In this paper, an algorithm using two biquad filters to suppress the wideband resonance for PMSM servo systems is proposed. This algorithm is based on the double biquad filters structure, so it is named, “double biquad filter.” The conventional single biquad filter method cannot suppress unexpected mechanical terms, which may lead to oscillations on the load side. A double biquad filter structure, which can cancel the effects of compliant coupling and suppress wideband resonance, is realized by inserting a virtual filter after the motor speed output. In practical implementation, the proposed control structure is composed of two biquad filters on both the forward and feedback paths of the speed control loop. Both of them collectively complete the wideband resonance suppression, and the filter on the feedback path can solve the oscillation on the load side. Meanwhile, with this approach, in certain cases, the servo system can be more robust than with the single biquad filter method. A step by step design procedure is provided for the proposed algorithm. Finally, its advantages are verified by theoretical analysis and experimental results.
I. INTRODUCTION
Permanent magnet synchronous motor servo drives are used in a wide range of industrial applications
[1]

[4]
. A response bandwidth and dynamic stiffness are two key ratings for its performance
[5]

[7]
. Therefore, the drive must be configured with high gains to achieve high performance. However, if a compliant coupling exists between the motor and the load, the high controller gains may lead to severe mechanical resonance and oscillation.
Researchers have proposed many solutions to suppress mechanical resonance. These include pole placement methods with a PID structure
[8]

[10]
, two degrees of freedom (DOF) robust controllers with a μSynthesis architecture
[11]
,
[12]
, additional state feedback methods
[13]

[17]
, detecting the resonant frequency (such as FFT) and designing the Notch filter
[18]

[22]
, oscillation suppression with fuzzy neural network wave controllers
[23]
,
[24]
, etc. Due to their simplicity and short convergence times, notch filters and acceleration feedback are widely used in industry applications at present. However, most research is focused on a specific condition, and cannot be applied in the case of wideband resonance suppression.
George Ellis divides resonance into two categories: highfrequency resonance and lowfrequency resonance
[15]
. Highfrequency resonance can occur only over a narrow range of frequencies and its resonance frequency is close to or higher than the crossover frequency. The machines that suffer the most from this kind resonance are those with stiff mechanical structures and low damping, such as lathes. The notch filter method can achieve a good effect in terms of highfrequency resonance suppression
[15]
. For the acceleration feedback method, it has less effect on highfrequency resonances suppression because the gain peak at the resonance frequency is usually very high
[17]
,
[20]
.
For lowfrequency resonance, it occurs over a wide range of frequencies and its resonance frequency is less than the crossover frequency
[15]
. Lowfrequency resonance often appears when the inertia ratio is large and the transmission component is flexible, such as in laser typesetters. The acceleration feedback method has good performance in terms of lowfrequency resonance suppression. However, the notch filter is not effective because it can only reduce amplitude gains in a narrow bandwidth
[15]
.
In some industrial applications such as winding machines and robotic arms, if the spring constant of the connecter is not very high, the mechanical resonance may change between a high and low frequency resonance according to the load inertia and system damping term variations, as shown
Fig. 1
. In
Fig. 1
, the category of the resonance is highfrequency resonance when the loadtoinertia ratio is 1:1. With the increasing of the inertia ratio to 10:1, the resonance is changed from highfrequency resonance to lowfrequency resonance.
Mechanical resonance change between highfrequency and lowfrequency resonance.
In order to solve these problems, servo systems often require the use of stepbystep engineering solutions. For example, acceleration feedback is used in lowfrequency resonance and the notch filter is adopted for highfrequency resonance. However, the switching condition between them is difficult to determine. Therefore, it is necessary to design a scheme for wideband resonance suppression.
In existing solutions, the biquad filter can eliminate the effects of compliant coupling, and correct the motor and load as an ideal rigidlycoupled system
[13]
,
[25]
. In a sense, the biquad filter can be applied to wideband resonance suppression. However, the biquad filter has two shortcomings, which constrain its application. Firstly, the load side may still oscillate after adding a biquad filter. Secondly, the filter is very sensitive to system parameters such as load inertia and spring constant
[13]
. Overall, the single biquad filter structure is the main cause of these shortcomings.
Therefore, in order to suppress wideband resonance, an advanced double biquad filter method is proposed in this paper. Its design procedure, robustness analysis and selfadaptive control strategy according to parameter changes are also investigated. Simulation and experimental results show that the proposed structure can be applied for wide bandwidth resonance suppression. The proposed double biquad filter can solve the problem of load side oscillation. If the stability margin is reduced by parameter errors but the system is still stable, the proposed structure can alleviate the damage caused by the parameter errors. If a change in the parameters makes the system unstable, parameter modification and reconstruction are necessary for both the single biquad filter and the proposed double biquad filter.
