Suppression Control Method of Torque Vibration for Brushless DC Motor Utilizing Repetitive Control with Fourier Transform

T. SICE Vol.E-2, No.1 (2002)

Trans. of the Society of Instrument and Control Engineers [ Vol.E-2, No.1, 42/53 (2002) ] Suppression Control Method of Torque Vibration for Brushless DC Motor Utilizing Repetitive Control with Fourier Transform

Satomi Hattori*, Muneaki Ishida** and Takamasa Hori***

Variable speed drive systems of brushless DC motor have been widely used for industry applications, home electric appliances, and so on, due to the progress of the power electronics, because of simple structure, easy maintenance, high efficiency, etc.. However, structural imperfectness of brushless DC motor and its control system produce torque ripple, and cause mechanical vibration by torque ripple and acoustic noise. In this paper, authors propose a suppression control method of vibration for brushless DC motor utilizing feedforward compensation control, and a generation method of compensating signals for the feedforward control by repetitive control with Fourier transform utilizing a vibration signal acquired by an acceleration sensor attached to the motor frame. Because the vibration signals detected by the acceleration sensor contain various complicated frequency components due to motor torque vibrations and mechanical resonant vibrations around the motor load system, we can not stabilize the repetitive control system and reduce the vibration by directly using the vibration signals from sensor itself. So we propose a generation method of the compensating signals of the repetitive controller consid- ering the periodicity of motor torque ripples, and using Fourier transformer which can select particular frequency components. In order to realize on-line generation of feedforward compensating signals to reduce the vibration, auto-tuning method of the repetitive control parameters is also presented. An experimental system to detect and reduce the motor torque vibration is constructed, and the effectiveness of the proposed method is confirmed by experimental results.

Key Words: Brushless DC motor, Torque ripple, Acceleration sensor, Repetitive control, Fourier transform

1. Introduction ever, structural imperfectness of brushless DC motor and its control system produce torque ripple, which causes Variable speed drive systems of brushless DC motors mechanical vibration and acoustic noise [1]-[3]. As the have been widely used for industry applications, home ripple torque suppression methods, the feedforward electric appliances, and so on, due to the progress of the compensation method of the compensation current esti- power electronics, because of simple structure, easy mated from the spatial harmonics of reluctance [4], the maintenance, high efficiency, etc.. Moreover, because method of using skewed rotor [5], [6] and so on are the brushless DC motor can realize the same high per- reported. formance as DC motor, it is used in the fields where the Also, it was well known that the torque ripple could high speed and high accuracy control is required. How- be reduced by addition of special feedforward compensation voltage/current to the usual voltage/current for the * Dept. of Electrical, Electronics and Information Engi- control input [4]. However, how do we obtain the feed- neering, Faculty of Science and Engineering, Shizuoka Institute of Science and Technology, 2200-2 Toyosa- forward compensation signals or data? Of course, we wa, Fukuroi, Shizuoka 437-8555, Japan can derive the compensation signals by analysis consid- ** Dept. of Electrical and Electronics Engineering, ering the structure of the motors, estimation from the Faculty of Engineering, Mie University, 1515 Kamiha- e.m.f. waveshape, etc. approximately, but cannot obtain ma-cho, Tsu, Mie 514-8507, Japan *** Dept. of Electronics and Information Engineering, the accurate signals. Finally for the individual motor, we Faculty of Engineering, Aichi University of Technolo- must make a fine adjustment of the compensation signal gy, 50-2 Manori, Nishihazama-cho, Gamagori, Aichi 443- in order to suppress the torque ripple sufficiently. 0047, Japan * Same to ** until March 2002 In earlier papers, from the viewpoint of direct reduc- *** Same to ** until March 2002 tion of the vibration of the motor frame caused by torque T. SICE Vol.E-2, No.1 (2002) ripple, we attached the acceleration sensor on the motor 2. Detection Method of Mechanical Vibration frame, and we have proposed some generation methods of compensating signals to reduce the torque ripple of 2.1 Detection System of Mechanical Vibration the brushless DC motor, 3-phase HB-type stepping motor and the induction motor by repetitive control. [7]- Fig.2 shows the detection system of the motor vibra- [10] (Fig.1). tion. This system is the usual case, where the motor base The purposes of this research is to construct the con- and the load base are separated each other. Here, the trol system utilizing the feedforward compensation sign- motor base is supported by vibration proof rubbers and als to reduce the torque ripple, and the generation the load (DC generator) base is fixed by bolts. method of compensating signals for the feedforward The ripple torque is caused by the imperfectness of control by repetitive control. Furthermore, in this paper, the motor, current sensors, driving source, etc. when the the auto-tuning method of the repetitive control parame- motor is rotating. The rotational speed of the brushless ters is considered for the reasons as shown below: DC motor is fluctuated due to the ripple torque. At the (1) Even when the characteristics of mechanical systems same time, the ripple torque is applied to the motor are not known clearly, the repetitive control must be frame by a reaction from the rotor and, as a result, the always performed stably. vibration of the motor frame occurs. Since the vibration (2) In the case of the operating point of the motor and is periodical in phase with the rotation of the motor, the the parameter of the controlled system change, on- repetitive control can be applied to the reduction of the line update of the compensation signal is necessary vibration. to reduce the vibration increased by those change. 2.2 Extraction of Specific Component of Vibration by In this paper, we propose a suppression control Fourier Transform method for torque vibration of brushless DC motors utilizing the acceleration sensor, the Fourier Transformer The periodical torque ripple occurs synchronously and the repetitive controller. In the proposed system, with the motor rotation. And also the repetitive con- only one frequency component of vibration signal from troller has large loop gain (basically infinity) only for the acceleration sensor is inputted to repetitive controller, fundamental repetitive frequency component and its har- the stability of the control system can be improved. monics. For the above reasons, the repetitive control sys- Approximate analysis is performed to study the stability tem is effective to reduce the vibration due to periodical of the repetitive control system. In order to realize on- torque ripples. line generation of feedforward compensation signals to However, because the vibration signals detected by reduce the vibration, auto-tuning method of the repeti- the acceleration sensor contain various frequency com- tive control parameters is also presented. An experimen- ponents and the mechanical system around the motor tal mechanical system to realize the mechanical vibra- and load has complicated resonant characteristics, we tion is constructed, and the effectiveness of the proposed can not stabilize the repetitive control system and reduce method is confirmed by experimental results. the vibration by directly using the vibration signals from sensor [8]. Then, we investigate the method which only

