the synchronous component in the control current, there still exists the synchronous component in the displacement stiffness. Consequently, the complete synchronous vibration suppression also has to compensate the synchronous vibration rising from the negative displacement stiffness existed in the PM-biased magnetic bearings. The compensation for the negative displacement stiffness is to add the compensation control current as
i′ = −
Kx Θ (t ), Ki
(19)
Substituting (19) into (18), the magnetic force is derived as
Fx = Kx (x + Θ) + Ki (ix + i′) = Kx x + Ki ix.
(20)
As we can observed from (20), the synchronous component has been eliminated completely in the magnetic force, including both the synchronous component elimination in control current, and the compensation for the negative displacement stiffness. That is, the synchronous vibration force is minimized and there is almost no vibration force transmitted to the housing. Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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Fig. 6. M − Δ loop for stability analysis with small gain theorem.
3.3. Stability analysis of the proposed control system In this section, the stability of the proposed control system which composed of the baseline controller and the microvibration suppression scheme is analyzed. Moreover, the selection of the appropriate compensation phase ϕ is discussed to ensure the global asymptomatic stability of the closed-loop system in the presence of varied rotational speed. The small gain theorem is a powerful method to analyze the stability characteristics of the systems subject to disturbances and model uncertainties as shown in Fig. 6. In order to obtain the exact stability region of the proposed control system with the application of small gain theorem, the baseline control system should be inverted into the equivalent system as presented in Fig. 7. According to Fig. 7(b), the actual plant Pn ( s ) is composed of Gr ( s ) and P ( s ), i.e.,
Pn ( s ) = P ( s ) + G r ( s )
(21)
Then the model uncertainty Δ( s ) is calculated as
Δ( s ) = Pn ( s ) − P ( s ) = G r ( s )
(22)
That is, the micro-vibration suppression scheme can be regarded as the uncertainty for the controlled plant. On the basis of the small gain theorem, a sufficient condition for the closed-loop stability is given by
Δ ( jω) M ( jω)
∞
<1∀ω∈
(23)
where M ( jω) is the complementary sensitivity function, which is described as
M ( jω) =
C ( jω) ⎡⎣ P ( jω) + G r ( jω) ⎤⎦ 1 + C ( jω) ⎡⎣ P ( jω) + G r ( jω) ⎤⎦
(24)
Fig. 7. Block diagram of the proposed control system. (a) Block diagram of the closed-loop control system. (b) Equivalent block diagram for M − Δ loop transformation.
Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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Fig. 8. Frequency magnitude response using small gain theorem under the rotational speed 2000 rpm with different compensation phase ϕ = 120° (red line), ϕ = 90° (black line), ϕ = 60° (blue line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
It is noted that the magnitude of the uncertainty Δ ( jω) is affected by the compensation phase ϕ , By selecting appropriate compensation phase, the inequality (23) can be satisfied and the global asymptomatic stability is achieved for the closedloop system in the presence of varied rotational speed. Fig. 8 shows the magnitude Bode plot with the rotational speed of 2000 rpm, different magnitude response is plotted with different compensation phases ϕ = 60°, 90°, 120°. It can be observed that with ϕ = 120°, the magnitude of the uncertainty is above the magnitude of the complementary sensitivity function all over the frequencies. That is, Eq. (24) is satisfied and the stability condition is guaranteed. The similar conclusion can be drawn from Fig. 9, when the rotational speed is 4000 rpm under the compensation phase ϕ = 60°. In addition, Fig. 10 shows the frequency magnitude response with adaptive compensation phase under different rotational speed. From Fig. 10, it can be observed that with the adaptive compensation phase, the magnitude of the uncertainty can be adjusted above the magnitude of the complementary sensitivity, the stability is guaranteed under different rotational speed.
