Active Vibration Control of a Rotating Mechanical System with a Rigid

Coupling using Active Disturbance Rejection Control

Tingyu Lin 1 , Dunant Halim* 2 , Liaoyuan Ran 3 Department of Mechanical, Materials and Manufacturing Engineering The University of Nottingham Ningbo China Ningbo, Zhejiang, China 315100 Zhuang Xu 4 Department of Electrical and Electronic Engineering The University of Nottingham Ningbo China Ningbo, Zhejiang, China 315100 Chung Ket Thein 5 School of Aerospace The University of Nottingham Ningbo China Ningbo, Zhejiang, China 315100

ABSTRACT The work aims to develop an active control method for mitigating torsional vibration of a rotating mechanical system with a rigid coupling, driven by an electric motor, in order to achieve good tracking and disturbance rejection performances. It is commonly known that torsional oscillations in a rotating system, which is normally used for various power transmission systems, can generate a significant level of vibration. In this work, therefore, an active control system that was based on the Active Disturbance Rejection Control (ADRC) approach was utilized to develop the control system that was robust against uncertainties associated with the internal dynamics and external disturbances affecting the system. The Extended State Observer (ESO) was used to estimate the generalized disturbance consisting of not only the external disturbances but also the disturbances associated with internal dynamics of the rotating system. The effectiveness of the developed control system was demonstrated, which showed that in the case of the varying coupling and load inertias, the control system could still perform well.

1 tingyu.lin@nottingham.edu.cn

* 2 dunant.halim@nottingham.edu.cn (Corresponding author)

3 liaoyuan.ran@nottingham.edu

4 zhuang.xu@nottingham.edu.cn

5 chungket.thein@nottingham.edu.cn

21-24 auGUST Scortsi event Cas La inter.noise | ee 2022

1. INTRODUCTION

With the rapid development of automation industry over the past few decades, the demand for high-performance rotating machineries has grown significantly. Most rotating machines rely on the use of coupling to connect an electric motor with a mechanical power transmission system. For such machines, excessive torsional vibration can cause failures to different components in a machine so it needs to be suppressed accordingly, such as by using motor torque through an active control implementation. In [1], Qian proposed a motor torque control method based on the backstepping algorithm to control torsional vibration by selecting a suitable asymmetric tangent barrier Lyapunov function. An adaptive notch filter with an FFT analyzer and sorting algorithm was proposed in [2], and the low-order torsional vibration associated with varying natural frequencies can be suppressed. Abouzeid developed a torsional vibration suppression method using a Proportional-Resonant (PR) controller, which was found to be insensitive to mechanical parameter differences [3].

However, uncertainties affecting the internal dynamics of a rotating system and the external disturbance can create difficulties for a model-based control system in achieving satisfactory vibration suppression. Active Disturbance Rejection Control (ADRC) was first proposed by Han in 1998 [4], which was found to be less dependent to an accurate dynamic model of the plan to be controlled. ADRC utilizes the Extended State Observer (ESO) to allow the real-time estimation of an extended state, that is the sum of internal and external disturbances. The estimated extended state is then utilized to decouple the system from the actual disturbances affecting the plant [5]. Non-linear ADRC was linearized to Linear Active Disturbance Rejection Control (LADRC) in [6], which also allowed ADRC parameters to be determined based on the consideration of the bandwidths of the observer and the controller. An artificial intelligence-based method [7] has also been used to automatically tune ADRC parameters, while another parameter tuning method for ESO has been proposed in [8]. ADRC has shown its potential in a wide range of industrial applications [9], including for vibration suppression of torsional systems [10, 11]. In the present work, therefore, an active vibration control system of a rotating mechanical system with a rigid coupling using ADRC is developed in this work. The performance of the developed motor control system will be investigated, including considering its robustness against uncertainties associated with the internal dynamics and external disturbances affecting the rotor system. 2. TORSIONAL VIBRATION OF A ROTATING SYSTEM WITH A RIGID COUPLING

A rotating mechanical system with a rigid coupling driven by an electric motor can be represented by a simplified three-inertia rotor system, as shown in Figure 1.

Figure 1: A three-inertia rotor system.

