PI controllers are one of the most widely used controllers in industrial control systems due to their simple algorithms and stability. The parameters
K_{p}
and
K_{i}
determine the performance of the system response. The response is expected to improve by increasing the gain of the PI controller. However, too large a gain will accelerate the speed response and cause vibrations, which is not what is expected. This paper proposes a way to suppress vibrations by detecting the vibration frequency and extracting the vibration signal as a compensation to the speed feedback. Additionally, in order to improve its disturbance rejection ability, a loworder disturbance observer is proposed. This paper also explains the operation principle of the proposed method by analyzing the transfer function and it describes the design of the controller parameters in detail. Simulation and experimental results are provided to verify the merits of the proposed method. These results also show the good performance of the proposed method. It has a rapid response and suppresses vibrations.
I. INTRODUCTION
AC servo systems are widely used in industrial applications due to their simple structure and easy maintenance. Common applications include CNC engraving machines and milling machine. Improving the performance of PMSM control is significant for many of its applications. A PI controller is one of the most advanced controllers. The control algorithm is simple, and the parameters are few and easy to tune. These advantages make PI controllers the most commonly used in industrial applications. It is reported that more than 90% of control systems use a PI control algorithm
[1]
. The parameters Kp and Ki determine the performance of the controller. Generally, these parameters are designed to make the response as quick as possible without reducing its stability. Increasing Kp can accelerate the speed response. However, a Kp that is too large can bring vibrations and make the system unstable. In order to ensure stable operation conditions and to reduce oscillations, many methods have been presented by scholars.
In
[2]
, Chengbin Ma designed and analyzed the performance of a number of PID structures including the integralproportional (IP), the modified IP (mIP), and the modified integralproportionalderivative (mIPD). It was shown that these IP and mIPD controllers can effectively suppress vibrations. However, these structures should be changed according to the inertia ration which decreases the generality of their application. In
[3]
,
[4]
, a fractional order controller was adopted to solve the problem of the contradiction between the stability margin loss and the strength of the vibration suppression. However, the design of the fractional order controller is difficult and the realization is somewhat problematic. Wen Li proposed a single neuronbased PI fuzzy controller and a fractionalorder disturbance observer for vibration suppression
[5]
. The PI fuzzy controller was used to obtain a desired steadystate precision and the observer was applied to generate a compensation signal. The results showed that this improved the speed loop control. However, while the robustness of the disturbance observer local loop to parameters variations was discussed, that of closed loop has not been considered. JulKi Seok combined P and PI controllers to achieve speed control
[6]
. However, the controller switching point was set manually, and the adjustable drive performance was seriously dependent on the approximate point. JongSun Ko presented a neural load torque compensation method to obtain better precision control
[13]
. WanCheng Wang proposed a systematic wavelet control algorithm to obtain faster command responses and better load disturbance responses for the speedloop control system or the position control system
[14]
. Experimental results verified the correctness of the proposed method. However, the experiments were only tested for a micro PMSM. Z. Zhiqiang built a relationship between the PID parameters and the oscillation characteristics of the closed loop response, and proposed a PID parameter tuning strategy according to the oscillation characteristics
[17]
. However, this study only provided simulation results, and experiments should be conducted. Hanifzadegan proposed a parallel structure of a feedback controller to suppress vibrations during operation
[18]
. One controller aims at improving the tracking performance and the other controller is used to reduce the peak amplitude at the resonance frequency. However, the performances of the controllers contradict each other.
In this paper, a strategy to improve the speed response while maintaining stability is presented without changing the classical PI structure. The method is based on detecting the speed vibration signal. The vibration frequency can be calculated by analyzing the speed error with a FFT. Then the speed vibration signal is extracted as compensation for speed feedback. Additionally, a loworder Luenberger observer is used to improve the antidisturbance ability. The effectiveness of the proposed control system is verified by simulation and experimental results.