Inverter Encoder Acceleration sensor one frequency component of the vibration signal is extracted from the vibration signals detected by the + SPMSM DCG acceleration sensor, and the repetitive control is performed for every frequency component of the vibration.

Motor frame Generally, since the period of the vibration may not Fixed base coincide with that of the power supply but may be multi- PWM Fourier controller transformer ple of that, we set the period of the vibration to Tr (= Nr T, T = f -1, f : motor driving frequency). For example, in Torque command + - Compensating signal Repetitive controller the case of Nr =2, the frequency components of the vibration are 0.5f, 1.0f, 1.5f, ... and 0.5f is fundamental Fig.1 Schematic diagram of control system wave component of the vibration. If the vibration signal

a S detected by the acceleration sensor is expressed as equation (1), the Fourier coefficients an, bn are given T. SICE Vol.E-2, No.1 (2002) equation (2) and (3), respectively, and a certain compon- 3. Configuration of Control System ent of the vibration a S nf (t) whose frequency is nf is given by equation (4), where, n = m / Nr (m=1, 2, 3, ...), Fig.4 shows a block diagram of the proposed system. w = 2pf. Therefore, we can extract the specific compon- Here, d axis coincides with a direction of the magnetic ent of the vibration from the vibration signal which con- flux produced by the field of SPMSM (Surface Perma- tains various frequency components by equations (2), nent Magnet Synchronous Motor), and a S is the mechan- (3), (4). ical vibration signal of the motor frame. The reduction ¥ ¥ control of the mechanical vibration is performed by the a0 S (t) = + å an cos(n t) + å bn sin(n t) (1) repetitive control system and/or the feedforward control 2 m =1 m =1 system. 2 Tr a = (t)cos( n t) n 0 S (2) Tr 3.1 Dynamic Equations of SPMSM

2 Tr b = (t)sin(n t) (3) Generally, in a d - q coordinate system, the dynamic n T 0 S r equations of SPMSM, used as the brushless DC motor, = + Snf(t) ancos(n t) bn sin(n t) (4) are described equation (5). Here, the coordinates trans- In this research, the above-mentioned calculations on formations from the (u, v, w) coordinate system to the (d real time are realized by the digital signal processor , q) coordinate system are described as equation (7) by

(denoted by DSP in the following). In this paper, a S nf using the transformation matrix [c] expressed as equa- (t) is called nf frequency component. tion (6). Fig.3 shows waveforms of the detected acceleration v i 0 signal a and 1.5f and 5f frequency components d Ra + pLa - reLa d S = + (5) extracted by Fourier transformation when the driving vq reLa Ra + pLa iq reF f frequency of the motor f is 12.5Hz. From Fig.3, we can confirm that the only one component of the vibration is cos sin 1 - 1 - 1 2 re re 2 2 extracted from the vibration signal which contains vari- c = (6) 3 - sin cos 0 3 - 3 ous frequency components. re re 2 2

Encoder l PWM DCG SPMSM Motor controller Acceleration sensor frame ql v q ref v d ref Encoder Current SPMSM DCG Acceleration sensor Bolt controller S Vibration proof rubber Acceleration sensor Fig.2 Detection system of motor frame vibration i d ref i d Coordinates AMP. and Low + - transfor- Pass Filter 20ms/div. i q ref a - mation S Vibration of Sw1 i q cmd + i q motor frame [0.05 G/div.] + q - i qc Timing signal re Fourier transformer Tr , qre 20ms/div. i qnc a Sw1 S nf k 2 + 1.5f frequency sTr - sTr N e k1 component e + + [0.015 G/div.] + Time leading Repetitive Proportional Sw2 element compensator compensator 20ms/div. (For Repetitive control) 5f frequency T , q component i qc’ r re [0.006 G/div.] + e - sTr Compensating signal generator + (For Feedforward control) Fig.3 Extraction method of specific component of vibration by Fourier transform Fig.4 Configuration of control system T. SICE Vol.E-2, No.1 (2002)

referred to the motor frame vu iu v i d d 3.3 Current Control System = c vv , = c iv (7) vq iq From equation (5), the motor currents (id, iq) are under vw iw

influence of the electromotive force wre F f and the reac- where i , i : d, q-axis components of armature cur- d q tance w re L d , w re L q. In order to control id and iq exactly rent by removing these influences, we adopt the non- v , v : d, q-axis components of armature vol- d q interference control [11]. The voltage command of the tage inverter is described as follows: R : armature winding resistance a L : armature winding self-inductance v a uref T vd ref w re : electrical angular speed of the rotor vvref = c (14) referred to the motor frame vqref F : maximum value of PM flux linkage v f w ref p : operator d /dt v =k (i - i ) - L i Also, the generated torque t of the brushless DC motor dref I dref d re a q (15) M is expressed as follows: v = k (i - i ) + (F + L i ) (16) qref I qref q re f a d