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Motor inertia ( 𝐽 𝑀 ) is connected to coupling inertia ( 𝐽 𝐶 ) by a torsional spring ( 𝐾 1 ) and a viscous damper ( 𝐶 1 ). In addition, another side of coupling inertia ( 𝐽 𝐶 ) is connected to the load inertia ( 𝐽 𝐿 ) by another torsional spring ( 𝐾 2 ) and a viscous damper ( 𝐶 2 ). A torque ( 𝜏 ) is acted on the motor part to drive the system. The angular positions, angular velocities, and angular accelerations of the motor, the coupling, and the load, are denoted as 𝜃 𝑀 , 𝜔 𝑀 , 𝛼 𝑀 , 𝜃 𝐶 , 𝜔 𝐶 , 𝛼 𝐶 and 𝜃 𝐿 , 𝜔 𝐿 , 𝛼 𝐿 , respectively. The equations of motion that govern the torsional vibration of the system can be obtained as:

𝐽 𝑀 𝜃 ̈ 𝑀 + 𝐶 1 𝜃 ̇ 𝑀 −𝐶 1 𝜃 ̇ 𝐶 + 𝐾 1 𝜃 𝑀 −𝐾 1 𝜃 𝐶 = 𝜏 𝐽 𝐶 𝜃 ̈ 𝐶 −𝐶 1 𝜃 ̇ 𝑀 + (𝐶 1 + 𝐶 2 )𝜃 ̇ 𝐶 −𝐶 2 𝜃 ̇ 𝐿 −𝐾 1 𝜃 𝑀 + (𝐾 1 + 𝐾 2 )𝜃 𝐶 −𝐾 2 𝜃 𝐿 = 0

(1)

{

𝐽 𝐿 𝜃 ̈ 𝐿 −𝐶 2 𝜃 ̇ 𝐶 + 𝐶 2 𝜃 ̇ 𝐿 −𝐾 2 𝜃 𝐶 + 𝐾 2 𝜃 𝐿 = 0

where 𝜃 ̇ 𝑀 = 𝜔 𝑀 , 𝜃 ̈ 𝑀 = 𝛼 𝑀 , 𝜃 ̇ 𝐶 = 𝜔 𝐶 , 𝜃 ̈ 𝐶 = 𝛼 𝐶 , 𝜃 ̇ 𝐿 = 𝜔 𝐿 and 𝜃 ̈ 𝐿 = 𝛼 𝐿 . Hence, the transfer function from 𝜏 to 𝜃 𝑀 is described by:

𝑏 0 𝑠 4 +𝑏 1 𝑠 3 +𝑏 2 𝑠 2 +𝑏 3 𝑠+𝑏 4 𝑎 0 𝑠 6 +𝑎 1 𝑠 5 +𝑎 2 𝑠 4 +𝑎 3 𝑠 3 +𝑎 4 𝑠 2 (2)

𝜃 𝑀

𝜏 =

while the transfer function from 𝜏 to 𝜔 𝑀 is given as:

𝑏 0 s 4 +b 1 s 3 +b 2 s 2 +b 3 s+b 4 𝑎 0 s 5 +𝑎 1 𝑠 4 +𝑎 2 𝑠 3 +𝑎 3 𝑠 2 +𝑎 4 𝑠 (3)

𝜔 𝑀

𝜏 =

where 𝑏 0 , … , 𝑏 4 , 𝑎 0 , … , 𝑎 4 are described in terms of 𝐽 𝑀 , 𝐽 𝐶 , 𝐽 𝐿 , 𝐶 1 , 𝐶 2 , 𝐾 1 and 𝐾 2 .

This three-inertia rotor system contains a rigid body mode and two flexible body modes. The frequency response that relates the motor velocity to the motor torque for a typical rotor system with a rigid coupling is shown in Figure 2, where two torsional resonances can be observed.

Figure 2: Frequency response from the motor velocity to the motor torque of a typical rotor

system with a rigid coupling.

3. MOTOR CONTROL DESIGN USING ADRC

In the present work, ADRC is used to suppress torsional vibration by using motor velocity feedback to achieve good tracking and disturbance rejection performances. ADRC is based on the real-time estimation of the generalized disturbance using the External State Observer (ESO), allowing

the cancelation of its effect using an appropriate control law [9]. Considering an external disturbance 𝑤 is applied to the rotor system, Eq. (3) can be re-written in the form of a differential equation. By integrating it several times, a first-order system can be obtained as:

