Fast Identification of Resonance Characteristic for 2-Mass System with

The 2014 International Power Electronics Conference

Fast Identification of Resonance Characteristic for 2-Mass System with Elastic Load Ming Yang, Liang Hao, Dianguo Xu

Harbin Institute of Techno[ogy Electrical Engineering and Automation Harbin, China [email protected]

Abstract-Servo mechanical

system

resonance.

is

The

greatly offline

influenced

by

suppression

of

identify the resonant frequency precisely. The offline suppression way firstly obtains system's Bode diagram by sweeping frequency or other methods, and then analyzes its characteristic of resonant frequencies to set filter parameters. This way takes much time and needs additional run at a time. Fixed-point arithmetic has some digital signal processing problems like delay, quantization error in [ 10]. Wider pass-band range of notch filter brings larger lag angle of phase. For increasing suppressed depth, narrow pass-band design will make resonant spectrum identification worse. Therefore, the design scheme adopting adaptive notch filter can identify resonance frequency and adjust filter parameter quickly [ 1 1]. Even through the equal numbers of notch filter's pole-zero will bring smaller lag angle of phase to pass-band range, the lag is still attached. In [ 12], using phase angle compensation to approach zero-lag angle achieves a certain effect. [n [ 13], traditional notch filter is replaced by FIR filter which is achieved by changing just one parameter. But this method is limited by resonance identification and can't eliminate multiply resonance frequency. So the actual effect is modest. Traditional method for acqmrmg frequency characteristic is sweeping frequency which is reliable and high-precision. But the operating time is long and it will bring frequency drift. [n [[4], PRBS and linear frequency modulation is considered. A power spectrum estimation method based on Welch is proposed in [ 15]. The offline suppression way needs the identification of characteristic. From the result of frequency characteristic, passive mode based on notch filter can be set intuitionistic and reasonable. And long operating time will increase the damage of system. So fast-identification of resonance characteristic is essential. This paper analyzes two methods based on PRBS signal and Chirp signal and uses the power spectrum method to process data. Accurate resonance characteristic can be acquired by offline resonance characteristic which is obtained by fast identification. [t provides accurate resonant parameters for suppression of mechanical resonance in passive mode.

mechanical resonance is determined by the identification of mechanical

resonance

characteristic.

The

resonance

characteristic can be obtained from the Bode diagram of current closed-loop. This paper discusses two methods of fast acquirement to get resonant characteristic based on pseudo random binary sequence (PRBS ) and Chirp signals.

Power spectrum method is used to process data. According to the identified frequency, the parameter of notch filter is determined.

The

demonstrate

the

results

of

accuracy

simulation of

this

and

experiment

method

and

the

suppression effect based on this method.

Keywords- Servo System; Identification of Resonance Characteristic;

Pseudo Random Binary

Sequence;

Chirp

Signals.

I.

INTRODUCTION

Servo drive system has some mechanical transmission devices to connect motor and load. Especially in the case of slender shaft, the factor of elasticity which cannot be ignored will engender mechanical resonance. At present, various research plans are aimed at measuring the position information of the drive motor only, and there is no sensors increased at the transmission device and load side. Suppressing resonance research strategies can be roughly divided into two categories: active mode and passive mode. Active mode is to take the initiative to change the controller parameters or controller structures to eliminate harmonic effects. Active mode can be divided into pure P[ control (two degree of freedom P[ control, RRC) [ 1-3], state feedback control based on P[ control [4-6], and other advanced algorithms. Passive mode is that notch filter will been inserted between the output of speed loop and current loop given, and other control system design unchanged. Notch filter can attenuate the amplitude of resonance frequency meanwhile has few effect on other frequency characteristics. Its parameter-design which is simple and practicable has explicit physical concepts. Many commercial servo systems generally adopt the precept that multiply notch filters are inserted to suppress multiply resonance frequencies [7-9]. But it requires an efficient resonant spectrum identification method to

978-1-4799-2705-0/14/$31.00 ©2014 IEEE

II.