This paper is arranged as follows. In Section II, the load side oscillation and parameters sensitivity of the single biquad filter are analyzed. To solve these problems and to realize wideband resonance suppression, a double biquad filter and its design procedure are proposed in Section III. In order to further improve the proposed double biquad filter, parameter sensitivity and a selfadaptive method are analyzed in Section IV. In Section V, experimental results verify the advantages of the proposed double biquad filter structure, and conclusions are made in the last section.
II. DISADVANTAGES OF THE SINGLE BIQUAD FILTER
For twomass systems, the transfer function from the drive torque,
T_{e}
, to the motor speed,
ω_{m}
, is:
Where,
J_{m}
and
J_{L}
are the motor and load inertias, respectively.
K_{s}
and
K_{w}
are the equivalent spring constant and the viscous damping constant, respectively.
G_{1}
(
s
) is a rigidlycoupled transfer function between the motor and the load, and
G_{2}
(
s
) is the effect of the compliant coupling.
G_{2}
(
s
) causes instability by altering the phase and gain of the lumped inertial plant. The viscous damping,
K_{w}
, for most machines is low so that both the numerator and denominator are lightly damped. The undamped values of the antiresonant frequency
f_{ares}
and the resonant frequency
f_{res}
are:
Fig. 2
shows a typical speed control system used in industry.
ω_{ref}
is the reference speed.
G_{nc}
(
s
) is equal to
G_{n}
(
s
)
G_{c}
(
s
).
G_{n}
(
s
) is a PI controller for the speed loop, which contains a proportionality factor,
K_{n}
, and an integral time
t_{n}
. The models used in this paper rely on a firstorder lowpass filter (Equ. (4)) acting as the current loop, which contains a torque coefficient,
K_{t}
, a speed detection delay time,
T_{f}
, and a current loop equivalent delay time
T_{ci}
. The motor speed,
ω_{m}
, is connected to the mechanical function
G_{3}
(
s
), and the load speed,
ω_{L}
, is obtained.
Industrial speed control system.
The single biquad filter (conventional biquad filter) is designed to cancel the effects of
G_{2}
(
s
). It has the ideal form:
If the ideal form is achieved, the single biquad filter may eliminate the effect of
G_{2}
(
s
), leaving
G_{1}
(
s
) as an ideal inertial load as shown in
Fig. 3
. This will enhance the response speed and dynamic stiffness as much as possible
[13]
.
Industrial speed system with ideal single biquad filter.
However, the single biquad filter has two major shortcomings. First, the motor speed may be controlled without oscillations. However, the load speed, which is connected to the motor through a compliant coupling, still resonates. The second shortcoming is that the servo system is very sensitive to parameter changes. If the parameters such as load inertia or spring constant are changed, the control loop may become unstable
[13]
.
Fig. 4
shows a simplified single biquad filter speed control system. Since the integration only affects the system response characteristics at low frequencies,
G_{n}
(
s
) is simplified as a Pcontroller. The control loop is corrected as a rigidlycoupled system. This indicates that no oscillations are to be expected on the motor side. However, in the transfer function
G_{3}
(
s
), the term (
J_{L}s
^{2}
+
K_{w}s
+
K_{s}
) produces a very high gain at the antiresonant frequency
f_{ares}
.
Simplified single biquad filter system.
The effect of this term is seen in the Bode plot from
ω_{ref}
to
ω_{L}
(
Fig. 5
), where the gain is maximized at
f_{ares}
. Consequently, an oscillation is expected on the load side, as shown in
Fig. 6
.
Bode plot from ω_{ref} to ω_{L}.
Motor damping and load oscillation with properly tuned single biquad filter.
III. DERIVATION OF THE DOUBLE BIQUAD FILTER TO ACHIEVE A VIRTUAL MECHANICAL CONNECTOR
Theoretically, the single biquad filter is designed to eliminate the effects of the compliant coupling. However, it has limitations as mentioned above. To overcome these problems, a double biquad filter is proposed in this section.
 A. Derivation of the Proposed Double Biquad Filter
When a single biquad filter is added in the forward path, a load oscillation is caused by the peaking of 1/(
J_{L}s
^{2}
+
K_{w}s
+
K_{s}
) in
G_{3}
(
s
). If the term (
J_{L}s
^{2}
+
K_{w}s
+
K_{s}
) can be cancelled or replaced, the oscillation on the load side may be suppressed. Consequently, a term
G_{mc}
(
s
) is added in front of
G_{3}
(
s
) to weaken the effects of the problem term (
J_{L}s
^{2}
+
K_{w}s
+
K_{s}
). In order to eliminate this problem term,
G_{mc}
(
s
) can be designed as in
Fig. 7
, where the denominator of
G_{3}
(
s
) is cancelled by the numerator of
G_{mc}
(
s
), and is replaced with a new term (
As
^{2}
+
Bs
+
K_{s}
). In this case, the oscillation of the load can be eliminated by tuning the parameters
A
and
B
in
G_{mc}
(
s
). The design of
A
and
B
is analyzed in next part. Finally, the oscillation on the load side is eliminated.
