A Double Bi-Quad Filter with Wide-Band Resonance Suppression for Servo Systems

Abstract

In this paper, an algorithm using two bi-quad filters to suppress the wide-band resonance for PMSM servo systems is proposed. This algorithm is based on the double bi-quad filters structure, so it is named, "double bi-quad filter." The conventional single bi-quad filter method cannot suppress unexpected mechanical terms, which may lead to oscillations on the load side. A double bi-quad filter structure, which can cancel the effects of compliant coupling and suppress wide-band resonance, is realized by inserting a virtual filter after the motor speed output. In practical implementation, the proposed control structure is composed of two bi-quad filters on both the forward and feedback paths of the speed control loop. Both of them collectively complete the wide-band 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 bi-quad 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 μ-Synthesis 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 wide-band resonance suppression.

George Ellis divides resonance into two categories: high-frequency resonance and low-frequency resonance [15]. High-frequency 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 high-frequency resonance suppression [15]. For the acceleration feedback method, it has less effect on high-frequency resonances suppression because the gain peak at the resonance frequency is usually very high [17], [20].

For low-frequency resonance, it occurs over a wide range of frequencies and its resonance frequency is less than the crossover frequency [15]. Low-frequency 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 low-frequency 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 high-frequency resonance when the load-to-inertia ratio is 1:1. With the increasing of the inertia ratio to 10:1, the resonance is changed from high-frequency resonance to low-frequency resonance.

Fig. 1.Mechanical resonance change between high-frequency and low-frequency resonance.

In order to solve these problems, servo systems often require the use of step-by-step engineering solutions. For example, acceleration feedback is used in low-frequency resonance and the notch filter is adopted for high-frequency resonance. However, the switching condition between them is difficult to determine. Therefore, it is necessary to design a scheme for wide-band resonance suppression.

In existing solutions, the bi-quad filter can eliminate the effects of compliant coupling, and correct the motor and load as an ideal rigidly-coupled system [13], [25]. In a sense, the bi-quad filter can be applied to wide-band resonance suppression. However, the bi-quad filter has two shortcomings, which constrain its application. Firstly, the load side may still oscillate after adding a bi-quad filter. Secondly, the filter is very sensitive to system parameters such as load inertia and spring constant [13]. Overall, the single bi-quad filter structure is the main cause of these shortcomings.

Therefore, in order to suppress wide-band resonance, an advanced double bi-quad filter method is proposed in this paper. Its design procedure, robustness analysis and self-adaptive 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 bi-quad 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 bi-quad filter and the proposed double bi-quad filter.

This paper is arranged as follows. In Section II, the load side oscillation and parameters sensitivity of the single bi-quad filter are analyzed. To solve these problems and to realize wide-band resonance suppression, a double bi-quad filter and its design procedure are proposed in Section III. In order to further improve the proposed double bi-quad filter, parameter sensitivity and a self-adaptive method are analyzed in Section IV. In Section V, experimental results verify the advantages of the proposed double bi-quad filter structure, and conclusions are made in the last section.

 

II. DISADVANTAGES OF THE SINGLE BI-QUAD FILTER

For two-mass systems, the transfer function from the drive torque, Te, to the motor speed, ωm, is:

Where, Jm and JL are the motor and load inertias, respectively. Ks and Kw are the equivalent spring constant and the viscous damping constant, respectively. G1(s) is a rigidly-coupled transfer function between the motor and the load, and G2(s) is the effect of the compliant coupling.

G2(s) causes instability by altering the phase and gain of the lumped inertial plant. The viscous damping, Kw, for most machines is low so that both the numerator and denominator are lightly damped. The un-damped values of the anti-resonant frequency fares and the resonant frequency fres are:

Fig. 2 shows a typical speed control system used in industry. ωref is the reference speed. Gnc(s) is equal to Gn(s)Gc(s). Gn(s) is a PI controller for the speed loop, which contains a proportionality factor, Kn, and an integral time tn. The models used in this paper rely on a first-order low-pass filter (Equ. (4)) acting as the current loop, which contains a torque coefficient, Kt, a speed detection delay time, Tf, and a current loop equivalent delay time Tci. The motor speed, ωm, is connected to the mechanical function G3(s), and the load speed, ωL, is obtained.

Fig. 2.Industrial speed control system.

The single bi-quad filter (conventional bi-quad filter) is designed to cancel the effects of G2(s). It has the ideal form:

If the ideal form is achieved, the single bi-quad filter may eliminate the effect of G2(s), leaving G1(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].

Fig. 3.Industrial speed system with ideal single bi-quad filter.