[NFORMATION

Typical two-inertia mechanical drive system has the following differential equations, shown as ([). The shaft

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K and damping coefficient c'v­ J1 and damping Cb and Actuators have inertia Jz and damping coefficient Cz. Electromagnetic has certain torsion stiffness

Motor has inertia

sv

-

PWM _

Inverter

torque Te and load torque T, are related by the shaft torque Tw. According to equation (1),

we

can obtain the mechanical

transmission device model as shown in Fig. 1 (damping coefficient is neglected). Park t

J/)j = r: - C j OJ Tw J/j2 = Tw C O '0 Tw = Cw (OJ - O2 ) + K (OJ °2 ) � =OJ lU2 =02 -

-

z

z

-

Position and

(1)

Speed detectlllg

Fig. 2 Structure block diagram of resonance characteristic identification in discrete system.

Basic sweeping frequency mainly provides continuous frequency signal with repetitive periodicity records the characteristics in different frequencies, and on this account, draws the curve of frequency characteristic. But this method is time-consuming, may loss the details of particular frequency characteristic and has DC offset when identifying open-loop system. A long time test on resonant system may even lead to mechanical damage, so it is necessary to study the other method that resonance characteristic can be obtained fast and accuracy. The main feature of reference signal in fast identification is that the signal contains all the frequency components, has wide range of spectrum, and presents aperiodic change severely in time domain. White noise is the most typical signal according to these features, but white noise signal is not easy to implement in actual digital system and the noise variance is large. So it should be replaced by other signals conforming to these features.

Fig. I Block diagram of mechanical transmission.

The transfer function between the motor speed and motor torque is shown as (2) when damping coefficient is neglected. The conjugate zero point refers to Anti­ Resonance Frequency (ARF) and the conjugate pole point refers to Natural Torsional Frequency (NTF). The existence of these points makes system response more intense at specific frequency.

Te

/2i+K (

)

Identification signal

A.

+

Gl ( S ) =� =

Rotor Position

UnIt

-

(2)

a.

J]Jzs + J] +Jz Ks

pseudo random sequence signal

Pseudo random sequence can fully embody the feature of fast identification signal. This paper uses M-sequence for pseudo random sequence. M-sequence is also called the maximum linear shift register sequence, which is generated by feedback shift register. There is a relation (3) between each element in sequence XIX2 XpXp 11 :

High performance servo system often need high stiffness, the gain of speed controller Kp is generally large. The resonance frequency has two cases that ARF or NTF frequency and mainly NTF frequency in discrete system. Suppression of resonance in passive mode uses notch filter to filtering the resonance. If system sustained oscillates at steady state caused by lack of stability margin, filter frequency is NTF frequency; conversely, there is damped oscillation instead of sustained oscillation at steady state, filter frequency is ARF frequency. This paper researches that obtaining Bode diagram of speed open-loop system to get resonance characteristic by offline mode. Fig. 2 is structure block diagram of resonance characteristic identification in discrete system. The reference of signal shouldn't too small; otherwise it will impact on identification accuracy. The signal amplitude of q-axis reference is one times of rated current value in this paper.

. • .

Xi

=

• . •

a1xi_1 EB a2xi_2 EB··· EB apxi_p' i = P + 1, p + 2,··· (3)

where ai is 0 or I, EB express modulo 2 operation after summation. Binary M-sequence has good pseudo random characteristic, which has the following features: 1. Balance: The difference of numbers of 0 and 1 is I in a cycle of binary M-sequence. 2. Binary autocorrelation function. For a ±1 level M-sequence with a period of T=2n-l, n is the order of M-sequence, its autocorrelation function is (4): (4) where u(k) is level value of sequence signal, r is integer. The autocorrelation characteristic is: R(r)=T when T is integral multiple of 0 or T, and weak correlation R(r)= -1

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The 2014 International Power Electronics Conference

y(jOJ)

when r is other value. When T is large, autocorrelation function of M-sequence is similar with white noise. This sharp autocorrelation characteristic is the reason that M­ sequence can replace white noise as a current reference. The number of 1 is one more than the one of - 1 in a cycle of pseudo random binary sequence. The mean value of sequence is small, avoids the problem of too large accumulated speed in low-speed open-loop actual system. Fig. 3 is waveform graph of PRBS as current reference.