II. PMSMMODEL AND PROBLEM FORMULATION
 A. PMSM Control System Modeling
The mechanical and electrical subsystem of a PMSM can be denoted as follows:
Where eq. (1a) and eq. (1b) describe the mechanical subsystem, and (1c) and (1d) show the electrical subsystem equations.
θ
,
ω
are the motor angle and speed;
i_{d}
,
i_{q}
are the dq axes stator currents;
u_{d}
,
u_{q}
are the dq axes stator voltages;
R
is the stator resistance;
L_{d}
,
L_{q}
are the dq axes stator inductances;
J_{m}
is the moment of inertia;
B
is the damping factor;
Ψ_{m}
is the flux linkage;
T_{L}
is the load torque; and
T_{m}
is the electrical torque.
 B. Speed PI Control Module
Fig. 1
describes the PI control structure under a speed loop. Where
ω_{ref}
is the speed command,
ω_{m}
is the rotor speed,
i_{qref}
is the current command,
i_{q}
is the practical current,
T_{f}
is the lowpass filter time constant of the speed feedback, and
K_{T}
is the torque coefficient.
The classical speed control structure.
The target of the speed PI control module is to generate
i
_{qref}
.
The transfer functions need to be analyzed:
ω_{m}
(
s
) /
ω_{ref}
(
s
) ,
ω_{m}
(
s
) /
T_{L}
(
s
) . First, the current closeloop transfer function should be obtained. Generally, the current transfer function is corrected as a traditional secondorder system. Since the bandwidth of the speed loop is far below the currentloop bandwidth, it is possible to neglect the highorder items in the transfer function without influencing the analysis of the whole system. Thus, it is possible to rewrite the current close loop as eq. (3).
Where
T_{cf}
is the current feedback filter time constant, and
T_{sf}
is the inverter switching period. Then it is possible to derive the transfer function
ω_{m}
(
s
) /
ω_{ref}
(
s
) ,
ω_{m}
(
s
) /
T_{L}
(
s
) as follows:
From eq. (7) and eq. (10), it can be seen that the speed response depends on how much of
ω_{ref}
can be passed by
H_{RR}
(
s
). It can also be seen that the system robustness depends on how much of
T_{L}
can be rejected by
H_{RL}
(
s
). Here the motor parameters in
Table I
are used to see the influence of different
K_{p}
values on the performance of the control system.
THE SIMULATION PARAMETERS
THE SIMULATION PARAMETERS
Generally, the speed loop is designed to be a traditional IIsystem. Therefore, according to the design specifications of the traditional IIsystem,
K_{p}
and
K_{i}
can be calculated
[23]
. The first step is the elimination of
H_{RR}
(
s
) and
H_{RL}
(
s
) by the motor parameters and controller parameters. Then it is possible to obtain the bode diagrams of
H_{RR}
(
s
) and
H_{RL}
(
s
) under different
K_{p}
values as shown in
Fig. 2
(a) and
Fig. 2
(b). The figures illustrate that the bode diagram of
H_{RR}
(
s
) peaks when
K_{p}
is big enough. According to the characteristics of a secondorder system, the resonance peak leads to speed vibrations. A larger
K_{p}
helps improve the disturbance rejection since the magnitude of disturbance frequency response is lower. However, it also peaks at the vibration frequency. In conclusion, increasing the gain of the speed PI controller can speed up the rapidity of the speed response. However, it may bring vibrations and weaken stability.
(a) The bode diagrams of H _{RR}(s) . (b) The bode diagrams of H _{RL}(s) .
 C. The Relationship between the Vibration Frequency and Kp
Assuming
T_{f}
=0, B=0. Therefore,
H_{RR}
(
s
) can be denoted by the following:
where
is the natural frequency and
ξ
is the damping coefficient. According to the secondorder system frequency response, the resonance frequency is defined as
. Therefore, so the resonance frequency of eq. (7) is
. Since
K _{i}
=
K _{p}
/
τ
, where
τ
is the time constant, it can be rewritten as
. It is well known that
τ
is a very small constant, and that
ω_{vib}
is close to
.
