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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 64, NO. 6, JUNE 2017
Combined FIR and Fractional-Order Repetitive Control for Harmonic Current Suppression of Magnetically Suspended Rotor System Peiling Cui, Qirui Wang, Sheng Li, and Qian Gao
Abstract—Magnetically suspended rotor has the characteristics of no friction and active vibration control. The mass unbalance and sensor runout of the rotor induce harmonic current, which will cause harmonic vibration. The ratio of the sampling rate to the fundamental frequency of harmonic current is often not an integer. The existing repetitive control (RC) for the suppression of harmonic current in magnetic bearing is applicable for the integer case. Harmonic current suppression performance is degraded drastically if the ratio is fractional. In this paper, combined finite-impulse response (FIR) and fractional-order RC is proposed. FIR filter and fractional-order RC are combined to perform harmonic current suppression at arbitrary frequency. Phase compensator is designed to enhance the stability of magnetically suspended rotor system. Experimental results are given to verify the effectiveness and superiority of the proposed method. Index Terms—Harmonic current, magnetically suspended rotor, mass unbalance, repetitive control (RC), sensor runout.
I. INTRODUCTION AGNETICALLY suspended rotor system has the characteristics of no friction, long lifespan, and active vibration control [1]. It has wide applications in the field of magnetically suspended flywheel, magnetically suspended control moment gyro, and magnetically suspended molecule pump [2]–[4]. For the limited manufacturing precision, there is unavoidable mass unbalance in magnetically suspended rotor, which will result in synchronous current. Besides, because of the nonuniform electronic or magnetic properties, there are synchronous and multifrequency signals in the sensor measurement [5]. This is called sensor runout which induces the harmonic current. The harmonic disturbances in control current will further
M
Manuscript received June 5, 2016; revised October 24, 2016; accepted November 19, 2016. Date of publication February 16, 2017; date of current version May 10, 2017. This work was supported in part by the National Natural Science Foundation of China under Grant 61673044, in part by the State Scholarship Fund under Grant 201606025036, and in part by the National Key Research and Development Plan under Grant 2016YFB0500804. (Corresponding author: Peiling Cui.) The authors are with the School of Instrumentation Science and Optoelectronics Engineering, Beihang University, Beijing 100191, China (e-mail:
[email protected];
[email protected];
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2017.2668985
lead to harmonic vibration, and it can be transmitted to the machine base. Thus, it is very important to suppress the harmonic current in magnetically suspended rotor system. In general, harmonic current suppression algorithms can be classified into two classes. One is for single-frequency interference suppression. The amplitude and phase of the signal at a specific frequency are estimated by an adaptive algorithm, and then the estimation signal is plugged into the original system by feedforward or feedback [6]. To eliminate multifrequency signals, several single-frequency algorithms are connected in [7]. Model reference adaptive control is put forward in [8] to suppress harmonic disturbance of a multiple-input multiple-output system, but it can be implemented only in a certain frequency range. Filter is widely used in many fields [9]. Considering the influence of rotation speed on compensation performance in [10] and [11], multiple notch filters are utilized to achieve highaccuracy compensation for harmonic components in magnetic bearing systems. However, the above methods are not conducive to practical applications. On one hand, the convergence rate of the algorithms at different frequencies is inconsistent, and the stability also differs [12]. On the other hand, the design complexity and computational burden rise dramatically with the increase of frequency components, so that the harmonic suppression performance will be badly degraded. The other class realizes multifrequency disturbance suppression by using one algorithm, e.g., self-optimizing control [13], the fast block least mean square algorithm [14], and repetitive control (RC) [15], etc. Based on the internal model principle, RC is found to be an effective solution among these methods to suppress periodic disturbance that has a known period, uncertain amplitude, and multiple frequency components. RC has the advantages of simple structure, low computational cost, and easy implementation in practice [16], [17]. In [18], a low-pass filter is cascaded with delay units to improve the robustness and stability of the RC system. But the amplitude attenuation and phase delay of high-frequency signals become obvious, which weaken the suppression performance of RC. In [19], a phase lead filter is introduced in RC to effectively compensate the phase lag caused by a low-pass filter in high frequencies. Aiming at operation features of magnetically suspended rotor systems, a phase compensator is added in [20] to widen the system bandwidth, and the control gain of RC is determined to meet the stability requirements. Nevertheless, the above-mentioned RC schemes are sensitive to frequency variations.