=

G(jOJ)' x(jOJ)

(6)

From FFT results, in-phase quadrature component of the same frequency Wr of the given and excitation signal is obtained, shown as (7). A is the amplitude of a certain phase. Rand Q are the real part and the image part of the FFT result.

R;(OJJ = 4 COS9i

Q(w,.) = Ai sin 9i

(7)

Ro(w,·) = Ao cos 90 Q}(OJr) = Ao sin 90

L-

-1 o

01 Time(s)

0.2

The frequency characteristic of system under test can be obtained by calculating the amplitude ratio and phase difference of every frequency. The calculation of amplitude-frequency and phase-frequency characteristic is shown in (8), and Bode figure can be draw in this way.

Fig. 3 Wavefonn graph of Pseudo random sequence signal.

(8)

Chirp signal

b.

Except of PRBS, Chirp signal also has wide range of spectrum and constant value of frequency components. Chirp signal which is frequency modulated pulse sweep signal is a kind of continuous cosine sweep signal. The following (5) is used in obtaining the frequency characteristic.

u(t) = A cos(21r(fJt2 + fat))

tan -I Qo tan -I Q R R/. o The method of FFT calculation IS still need to calculate the characteristic of every frequency. It will cost much resource of system. In contrast, power spectrum method is fast and fit fast signals well. The frequency characteristic H(w) can be obtained by the cross-power spectrum from input and output G,y(w)and the auto-power spectrum from input Glw), shown as (9). m( (j) ) 'I' r

(5)

where A is amplitude of sweeping frequency, fJ is variation rate of frequency, fo is initial frequency. From (5), Chirp signal is the cosine function varies linearly with time. Fig. 4 is waveform graph of Chirp signal as current reference.

0

III.

.� -0.5

A.

-1 01. Time(s)

m '1'/

=

_

SIMULATION AND EXPERIMENT

Simulation results

The parameters of servo system used in simulation which is in MATLAB/Simulink are listed in Table I.

0.2

TABLE I MAIN PARAM ETER OF PMSM SERVO

Fig. 4 Waveform graph of Chirp signal.

This signal has uniform frequency component and wide range of spectrum which is similar with white noise. The signal band of Chirp has cut-off range which depends on end-frequency of high frequency, higher end­ frequency and larger band. The above two signals have rapidity and accuracy, only need 1s complete the reference signal in discrete system that sampling frequency is 1 kHz. B.

_

If the result has much noise component, the improved method with spectral window, like Bartlett and Welch, will have desired result.

� 05.

'0

m '1'0

(9)

"

B ." �

=

Parameter

Data process

The basic data processing method is FFT calculation. The excitation signal yet) caused by reference x(t) has Fourier transform y(jw) which equals to the product of system transfer function C(jw) and x(jw), shown as (6):

SYSTEM

Value

Rated power

750 W

Rated torque

2.39 N'm

Rated speed

300 r/min

Rated current

4.4A

Number of pole-pairs

4

Inertia of motor

I.I x 10-3 N . m2

Inertia of load

3 X 1O-JN' m2

Elasticity of shaft

626 N' m/rad

From this parameter, ARF=72 Hz and NTF= 140 Hz. Fig. 5 is simulation contrast figure between M-sequence and Chirp signals. From the results, two reference signals

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The 2014 International Power Electronics Conference

can both acquire the characteristic of open-loop system (only amplitude-frequency in this paper), but the result of Chirp signal is better than the one of M-sequence, and noise variance is smaller.