M = Mi + rip1 (8) where k : proportional gain of the current controller I = F Here, the PWM inverter, which drives the SPMSM, is Mi n p f iq (9) assumed also an ideal voltage source (vd = vdref , vq = vqref where t Mi : torque component proportional to iq ). Furthermore, if kI is large, id =idref and iq = iqref, and

t rip1 : torque ripple component caused by the therefore t Mi is expressed as follows: imperfectness of the motor = F + Mi n p f iqref rip2 (17) n p : number of pole-pairs where t : torque ripple caused by the imperfect- 3.2 Motor Equations of the Mechanical System rip2 ness of the current controller The mechanical system shown in Fig.2 is approxi- Finally, the generated torque of the brushless DC motor mated by one resonance system in order to express the t M is expressed by: fundamental characteristics of the repetitive control sys- M = n pF f iqref + rip (18) tem. In this case, motion equation of the rotor and the = + motor frame are expressed as follows: rip rip1 rip2 (19)

ML º M - L = J ML p re (10) 3.4 Repetitive Control System and Compensation Signal

Generator F = p F , F = p F , S = (lcos l ) F (11) 2 Since the periodical vibration of the brushless DC M =( J F p + DF p + KF ) F (12) motor in phase with the rotation of the motor is dealt = + re ( rm F )n p (13) with in this paper, the repetitive control can be applied to

the reduction of the periodical vibration. In the repetitive where t : load torque (constant torque here) L controller in Fig.4, the compensation signal i for the w , a : mechanical angular speed and acceler- qc F F torque ripple t is produced from the vibration signal ation of the motor frame, respectively rip detected by the acceleration sensor attached to the motor q : mechanical rotational angle of the F frame during the repetitive control. motor frame In the continuous system, the repetitive compensator l, : refer to Fig.2 l is the compensator which adds an input from the sensor J : total inertia moment of the motor frame F to its output which has only the cycle T time-lag behind K , D : spring constant and viscosity coeffi- r F F the input and it has large loop gain (basically infinity) cient of the mechanical system to sup- only for the fundamental repetitive frequency component port the motor frame, respectively and its harmonics. Therefore, it can reduce the error w rm : mechanical angular speed of the rotor T. SICE Vol.E-2, No.1 (2002)

(input signal) to zero, and it has the memory characteris- T T '= Tr/N r tics to those components. From the above-mentioned ・・・・・・memory characteristics, even when the input is set to zero (Sw1 in Fig.4 : OFF), the compensation signal Input signal ＝ acquired during Sw1 ON can be outputted continuously. t one component of vibration 0 Using these characteristics, the repetitive compensator 2p (implemented with memories) is also used as the com- Calculation of average t pensation signal generator, which stores the acquired value of T ' seconds 0 2p compensation signal, and outputs it as the feedforward addition compensation signal as shown in Fig.4. M1 M2 ・・・ M i ・・・ MN-1 MN LPF is a low pass filter to cut-off the higher frequency Repetitive compensator with N memories sensor noise. The Fourier transformer is a kind of filter when k2=1 to cut-off all higher and lower order harmonics except a Output signal ＝ t particular vibration frequency, the parameter k1 is a pro- Compensating signal 0 portional compensator for adjusting control loop gain, 2p Fig.5 Fundamental operation of repetitive controller and the time leading element (the parameter: k2 ) is that to compensate the phase lag in the loop. The parameter k1 and k2 are determined so that the repetitive control by k2 rather than the memory added the input signal. system becomes stable for the individual vibration fre- In the proposed system, as the previous section quency component. explained, the repetitive compensator (implemented with memories) is also used as the compensation signal gen- 3.5 Algorithm of Repetitive Control and Feedforward erator, which stores the learned compensation signal, Compensation Control and outputs it as feedforward compensation signal. The repetitive compensator is the compensator which When the switch Sw1 in Fig.4 closes, the repetitive adds the input signal to the output one which delays the control is carried out and at the same time the signal for input one by only one cycle Tr. In this research, in order the torque ripple compensation is renewed by the input to realize the repetitive compensator by DSP, we have to signal and restored in the controller. When the selected design it by the discrete system. Fig.5 shows the funda- frequency component of the vibration is reduced negligi- mental operation of the repetitive compensator and the bly small, the switch Sw1 opens and the signal input to time leading element in the discrete system. As shown in the repetitive controller stops. After that instant, the Fig.5, the repetitive compensator consists of N memo- open-loop feedforward control is carried out repetitively ries, and in the case the control period is T', and the rela- using the stored signal in the repetitive controller as a tion of Tr = N T' is derived. As this control system, in the feedforward compensation signal. It should be noted that case the period of the input signal (ripple torque) Tr syn- the repetitive control with the Fourier transformer could chronizes with the rotation speed of the brushless DC be carried out for one frequency component, even motor, the number of N also change with change of Tr, including the previously acquired compensation signals in the system fixed the control period T' inconveniently. for the other frequency components. Then, T' is allowed to change and the repetitive compen- After finishing compensation signal acquisition, this sator with N memories in one period Tr is realized by compensation signal are stored in the other memories synchronizing the control period T' with the rotation when the switch Sw1 opens and Sw2 closes. angle of the rotor qre. The input signal to the repetitive compensator (= the output signal from the proportional 4. Stability of Repetitive Control System compensator) is averaged within the period of T' and added to the data in the corresponding memory. The out- The gain of the repetitive controller becomes infinity put signal from the repetitive compensator is obtained by at multiple of the repetitive frequency. Therefore the reading data form the memory before input signal is repetitive control system may become unstable, i.e., the added to the data in the memory, and holding it for T'. vibration diverges. Here, we examine the transient char-