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CUI et al.: COMBINED FIR AND FRACTIONAL-ORDER REPETITIVE CONTROL FOR HARMONIC CURRENT SUPPRESSION
In engineering applications, the conventional RC needs to be discretized with N delay units (N is the ratio of sampling rate to the fundamental frequency), and N is ordered to be an integer for actual implementation and accurate suppression. However, in fractional cases, N has to be rounded to its nearest integer and the suppression frequencies provided by RC are not equal to the interested harmonic frequencies. Hence, the elimination precision is greatly decreased and the system might even be destabilized by variable harmonic disturbance [21]. According to the properties of linear phase, the general low-pass filter in [22] is substituted with an improved finite-impulse response (FIR) filter to compensate the fractional part of delay by changing the filter’s coefficients, which enhances the system frequency adaptability. But this scheme requires a low normalized cut-off frequency so that it cannot apply to high-frequency signals such as model error. In [23], an adaptive fuzzy control approach is presented to deal with unmodeled dynamics. In [24], a fractional delay filter is adopted to well approximate the fractional delay. Compared with the general RC, the harmonic disturbance at arbitrary frequency can be suppressed by the new RC at a fixed sampling rate. Due to the phase delay and amplitude attenuation caused by a low-pass filter, the harmonic suppression performance of the system is rather limited particularly in high frequencies. Besides, the more accurate approximation implies more control branches and heavier computational burden. It is necessary to make a tradeoff between control accuracy and dynamic response. Nevertheless, the aforementioned RCs are mainly used in power systems. They are not applicable for harmonic current suppression in magnetically suspended rotor system, which is open-loop unstable with high rotation speed [25]. To address this issue, a combined FIR and fractional-order RC is proposed in this paper. It comprises an integer-order delay element, an FIR filter, and a fractional-order delay element, which are connected in series. The novel control scheme can not only cancel harmonic current at arbitrary frequency, but overcome the drawbacks of a low-pass filter to achieve effective suppression at full speed range of the rotor. A phase compensator is designed to stabilize the entire system. Furthermore, experimental results are given to show the effectiveness of the proposed RC. This paper is organized as follows. In Section II, the dynamic model of magnetically suspended rotor is built. In Section III, the combined FIR and fractional-order RC is introduced and its stability performance is analyzed. In Section IV, experimental results are provided to verify the validity and superiority of the proposed method. Section V gives the conclusion. II. DYNAMIC MODEL FOR MAGNETICALLY SUSPENDED ROTOR SYSTEM WITH MASS UNBALANCE AND SENSOR RUNOUT The structure of magnetically suspended rotor system is shown in Fig. 1. Two radial translation degrees of freedom are controlled by active magnetic bearing. W denotes the geometric center of the stator, O is the geometric center of the rotor, and C is the mass center of the rotor. Inertial coordinate WXY is built with W as the origin. Rotational coordinate Oεη at the
Fig. 1.
Structure of magnetically suspended rotor system.
Fig. 2.
Schematic diagram of the coordinate system.