𝑦̇ 𝑀 = 𝑏 0

𝑢+ (− 𝑎 1

𝑦 𝑀 − 𝑎 2

∫𝑦 𝑀 − 𝑎 3

∬𝑦 𝑀 − 𝑎 4

∭𝑦 𝑀 + 𝑏 1

∫𝑢+ 𝑏 2

∬𝑢+ 𝑏 3

∭𝑢

𝑎 0

𝑎 0

𝑎 0

𝑎 0

𝑎 0

𝑎 0

𝑎 0

𝑎 0

+ 𝑏 4

∬∬𝑢+ 1

∬∬𝑤)

𝑎 0

𝑎 0

𝑏 0 𝑎 0 𝑢+ 𝑓(𝑦 𝑀 , ∫𝑦 𝑀 , ∬𝑦 𝑀 , ∭𝑦 𝑀 , ∫𝑢, ∬𝑢, ∭𝑢, ∬∬𝑢, ∬∬𝑤) (4)

=

where 𝑦 𝑀 is the motor velocity; 𝑢 is the torque acting on the motor; and 𝑓(·) is the generalized disturbance that includes uncertainties associated with the internal dynamics and the external disturbance. Here, the output 𝑦 𝑀 and the generalized disturbance 𝑓 are respectively described as the first state 𝑧 1 and the second state 𝑧 2 , leading to the state space system that can be described by:

{ 𝑧̇ = 𝐴 𝑐 𝑧+ 𝑏 𝑎 𝐵 𝑐 𝑢+ 𝐸 𝑐 𝑓̇

𝑦 𝑀 = 𝐶 𝑐 𝑧+ 𝐷 𝑐 𝑢 (5)

𝑏 0 𝑎 0 , 𝐴 𝑐 = [0 1 0 0 ] , 𝐵 𝑐 = [1

0 ] , 𝐸 𝑐 = [0

where 𝑏 𝑎 =

1 ] , 𝐶 𝑐 = [1 0], and 𝐷 𝑐 = [0].

The second order ESO for estimating the generalized disturbance can then be expressed as:

{ 𝑧̂̇ = 𝐴 𝑐 𝑧̂ + 𝑏̂ 𝑎 𝐵 𝑐 𝑢+ 𝐿 𝑐 𝑒

𝑧̂ 1 = 𝐶 𝑐 𝑧̂ + 𝐷 𝑐 𝑢 (6)

where 𝑧̂ is the estimate of 𝑧 , while 𝑏 ̂ 𝑎 is the estimated value of 𝑏 𝑎 . The observer error is defined by 𝑒= 𝑧 1 −𝑧̂ 1 , while the observer gain vector is 𝐿 𝑐 = [𝛽 1 𝛽 2 ] 𝑇 . According to the ESO parameter tuning method in [6], the observer gains can be determined as 𝛽 1 = 2𝜔 𝑜 , 𝛽 2 = 𝜔 𝑜

2 , where the observer states 𝑧̂ 1 and 𝑧̂ 2 are expected to accurately track the respective system’s states, 𝑦 𝑀 and 𝑓 .

The control law that takes into account the estimated generalized disturbance can be defined as:

𝑢 0 −𝑧 ̂ 2

𝑢(𝑡) =

𝑏 ̂ 𝑎 (7)

𝑢 0 = 𝑘 𝑝 (𝑟(𝑡) −𝑦 𝑀 ) (8)

where 𝑟(𝑡) is the setting value and 𝑘 𝑝 is the controller gain. Eq. (7) can be substituted into Eq. (4) to give:

𝑦̇ 𝑀 ≈𝑢 0 (9) where the system becomes a pure integrator that can be controlled by using a proportional controller in Eq. (8). 4. ANALYSIS OF CONTROL PERFORMANCE

In this section, simulations are carried out to verify the effectiveness of the developed control system.

4.1 Selection of controller and observer bandwidths

The three-inertia rotor system is tested in simulation using some system parameters used in [10], with nominal values of the torsional system being 𝐾 1 = 372 N ∙m/rad, 𝐾 2 = 1.3 N ∙m/rad, 𝐶 1 = 0.002 N ∙m ∙s/rad, 𝐶 2 = 0.0015 N ∙m ∙s/rad, 𝐽 𝑀 = 0.0037 kg ∙m 2 , 𝐽 𝐶 = 0.0011 kg ∙m 2 , 𝐽 𝐿 = 0.0025 kg ∙m 2 . The torsional natural frequencies of the system are 28.1 rad/s and 663.0 rad/s.