Single biquad filter system with G_{mc}(s).
In
Fig. 7
, the frames after
ω_{m}
are not included in the control loop. The effort of changing
G_{mc}
(
s
) to alter
G_{3}
(
s
) is equivalent to redesigning the mechanical coupling. However, this method is not always practical and may increase the cost. Instead,
G_{mc}
(
s
) can be realized by adding a digital filter in the control loop.
According to
Fig. 7
, the lowfrequency gain of
G_{mc}
(
s
) is 1, which guarantees that
ω_{m}
=
ω_{m}
^{'}
in the steady state. As a result,
G_{mc}
(
s
) can be included directly into the control loop, which changes the dynamic response of
ω_{m}
without a static error as shown in
Fig. 8
.
Move filter G_{mc}(s) into control loop.
Considering the realization of the digital control, the control loop needs to be further modified. The filter
G_{mc}
(
s
) can be transferred to the forward and feedback paths of the closedloop as shown in
Fig. 9
.
Industrial speed system with ideal double biquad filter.
After modifying the system frame, the filter on the forward path is
G_{BQa}
(
s
)=
G_{BQ}
(
s
)
G_{mc}
(
s
) while the filter on the feedback path is
G_{BQf}
(
s
)=
G_{mc}
^{1}
(
s
). The biquad filters on the forward path and the feedback path are given by:
The transfer function of the new system's openloop is:
The transfer function from the speed reference,
ω_{ref}
, to the load speed,
ω_{L}
, is:
G_{mo}
(
s
) shows that the new system's openloop characteristic is the same as that in the rigidly connected system, while (
J_{L}s
^{2}
+
K_{w}s
+
K_{s}
) in the denominator of
G_{L}
(
s
) is changed to (
As
^{2}
+
Bs
+
K_{s}
). The suppression of the load side oscillation can be obtained by tuning
A
and
B
, which makes 1/(
As
^{2}
+
Bs
+
K_{s}
) a low pass filter without peaking. When compared with the single biquad filter scheme, there are two biquad filters in the control loop. Therefore, it is called a “double biquad filter” in this paper.
 B. Design of the Double BiQuad Filter
The design of the double biquad filter requires several parameters, such as the gains of the speed and current loops, the delay times,
T_{f}
and
T_{ci}
, the spring constant,
K_{s}
, the damping constant,
K_{w}
, the motor inertia,
J_{m}
, the load inertia,
J_{L}
, and the filter parameters
A
and
B
.
Among them, the PI controllers can be designed on the basis of
[31]
. In general, mechanical resonance occurs in high controller gains which can achieve a high servo performance. In order to simulate the real resonance condition, the speed loop gain is high.
T_{f}
and
T_{ci}
can be calculated from an actual system.
J_{m}
can be obtained from the manufacture.
J_{L}
can be deduced by the motor speed response
[26]
, the model reference adaptive system
[27]
,
[28]
, and by using a disturbance observer to identify the inertia
[29]
,
[30]
. Then,
K_{s}
can calculate by the method in
[22]
.
K_{w}
can be calculated from the ratio between the openloop gains at the resonant frequency of the motor under the noload, rigidlycoupled load and compliantlycoupled load conditions.
When compared with the single biquad filter, the role of tuning
A
and
B
in the proposed double biquad filter is crucial. This can resolve the load side oscillation and affect the performance of the servo system. However, it is difficult to design
A
and
B
due to a high order of
G_{L}
(
s
). The Pade Approximation is a common method to reduce the function order. Because the Pade Approximation has big effect on the high order part, the response characteristics of
G_{L}
^{*}
(
s
) (
G_{L}
(
s
) by approximation) is very different from
G_{L}
(
s
) at high frequencies. On the other hand, Pade Approximation has less effect on the low order part, so the response characteristic of
G_{L}
^{*}
(
s
) is similar to that of
G_{L}
(
s
) on low and medium frequencies. Therefore, since resonance usually affects the response characteristics of servo systems at low and medium frequencies, the Pade Approximation can be used to reduce the order of
G_{L}
(
s
). The original system can be reduced to a secondorder system.