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rotation speed Ω is built with O as the origin. The schematic diagram of the coordinate system is given in Fig. 2. Because the rotor is symmetrical in structure, the analysis of harmonic disturbance and the design of control algorithm are given by taking the X-axis as an example. The dynamic equation of magnetically suspended rotor system is m¨ x = fx + fu
(1)
where m denotes the rotor mass, fx denotes the magnetic bearing force in the X direction. fu is fu = meΩ2 cos(Ωt + φ)
(2)
where e is the eccentricity between the geometric center O and the mass center C. Ω denotes the rotor rotation speed. φ denotes the phase of the rotor mass unbalance. When the rotor is suspended almost in the center, the magnetic bearing force fx can be linearized at the equilibrium point [26] fx ≈ Kx x + Ki i
(3)
where Kx and Ki denote the displacement stiffness and the current stiffness of magnetic bearing. i denotes the control current in the magnetic bearing coil. The sensing surface of the displacement sensor has nonuniform electrical or magnetic properties. The output of the displacement sensor contains harmonic signal xs (t) = x(t) + xd (t)
(4)
where xd (t) is the sensor runout and can be expressed by the form [27] xd (t) =
p h=1
ch sin (hΩt + θh )
(5)
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 64, NO. 6, JUNE 2017
where Ts denotes the sampling period. N1 denotes the nearest integer that is around or equal to N . From Fig. 4, the sensitivity function from R(s) and D(s) to I(s) is S1 (s) =
I(s) R(s) − D(s)Ks
= S0 (s)
1 − Q(s)e−N 1 T s s 1 − Q(s)e−N 1 T s s −
S 0 (s) −N 1 T s s C(s) G c (s) Q(s)e
Fig. 3. Block diagram of the magnetically suspended rotor control system.
Fig. 4.
General plug-in repetitive control system.
where h is the order of harmonics (h = 1, 2, 3 . . .), ch is the amplitude of the sensor runout, and θh is the initial phase. The block diagram of the magnetically suspended rotor control system in the X channel is given in Fig. 3. Gc (s) is the feedback controller. Gw (s) is the transfer function of the amplifier. Gp (s) is the transfer function of the magnetically suspended rotor. Ks is the sensor gain. R(s) denotes the reference signal. I(s) is the current of the active magnetic bearing. From Fig. 3, the transfer function of the magnetic bearing current I(s) is I(s) =
Gc (s)Gw (s) R(s) 1 + Ks Gc (s)Gw (s)Gp (s) +
−Ks Gc (s)Gw (s) Xd (s) 1 + Ks Gc (s)Gw (s)Gp (s)
+
−Ks Gc (s)Gw (s)Gp (s) Fu (s). Ki (1 + Ks Gc (s)Gw (s)Gp (s))
(6)
III. HARMONIC CURRENT SUPPRESSION OF MAGNETICALLY SUSPENDED ROTOR A. General RC The block diagram of the general plug-in RC is shown in Fig. 4. Q(s) is a low-pass filter with the cut-off frequency ωc . e−N 1 ·T s s is a delay element that is an integer multiple of sampling period. C(s) is designed as a phase compensator to ensure the stability of the RC system and to increase the bandwidth. Gr c (s) is the general plug-in RC. D(s) is the equivalent periodic disturbance of Xd (s) and Fu (s). From Fig. 4, the transfer function of the plug-in RC is Q(s)e−N 1 T s s C(s) 1 − Q(s)e−N 1 T s s
G c (s)G w (s) is the sensitivity function where S0 (s) = 1+G c (s)G w (s)G p (s)K s of the original system without suppression algorithm. Under ideal conditions, it can be obtained that |Q(s)|s=j ω = 1 and arg [Q(s)]s=j ω = 0 for ω ∈ (0, ωc ). By using ωn = 2π T · k to denote the frequencies of harmonic disturbance with T being the fundamental period and k = 1, 2, 3 . . ., the sensitivity function is equal to zero when ω = ωn and N1 = N , i.e. |S1 (s)|s=j ω n = 0, and arg [S1 (s)]s=j ω n = 0. Thus, the harmonic disturbance can be rejected by the general plug-in RC. As can be seen from (8), the above results cannot be achieved by general RC in practice. First, for ω ∈ (0, ωc ), the low-pass filter has amplitude attenuation and phase delay. The amplitude attenuation can be ignored by increasing the cut-off frequency or reducing the transition bandwidth. But the phase delay of the low-pass filter will become larger as the frequency increases, and the phase variation trend cannot be accurately acquired. Second, only if the ratio N of synchronous period T to sampling period Ts is an integer, then N1 = N . This shows that the general RC can perform harmonic suppression at specific fixed frequency, but not at arbitrary frequency.