On the other hand, the time leading operation of (k2/N)Tr acteristics of the repetitive control system and describe is realized by reading data from the memory which leads how to determine the parameters k1 and k2. T. SICE Vol.E-2, No.1 (2002)

D I DT 4.1 System Equations and Block Diagrams q cmd rip + - + - + - s Tr (k / N ) s T From the torque equations (8) and (18) and the motion e e 2 r G1 (s) equations of the mechanical system (11) and (12), a + D Iqc D Iq ref DTM block diagram of repetitive control system as shown in D AS' D AS D AF D WF Fig.6. Here, the ripple torque DT is regarded as a dis- rip k1 GLPF (s) G3 (s) s G2 (s) turbance, and the Fourier transformer is not included. Fig.6 Block diagram of linearized system The transfer functions shown in Fig.6 are expressed as follows: DTrip DAS -s T G (s) r 1 - C(s) P(s) DT 1 - e - G (s) = Mi = n F (20) 1 p f + DIqref - s T e r DWF s Fig.7 Transformed equivalent block diagram G2 (s) = = 2 (21) DTM JF s + DFs + KF

DAS lcos l 1- C( j )P( j ) < 1 (27) G3 (s) = = (22) n n DA F 9.8 where w n: angular frequency of vibration extracted by the Fourier transformer 4.2 System Stability Fig.8 shows the Nyquist loci of 1-C(j )P(j ) and the

With respect to DAS, the block diagram shown in unit circle in the case without the time leading element Fig.6 can be transformed to that shown in Fig.7. The (k2=0). Here, the marked point on the loci labeled nf is transfer functions shown in Fig.7 are expressed as fol- called Nyquist point of wn (=2pnf). System parameters lows: for analysis are shown in Table 1. When the frequency is increased from zero to infinite, the Nyquist loci starts P (s) = G1(s)G2 (s)G3(s)GLPF (s) (23)

k from a point 1 + j0, and it returns to the same point due 2 sT N r (24) to the high-pass nature of the acceleration sensor and the C(s) = k1 ×e low-pass one of the mechanical system after drawing a = G(s) G2 (s) G3(s) s (25) large circle corresponding to the vibration mode (near where P(s) : transfer function of motor and load ( the natural frequency of the mechanical system : reso-

from Diqref to DAS') nance point). In the proposed system, if only Nyquist C(s) : transfer function of proportional com- point of wn extracted by the Fourier transformer stays pensator and time leading element inside the unit circle, the stability of the repetitive con-

G(s) : transfer function from DTrip to DAS' trol system is guaranteed. GLPF(s) : transfer function of a noise filter (low Most Nyquist points (operating points) other than the pass filter) resonance point exist near the point 1 + j0. We can see in

The stability of the repetitive control system is deter- Fig.8 that if the value of k1 is large, a loci of vector will mined by the frequency response of the transfer function be expanded centering on the point 1 + j0. In the case 1-C(s)P(s) in Fig.7. According to the small gain theory operating points exist inside the unit circle, so the oper- [12], the following condition must be satisfied in order ating points can be then apart from the circumference of to stabilize the system. the unit circle, convergence speed of the vibration can be made quick (for example, 4.5f, 5f, 6f in Fig.8). On the 1- C( j )P( j ) < 1 " (26) other hand, in the case operating points exist outside the Equation (26) means that all points on the Nyquist loci unit circle, the operating points are apart from the unit of 1-C(jw)P(jw) must stay inside the unit circle for stable circle greatly and divergence speed of the vibration will operation, but impossible. On the other hand, because become quick (for example, 3.5f, 4f in Fig.8). Therefore, the frequency region of the vibration signal is limited by it is advisable to set small value on k1, unless it causes the Fourier transformer before inputted to the repetitive inconvenience for the convergency of the vibration. controller in the proposed system, the stability criterion As mentioned above, in the case operating points exist of the system can be expressed by outside the unit circle, stabilization of the repetitive con- T. SICE Vol.E-2, No.1 (2002)

2 0.15 Unit circle Unit circle k1 = 0.05 0 0.1 k2 = 0

-2

4f 0.05 -4 1.5f 4f 3f '

Im 0 Im Im -6 Im 3f

-8 -0.05 k1 = 0.05 -10 -0.1 k1 = - 0.05 -12 k1 = 0.1 k2 = 0 -0.15 -14 -10 -5 0 5 10 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 ReRe ReRe