In which:
Compared with a standard secondorder system (Equ. (12)), Equ. (13) can be derived.
The expressions of A and B is shown as:
Simulations have been carried out and the results are displayed in
Table I
. In these simulations,
K_{n}
=500Hz,
T_{f}
=1
ms
,
K_{s}
=1500
Nm/rad
,
K_{w}
=0.1
Nm/(rad/s)
and
K_{t}
=1.0
Nm/A
.
EFFECT OF ΩN AND Ξ
From
Table I
, it is found that
ω_{n}
and
ξ
have the same effect as an undamped natural frequency and the damping ratio of a standard secondorder system. When
ξ
is increased, the overshoot gets smaller. In addition, the speed loop response gets faster when
ω_{n}
increases.
Table I
may serve as a reference for the selection of
ω_{n}
and
ξ
. In general,
ω_{n}
and
ξ
decide the bandwidth of servo systems to some extent, and
A
,
B
are decided by
ω_{n}
and
ξ
. It should be noted that there is a mutual restriction between the speed loop response and the overshoot in the system. If both
ξ
and
ω_{n}
are designed too large,
A
and
B
will not be calculated correctly. As a result, the system will be unstable. In order to obtain a good performance, the selection of
ω_{n}
and
ξ
should make a compromise between the response and the overshoot.
 C. Simulation Comparison
With the designed parameters, a performance comparison between the conventional single biquad filter and the proposed double biquad filter is carried out. For the simulation system,
J_{m}
is 1.0×10
^{3}
kg·m
^{2}
,
J_{L}
is 1.0×10
^{3}
kg·m
^{2}
,
K_{s}
is 3500
Nm/rad
, and
K_{w}
is 0.02
Nm/(rad/s)
.
In order to make a comprehensive consideration for the speed step response, the equivalent parameters are set as
ω_{n}
=600 and
ξ
=0.707, then
A
=0.00011502 and
B
=4.76833.
Motor side open loop Bode diagrams of three different conditions (without a filter, with a single biquad filter and with the proposed double biquad filter) are plotted in
Fig. 10
(a). With the designed parameters, the motor side openloop Bode plots with the single biquad filter and the proposed double biquad filter are the same. As a result, the comparison is fair. Furthermore, load side openloop Bode diagrams without a filter, with a conventional single biquad filter and with the proposed double biquad filter are given in
Fig. 10
(b). It can be found that on the motor side, the method with a single biquad filter and the proposed double biquad filter are both able to correct the system to a rigidlycoupled load and suppress resonance. However, on the load side, only the proposed double biquad filter can eliminate the gain peak at the antiresonant frequency and suppress potential oscillations. With the single biquad filter, there exists a gain peak at the antiresonant frequency which makes it easy for oscillations to occur. These results can be verified by simulation waveforms of the speed loop step response as shown in
Fig. 11
.
Motor side openloop Bode diagram. (b) Load side Bode diagram.
(a) Motor side speed step response of two schemes. (b) Load side speed step response of two schemes.
IV. ANALYSIS OF PARAMETER SENSITIVITY
In the proposed double biquad filter, there are four parameters
J_{m}
,
J_{L}
,
K_{s}
and
K_{w}
decided by mechanical structures. Among them,
J_{m}
is a constant, but the parameter sensitivity of the others should be analyzed. In addition, the change of
T_{L}
needs to be investigated to determine whether it affects system stability.
 A. Damping Constant Kw
In industry applications, the error of
K_{w}
comes from the parameter identification and the influence of the environment.
Fig. 12
shows a motor side open loop Bode plot without filters. It can be found that
K_{w}
mainly affects the peak in the amplitude near the resonant frequency and the antiresonant frequency. However, the system's phase, resonant frequency and antiresonant frequency do not deviate. The simulation result shows a change of
K_{w}
does not have a large influence on the resonance suppression of the double biquad filter.
Under different conditions, Bode plots of G2(s). In which, K_{s}=3000Nm/rad, J_{m}=J_{L}=1.0×10^{3}kg·m^{2}.
In fact, the damping constant
K_{w}
has so little effect on the performance of a servo system so that many scholars ignore it in the analysis of resonance problems.
 B. Spring Constant Ksand Load Inertia JL
Changes of
K_{s}
and
J_{L}
cause the deviation of the resonant frequency and the antiresonant frequency, and have a bigger influence on systems. The parameter sensitivity analysis of
K_{s}
and
J_{L}
should be divided into two parts: 1) the changes of
K_{s}
and
J_{L}
that reduce the stability margin, while the system remains stable; 2) the changes of
K_{s}
and
J_{L}
that make the system unstable.
In the first case, on the motor side, the double biquad filter is equivalent to adding a filter
1/G_{BQf}
(
s
) on the motor speed output side.