B. Combined FIR and Fractional-Order RC
Equations (2), (5), and (6) show that the harmonic current in magnetic bearing will be caused by the mass unbalance and sensor runout, which will lead to harmonic vibration.
Gr c (s) =
(8)
(7)
There are two challenges in harmonic current suppression of magnetic bearing system using RC. First, the rotation speed of magnetically suspended rotor system is high, and the frequencies of resulting harmonic disturbance is high correspondingly. RC is required to have high-frequency harmonic suppression ability. Second, the rotation speed is not fixed, and the rotor often operates at various speed. RC is required to have frequency adaptability to realize harmonic current suppression at arbitrary fixed rotation frequency. 1) Introduction of Combined FIR and FractionalOrder RC: According to the shortcomings of general RC and the current suppression characteristics of magnetically suspended rotor systems, the combined FIR and fractional-order RC is proposed by connecting an integer-order delay element, an FIR filter and a fractional-order delay element in series. The block diagram of combined FIR and fractional-order RC is shown in Fig. 5. e−N 2 ·T s s is a delay element that is an integer multiple of sampling period. F (s) is an FIR filter with linear phase. Its phase delay ϕ is proportional to frequency ω, and can be accurately obtained and compensated to make up the drawbacks of low-pass filter. e−A ·T s s is the fractional delay element obtained by fractional delay filters. By using the above three links in series form, the error resulting from the phase delay of low-pass filter can be eliminated, and the system bandwidth
CUI et al.: COMBINED FIR AND FRACTIONAL-ORDER REPETITIVE CONTROL FOR HARMONIC CURRENT SUPPRESSION
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The transition bandwidth of the FIR filter should be as small as possible. In the design of FIR, the least transition bandwidth can be implemented by triangular window function [22]. Hence, triangular window function is utilized in this paper to design the FIR filter. The transition bandwidth should not be more than π5 , and the order of the corresponding FIR filter is M = 9. 3) Fractional Delay Element: In engineering applications, the fractional delay element e−A ·T s s cannot be used directly. It can be approximated by Lagrange interpolation polynomial [24] Fig. 5.
e−A ·T s s ≈
Block diagram of combined FIR and fractional-order RC.
n
Dl e−l·T s s
(12)
l=0
is augmented. In addition, the error caused by the noninteger ratio of sampling frequency to the fundamental frequency of harmonic disturbance can be rejected as well. Therefore, the suppression of harmonic current in magnetic bearing at arbitrary fixed rotation frequency can be realized. Taking the reference signal R(s) and the equivalent harmonic disturbance D(s) as input, the current I(s) in magnetic bearing as output, the sensitivity function S2 (s) can be obtained as follows:
where the coefficient Dl is Dl =
n A−r . l−r r =0
(13)
r = l