III. PROPOSED METHOD FOR THE CONTROL MODULE
To reduce the speed vibration in the control system, a control strategy is proposed, as shown in
Fig. 3
, where
k
and
k_{vib}
are the control parameters of the disturbance torque observer and speed vibration observer, respectively,
k_{lf}
is the cutoff frequency of the lowpass filter, LF represents the lowpass filter,
k_{j}
is the reciprocal value of the inertia, and
ζ
is the measurement noise. Through the analysis of how
K_{p}
affects the performance of the control system in section II.B, it can be seen that the key to suppressing the oscillation induced by a large
K_{p}
is to reduce the magnitude at the resonance frequency. The following sections will describe how to implement the method to achieve the target.
The proposed method’s structure.
 A. Disturbance Torque Observer
A disturbance observer can help improve the robustness of the control system. Here, a Luenberger observer is used to estimate the disturbance torque.
Fig. 4
shows the structure of the proposed disturbance observer. Based on
Fig. 4
, it is possible to describe the transfer function
ω_{m}
/
T_{L}
as eq. (11). Compared with the transfer function under the classical PI control, i.e., eq. (12), it can be found that eq. (11) is equal to eq. (12) multiplied by
s
^{2}
/ 2
ks
+
k
^{2}

s
^{2}
, which has highfilter characteristics. Therefore, disturbance effects are more damped below the frequency
k
(
rad
/
s
) . Then, considering the transfer functions of
ω_{m}
/
ζ
, i.e., eq. (13–14), the influence of the measurement noise with the disturbance compensation is declined because of a lowpass filter factor 1 
J_{m}
(2
ks
+
k
^{2}
)
s
/[
K_{T}
(
s
+
k
)
^{2}
(
k_{p}s
+
k_{i}
)]. Therefore, the influence of the measurement noise can be reduced.
The disturbance observer.
 B. Feedback of the Estimated Speed Vibration
A large K
_{p}
may bring vibration to the operation portion, and this vibration is transmitted to the feedback of the speed loop. If the speed vibration signal is extracted, then it is possible to offset the vibration by adding it to the speed feedback.
A bandpass filter is appropriate to deal with certain frequency vibrations concerned with speed. Assuming
ω_{lf}
,
ω_{hf}
is the cutoff frequency of the lowpass filter and highfrequency filter, and the bandwidth of the bandpass filter is
ω_{hf}

ω_{lf}
.
Fig. 5
shows the frequency response of the highpass+lowpass filter, the bandpass filter, and the highpass+bandpass filter, where
ω_{lf}
= 628
rad
and
ω_{hf}
= 1000
rad
. This figure also illustrates that the highpass+lowpass filter has the same characteristics as the bandpass filter. Furthermore, from the
Fig. 3
, it can be seen that the transfer function
, is actually a highpass filter, which is denoted by eq. (15). Therefore, it is only necessary to design an additional lowpass filter to extract the vibration signal. The estimation of the whole speed vibration is simple within this structure.
Frequency response of three different types of filters.
 C. Control Parameters Design
1) Online Spectrum Analysis:
Since the proposed method needs to extract the vibration signal, the frequency of the vibration should be known. The DFT (discrete Fourier transform) is applied to analyze the speed error between the motor speed and the speed reference. When an ADC (analogtodigital conversion) is applied to the motor speed error at an approximate sampling rate over the observation interval T=N*Ts, where N is the number of acquired samples and Ts is the sampling period, the finite discretetime sequence of the speed error can be obtained:
Spderr
(
kT_{s}
) is the
kth
harmonics. In order to shorten the DFT calculation time in practice, the FFT is used. Actually, the FFT is widely used in control systems. Q. G.Wang and J. J. Nelson processed signals by using the FFT, and frequency spectra are added to the feedback signal
[7]
,
[8]
. R. BlascoGimenez used the FFT to measure motor speed
[9]
, and N. Baoliang suppressed the vibration magnitude by analyzing the FFT spectrum of vibration signals
[10]
. The speed error acquisition uses the DMA with buffering for date collection and processing. Then the discrete date is transformed to the frequency domain according to the FFT algorithm. Thus, the power spectrum density can be calculated by squaring the amplitude of the FFT output. The vibration frequency
ω_{vib}
corresponds to the largest PSD.