Fig. 13
shows a Bode plot of
1/G_{BQf}
(
s
) and two schemes from
ω_{ref}
to
ω_{m}
. It can be seen that
1/G_{BQf}
(
s
) can reduce gains near the resonance frequency, and smooth the motor speed.
Bode plot of 1/G_{BQf}(s) and Bode plot of two schemes from ω_{ref} to ω_{m}.
Consequently, the motor steady state speed of the proposed double biquad filter is more stable than the single biquad filter, when changes of
K_{s}
and
J_{L}
only cause a reduction in the stability margin, as shown in
Fig. 14
and
Fig. 15
.
When system is stable, K_{s} is changed 20%. (a) Speed step response on motor side. (b) Speed step response on load side.
When system is stable, J_{L} is change 20%. (a) Speed step response on motor side. (b) Speed step response on load side.
In the second case, the closed loop system is unstable on the motor side.
1/G_{BQf}
(
s
) does not work anymore. Therefore, parameter modification and reconstruction are necessary for the double biquad filter. Reconstruction schemes can be classified into two different branches as follows:
1) Slight Change of K_{s}:
The
K_{s}
of some connecter parts such as solid coupling, gears and reduction boxes cannot change if the mechanical lifetime is very long and plastic deformation does not occur.
In addition, by using the proposed method, the accuracy of the
K_{s}
measurement is pretty high (if the error of
f_{res}
is very small, and the offline measurement of
J_{L}
is constant). In this situation,
K_{s}
can be treated as a constant, and the system resonance frequency changes mainly due to changes of
J_{L}
.
Under this kind of circumstance, when system resonance reoccurs,
J_{L}
can be recalculated by online measurement of the resonance peak frequency
f_{res}
. With the updated
J_{L}
, the proposed double biquad filter is reconstructed, and can be used to replace former double biquad filter.
2) Slight Change of J_{L}:
The
J_{L}
of some equipment does not change frequently, such as laser phototypesetters, laser engraving machines, etc. For these cases,
J_{L}
can be treated as a constant to solve the resonance problem. A change of
K_{s}
should be considered as the main cause of
f_{res}
modification. When system resonance reoccurs,
K_{s}
can be recalculated by online measurement of the resonance peak frequency
f_{res}
.
 C. Load Torque TL
When studying compliant coupling,
T_{L}
is generally neglected. If the effects of
T_{L}
are considered,
Fig. 9
becomes more complicated, as shown in
Fig. 16
.
double biquad filter system with T_{L}.
Fig. 16
can be transformed as
Fig.17
.
Transformed system from Fig. 16.
In
Fig. 17
:
Finally, the effect of
T_{L}
can be transformed to the input side, which is shown in
Fig. 18
.
Transform T_{L} effect to input side.
In
Fig. 18
:
G_{B}
(
s
) is the transfer function from
ω_{ref}
^{'}
to
ω_{m}
^{'}
.
When compared with
Fig. 9
, the effect of
T_{L}
is equivalent to adding an extra speed reference at the input side, and it has no effects on the system closedloop characteristics. Thus, the proposed double biquad need not be reconstructed regardless of whether
T_{L}
changes or not. In
[25]
, the experimental results have verified that changes of the load torque have no effect on the filter suppression methods. Therefore, most of the experiments in previously published papers are implemented under no the load condition
[19]
,
[32]
,
[33]
.
V. EXPERIMENTAL RESULTS
In this section, several experiments have been implemented to validate the proposed double biquad filter. The system setup for experimental testing is shown in
Fig. 19
. The control algorithm is implemented through a TI TMS320F28035 DSP. The specifications and parameters of the testing PMSM are listed as follows: the rated power is 1.3
KW
, rated speed is 2500
rpm
, rated current is 5
A
, torque coefficient is 1.0
N.m/A
, and rotor inertia is 1.03×10
^{3}
kg·m
^{2}
. A Yaskawa SGMGV 13ADC61 motor acts as the load. Its rated power is 1.3
KW
, rated speed is 1500
rpm
, rated current is 10.7
A
, torque coefficient is 0.89
N.m/A
, and rotor inertia is 1.99×103
^{3}
kg·m
^{2}
. Metal plates can be added to simulate load inertia.
Experimental setup used to test proposed methods.
The motor speed is output by the experimental servo’s DA module, and the load’s speed is output by the Yaskawa’s DA module. In the oscilloscope, 1
V
represents 30
rpm
and 1
A
, respectively. The delay times
T_{f}
and
T_{ci}
are calculated as 1
ms
and 33
us
, respectively.