4. Experimental results 4.1. Experimental setup To evaluate the effectiveness of the proposed control method, experiments were carried out on the MSFW as shown in Fig. 11. The micro-vibration forces are tested by the Kistler table, which is a standard forces measurement equipment. The Kistler table can monitor the forces in 3-DOF. The interface table as shown in Fig. 11 is to isolate the noise and ensure the rigid contact of the MSFW with the Kistler table. During the test, the MSFW operates in its operational speed range from 0– 4000 rpm. For every 50 rpm, the micro-vibration data is collected by the data acquisition system and transferred to real-
Fig. 9. Frequency magnitude response using small gain theorem under the rotational speed 4000 rpm with different compensation phase ϕ = 60° (red line), ϕ = 90° (black line), ϕ = 120° (blue line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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Fig. 10. Frequency magnitude response with adaptive compensation phase under different rotational speed, Ω = 1000 rpm (pink line), Ω = 2000 rpm (green line), Ω = 3000 rpm (black line), Ω = 4000 rpm (blue line), Ω = 5000 rpm (red line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 11. The experimental set-up.
time multi-analyzer for further analysis. It is noted that the instability possibility aggravates especially in the middle speed range, as revealed in the bode plot of the sensitivity function shown in Fig. 12. For the high speed range, the slope of the sensitivity function converges to zero and the compensation phase almost unchanged with chosen as zero. The proposed controller model can be scaled up and be suitable for even higher rotational speed. In the experiment, the current stiffness for radial and axial magnetic bearings are 4.3 105 N/m and 1.3 105 N/m,
Fig. 12. Bode plot of the sensitivity function of the baseline control systems.
Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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Fig. 13. Current fluctuations without/with synchronous suppression. (a1) Ω = 1200 rpm , (a2) Ω = 2400 rpm , (a3)Ω = 3600 rpm .
respectively. The radial displacement stiffness is 230 N/A and the axial displacement stiffness is 128 N/A. The mass of the rotor is 4.08 kg. The moments of inertia are Jx ¼Jy ¼0.062 kg m2 and Jz ¼0.102 kg m2. The control parameters for the baseline PID controller are listed as kP = 0.288, kI = 4 , kD = 0.00085. In addition, the error between the simulated model and the experimental model is small, especially in the nominal operating point. 4.2. Synchronous current fluctuation suppression First, the proposed resonant controller is used to suppress the synchronous current fluctuations in the coils. Control current fluctuations with and without the proposed resonant controller are compared. A portion of the experimental results at the constant rotational speed of 1200 rpm, 2400 rpm, and 3600 rpm are shown in Figs. 13–15 with different compensation phases and the optimal convergence gain γ . Fig. 13 shows the control current fluctuations in time domain with the synchronous current suppression triggering at about 0.5 s. Fig. 14 presents the frequency spectrum of the recorded control current (a1)–(a3) without synchronous current suppression and (b1)–(b3) with the proposed synchronous current suppression at the rotational speed of 1200 rpm, 2400 rpm, 3600 rpm, respectively. Fig. 15 (a1)–(a3) shows the control current orbits of the radial translational magnetic bearings without synchronous current suppression, while Fig. 15 (b1–b3) presents the control current orbits of the radial translational magnetic bearings with synchronous current suppression at the rotational speed of 1200 rpm, 2400 rpm, and 3600 rpm, respectively. It can be observed from Fig. 13, the peak-peak amplitude of the current fluctuations reduces to less than 0.01 A, which is almost 20% of that without the synchronous current suppression. This is also demonstrated by the current frequency spectrum as shown in Fig. 14. The peaks at 20 Hz, 40 Hz, and 60 Hz in the frequency spectrum completely vanish when the suppression method on. Moreover, the control current orbit converges toward its center quickly after suppressing the synchronous current fluctuations, as shown in Fig. 15. The significant reduction in the coil current fluctuations in different constant rotational speed proves that the proposed resonant controller is effective to the frequency varying synchronous disturbance. 4.3. Micro-vibration forces suppression performance To comprehensively evaluate the performance of the proposed method in attenuating the synchronous micro-vibrations, the micro-vibration forces of the MSFW in the full operational speed range are measured by Kilster table, as illustrated in the experimental set-up.
Fig. 14. Current frequency spectrum without/with suppression. (a1)–(b1) Ω = 1200 rpm , (a2)–(b2) Ω = 2400 rpm , (a3)–(b3) Ω = 3600 rpm .
Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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Fig. 15. Current orbits without/with suppression. (a1)–(b1) Ω = 1200 rpm , (a2)–(b2) Ω = 2400 rpm , (a3)–(b3) Ω = 3600 rpm .
Fig. 16. Experimental results for synchronous micro-vibration forces without suppression.
Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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Fig. 17. Experimental results for synchronous micro-vibration forces with synchronous current fluctuation suppression.
The micro-vibration forces of the MSFW are presented by waterfall plots, which are widely used for evaluating the forces with both frequency and rotational speed. The comparative results are given in Figs. 16–18 for demonstrating the effectiveness of the proposed synchronous micro-vibration suppression method. Fig. 16 shows the waterfall plots of the microvibration forces without any micro-vibration suppression method, Fig. 17 shows the waterfall plots with only the synchronous current fluctuations elimination and Fig. 18 shows the waterfall plots with both synchronous current elimination and residual synchronous vibration compensation. From Fig. 16, it can be seen that there are diagonal ridges of micro-vibration forces. As these micro-vibration forces slide along the frequency axis with the increasing of the wheel speed, the frequency of these micro-vibrations are linearly depended on the wheel speed. They are the wheel harmonics. The synchronous harmonic is the largest of these diagonal ridges, which is highlighted with the red line. As we can observed from Fig. 16, the significant amplification of the synchronous micro-vibration forces illustrates it as the dominant vibration to the housing. Fig. 17 presents the results with eliminating the synchronous current fluctuations. From Fig. 17, it can be observed that the synchronous vibration force does not increase with the rotational speed, it remains at a relatively small value. Specifically, take Fx as an example, the maximum value of the synchronous vibration force without any suppression, as shown in Fig. 16(a), reaches almost 9.1 N. While when the synchronous current elimination brought into effect, as shown in Fig. 17(a), the maximum synchronous vibration force is less than 1.8 N at the maximum rotational speed 4000 rpm. About 80% reduction of the micro-vibration force proves that the proposed synchronous current reduction method is effective in suppressing the synchronous micro-vibration forces. Fig. 18 shows the results with not only eliminating the synchronous current fluctuations but also suppressing the residual synchronous disturbance in the displacement stiffness. Also take Fx as an example, the maximum value of the microvibration force Fx is 0.8 N at the rotational speed 4000 rpm with the complete suppression. Comparing Fig. 18 with Fig. 17, the complete suppression method has a smaller micro-vibration forces, which decrease to 33% of that with only synchronous current fluctuation suppression. The comparative results of the micro-vibration forces are concluded in Table 1. The micro-vibration forces in other DOFs have the similar performance with Fx. Moreover, the proposed suppression method is tested in continues speed run-up condition, and it achieves the good synchronous vibration forces suppression performance with the time-varying rotational speed. It indicates that, by the compensation phases adjustment for the proposed resonant controller when it plugged into the baseline control system under different rotational speed, the proposed method achieves a good robustness stability performance in the presence of frequency varying disturbances. Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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Fig. 18. Experimental results for synchronous micro-vibration forces with both synchronous current suppression and residual synchronous vibration forces compensation in displacement stiffness.
Table 1 Experimental results for micro-vibration forces comparison. Synchronous vibration force (maximum value)
Fx (N)
Fy (N)
Fz (N)
Without suppression With synchronous current suppression Complete suppression
9.1 1.8 0.6
8.8 1.2 0.5
4.1 0.6 0.2
Consequently, these experimental results demonstrate that the proposed method has significant effectiveness in rejecting the frequency varying synchronous micro-vibrations for the MSFW and retaining the good nominal performance of the baseline controller.