(a) Overall Nyquist loci (k2 = 0) (a) When operating point exists out of unit circle (case 1) 0.5 0.15 k1 = 0.1 Unit circle k = 0.05 0.4 4.5f 1 22f k = 0 0.3 2f 0.1 2 5f 0.2 6f k1 = - 0.05 6f 2.5f 0.05 0.1 k2 = 0 4.5f 3f

Im 0 5f 3.5f Im Im Im 0 -0.1 k = 0.05 1 3.5f 22f '' -0.2 -0.05 1.5f 2f 2.5f -0.3 -0.1 k1 = -0.05 -0.4 Unit circle 22f ' k2 = N/4n -0.5 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Re 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 Re ReRe (b) Expanded loci of (a) around (1+j 0) (k2 = 0) (b) When operating point exists out of unit circle (case 2) Fig.8 Nyquist loci of transfer function (1-C(s)P(s))：f =14Hz Fig.9 Nyquist loci of transfer function (1-C(s)P(s))：f =14Hz (Without time leading compensation) (With time leading compensation)

Table 1 System parameters ing point is at the place by 22f as shown in Fig.9(b), it

SPMSM cannot be moved into the unit circle by only reversing Rated voltage : 80[V] Rated current : 7.2[A] the sign of k1 (22f ® 22f '). In such a case, the operating Rated torque : 1.27[N×m] point can be moved into the unit circle by compensating R : 0.8[W] L : 0.92mH a a only 90 degrees phase (by using the time leading com- F f : 0.051[Vsec/rad] np : 2[pole pairs] Load : DC Generator pensator k2) (22f ' ® 22f ''). Here, the value of k2 is a nat- Rated voltage : 145[V] Rated current : 11.9[A] Rated torque : 8.2[N×m] ural number corresponding to about 90 degrees phase of Motor + Load nf frequency component. J =8.06×10-2[kg×m2/rad] F From the above results, for the selected frequency -1 DF =4.10×10 [N×m sec/rad] 3 component of the vibration, which we want to reduce, KF =9.84×10 [N×m/rad] f0 =55[Hz] (Natural frequency of mechanical system) we can always adjust the control parameters k1 and k2 so LPF : 8th order Butterworth low pass filter that the selected component stays inside the unit circle. Cut off frequency : 1[kHz] As a result, the control system can be always stabilized. trol system using only the proportional compensator is Here, although we examined the stability of the system impossible. Then, we think that the operating point can based on the control model shown in Fig.6, theoretically, be moved into the unit circle by the time leading com- we think that the same method can also apply to any pensation (k2) or reverse of the sign of k1 as shown in motors and mechanical system. However, in the case the Fig.9. 3f and 22f, 3f ' and 22f '' in Fig.9 denote the oper- operating point exist near the resonance point of the ating points without and with the phase control, respec- mechanical system, which shows sharp resonance char- tively. In the case the operating point is at the place acteristic, like 4f frequency component shown in Fig.8 denoted by 3f as shown in Fig.9(a), it can be moved into (a), so the gain of the transfer function 1-C(s)P(s) is very the unit circle by reversing the phase (by setting the sign high, it is necessary to set the value of k1 quite small. of k1 is negative) (3f ® 3f '). But, in the case the operat- Furthermore, when the operating point moves consider- T. SICE Vol.E-2, No.1 (2002) ably on the locus in Fig.8(a) for the slight change of the Set : n vibration frequency by change of the rotational speed of FT : Start the motor, it is difficult to guarantee the stability of the Set : k, delay, Max, Min repetitive control system by adjustment of the above- mentioned phase compensation. In this case, it is desir- Repetitive control : Start able to perform the feedforward compensation control k = k using the data obtained by the repetitive control around *1 1 k = 0 the resonance point. But, it is rare case that the vibration 2 frequency coincides with one of the steep resonance No point. An≧ Max delay Yes No 4.3 Auto-tuning Method of Control Parameters An ≧ Min Memory k1 = - k clear k = 0 Yes *3 As mentioned in the previous section, the stability 2 condition of the repetitive control system is to satisfy the *2 No A ≧ Max delay inequation (27), and if the operating point of wn exists in n the unit circle, the stability of the system will be guar- Yes No An ≧ Min anteed. However, since the vibration signal caused by Memory k1 = k clear k = N/4n the ripple torque of the brushless DC motor include 2 Yes *4 *3 many components caused by the imperfectness of the No A ≧ Max motor, the repetitive control system may become n delay unstable with the control parameters obtained by the Yes No k = - k An ≧ Min approximate system modeling. Also, from the analysis Memory 1 clear k = N/4n results in the previous section, it is confirmed that a set 2 Yes *1 of the control parameters which makes the repetitive *4 No A ≧ Max control system stable certainly exists among the four n delay kinds of parameters. Therefore, here, we propose the Yes No A ≧ Min Memory n auto-tuning method of the repetitive control parameters *1 clear Yes *2 (k1, k2) based on the approximate analysis. Fig.10 shows a flow chart of the auto-tuning algorithm of the parame- Fig.10 Flow chart of auto-tuning of parameters k1, k2 ters. where An : amplitude of a S nf vibration signal is reduced. On the other hand, if the ini- Max, Min : threshold value to determine whether tial parameters are not suitable for it, the vibration signal

An diverges or converges either tends to diverge or does not change so much. In delay : time delay the latter case, the parameters should be changed. Then,