K_{n}
is set as 450
Hz
. According to Section IV, Part C, the magnitude of load torque has no effects on the double biquad filter. Thus, for ease of operation, all of the experiments are implemented under the no load condition.
 A. Highfrequency Resonance Experiment
The highfrequency resonance performance and system performance with the proposed double biquad filter is verified in this subsection. A NBK’s XGT244C is used as the connected part and its specific parameters are shown in
Table II
. The mechanical damping constant
K_{w}
≈0.11
Nm
/(
rad/s
).
XGT244C’S PARAMETERS
Fig. 20
(a) shows the whole process of resonance suppression when the load inertia does not increase. With a step reference of 150
rpm
, the system oscillates with a frequency of 248
Hz
, as shown in
Fig. 20
(b). Then, the system automatically starts resonance suppression at time T1 as shown in
Fig. 20
(a). This process is divided into two stages.

1) Adding the first and second low pass filter at time T2 and T3, respectively. The frequencyfrescan be identified as 232Hzby the change in the speed oscillation amplitude.Ksis calculated as 1412Nm.rad, as shown inFig. 20(c).

2) According toTable I, comprehensively consider the speed loop response, overshoot and system stability, while settingωn=400 andξ=0.707. CalculateA=0.005363,B=4.3837 from Equ. (14). The proposed Double biquad filter is built and then added to the system at time T4. Finally complete the resonance suppression as shown inFig. 20(a) and (c).
(a) Whole resonance suppressing process. (b) A partial view of oscillation without suppression. (c) Automatically resonance suppression. (d) Speed increase after adding double biquad filter.
Fig. 20
(d) shows that by adding the double biquad filter, the waveform’s speed increases. The waveform shows that the proposed double biquad filter achieves the desired goal since the resonance on both the motor and load sides are suppressed.
In order to simulate load inertia variations, a metal plate (its inertia is 0.01
kg·m
^{2}
) is added on the load side, and
f_{res}
is changed. The original double biquad filter suppressing effect cannot be achieved. Therefore, the filter needs to be reconstructed. The whole process is shown in
Fig. 21
(a).
(a) Whole resonance suppressing process. (b) A partial view of oscillation without suppression. (c) Reconstruct a new double biquad filter. (d) Speed increase after adding new double biquad filter.
At T5, the original double biquad filter is added, but the system still oscillates. At T6, the original double biquad filter is removed. At T7, the system starts to reconstruct the double biquad filter.
Fig. 21
(b) shows that the speed oscillation frequency is changed to 224
Hz
.
Fig. 21
(c) shows the details of reconstruction process. At T8 and T9, the first and second lowpass filter are added resulting in
f_{res}
=196
Hz
. According to the principle that
K_{s}
does not change, the calculated new load inertia is 0.0137
kg·m
^{2}
, where
A
=0.0029606 and
B
=2.00223. At T10, a new filter is reconstructed and the resonance suppression is complete.
Fig. 21
(d) shows the details of the system speed increase.
Fig. 21
shows that the double biquad filter can be achieved by online reconstruction to offset the system parameter changes.
 B. Lowfrequency Resonance Experiment
During the second experiment, a selfmade low
K_{s}
coupling is used as the connector. Its aim is to verify the effect of the double biquad filter suppression for lowfrequency resonance. The coupling’s damping constant is 0.1
Nm
/(
rad/s
). It is noted that since the spring constant is low, the load side does not oscillate much.
Fig. 22
(a) shows the whole process of the lowfrequency resonance, in which the adaptive procedure starts at T11.
Fig. 22
(b) shows that the oscillation frequency is 212
Hz
.
Fig. 22
(c) shows the process of automatic resonance suppression, where two low pass filter are added at T12 and T13. This results in
f_{res}
≈48
Hz
and
K_{s}
≈60.4
Nm.rad
. Set
ω_{n}
=400 and
ξ
=0.707, and tune
A
=0.0002543 and
B
=0.18094. Finally, the double biquad filter is constructed at T14, and the resonance suppression is complete.
Fig. 22
(d) shows the details of the system speed up.
(a) Whole resonance suppressing process for lowfrequency. (b) A partial view of oscillation without suppression. (c) Automatically resonance suppression. (d) Speed increase after adding double biquad filter.
Fig. 22
shows the experiment reaching its intended goal since the proposed double biquad filter for lowfrequency resonance has a good suppression of resonance oscillations. Since
K_{s}
is small, the effect of
J_{L}
changing is not obvious. As a result, there are no further reconstruction experiments.