From Section 3, the observer gains are determined by 𝜔 𝑜 in the second-order ADRC, while the proportional controller 𝑘 𝑝 can be determined by the controller bandwidth 𝜔 𝑐 . The bandwidth selection for the observer and controller can be considered as follows [11]:

 𝜔 𝑜 < (5~10) × the sampling rate  𝜔 𝑜 > (2~5) × 𝜔 𝑐  𝜔 𝑜 > the required bandwidth

4.2 Estimation performance of the observer

As shown in Figure 3, a ramp function that rises from 0.5 s to 0.6 s is selected as the reference motor velocity input, and a step torque of 1 Nm is applied to the motor as an external disturbance at 1 s. The controller bandwidth is set as 1600 Hz, while the bandwidth of the observer is set as 6400 Hz. The observer’s estimation performance is shown in the figure. It can be observed that the estimation errors for states 1 and 2 immediately increase to 0.0098 rad/s and 1 Nm respectively, at 1 s when the external disturbance is applied to the system. The estimation errors then decrease rapidly and finally converge to zero, accurately tracking the system’s states. This shows an excellent estimation performance by using the extended state observer to estimate the system’s states, particularly the unknown generalized disturbance affecting the rotor system.

‘Vetoolty (rect) State tym),

(a) (b) Figure 3: State estimation performance, showing the comparison of the system’s states with the estimated states from the observer: (a) State 1: motor velocity; (b) State 2: generalized disturbance.

4.3 Control performance using different controller and observer bandwidths

In this section, the control systems with different controller and observer bandwidths, 𝜔 𝑐 and 𝜔 𝑜 , are compared. The control systems are described as three simulation cases denoted as, ADRC 1: 𝜔 𝑐 =

hs Laand State 210) ene

Tor a

400 𝐻𝑧, 𝜔 𝑜 = 1600 𝐻𝑧 ; ADRC 2: 𝜔 𝑐 = 800 𝐻𝑧, 𝜔 𝑜 = 3200 𝐻𝑧 ; and ADRC 3: 𝜔 𝑐 = 1600 𝐻𝑧, 𝜔 𝑜 = 6400 𝐻𝑧 , respectively. The control performance comparison is described in Figure 4 with the tracking and disturbance rejection performances under different load and coupling inertias.

(a)

(b)

(c) Figure 4: Motor responses with different control and observer bandwidths: (a) Tracking performance; (b) Disturbance rejection performance; (c) Estimated generalized disturbance. As observed from the figure, the tracking and disturbance rejection performances can generally be improved as the controller and observer bandwidths are extended. Figure 4(c) also indicates that ADRC with higher bandwidths can provide a more accurate estimation of the generalized disturbance. Based on the results, the controller and observer bandwidths of ADRC 3 are selected to further analyze the control performance.

4.4 Control performance for the rotor system with different coupling and load inertias

To evaluate the robustness of the developed control system, the control performance comparison is undertaken for a rotor system with different sets of coupling and load inertias. Two simulation cases are designed, which are denoted as: ADRC 4: 𝐽 𝐶 = 0.0011 kg ∙m 2 , 𝐽 𝐿 = 0.005 kg ∙m 2 ; and ADRC 5: 𝐽 𝐶 = 0.0022 kg ∙m 2 , 𝐽 𝐿 = 0.0025 kg ∙m 2 . The comparison of control performance for ADRC 3, 4, and 5 are shown in Figure 5. By comparing the control system with different coupling and load inertias, it is observed that the changes in the tracking and disturbance rejection performances are relatively small. It is thus observed that the control system performs well against variations in the coupling and load inertias, demonstrating satisfactory control robustness.

(a)

(b)

(c) Figure 5: Motor response for a rotor system with different coupling and load inertias: (a) Tracking

performance; (b) Disturbance rejection performance; (c) Estimated generalized disturbance. 5. CONCLUSIONS

In this paper, an active control method based on ADRC has been proposed for torsional vibration mitigation of a rotating system with a rigid coupling, in order to achieve good tracking and disturbance rejection performances. An ESO was designed to accurately estimate the generalized disturbance, which can be contributed by uncertainties in internal dynamics due to varying coupling and load inertias, and the external disturbance. Excellent robustness of the developed motor control system that was based on ADRC, has been demonstrated through simulation studies. It was shown that the control system was able to achieve effective tracking control and vibration suppression for a rotor system with varying coupling and load inertias. 6. ACKNOWLEDGEMENTS

This work was supported by Ningbo Science and Technology Bureau under Natural Science Programme (Project code 202003N4183) China.

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