In this paper, a STM32F302 is used as the control unit. Its powerful calculation ability makes the realization of the FFT analysis easy. It only takes about 1.3ms when N=1024. Therefore, it is suitable for the real time control of high dynamic PMSM servo systems. In the experiments, N=1024 is adopted.
2) The Design of the Disturbance Observer Parameters k:
According to the discussion about the disturbance observer design in the previous section, we can see that a large k is helpful for interference rejection, while a small
k
can improve the measurement noise rejection. As the system cutoff frequency under PI control is near
, we recommend that k be no bigger than
, considering the reliability of both.
3) The Design of the Vibration Suppression Parameters k_{vib} and k_{lf}:
After finding the vibration frequency using the FFT, it is possible to define
k_{vib}
=
ω_{vib}
in order to extract the vibration signal and to simplify the calculation of the design processes. Then the design of
k_{lf}
is analyzed. Firstly, the transfer function between
ω_{ref}
and
ω_{m}
is considered with the speed vibration compensation
ω_{cmp}
as follows (T
_{f}
=0 and B=0 are assumed for the sake of simplicity) :
The characteristic polynomial in the closed loop from the above equations can be written as:
and the numerator polynomial is:
In order to offset the peak of eq. (16c) at the resonance frequency, G
_{rm}
(s)=1 is expected at the resonance frequency. Then the following equations can be obtained:
Since it is known that
ω_{vib}
is close to
as discussed in section II.C, it can be supposed that
. In addition, since
K_{i}
is equal to
K _{p}
/
τ
, it is possible to further simplify
a
_{1}
,
a
_{0}
,
b
_{1}
,
b
_{0}
. Assuming that
k_{lf}
=
mω_{vib}
, it is possible to obtain the following equations:
From eq. (17), there are two solutions,
m
= 6.8 and
m
= 1.17 . However, in order to guarantee the phase margin at the resonance frequency,
a
_{1}
> 0 should be satisfied as
b
_{1}
> 0,
b
_{0}
< 0,
a
_{0}
< 0 . As a result, only
m
= 6.8 can be the solution. Therefore:
4) The Proof of the Vibration Suppression by Using the Proposed Method:
So far, all the control parameters have been designed. Now the merits of the proposed method can be analyzed by comparing the speed command response and disturbance resistance under the proposed method and the classical PI control.
The simulation parameters in
Table I
are used to test the characteristics of both transfer functions. Firstly, the PI control structure in
Fig. 1
is built under the Simulink environment. The speed error data is transfered to the frequency domain by the FFT algorithm. The frequency spectrum of
ω_{err}
when
K_{p}
=3.45 is shown in
Fig. 6
(a). From this picture, it can be seen that the vibration frequency is rad s
ω_{vib}
= 392.5
rad
/
s
, which is close to
. Based on
ω_{vib}
= 392.5
rad
/
s
and eq. (18), it is possible to calculate
k_{vib}
= 392.5
rad
/
s
k_{lf}
= 2669
rad
/
s
. Then bode diagrams of the disturbance response and speed command response can be plotted under the proposed method and the classical PI control. The results are shown in
Fig. 6
(b) and
6
(c). From
Fig. 6
(b), it can be seen that the lowfrequency magnitude of the disturbance response is much more dampened with the proposed method than with the PI control. In addition, the resonance peak of the disturbance response at the resonance frequency is canceled by the proposed method. Similarly,
Fig. 6
c compares the speed command responses under the two control methods. It is obvious that the magnitude of the classical PI controller is higher than that of the proposed method at the resonance frequency. Actually, the magnitude is near 0dB under the proposed method which is what is expected according to the design in section III.C. In conclusion, the proposed method has better speed command and disturbance rejection performance than the PI control. Therefore, the concept of reducing the resonance peak to suppress vibrations can be achieved by the proposed method.