N : number of memories of repetitive in the case the vibration signal tends to diverge and An compensator ( = number of partition exceeds the threshold value Max (i.e., An ³ Max), the of vibration signal ) learned data in a memory of the repetitive controller is N/4n : a natural number corresponding to cleared and the compensation signal is relearned. On the about 90 degrees phase of nf fre- other hand, in the case the vibration signal does not

quency component of vibration change so much (i.e., An ³ Min), the phase of the learned In the first place, the value of k is set so that all oper- compensation signal is changed 90 degrees. Thus, in the ating points of wn except sharp resonance point can be proposed auto-tuning method, the parameters of the put inside the unit circle by appropriate selection of the repetitive control are changed according to the change of values of k1 and k2 from {-k, k} and {0, N/4n}, respec- An. tively. If the switch Sw1 in Fig.4 closes, the repetitive It can also be confirmed from Fig.9 that, even when control is carried out using the initial parameters k1 = k the characteristics of the motor and mechanical system and k2 = 0. If the initial parameters are suitable for are not enough grasped, it can be expected to put the selected frequency component of the vibration a S nf, the operating point of wn in the unit circle, and to stabilize the repetitive control system according to the proposed T. SICE Vol.E-2, No.1 (2002) method In addition, in the case the operating point exist by mutual interference among the motor structure, the near the sharp resonance point of the mechanical system, harmonics currents caused by the imperfectness of the the vibration signal diverge in any loop of the proposed current controller, the mechanical characteristics of the algorithm are brought, therefore k1 must be set small. experimental system and so on. Especially, 1.5f and 3f However, in that case, k1 reduces to zero and as a result frequency components are considered to be amplified by it is almost same to not learning the compensation sign- the mechanical resonance phenomenon. al, the feedforward compensation control using the data Fig.13 shows an example of the auto-tuning process obtained by the repetitive control around the resonance of the parameters for 3.5f frequency component. point must be carried out. #1 : repetitive control (compensation signal learn-

ing) started ; k1 = k, k2 = 0 #1 - #2 : delay ® vibration did not change so much ® 5. Experimental Results repetitive control stopped In experiment, we use the mechanical system shown #2 : changed of parameter to adjust 90 degrees in Fig.2, the control system in Fig.11 and the system phase shift of compensation signal ® repetitive parameters in Table2. Here, Tc is the period of the control restarted ; k1 = k, k2 = N/4n inverter and Ts is that of the speed control. And we use #2 - #3 : vibration diverged (selected parameters at #2 the encoder, in order to detect the motor magnetic pole were not suitable) ® repetitive control stopped position information which is necessary for the Fourier #3 : memory clear ® changed of parameters ® transformation, the motor driving and the repetitive con- repetitive control restarted ; k1 = - k, k2 = N/4n trol. #4 : vibration reduced (suitable parameters were Fig.12 shows the vibration waveform when the brush- selected ® repetitive control continued less DC motor drive, and its FFT spectrum. From Fig.12, The auto-tuning algorithm of the control parameters for the mechanical vibration (acceleration) contains various each vibration frequency components was carried out by frequency components. It is considered to be generated experiment. In Table2, the final values of k1 and k2 determined by the auto-tuning algorithm are shown. Inverter From these results shown in Table 2, the parameters of Encoder 3 + 1.5f frequency component differs from the value esti-

SPMSM DCG 50 ms/div.

Vibration of motor frame a [0.05G/div.] Current command S

( Torque command ) -3 DSP control AMP. and Low ´ 10 G 12 system Sw Pass Filter 3.5f 1.5f 9 Fig.11 Configuration of experimental system Spectrum of 2.5f 4.5f 6 Table 2 Parameters for experimental system vibration 4f 2f 3f 5f 3 INV. source : 120[V], Tc =100[ms] 6f Current controller 0 iq cmd = 4.6[A], id ref = 0.0[A], f = 14[Hz], kI = 2.4 3.0 100 200Hz Repetitive controller Fig.12 Vibration of motor frame Final values of parameters by auto-tuning algorithm

1.5f : k1 = k , k2 = N/4n 2f : k1 = - k , k2 = 0 #1 #2 #3 #4 2.5f : k1 = - k , k2 = 0 3f : k1 = - k , k2 = 0 3.5f : k1 = - k , k2 = N/4n 4f : k1 = - k , k2 =N/4n 4.5f : k1 = k , k2 = 0 5f : k1 = k , k2 = 0 6f : k = k , k =N/4n 1 2 3.5f frequency Initial values component Memory : N =420 k = 0.05[A/G] delay : 15[sec] [0.006G/div.] Speed controller Rotary encoder : 6000[pulses/rotate]