5. Conclusion In this paper, an adaptive and robust control method for frequency varying synchronous micro-vibration suppression is proposed for a MSFW. In order to completely minimize the synchronous micro-vibration forces in the presence of the timevarying rotational speed, an improved resonant controller with phase compensation is proposed to eliminate the synchronous current fluctuations and the compensation for the synchronous vibration forces remains in negative displacement stiffness is considered. The stability of the closed-loop system is provided by using the small gain theorem. Finally, the performance and effectiveness of the proposed synchronous micro-vibration suppression method are verified by the direct vibration forces testing experiments for the MSFW via a standard forces measurement equipment. Through the experimental results, the synchronous micro-vibration forces with the proposed suppression method can be decreased to about 6.5% of that without any suppression. In addition, experimental results are tested in the full operational speed range, which demonstrates the robustness of the proposed method when it plugged into the baseline control system. The future work will focus on extending the proposed resonant controller to the multi-frequency micro-vibration suppression. Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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Acknowledgment The work was supported by the National Natural Science Foundation of China under Grant 61403392, the Fundamental Research Funds for the Central Universities under Grant YWF-16-BJ-J-24, and the National Natural Science Foundation of China under Grant 61573032.
References [1] W. Zhou, D. Li, Experimental research on a vibration isolation platform for momentum wheel assembly, J. Sound Vib. 332 (5) (2013) 1157–1171. [2] V. Nagabhushan, N.G. Fitz-Coy, On-orbit jitter control in momentum actuators using a three-flywheel system, Acta Astronaut. 95 (2014) 61–81. [3] H.U. Oh, S. Taniwaki, N. Kinjyo, K. Izawa, Flywheel vibration isolation test using a variable-damping isolator, Smart Mater. Struct. 15 (2) (2006) 365–373. [4] Q. Luo, D. Li, W. Zhou, Studies on vibration isolation for a multiple flywheel system in variable configurations, J. Vib. Control 21 (2015) 105–123. [5] C. Peng, J. Fang, X. Xu, Mismatched disturbance rejection control for voltage-controlled active magnetic bearing via state-space disturbance observe, IEEE Trans. Power Electron. 30 (5) (2015) 2753–2762. [6] D. Gerada, A. Mebarki, N.L. Brown, C. Gerada, A. Cavagino, A. Boglietti, High-speed electrical machines: technologies, trends, and developments, IEEE Trans. Ind. Electron. 61 (6) (2014) 2946–2959. [7] Y. Ren, D. Su, J. Fang, Whirling modes stability criterion for a magnetically suspended flywheel rotor with significant gyroscopic effects and bending modes, IEEE Trans. Ind. Electron. 28 (12) (2013) 5890–5901. [8] C. Peng, J. Fang, Composite anti-disturbance controller for magnetically suspended control moment gyro subject to mismatched disturbances, Nonlinear Dyn. 79 (2015) 1563–1573. [9] L.L. Sun, C.H. Hansen, C. Doolan, Evaluation of the performance of a passive-active vibration isolation systems, Mech. Syst. Signal Process. 50–51 (2015) 480–497. [10] W. Wu, X. Chen, Y. Shan, Analysis and experiment of a vibration isolator using a novel magnetic spring with negative stiffness, J. Sound Vib. 333 (13) (2014) 2958–2970. [11] W. Sun, H. Gao, O. Kaynak, Vibration isolation for active suspensions with performance constraints and actuator saturation, IEEE/ASME Trans. Mechatron. 20 (2) (2015) 675–683. [12] C. Liu, X. Jing, S. Daley, F. Li, Recent advances in micro-vibration isolation, Mech. Syst. Signal Process. 56–57 (2015) 55–80. [13] G. Schweitzer, E.H. Maslen, Magnetic Bearings: Theory, Design, and Application to Rotating Machinery, Springer, Berlin, 2009. [14] K. Jiang, C. Zhu, L. Chen, X. Qian, Multi-DOF rotor model based measurement of stiffness and damping for active magnetic bearing using multifrequency excitation, Mech. Syst. Signal Process. 60–61 (2015) 358–374. [15] K. Jiang, C. Zhu, Multi-frequency periodic vibration suppressing in active magnetic bearing-rotor systems via response matching in frequency domain, Mech. Syst. Signal Process. 