A Review on Self-Recovery Regulation (SR) Technique for Unbalance Vibration of High-End Equipment

On the basis of studying the principle of FSR, the SR of vibration and noise is proposed referring to the methods of “molecular targeted therapy” in contemporary medical science, which endows the machine with the ability to maintain a healthy state spontaneously. It is a new direction of bionic self-recovery principle and FSR system of equipment [17].

The typical FSR system takes the complex system of high-end equipment as the research object, and aims at “fast, accurate, precise and stable”. As shown in Figure 1, the system should be composed of three parts, including Data collector, Controller and Actuator. (1) Data collector is used to achieve the function of information acquisition, including monitoring of sensitive parameters, data communication and signal pre-processing. (2) Controller is used to realize the functions of information cognition and information decision-making, including early warning of faults, real-time diagnosis, fast accurate traceability, parameter adaptation, and structural adaptation. (3) Actuator is used to realize the function of information execution, including intelligent interlocking, fine and stable regulation, etc. During the operation of the equipment, all parts of the SR system work together to quickly capture fault information, accurately locate the fault location and form control instructions, and generate a self-recovery force by the robust action of the actuator. There are no conditions to form faults or the faults are eliminated in the bud by the regulation system. Therefore, the equipment can run without fault or can be restored from an abnormal state to a normal state without stopping, which ensures the long-term healthy operation of the equipment.

The vibration differential equation of the rotor system may be shown as: where M, C, G and K are the matrix of the rotor, representing mass, damping, gyroscopic moment and stiffness, respectively. When the unbalance fault force on the rotor system is F₀, the differential equation can be solved, and the vibration amplitude caused by the unbalance fault is obtained as shown:

In order to control the unbalance vibration, a balancing actuator needs to be pre-installed on the rotor and rotates synchronously with the rotor during a run. Assuming that the mass of the actuator is m and the unbalance self-recovery force generated by the actuator is F₁, Eq. (1) may be revised as:

Therefore, when the controller of the SR system changes the internal mass distribution of the balancing actuator, the self-recovery force F1 generated by the balancing actuator gradually counteracts the unbalance failure force F0 of the system. When the two forces are approximately equal, the vibration of the rotor system can be decreased to the allowable range.

The principle block diagram of the SR system for unbalance vibration is shown in Figure 2. Taking the unbalance vibration of high-end equipment as the control target, the system can be divided into four layers from the view of information exchange. (1) Condition monitoring layer is used to analyze the data requirements of the SR system, select the appropriate sensors to measure the required physical quantities at the appropriate locations, and output the data according to the defined signal format. (2) Fault diagnosis layer is used to extract features of multi-dimensional signals, especially unbalance vibration components, through signal processing technique such as filtering, spectral analysis and principal component analysis, and online locate the unbalance vector when the unbalance vibration beyond the preset threshold. (3) Taking non-overshoot as the goal, target control layer is used to optimize the control path and output control commands, based on the running state, load conditions and historical data of the equipment. (4) According to the dynamic characteristics of the equipment, self-recovery execution layer is used to complete the design and layout of the balancing actuator, accurately execute the control commands of the controller, and online suppress the unbalance vibration of the equipment. Finally, the SR system for unbalance vibration of high-end equipment with four characteristics is formed, that is, state self-perception, real-time self-analysis, intelligent self-decision and precise self-execution.

For aero-engines, excessive unbalance vibration will significantly increase engine parts damage and fuel consumption, and reduce engine operation efficiency and service life. At present, there are three commonly used solutions [18]. (1) The engine is disassembled back to the factory, and the parts are reassembled after the dynamic balance is completed on the dynamic balancing machine [19‐21]. (2) There is no need to disassemble the engine, the trial operation of the whole engine is accomplished on the engine test bench and the vibration parameters are measured, and the dynamic balance is carried out on the bench [18]. (3) The vibration parameters of the engine are monitored during the flight, and the unbalance vector calculation and balancing operation are carried out after the aircraft has landed, then the balancing process can be completed on the aircraft [22, 23]. Among them, both solution 1 and 2 require a long maintenance cycle, which is time-consuming and laborious, and seriously affect the service time of the engine. Although solution 3 has simplified the dynamic balance operation, there is still a difference between the balance condition and the actual operation condition of the aero-engine, hence the balance effect is limited.

Therefore, the in-flight propeller balancing system (IPBS) is developed to solve the above problems by a company of USA. The system can surveil the vibration parameters of the turboprop engine in real time while the aircraft is flying, and restrain the unbalance vibration of the propeller in real time under a variety of operating conditions, such as aircraft start-up, normal operation or parking. As shown in Figure 3, IPBS is mainly made up of three parts, that is, the sensor, the controller and the balancing actuator. The balancing actuator is pre-installed on the engine rotating shaft, and the vibration of the rotating shaft is monitored in real-time by the accelerometer. When the measured vibration value of the rotor exceeds the prescribed safety range, the system calculates the corresponding unbalance through the monitoring data of the sensor, then the controller transmits instructions to adjust the balancing actuator to automatically balance the rotor. In addition, the system can diagnose the engine through monitoring data, and the diagnostic information can be used to forecast the health condition of the engine system and optimize maintenance [11, 24].