VI. CONCLUSION
When compared with a single biquad filter, the proposed double biquad filter method is actually equivalent to add a virtual filter after the motor speed output. It contains two biquad filters on both the forward and feedback paths of the speed control loop. The two filters collectively complete the wideband resonance suppression, and the filter on the feedback path can solve the problem of oscillation on the load side. Meanwhile, if the stability margin is reduced by parameter errors, but the system is still stable, the proposed structure can alleviate the damage caused by parameter errors. Simulation and experimental results confirm the validity of the theoretical conclusions and the advantageous performance of the improved control algorithm.
BIO
Xin Luo received his B.S. and M.S. degrees from the Huazhong University of Science and Technology, Hubei, China, in 2007 and 2010, respectively, where he is presently working towards his Ph.D. degree. His current research interests include PMSM drive systems and power conversion circuits.
Anwen Shen received his B.S. and M.S. degrees from Zhejiang University, Zhejiang, China, in 1991 and 1994, respectively, and his Ph.D. degree in Electric Drives and Automation from the Department of Control Science and Engineering, Huazhong University of Science and Technology (HUST), Wuhan, China, in 1997. He is presently a Professor in the School of Automation, HUST. His current research interests include power electronics, electrical drives, and intelligent control.
Renchao Mao received his B.S. degree in Automation from the Department of Control Science and Engineering, Huazhong University of Science and Technology (HUST), Wuhan, China, in 2012, where he is presently working toward his M.S. degree in Control Science and Engineering. His current research interests include power electronics and motion control.
Zhao W. L.
,
Lipo T. A.
,
Kwon B.I.
2014
“Materialefficient permanentmagnet shape for torque pulsation minimization in SPM motors for automotive applications,”
IEEE Trans. Ind. Electron.
61
(10)
5779 
5787
DOI : 10.1109/TIE.2014.2301758
Li X. D.
,
Li S. H.
2014
“Speed control for a PMSM servo system using model reference adaptive control and an extended state observer,”
Journal of Power Electronics
14
(3)
549 
563
DOI : 10.6113/JPE.2014.14.3.549
Na J.
,
Chen Q.
,
Ren X. M.
,
Guo Y.
2014
“Adaptive prescribed performance motion control of servo mechanisms with friction compensation,”
IEEE Trans. Ind. Electron.
61
(1)
486 
494
DOI : 10.1109/TIE.2013.2240635
Park O.S.
,
Park J.W.
,
Bae C.B.
,
Kim J.M.
2013
“A dead time compensation algorithm of independent multiphase PMSM with threedimensional space vector control,”
Journal of Power Electronics
13
(1)
77 
85
DOI : 10.6113/JPE.2013.13.1.77
Elsayed M. T.
,
Mahgoub O. A.
,
Zaid S. A.
2012
“Simulation study of a new approach for field weakening control of PMSM,”
Journal of Power Electronics
12
(1)
136 
144
DOI : 10.6113/JPE.2012.12.1.136
Choi H. H.
,
Vu N. T.T.
,
Jung J.W.
2012
“Design and implementation of a Takagi–Sugeno fuzzy speed regulator for a permanent magnet synchronous motor,”
IEEE Trans. Ind. Electron.
59
(8)
3069 
3077
DOI : 10.1109/TIE.2011.2141091
Torregrossa D.
,
Khoobroo A.
,
Fahimi B.
2012
“Prediction of acoustic noise and torque pulsation in PM synchronous machines with static eccentricity and partial demagnetization using field reconstruction method,”
IEEE Trans. Ind. Electron.
59
(2)
934 
944
DOI : 10.1109/TIE.2011.2151810
Zhang G.
2010
“Speed control of twoinertia system by PI/PID control,”
IEEE Trans. Ind. Electron.
47
(3)
603 
609
DOI : 10.1109/41.847901
Pera M.C.
,
Candusso D.
,
Hissel D.
,
Kauffmann J. M.
2007
“Enhanced servocontrol performance of dualmass systems,”
IEEE Trans. Ind. Electron.
54
(3)
1387 
1399
DOI : 10.1109/TIE.2007.893048
Vukosavic S. N.
,
Stojic M. R.
1998
“Suppression of torsional oscillations in a highperformance speed servo drive,”
IEEE Trans. Ind. Electron.
45
(1)
108 
117
DOI : 10.1109/41.661311
Itoh K.
,
Iwasaki M.
,
Matsui N.
2004
“Optimal design of robust vibration suppression controller using genetic algorithms,”
IEEE Trans. Ind. Electron.
51
(5)
947 
953
DOI : 10.1109/TIE.2004.834943
Ito K.
,
Iwasaki M.
2001
“GAbased practical compensator design for a motion control system,”
IEEE/ASME Trans. Mechatron.
6
(2)
143 
148
DOI : 10.1109/3516.928728
Ellis G.
,
Lorenz R. D.