� ]

"2

20

0

f

M-seq Signal

20

- Chirp Signal

-

...... ,,/

-30

\.

10

\i

Hz

10

'

Because of the sustained oscillation phenomenon in discrete system, proving deficiency of stability margin, the resonance frequency is NTF frequency when discrete system is oscillation at high stiffness state. Notch filter is set after the output of speed controller. The filter frequency is set by NTF frequency read from amplitude­ frequency characteristic. The additional pole can be changed by adjusting the bandwidth of filter. The additional pole can be set to real axis when the parameter of bandwidth is 2 if the damping is ignored. Fig. 9 is motor speed step contrast before and after filtering. Experiment shows that the parameter of notch filter can be determined according to the identification characteristic, and suppression of mechanical resonance can be achieve by this method.

Hz

Fig. 5 Simulation results between two signals with power spectrum method.

Experiment results

Fig. 6 is real photo of 2-mass system. There is a transmission shaft between motor and load, which produces the mechanical resonance. The resonance frequency can be changed by attaching inertia load to any sides or by changing other shafts.

rr====: � 1000 8. '"

Fig. 6 Real plant of 2-mass system in laboratory. The left side is drive motor, and right side is load motor. The inertia of drive motor is 1. 1 X 10-3 N' m2Rated parameters meet with the simulation.

0

o

r=:r-

� . 1 0

------, o.c4 ,' -----,O.L.6,---------o'O.8°----

2

Time(s)

Time(s)

0.6

0.8

1

(a)

- Chirp Signal

-1 oL-------,o'"'.

0.4

Time(s)

-,-,. M-seq Signal

10

0.2

400 �---�---�---�---�---�

Use the above methods to obtain mechanical resonance characteristics on physical platform. The speed sampling frequency of two reference signals is I kHz, the length of signals are I024, so it only need 1.07 s which is the total time with FFT operation to get the Bode figure of resonance characteristic. Fig. 7 is motor speed of two methods. Two methods both have the advantages of short operation time, no accumulated speed, avoiding the mechanical damage of long operation time and high speed on resonance system.

o

'

Fig. 8 Real results between two signals with power spectrum method.

-20

B.

0

1-10

10

tiP-I0

�

-,_ ..

�

• • • · t=:�....... : o

0.2

0.4

o

0.2

0.4

Time(s)

Time(s)

0.6

0.8

1

0.6

0.8

1

(b)

Fig. 9 Real comparison before and after setting notch filter according to identification of characteristics. (a) Notch filter is not set. (b) Notch filter is set according to identification result.

Fig. 7 Wavefonn of motor speed between two methods.

Fig. 8 is contrastive result of two signals with power spectrum method. Two methods can also obtain the resonance characteristic. The feasibility and practicability are proved in this experiment. And Chirp signal which has less noise than M-sequence is more practical.

IV.

CONCLUSIONS

The method of offline suppressing mechanical resonance is determined by the identification of mechanical resonance characteristic. This paper proposes two practical fast identification methods. Bode figure of

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The 2014 International Power Electronics Conference

IEEE Trans. Ind. Electron., Vo1.45, no.I, pp.108-117, FEB 1998. [14] Jennison B.K, "Performance of a linear frequency modulated signal detection algorithm". The Record of the IEEE2000 International. 2000: 447-450. [15] Villwock S,Pacas M. "Application of the Welch-Method for the Identitication of Two- and Three-Mass-System". IEEE Trans on Industrial Electronics. 2008:457-466.

resonance characteristic can be obtained by the reference signal of PRBS and Chirp signal. This identification method has a feature of rapidity and small rotation which can reduce the mechanical damage to resonant system and improves the safety of test. ACKNOWLEDGMENT

The paper is supported by National Natural Science Foundation of China (Project 6 1273 147). This paper and its related research are supported by grants from the Power Electronics Science and Education Development Program of Delta Environmental & Educational Foundation. REFERENCES [1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

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