(a) The frequency spectrum of the speed error. (b) The bode of disturbance response. (c) The bode of speed command response.
IV. SIMULATION AND EXPERIMENTAL RESULTS
 A. Simulation Results
In this section, the classical PI control and the proposed control method are compared. The simulation parameters are the same as those in
Table I
.
Fig. 7
(a)(e) show the motor speed response and current response with different values of
K_{p}
under the PI control and the proposed method. In
Fig. 7
(a), it can be seen that the speed overshoot under the PI control is 20% compared with 16% under the proposed method. It can also be seen in
Fig. 7
(b)(c) that vibrations and a larger overshoot occur when
K_{p}
increases under the PI controller, while the current and speed response using the proposed method has a shorter settling time and a smaller overshoot. In
Fig. 7
(d)(e), it can be seen that K
_{p}
is too large. As a result, the speed and current vibrations last for a long time under the PI control. Meanwhile the system is still stable under the proposed method.
Fig. 7
(f) shows the frequency spectrum of the speed error under the proposed method. Compared with
Fig. 6
(a), the content of the vibration frequency is significantly decreased. Since disturbance rejection is also an important performance index, the disturbance response is also tested. A step disturbance torque
T_{L}
= 1
N
.
m
is input at t = 0
s
. The results are shown in
Fig. 8
, and it can be seen that the speed loss under the proposed method is smaller than that under the PI control, and the recovery time is also smaller.
(a) The motor speed response under different K_{p}=0.69. (b) The motor speed response with Kp=3.45. (c). Current response with Kp=3.45. (d). The motor speed response under different Kp=20.7 (e). Current response with Kp=20.7. (f). The Speed error frequency spectrum under the proposed method.
The disturbance response of the motor speed under the PI control and proposed method.
 B. Experimental Results
To show that the proposed method is valid in practice, a general diagram of the proposed control is declared as in
Fig. 9
(a). An experimental platform is set up as in
Fig. 9
(b). The platform consisted of a PMSM, a servo driver, and a PC for recording the results. The PMSM is a 80MSL02430 made by Maxsine Company, China. The motor parameters are J
_{m}
=3.5*10e4 and I
_{N}
=4.4A. Both the PI control system and the proposed control system are tested for comparison.
(a) Software structure and experimental platform. (b) Experimental platform.
The speed step response (
n
= 100
r
/ min ) results are shown in
Fig. 10

12
. The
K_{p}
unit is Hz, and it is the inside unit in the DSP.
Fig. 10

11
describe the speed response with different values of
K_{p}
. At the same time the corresponding show the motor speed results when the disturbance torque is added suddenly under the two control methods. From Fig. current signals are also shown in
Fig. 12
(a)(d). The current responses and the speed responses coincided.
Table II
shows a comparison of the speed response between the PI control and the proposed method.
(a) Speed step response with Kp=174Hz under PI control. (b) Speed step response with Kp=174Hz under proposed method. (c) Speed step response with K_{p}=260Hz under PI control. (d) Speed step response under the proposed method Kp=260Hz.
(a) Speed step response with Kp=420Hz under the PI control. (b) Speed step response with Kp=420Hz under the proposed method.
(a) Current response with Kp=260Hz under PI control. (b) Current response under proposed method Kp=260Hz. (c) Current response with Kp=420Hz under PI control. (d) Current response under proposed control Kp=420Hz.