25 (2011) 1417–1429. [16] Q. Chen, G. Liu, S. Zheng, Suppression of imbalance vibration for AMBs controlled driveline system using double-loop structure, J. Sound Vib. 337 (2015) 1–13. [17] J. Fang, S. Zheng, B. Han, AMB vibration control for structural resonance of double-gimbal control moment gyro with high-speed magnetically suspended rotor, IEEE/ASME Trans. Mechatron. 18 (1) (2013) 32–43. [18] C.R. Knospe, S.M. Tamer, S.J. Fedigan, Robustness of adaptive rotor vibration control to structured uncertainty, ASME J. Dyn. Syst. Meas. Control 119 (1997) 243–250. [19] F. Matsumura, M. Fujita, K. Okawa, Modeling and control of magnetic bearing systems achieving a rotation around the axis of inertia, in: Proc. of the 2nd Int. Symp. Magnetic Bearings, Tokyo, Japan, 1990, pp. 273–280. [20] T. Mizuno, T. Higuchi, Design of magnetic bearing controllers based on disturbance estimation, in: Proc. of the 2nd Int. Symp. Magnetic Bearings, Tokyo, Japan, Jul. 12–14, 1990, pp. 281–288. [21] M. Ahrens, L. Kučera, R. Larsonneur, Performance of a magnetically suspended flywheel energy storage device, IEEE Trans. Control Syst. Technol. 4 (5) (1996) 494–502. [22] R. Herzog, P. Buhler, C. Gahler, R. Larsonneur, Unbalance compensation using generalized notchfilters in the multivariable feedback of magnetic bearings, IEEE Trans. Control Syst. Technol. 4 (5) (1996) 580–586. [23] J.D. Setiewan, R. Mukherjee, E.H. Maslen, Synchronous sensor runout and unbalance compensation in active magnetic bearings using bias current excitation, ASME J. Dyn. Syst. Meas. Control 124 (1) (2002) 14–24. [24] B. Shafai, S. Beale, P. LaRocca, E. Cusson, Magnetic bearing control systems and adaptive forced balancing, IEEE Control Syst. Mag. 14 (2) (1994) 4–13. [25] C. Bi, D. Z. Wu, Q. Jiang, Z. J. Liu, Optimize control current in magnetic bearings using automatic learning control, in: Proc. of the IEEE Int. Conf. Mechatronics, Istanbul, Turkey, Jun. 3–5, 2004. [26] J. Shi, R. Zmood, L. Qin, Synchronous disturbance attenuation in magnetic bearing systems using adaptive compensation signals, Control Eng. Pract. 12 (3) (2004) 283–290. [27] C. Bi, D. Wu, Q. Jiang, Z. Liu, Automatic learning control for unbalance compensation in active magnetic bearings, IEEE Trans. Magn. 41 (7) (2005) 2270–2280. [28] L. Li, T. Shinshi, C. Iijima, X. Zhang, A. Shimokohbe, Compensation of rotor imbalance for precision rotation of a planar magnetic bearing rotor, Precis. Eng. 27 (2003) 140–150. [29] J. Tang, B. Liu, J. Fang, S.S. Ge, Suppression of vibration caused by residual unbalance of rotor for magnetically suspended flywheel, J. Vib. Control 19 (13) (2012) 1962–1979. [30] X.M. Sun, D. Wu, G. Liu, W. Wang, Input-to-state stability for networked predictive control with random delays in both feedback and forward channels, IEEE Trans. Ind. Electron. 61 (7) (2014) 3519–3526. [31] G. Escobar, M. Hernandez-Gomez, P.R. Martinez, M.F. Martinez-Montejano, A repetitive-based controller for a power factor precompensator, in: IEEE Transactions on Circuits and Systems I, Regular Paper, Vol. 54(9), 2007, pp. 1968–1976. [32] J.H. Su, B.C. Hsu, Application of small-gain theorem in the dead-time compensation of voltage-source-inverter drives, IEEE Trans. Ind. Electron. 52 (5) (2005) 1456–1458. [33] J. Fang, J. Sun, H. Liu, J. Tang, A novel 3-DOF axial hybrid magnetic bearing, IEEE Trans. Magn. 46 (12) (2010) 4034–4045. [34] J. Sun, J. Fang, A novel structure of permanent-magnet-biased radial hybrid magnetic bearing, J. Magn. Magn. Mater. 323 (2) (2011) 202–208. [35] J. Fang, Y. Ren, Y. Fan, Nutation and precession stability criterion of magnetically suspended rigid rotors with gyroscopic effects based on positive and negative frequency characteristics, IEEE Trans. Ind. Electron. 61 (4) (2014) 2003–2014. [36] C. Peng, J. Fang, P. Cui, Dynamics modeling and measurement of the microvibrations for a magnetically suspended flywheel, IEEE Trans. Instrum. Meas. 64 (12) (2015) 3239–3252. [37] Y. Yang, K. Zhou, M. Cheng, B. Zhang, Phase compensation resonant controller for PWM converters, IEEE Trans. Ind. Inform. 9 (2) (2013) 957–964. [38] K. Ogata, Discrete-Time Control Systems, 2nd edition, Prentice Hall, 1994.
Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i
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Cong Peng received the B.S. degree from Southeast University, Nanjing, China, in 2010. She is currently working toward the Ph.D. degree in the School of Instrumentation Science and Opto-electronics Engineering, Beihang University, Beijing, China. She is currently a Research Member with Fundamental Science on Novel Inertial Instrument & Navigation System Technology Laboratory, Beihang University. Her research interests include robust control of the magnetic bearing system and the active vibration control of the magnetically suspended actuators.
Yahong Fan was born in Xi’an, China, in 1973. He received the B.S. degree in electronic engineering from Shaanxi Institute of Technology (now Shaanxi University of Technology), Hanzhong, China, in 1996 and the Ph.D. degree in precision instrument and machinery from Beihang University, Beijing, China, in 2011. He is currently a Researcher with Beijing Institute of Control Engineering, China Academy of Space Technology, Beijing. His research interests include spacecraft attitude dynamics and control, active vibration control, and novel inertial actuators like the magnetically suspended flywheel and control moment gyroscope. Dr. Fan received the first-class National Invention Award of China as the fourth inventor in 2007.
Ziyuan Huang received the M.S. degree in mechanical engineering from Xi’an University of Technology, Xi’an, China, in 2012. He is currently working toward the Ph.D. degree in instrument science and technology at the School of Instrumentation Science and Optoelectronics Engineering, Beihang University, Beijing, China. His research interests include high-speed permanent-magnet electrical machine structural design, rotor dynamics and strength analysis, motor losses, and thermal analysis.
Bangcheng Han was born in February 1974. He received the M.S. degree from Jilin University, Changchun, China, in 2001, and the Ph.D. degree from Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun, China, in 2004. In 2004, he was a Postdoctoral Research Fellow in the School of Instrumentation Science and Optoelectronics Engineering, Beihang University, Beijing, China. In 2006, he joined Beihang University, where he is currently a Professor in the School of Instrumentation Science and Optoelectronics Engineering. He has over 20 journal and conference publications. His research interests include mechatronics, magnetic suspension technology, and attitude control actuator of spacecraft.
Jiancheng Fang received the B.S. degree from Shandong University of Technology, Jinan, China, in 1983, the M.S. degree from Xi’an Jiaotong University, Xi’an, China, in 1988, and the Ph.D. degree from Southeast University, Nanjing, China, in 1996. His current research mainly focuses on the magnetic bearing technology and novel inertial instrument and equipment technology.Prof. Fang has the special appointment professorship with the title of “Cheung Kong Scholar,” which has been jointly established by the Ministry of Education of China and the Li Ka Shing Foundation. He is in the first group of Principal Scientists of the National Laboratory for Aeronautics and Astronautics of China. He received the first-class National Invention Award of China as the first inventor, and the second-class National Science and Technology Progress Award of China as the first contributor in 2007.
Please cite this article as: C. Peng, et al., Frequency-varying synchronous micro-vibration suppression for a MSFW with application of small-gain theorem, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.033i