However, the single bi-quad 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 bi-quad filter speed control system. Since the integration only affects the system response characteristics at low frequencies, Gn(s) is simplified as a P-controller. The control loop is corrected as a rigidly-coupled system. This indicates that no oscillations are to be expected on the motor side. However, in the transfer function G3(s), the term (JLs2+Kws+Ks) produces a very high gain at the anti-resonant frequency fares.

Fig. 4.Simplified single bi-quad filter system.

The effect of this term is seen in the Bode plot from ωref to ωL (Fig. 5), where the gain is maximized at fares. Consequently, an oscillation is expected on the load side, as shown in Fig. 6.

Fig. 5.Bode plot from ωref to ωL.

Fig. 6.Motor damping and load oscillation with properly tuned single bi-quad filter.

 

III. DERIVATION OF THE DOUBLE BI-QUAD FILTER TO ACHIEVE A VIRTUAL MECHANICAL CONNECTOR

Theoretically, the single bi-quad filter is designed to eliminate the effects of the compliant coupling. However, it has limitations as mentioned above. To overcome these problems, a double bi-quad filter is proposed in this section.

A. Derivation of the Proposed Double Bi-quad Filter

When a single bi-quad filter is added in the forward path, a load oscillation is caused by the peaking of 1/(JLs2+Kws+Ks) in G3(s). If the term (JLs2+Kws+Ks) can be cancelled or replaced, the oscillation on the load side may be suppressed. Consequently, a term Gmc(s) is added in front of G3(s) to weaken the effects of the problem term (JLs2+Kws+Ks). In order to eliminate this problem term, Gmc(s) can be designed as in Fig. 7, where the denominator of G3(s) is cancelled by the numerator of Gmc(s), and is replaced with a new term (As2+Bs+Ks). In this case, the oscillation of the load can be eliminated by tuning the parameters A and B in Gmc(s). The design of A and B is analyzed in next part. Finally, the oscillation on the load side is eliminated.

Fig. 7.Single bi-quad filter system with Gmc(s).

In Fig. 7, the frames after ωm are not included in the control loop. The effort of changing Gmc(s) to alter G3(s) is equivalent to redesigning the mechanical coupling. However, this method is not always practical and may increase the cost. Instead, Gmc(s) can be realized by adding a digital filter in the control loop.

According to Fig. 7, the low-frequency gain of Gmc(s) is 1, which guarantees that ωm = ωm' in the steady state. As a result, Gmc(s) can be included directly into the control loop, which changes the dynamic response of ωm without a static error as shown in Fig. 8.

Fig. 8.Move filter Gmc(s) into control loop.

Considering the realization of the digital control, the control loop needs to be further modified. The filter Gmc(s) can be transferred to the forward and feedback paths of the closed-loop as shown in Fig. 9.

Fig. 9.Industrial speed system with ideal double bi-quad filter.

After modifying the system frame, the filter on the forward path is GBQa(s)=GBQ(s)Gmc(s) while the filter on the feedback path is GBQf(s)=Gmc-1(s). The bi-quad filters on the forward path and the feedback path are given by:

The transfer function of the new system's open-loop is:

The transfer function from the speed reference, ωref, to the load speed, ωL, is:

Gmo(s) shows that the new system's open-loop characteristic is the same as that in the rigidly connected system, while (JLs2+Kws+Ks) in the denominator of GL(s) is changed to (As2+Bs+Ks). The suppression of the load side oscillation can be obtained by tuning A and B, which makes 1/(As2+Bs+Ks) a low pass filter without peaking. When compared with the single bi-quad filter scheme, there are two bi-quad filters in the control loop. Therefore, it is called a “double bi-quad filter” in this paper.

B. Design of the Double Bi-Quad Filter

The design of the double bi-quad filter requires several parameters, such as the gains of the speed and current loops, the delay times, Tf and Tci, the spring constant, Ks, the damping constant, Kw, the motor inertia, Jm, the load inertia, JL, 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. Tf and Tci can be calculated from an actual system. Jm can be obtained from the manufacture. JL 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, Ks can calculate by the method in [22]. Kw can be calculated from the ratio between the open-loop gains at the resonant frequency of the motor under the no-load, rigidly-coupled load and compliantly-coupled load conditions.

When compared with the single bi-quad filter, the role of tuning A and B in the proposed double bi-quad 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 GL(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 GL*(s) (GL(s) by approximation) is very different from GL(s) at high frequencies. On the other hand, Pade Approximation has less effect on the low order part, so the response characteristic of GL*(s) is similar to that of GL(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 GL(s). The original system can be reduced to a second-order system.

In which:

Compared with a standard second-order 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, Kn=500Hz, Tf=1ms, Ks=1500Nm/rad, Kw=0.1Nm/(rad/s) and Kt=1.0Nm/A.