With the increase of the order n, the difference Rn between the polynomial nl=0 Dl e−l·T s s and the fractional delay element e−A ·T s s becomes smaller, but the computational cost is increased drastically. Thus, Rn and the computational cost should be considered simultaneously. The magnitude responses of Lagrange I(s) S2 (s) = interpolation polynomial are given in Fig. 6. R(s) − D(s)Ks 4) Stability Analysis of Combined FIR and = S0 (s) Fractional-Order RC: From the analysis of FIR filter and fractional delay element, it can be seen that the phase of the 1 − F (s)e−A ·T s s e−N 2 ·T s s . FIR filter is ϕ = −εT ω. The magnitude is |F (z)| ≈ 1 with S (s) s 1 − F (s)e−A ·T s s e−N 2 ·T s s − G0c (s) F (s)e−A ·T s s e−N 2 ·T s s C(s) ω ∈ (0, ωc ). The fractional delay element is approximated by (9) Lagrange interpolation polynomial. The phase is−A · Ts ω, and the magnitude is | nl=0 Dl e−l·T s s |s=j ω ≈ 1. For the stability The cut-off frequency ωc of low-pass filter is larger than analysis, the following can be obtained: the maximal frequency ωm ax of harmonic disturbance. For ω ∈ n (0, ωm ax ), the amplitude attenuation of F (z) is very small which F (z)e−N 2 ·T s s Dl e−l·T s s = |F (z)| · e−N 2 ·T s s can be approximated as |F (z)| = 1. l=0 2) FIR Low-Pass Filter: FIR filter satisfies the charac n teristics of low-pass filter, and its phase is proportional to the −l·T s s −(ε+N 2 +A )·T s s · D e e l frequency. Without loss of generality, FIR filter can be expressed l=0 as follows to replace the conventional low-pass filter: ≈ 1 · e−(ε+N 2 +A )·T s s M −1 ai z −i (10) F (z) = =1 · e−T s . (14) i=0
where M denotes the order of the FIR filter. ai denotes the coefficient. If ai is even symmetric, the phase ϕ of the FIR filter is ϕ = arg [F (z)] = arg F (eT s s ) s=j ω = −εωTs (11) where ε = (M2−1) . Ts = 0.0002s is the sampling period. For assuring stability margin and harmonic suppression precision, the cut-off frequency ωc of the FIR filter should be larger than the maximal frequency of harmonic disturbance. But if the cut-off frequency ωc is too large, the system stability will be destroyed. Considering the rotation speed of magnetically suspended rotor is high, ωc = 15 ωs is chosen. ωs = 2π T s denotes the sampling frequency. ωc = 6283 rad/s.
For ω ∈ (0, ωc ), Fig. 5 can be simplified as Fig. 7 according to (14). The phase compensation function C(s) is C(s) = Kr c Kf (s)eN h ·T s s
(15)
where Kr c is the gain. Kf (s) is the phase compensation function in low- and middle frequencies. eN h ·T s s is the phase compensation function in high frequencies. The closed-loop characteristic equation in Fig. 7 is given by M (s) − N (s)e−T s = 0
(16)
where M (s) = 1 + Gc (s)Gw (s)Gp (s)Ks
(17)
N (s) = 1 + C(s)Gw (s) + Gc (s)Gw (s)Gp (s)Ks . (18)
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Fig. 8.
Phase response of G(s).
the new system with combined FIR and fractional-order RC is stable. Define G(s) as follows: G(s) =
Gw (s) 1 + Gc (s)Gp (s)Gw (s)Ks
(20)
where G(s)|s=j ω = L(ω)eiθ (ω ) . By substituting (20) into (19), the stability condition can be rewritten as (21) Kr c L(ω) · Kb (ω)ej [θ (ω )+θ b (ω )+N h ·T s ω ] + 1 < 1 where Kf (s)|s=j ω = Kb (ω)ej θ b (ω ) . Denoting λ(ω) = θ(ω) + θb (ω) + Nh Ts ω, then |Kr c L(ω) · Kb (ω) cos λ(ω) + jKr c L(ω) ·Kb (ω) sin λ(ω) + 1| < 1.
(22)
Because Kr c > 0, L(ω) > 0, Kb (ω) > 0, (22) can be simplified as Fig. 6. Magnitude responses of Lagrange interpolation polynomial. (a) n = 3. (b) n = 2.
Kr c L(ω) · Kb (ω) < −2 cos λ(ω).
(23)
For satisfying (23), cos λ(ω) < 0, namely, 90◦ < λ(ω) < 270◦ .
(24)
As can be seen from (23) and (24), the stability of the closedloop system can be guaranteed if the following condition is satisfied: Kr c