Ts =400[ms] Resolution of A-D converter : 12[bit] 50 sec/div. - 4 Input : -0.51 ~ +0.51[G] , ±1.25´ 10 [G/resolution] Fig.13 Auto-tuning process of repetitive control parameters T. SICE Vol.E-2, No.1 (2002) mated from Fig.9(a). Although it is considered that the 50 ms/div. 50 ms/div. resonance mode omitted by approximation of equation (12) has influenced near 1.5f frequency component, and Vibration of motor frame the characteristics of the mechanical system are not [0.05G/div.] enough grasped in this experiment, we confirmed that q - axis current compensating signal stable vibration suppression control was realized by pro- [5A/div.] posed auto-tuning algorithm of the control parameters. u - phase current Fig.14 shows experimental results of the repetitive con- waveform [5A/div.] trol for 1.5f frequency component. Fig.14(a) and (b) (a) (b) show the case without and with the repetitive control, (a) Without vibration suppression control respectively. Fig.14(b) shows FFT spectrum of the (b) With vibration suppression control motor frame vibration with the repetitive control. From (1.5f, 2f, 2.5f, 3f, 3.5f, 4f, 4.5f, 5f, 6f only) Fig.14, only 1.5f frequency component. ´ 10-3G 12 Next, we carried out the auto-tuning of the parameter 9 and the repetitive control for 1.5f to 6f frequency com- 3.5f Spectrum of 2.5f 6 ponents, respectively. And the compensation signals vibration 3f 4.5f 1.5f 2f 4f learned in each operations were compounded to obtain 3 5f the compensation signal for feedforward compensation 6f 0 control in order to reduce from 1.5f to 6f frequency com- 3.0 100 200Hz ponents simultaneously. Fig.15 shows experimental (c) FFT spectrum of vibration with vibration results of the feedforward compensation control using suppression control the compounded compensation signal by open-loop (in Fig.15 Experimental results for repetitive control and feedfor- Fig.11 : Sw ; OFF) for the brushless DC motor on-load in ward control steady state. Fig.15(a) shows the case without the repetitive control nor feedforward compensation control. employed as the feedforward control. Fig.15(c) shows Fig.15(b) shows the final result of the vibration suppres- FFT spectrum of the motor frame vibration with the sion control where the repetitive control is performed for feedforward compensation control. From Fig.15, we can each of 1.5f, 2.5f, 3f, 3.5f, 4f, 4.5f, 5f, 6f frequency com- consider that the 1.5f, 2.5f, 3f, 3.5f, 4f, 4.5f, 5f, 6f fre- ponents one by one with the acquired compensation data quency components are considerably reduced. Table 3 shows the vibration suppression control effect 50 ms/div. 50 ms/div. each frequency components of the vibration. The values 1.5f frequency component in ( ) show the equivalent ripple torque t n fe converted [0.015G/div.] from the acceleration a S nf by equation (29), where Tn is q - axis current compensating signal amplitude of t n fe, and expressed the amplitude Tn as a [2.5A/div.] rate to the steady state torque T0 in order to know the u - phase current rough magnitude of the ripple torque. Here, natural waveform [5A/div.] (resonance) frequency of the mechanical system decided

(a) (b) by JF and KF is set to f0, in the domain of f >> f0, so (a) Without vibration suppression control equation (25) turn into equation (28), the amplitude of (b) With vibration suppression control (1.5f only) the equivalent ripple torque Tn will be converted by ´ 10-3G 12 equation (29).

9 lcos l G (s) = G2 (s)G3(s)s @ Spectrum of 9.8×JF 6 1.5f vibration (28) KF f ññf0 = /2 3 Q J F 0 9.8×J 3.0 100 200Hz T = F A (29) n lcos n (c) FFT spectrum of vibration in the case of (b) l Fig.14 Experimental results for the repetitive control In addition, the equivalent ripple torques may be seem- T. SICE Vol.E-2, No.1 (2002)

Table 3 Experimental results pensation signals generated by the repetitive con-

without vibration with vibration trol. suppression control suppression control nf (4) Examples of application of the proposed control -3 -3 method is shown below: An [´ 10 G] Tn / T0 ´ 100[%] An [´ 10 G] Tn / T0 ´ 100[%] 1.5f 9.34 （11.9） 1.28 （1.63） (i) In the case that the changes in the motor parame- 2f 3.98 （5.06） 1.58 （2.01） ters and mechanical characteristics are small, the 2.5f 4.74 （6.03） 1.17 （1.49） compensation signals for the feedforward compen- 3f 3.52 （4.48） 1.17 （1.49） sation control are learned or updated at the factory 3.5f 10.5 （13.4） 1.48 （1.88） shipments or regular maintenance. 4f 4.44 （5.65） 1.07 （1.36） (ii) In the case that the characteristics of the motor and 4.5f 1.43 （1.82） 0.459 （0.57） the mechanical system depend on the environment 5f 1.33 （1.69） 0.408 （0.52） considerably, the acceleration sensor is incorpo- 6f 0.765 （0.97） 0.204 （0.26） rated into the motor control system and the feedforward compensation control using the compen- [steady state torque ：To = 0.54N×m] [rated torque : 1.27N×m]