TABLE IEFFECT OF ΩN AND Ξ

From Table I, it is found that ωn and ξ have the same effect as an un-damped natural frequency and the damping ratio of a standard second-order system. When ξ is increased, the overshoot gets smaller. In addition, the speed loop response gets faster when ωn increases. Table I may serve as a reference for the selection of ωn and ξ. In general, ωn and ξ decide the bandwidth of servo systems to some extent, and A, B are decided by ωn 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 ωn 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 ωn 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 bi-quad filter and the proposed double bi-quad filter is carried out. For the simulation system, Jm is 1.0×10-3kg·m2, JL is 1.0×10-3kg·m2, Ks is 3500Nm/rad, and Kw is 0.02Nm/(rad/s).

In order to make a comprehensive consideration for the speed step response, the equivalent parameters are set as ωn=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 bi-quad filter and with the proposed double bi-quad filter) are plotted in Fig. 10 (a). With the designed parameters, the motor side open-loop Bode plots with the single bi-quad filter and the proposed double bi-quad filter are the same. As a result, the comparison is fair. Furthermore, load side open-loop Bode diagrams without a filter, with a conventional single bi-quad filter and with the proposed double bi-quad filter are given in Fig. 10 (b). It can be found that on the motor side, the method with a single bi-quad filter and the proposed double bi-quad filter are both able to correct the system to a rigidly-coupled load and suppress resonance. However, on the load side, only the proposed double bi-quad filter can eliminate the gain peak at the anti-resonant frequency and suppress potential oscillations. With the single bi-quad filter, there exists a gain peak at the anti-resonant 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.

Fig. 10.Motor side open-loop Bode diagram. (b) Load side Bode diagram.

Fig. 11.(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 bi-quad filter, there are four parameters Jm, JL, Ks and Kw decided by mechanical structures. Among them, Jm is a constant, but the parameter sensitivity of the others should be analyzed. In addition, the change of TL needs to be investigated to determine whether it affects system stability.

A. Damping Constant Kw

In industry applications, the error of Kw 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 Kw mainly affects the peak in the amplitude near the resonant frequency and the anti-resonant frequency. However, the system's phase, resonant frequency and anti-resonant frequency do not deviate. The simulation result shows a change of Kw does not have a large influence on the resonance suppression of the double bi-quad filter.

Fig. 12.Under different conditions, Bode plots of G2(s). In which, Ks=3000Nm/rad, Jm=JL=1.0×10-3kg·m2.

In fact, the damping constant Kw 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 Ks and Load Inertia JL

Changes of Ks and JL cause the deviation of the resonant frequency and the anti-resonant frequency, and have a bigger influence on systems. The parameter sensitivity analysis of Ks and JL should be divided into two parts: 1) the changes of Ks and JL that reduce the stability margin, while the system remains stable; 2) the changes of Ks and JL that make the system unstable.

In the first case, on the motor side, the double bi-quad filter is equivalent to adding a filter 1/GBQf(s) on the motor speed output side. Fig. 13 shows a Bode plot of 1/GBQf(s) and two schemes from ωref to ωm. It can be seen that 1/GBQf(s) can reduce gains near the resonance frequency, and smooth the motor speed.

Fig. 13.Bode plot of 1/GBQf(s) and Bode plot of two schemes from ωref to ωm.

Consequently, the motor steady state speed of the proposed double bi-quad filter is more stable than the single bi-quad filter, when changes of Ks and JL only cause a reduction in the stability margin, as shown in Fig. 14 and Fig. 15.

Fig. 14.When system is stable, Ks is changed 20%. (a) Speed step response on motor side. (b) Speed step response on load side.

Fig. 15.When system is stable, JL 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/GBQf(s) does not work anymore. Therefore, parameter modification and reconstruction are necessary for the double bi-quad filter. Reconstruction schemes can be classified into two different branches as follows:

1) Slight Change of Ks: The Ks 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 Ks measurement is pretty high (if the error of fres is very small, and the off-line measurement of JL is constant). In this situation, Ks can be treated as a constant, and the system resonance frequency changes mainly due to changes of JL.

Under this kind of circumstance, when system resonance reoccurs, JL can be recalculated by online measurement of the resonance peak frequency fres. With the updated JL, the proposed double bi-quad filter is reconstructed, and can be used to replace former double bi-quad filter.

2) Slight Change of JL: The JL of some equipment does not change frequently, such as laser phototypesetters, laser engraving machines, etc. For these cases, JL can be treated as a constant to solve the resonance problem. A change of Ks should be considered as the main cause of fres modification. When system resonance reoccurs, Ks can be recalculated by online measurement of the resonance peak frequency fres.

C. Load Torque TL

When studying compliant coupling, TL is generally neglected. If the effects of TL are considered, Fig. 9 becomes more complicated, as shown in Fig. 16.