“Resonant load control methods for industrial servo drives,”
in Proc. IAS
2000
1438 
1445
Sugiura K.
,
Hori Y.
1996
“Vibration suppression in 2 and 3mass system based on the feedback of imperfect derivative of the estimated torsional torque,”
IEEE Trans. Ind. Electron.
43
(1)
56 
64
DOI : 10.1109/41.481408
Ellis G.
,
Gao Z. Q.
“Cures for lowfrequency mechanical resonance in industrial servo systems,”
in Proc. IAS
2001
252 
258
Szabat K.
,
OrlowskaKowalska T.
2007
“Vibration suppression in a twomass drive system using PI speed controller and additional feedbacks—comparative study,”
IEEE Trans. Ind. Electron.
54
(2)
1193 
1206
DOI : 10.1109/TIE.2007.892608
Makkapati V. P.
,
Reichhartinger M.
,
Horn M.
“Performance improvement of servo drives with mechanical elasticity via Extended Acceleration Feedback,”
in Proc. CCA
2012
1279 
1284
Schmidt P.
,
Rehm T.
“Notch filter tuning for resonant frequency reduction in dual inertia systems,”
in Proc. IAS
1999
1730 
1734
Valenzuela M. A.
,
Bentley J. M.
,
Villablanca A.
,
Lorenz R. D.
2005
“Dynamic compensation of torsional oscillation in paper machine sections,”
IEEE Trans. Ind. Appl.
41
(6)
1458 
1466
DOI : 10.1109/TIA.2005.858317
Kang J.
,
Chen S. L.
,
Di X. G.
“Online detection and suppression of mechanical resonance for servo system,”
in Proc. ICICP
2012
16 
21
Liu P.
,
Zhao D. B.
,
Zhang L.
,
Zhang W.
“Design of notch filter applied in miniaturized NC micromilling machine tool,”
in Proc. IEEE Knowledge Acquisition and Modeling Workshop
2008
928 
931
Wang W. Y.
,
Xu J. B.
,
Shen A. W.
“Detection and reduction of middle frequency resonance for industrial servo,”
in Proc. ICICP
2012
153 
160
OrlowskaKowalska T.
,
Szabat K.
2008
“Damping of torsional vibrations in twomass system using adaptive sliding neurofuzzy approach,”
IEEE Trans. Ind. Informat.
4
(1)
47 
57
DOI : 10.1109/TII.2008.916054
OrlowskaKowalska T.
,
Dybkowski M.
2010
“Stator current based MRAS estimator for a wide range speedsensorless inductionmotor drive,”
IEEE Trans. Ind. Electron.
57
(4)
1296 
1308
DOI : 10.1109/TIE.2009.2031134
Brock S.
,
Luczak D.
“Speed control in direct drive with nonstiff load,”
in Proc. ISIE
2011
1937 
1942
Zhang B.
,
Li Y. H.
,
Zuo Y. S.
“A DSPbased fully digital PMSM servo drive using online selftuning PI controller,”
in Proc. IPEMC
2000
1012 
1017
Guo L.
,
Chen W. H.
2005
“Disturbance attenuation and rejection for a class of nonlinear systems via DOBC approach,”
Int. J. Robust Nonlinear Control
15
(3)
109 
125
DOI : 10.1002/rnc.978
Guo Y. J
,
Huang L. P.
,
Qiu Y.
,
Maramatsu M.
“Inertia identification and autotuning of induction motor using MRAS,”
in Proc. IPEMC
2000
1006 
1011
Garrido R.
,
Concha A.
2014
“Inertia and Friction estimation of a velocitycontrolled servo using position measurements,”
IEEE Trans. Ind. Electron.
61
(9)
4759 
4770
DOI : 10.1109/TIE.2013.2293692
Yang S. M.
,
Deng Y. J.
“Observerbased inertial identification for autotuning servo motor drives,”
in Proc. IAS
2005
968 
972
Harnefors L.
,
HansPeter N.
1998
“Modelbased current control of AC machines using the internal model control method,”
IEEE Trans. Ind. Appl.
34
(1)
133 
141
DOI : 10.1109/28.658735
Lee D. H.
,
Lee J. H.
,
Ahn J. W.
2012
“Mechanical vibration reduction control of twomasspermanent magnet synchronous motor using adaptivenotch filter with fast Fourier transform analysis,”
IET Electric Power Appl.
61
(7)
455 
461
DOI : 10.1049/ietepa.2011.0322
Wang H. J.
,
Lee D. H.
,
Lee Z. G.
,
Ahn J. W.
“Vibration rejection scheme of servo drive system with adaptive notch filter,”
in Proc. PESC
Jun. 2006
1 
6