[Tn / An = 6.87 N×m / G] by (28) sation signals learned before is carried out moni- toring the change of the vibration. However, when ingly larger than actual ones for mechanical resonance. the compensation signal becomes inadequate and From experimental results shown Fig.15 and Table 3, we some frequency components of the vibration can confirm usefulness of proposed method. becomes large by the characteristic change etc., the repetitive control is carried out to update the compensation signal for the frequency compon- 6. Conclusion ents. We proposed a suppression control method of the In this paper the proposed methods were examined in torque vibration of the brushless DC motors utilizing the case of low speed drive of the brushless DC motor. feedforward compensation control, and a generation However, for the higher order frequency components of method of compensation signals for the feedforward the vibration and those near the sharp resonance fre- control by the repetitive control system with the Fourier quency of the mechanical system, the effectiveness of transformer utilizing a vibration signal acquired by an the proposed vibration suppression method was small. acceleration sensor attached to the motor frame. Useful- This problem will be investigated further. ness of proposed method was confirmed by the experimental results. References The results are summarized as follows: (1) We showed that the stable vibration suppression [ 1 ] K. Kamiya, M. Shigyo, T. Makino and N. Matsui, "DSP- control system was realized for various frequency Based High Precision Torque Control of Permanent Mag- net DD Motor," T. IEE Japan, Vol.110-D, No.2, 1990, components of the vibration by incorporating the pp.117-124 Fourier transformer into the repetitive control sys- [ 2 ] T. Hanamoto, H. Ikeda, Y. Tanaka and T Mochizuki, tem. "Small Vibration Suppression Control of Brushless DC (2) We showed that stable repetitive control always Motors with Periodical Disturbance," T. IEE Japan, could carry out, even when the characteristics of the Vol.117-D, No.3, 1997, pp.335-341 mechanical system are not enough grasped, by [ 3 ] T. Yokozuka, E. Baba, J. Shibayama, H. Sekiguchi and T. Kobayashi, "Oscillation Frequency in Hybrid Stepping applying the auto-tuning algorithm for the repetitive Motors", T. IEE Japan, Vol.109-D, No.2, 1989, pp.114- control parameters. Also, the control parameters can 121 be changed automatically for change of the motor [ 4 ] T. Kosaka, N. Matsui, Y. Taniguchi and H. Do-meki, operating point or the controlled system by this pro- "Some Considerations on Torque Ripple Suppression of posed method, and the repetitive control system can Reluctance Motors", T. IEE Japan, Vol.118-D, No.2, behave stably. 1998, pp.150-157 [ 5 ] M. Nashiki, A. Satake, Y. Kawai, T. Yokochi and S. Oku- (3) We showed that two or more vibration frequency ma, "Torque Ripple Reduction of Reluctance Motor with components can be simultaneously suppressed by Slit Rotor", T. IEE Japan, Vol.117-D, No.8, 1997, the feedforward compensation control using com- pp.1008-1014 T. SICE Vol.E-2, No.1 (2002)

[ 6 ] S. Kitamura, Y. Ishihara and T. Todaka, "Magnetic Field Muneaki Ishida was born in Aichi, Japan Analysis of DC Brushless Motor with Skewed Magnet by on April 12, 1952. He received the B.S., 2-D FEM", T. IEE Japan, Vol.118-D, No.2, 1998, pp.253- M.S., and Ph.D degrees from Nagoya Uni- 259 versity in 1975, 1977 and 1980 respectively, [ 7 ] S. Higuchi, M. Ishida and T. Hori, "Reduction Control of Vibration of Induction Motor," T. IEE Japan, Vol.115-D, all in Electrical and Electronic Engineering. No.9, 1995, pp.1123-1130 He was with Nagoya University as a Research Associate [ 8 ] T. H. Su, M. Ishida, and T. Hori, "Reduction Control of in the Department of Electrical Engineering from 1980 Vibration of Three-Phase HB-Type Stepping Motor by to 1987. From 1987, he was with Mie University, ini- Repetitive Control Utilizing Acceleration Sensor," 1998 tially as an Associate professor and since October 1996 National Convention Record I.E.E. Japan Industry Appli- he has been with the same university as a professor. He cations Society- No.209 [ 9 ] S. Hattori and M. Ishida, "Suppression Control of Vibra- is engaged in research on static power converter, and AC tion for Brushless DC Motor with Acceleration Sensor - motor drive systems. Repetitive Control Using Fourier Transform-," 1999 National Convention Record I.E.E. Japan Industry Appli- Takamasa Hori was born in Toyama, Ⅲ cations Society- No.358, pp. 461-464 Japan on October 30, 1938. He received the [10] T. H. Su, M. Ishida, and T. Hori, "Repetitive Control for B.S. degrees in 1961 and the Ph.D degree in Vibration Reduction of 3-Phase HB-Type Stepping Motor Utilizing Accelerometer and Fourier Transformer," in 1970 from Osaka University both in Electri- Proceedings of the IEEE 1999 International Conference cal Engineering. From 1961 to 1986, he was on Power Electronics and Drive Systems (PEDS '99), with the Hitachi Research Laboratory of Hitachi Ltd. 1999, pp. 815-820. engaging in engineering research and development on [11] F. Blaschke, "The Principle of Field Orientation as Power Electronics. From January 1987 to March 2002, Applied to the New TRANSVECTOR Control System for he was a professor of Mie University. From April 2002, Rotating Field Machine," Siemens Review, Vol.34, pp.217 he is a professor of Aichi University of Technology. His [12] M. Nakano, T. Inoue, Y. Yamamoto, and S.Hara, Repeti- research interests include resonant converters, servo tive Control, The Society of Instrument and Control motor control, inverter drive system and small actuator Engineers of Japan, 1989, pp.82-89. for robotics and information equipment. Dr. Hori has received two prize paper awards from IEE of Japan and is a Fellow of IEEE and SICE Japan. He is also an Satomi Hattori was born in Aichi, Japan AdCom member in IEEE/IES since 1984. on December 28, 1974. She received the B.S., M.S. and Ph.D degrees from Mie Uni- versity in 1997, 1999 and 2002 respectively, Reprinted from Trans. of the SICE, Vol.36 No.5, all in Electrical and Electronic Engineering. 438/447 (2000) Since 2002, she has been with Shizuoka Institute of Sci- ence and Technology as a Research Associate in the Department of Electronic Engineering. Her research interests include vibration suppression control of AC servo motor system and DSP-based control system.