Fig. 16.double bi-quad filter system with TL.

Fig. 16 can be transformed as Fig.17.

Fig. 17.Transformed system from Fig. 16.

In Fig. 17:

Finally, the effect of TL can be transformed to the input side, which is shown in Fig. 18.

Fig. 18.Transform TL effect to input side.

In Fig. 18:

GB(s) is the transfer function from ωref' to ωm'.

When compared with Fig. 9, the effect of TL is equivalent to adding an extra speed reference at the input side, and it has no effects on the system closed-loop characteristics. Thus, the proposed double bi-quad need not be reconstructed regardless of whether TL 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 bi-quad 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 1.3KW, rated speed is 2500rpm, rated current is 5A, torque coefficient is 1.0N.m/A, and rotor inertia is 1.03×10-3kg·m2. A Yaskawa SGMGV 13ADC61 motor acts as the load. Its rated power is 1.3KW, rated speed is 1500rpm, rated current is 10.7A, torque coefficient is 0.89N.m/A, and rotor inertia is 1.99×10-3-3kg·m2. Metal plates can be added to simulate load inertia.

Fig. 19.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, 1V represents 30rpm and 1A, respectively. The delay times Tf and Tci are calculated as 1ms and 33us, respectively. Kn is set as 450Hz. According to Section IV, Part C, the magnitude of load torque has no effects on the double bi-quad filter. Thus, for ease of operation, all of the experiments are implemented under the no load condition.

A. High-frequency Resonance Experiment

The high-frequency resonance performance and system performance with the proposed double bi-quad filter is verified in this sub-section. A NBK’s XGT2-44C is used as the connected part and its specific parameters are shown in Table II. The mechanical damping constant Kw≈0.11Nm/(rad/s).

TABLE IIXGT2-44C’S PARAMETERS

Fig. 20(a) shows the whole process of resonance suppression when the load inertia does not increase. With a step reference of 150rpm, the system oscillates with a frequency of 248Hz, 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.

Fig. 20.(a) Whole resonance suppressing process. (b) A partial view of oscillation without suppression. (c) Automatically resonance suppression. (d) Speed increase after adding double bi-quad filter.

Fig. 20(d) shows that by adding the double bi-quad filter, the waveform’s speed increases. The waveform shows that the proposed double bi-quad 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 0.01kg·m2) is added on the load side, and fres is changed. The original double bi-quad filter suppressing effect cannot be achieved. Therefore, the filter needs to be reconstructed. The whole process is shown in Fig. 21(a).

Fig. 21.(a) Whole resonance suppressing process. (b) A partial view of oscillation without suppression. (c) Reconstruct a new double bi-quad filter. (d) Speed increase after adding new double bi-quad filter.

At T5, the original double bi-quad filter is added, but the system still oscillates. At T6, the original double bi-quad filter is removed. At T7, the system starts to reconstruct the double bi-quad filter.

Fig. 21(b) shows that the speed oscillation frequency is changed to 224Hz. Fig. 21(c) shows the details of reconstruction process. At T8 and T9, the first and second low-pass filter are added resulting in fres=196Hz. According to the principle that Ks does not change, the calculated new load inertia is 0.0137 kg·m2, 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 bi-quad filter can be achieved by online reconstruction to offset the system parameter changes.

B. Low-frequency Resonance Experiment

During the second experiment, a self-made low Ks coupling is used as the connector. Its aim is to verify the effect of the double bi-quad filter suppression for low-frequency 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 low-frequency resonance, in which the adaptive procedure starts at T11. Fig. 22(b) shows that the oscillation frequency is 212Hz. Fig. 22(c) shows the process of automatic resonance suppression, where two low pass filter are added at T12 and T13. This results in fres≈48Hz and Ks≈60.4Nm.rad. Set ωn=400 and ξ=0.707, and tune A=0.0002543 and B=0.18094. Finally, the double bi-quad filter is constructed at T14, and the resonance suppression is complete. Fig. 22(d) shows the details of the system speed up.

Fig. 22.(a) Whole resonance suppressing process for low-frequency. (b) A partial view of oscillation without suppression. (c) Automatically resonance suppression. (d) Speed increase after adding double bi-quad filter.

Fig. 22 shows the experiment reaching its intended goal since the proposed double bi-quad filter for low-frequency resonance has a good suppression of resonance oscillations. Since Ks is small, the effect of JL changing is not obvious. As a result, there are no further reconstruction experiments.

 

VI. CONCLUSION

When compared with a single bi-quad filter, the proposed double bi-quad filter method is actually equivalent to add a virtual filter after the motor speed output. It contains two bi-quad filters on both the forward and feedback paths of the speed control loop. The two filters collectively complete the wide-band 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.