COMPARISON UNDER PI CONTROL AND PROPOSED METHOD
COMPARISON UNDER PI CONTROL AND PROPOSED METHOD
From
Table II
it can be seen that the performance of the PI controller becomes worse as Kp increases, while the proposed method still has good features when Kp is high. Compared with the simulation results, the same trends are seen. The overshoot of the speed responses with the proposed method is reduced by two times compared with the corresponding results of the PI controller in the simulation and experiments. In
Fig. 11
(a) and
Fig. 12
(c), it can be seen that the speed and current responses vibrates for a long time just like those in the simulation in
Fig. 7
(d)(e). However, by using the proposed method, the vibration time is reduced remarkably. The simulation results in
Fig. 7
(d) vibrated for 12 periods under the proposed method which is basically same as the experimental results in
Fig. 11
(b). Additionally,
Fig. 13
(a)(b)
13
(a), it can be seen that speed vibrations happen because of disturbances. In addition, the vibration frequency is the same as the speed step response in
Fig. 11
(a). However, under the proposed method, the vibration is suppressed in
Fig. 13
b. This is also consistent with the bode diagram of the transfer function between the disturbance in the motor speed in
Fig. 6
b. Actually, the load disturbance may cause speed vibrations. However, this does not change the resonance frequency if Kp is invariable. The analysis in section II.C also shows the relative parameters which affects the oscillation frequency. Therefore, if the speed oscillation frequency is detected accurately, the proposed method is effective even if the load disturbance is variable. To verify this, a variable disturbance torque based on the previous disturbance experiment was added.
Fig. 14
shows the speed response result. It can be see that although the speed has some fluctuations, the resonance induced by the high Kp is suppressed. In conclusion, through the experimental results, it can be seen that the corresponding characteristics of the PI control and the proposed method are consistent with the simulation performance. It can also be seen that the proposed method is efficient in terms of vibration suppression which is induced by improving the Kp. It also performs well under disturbances.
(a) Disturbance results with Kp=420Hz under the PI control. (b) Disturbance results with Kp=420Hz under proposed method.
The speed response with Kp=420Hz and variable disturbance.
V. CONCLUSION
In this paper, the essential reasons for vibrations with a high speedloop gain are described. Then the vibration suppression method is presented. The basic idea of the method is to counteract the resonance peak at the resonance frequency. An online spectrum analysis FFT was utilized to detect the vibration frequency
ω_{vib}
. Then the vibration signal was extracted as the compensation for the speed feedback by using a loworder speed observer and a lowpass filter. The disturbance observer is a loworder Luenberger observer, which helps improve the antidisturbance ability. Additionally, the design of the controller parameters is an important key to implementing the proposed method in practice. Design processes that are too complex are tough for realization. In this paper, simply design rules were obtained by analyzing the system transfer functions. The parameter k is relative to
and
k_{lf}
= 6.8
ω_{vib}
. Bode diagrams of the frequency response under the classical PI controller and the proposed method illustrate the vibration suppression principle in theory and show that the resonance peak can be successfully reduced by the proposed method. The experimental results also show that the proposed method can effectively suppress vibration in a PMSM servo driver. The overshoot is smaller and the settling time is reduced.
BIO
Qiong Li was born in China, in 1989. She received her B.S. degree in Electrical Engineering from East China Jiaotong University, Jiangxi, China. She is presently working towards her Ph.D degree in Electrical Engineering at the Huazhong University of Science and Technology, Wuhan, China. Her current research interests include highperformance servo drivers and the control of robots in industrial fields.
Qiang Xu was born in China, in 1968. He received his B.S., M.S., and Ph.D. degrees in Electrical Engineering from the Huazhong University of Science and Technology, Wuhan, China. He is presently working as a professor in the Department of Electrical Engineering, Huazhong University of Science and Technology. His current research interests include highperformance servo drivers and the control of robots in industrial fields.
Shenghua Huang was born in China, He is presently working as a professor in the Department of Electrical Engineering, Huazhong University of Science and Technology. His current research interests include highperformance servo drivers .
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