The system has been successfully applied to four turboprop engines of a large transport aircraft. As shown in Figure 4, each of the turboprop engines is equipped with a balancing actuator, the four balancing actuators share a controller, and the power supply of the balance system is provided by the cockpit. The installation of the vibration sensor and the actuator, and the wiring of the flight deck and cockpit, are shown in Figure 5, the performance parameters of the balancing actuator are shown in Table 1, and the vibration suppression effect of the system under several flight conditions is shown in Figure 6.

According to the actual test results, the system can restrain the unbalance vibration of turboprop engines in real time under all flight conditions, so that the vibration of the four engines can always be kept below 0.05 inch/s, and the maximum vibration reduction can reach more than 90%. At the same time, during the testing process of 250 flight-hour, the balancing system performed a total of 9000 dynamic balancing operations, that is, in order to ensure that the engine vibration is always below the safety setting value, the engine needs to be dynamically balanced once per 1.6 min. The data further validates the necessity of installing the balancing system for aero-engine.

As the last process of machining, grinding have an important impact on the overall level of the machining industry. Due to the inevitable problem of uneven adsorption of grinding fluid or uneven wear of grinding wheel during the grinding process, even if the grinding wheel has been pre-balanced before installation, the balance state of grinding wheel will still be destroyed after a period of use, resulting in excessive vibration of the machine. If the parts continue to be processed, the surface quality of the subsequent parts cannot be guaranteed. While in the conventional dynamic balance process of grinding wheel, the grinding wheel often has to be disassembled and installed repeatedly, which is time-consuming and laborious, and it is not easy to ensure the machining consistency of batches of parts.

With the intention to resolve the above problems, domestic and foreign scholars have carried out continuous research on on-line automatic balancing technique of grinding wheels. Several world-leading manufacturing countries have formed the related products, such as USA, Germany and Italy.

As shown in Figure 7, the typical automatic balancing system for grinding wheel is mainly composed of vibration sensors, controllers and balancing actuators. Among them, the vibration sensor is a piezoelectric sensor, which is used to monitor the vibration of the grinding wheel. The controller diagnoses the vibration data monitored by the sensor, and displays the diagnosis results on the electronic screen, then sends control instructions to control the actuator. The actuator is installed on the side of the grinding wheel through the flange and rotates synchronously with the grinding wheel, and after receiving the balance instruction, the compensation quality is formed by changing its own mass distribution [12]. The performance parameters of commonly used balancing actuators are shown in Table 2. In actual applications, the actuator can be selected according to the dynamic balance requirements of the grinding wheel.

When the balancing system is installed on the grinding machine, the vibration level of the grinding wheel can always be controlled to less than 0.4 μm. Compared with 2.0 μm of the traditional dynamic balance method of grinding wheel, the control effect is improved by more than 5 times. As shown in Figure 8, the mechanical wear of the grinder and the start and stop times of machining can be significantly reduced, and the linear velocity of the grinding wheel can be further increased, resulting in higher machining efficiency and product quality.

The feature extraction based on the method is mainly statistical analysis. The time domain features of vibration signals based on statistical analysis can be divided into dimensional and dimensionless characteristic parameters. The dimensional characteristic parameters mainly include mean value, peak value, and root mean square of the vibration signal, while the dimensionless characteristic parameters are obtained by the combination of dimensional parameters, including kurtosis factor, kurtosis index, pulse index, and so on.

Deng et al. [30] proposed a complete fault characteristic extraction approach. One basis for this approach is local mean decomposition (LMD) which can be used to process the original vibration to generate the product functions (PFS), and the other is Teager energy kurtosis (TEK) which can be used to calculate at PFS. The corresponding TEK of each Teager energy data can be used to identify the failure by this method. The effectiveness of the method for rotor-bearing system was verified by experiments.

Imbalance and shaft bending are the two most common types of fault in rotary machinery, and the two types of faults display similar characteristics. In order to extract and identify the features of these two faults, Sanches et al. [31] put forward a time domain recognition method based on correlation analysis and model reduction. Since the minimum entropy deconvolution (MED) can not identify multiple faults of rotating machinery, He et al. [32] proposed a multi-fault detection technique which based on spectral kurtosis (SK) and MED. The layout of the vacuum pump test-bed is shown in Figure 9, and the flowchart of this technique is shown in Figure 10. In this technique, the signal is decomposed via SK, and each decomposed signal is refined by MED, so as to identify a variety of faults.

However, if the fault component contained in the signal is complex, it is difficult to make an accurate judgment of the fault only through the time domain eigenvalues, so the frequency domain analysis is accomplished for further analysis.

This method refers to the processing of the time domain signal through Fourier transform, and the signal is reflected in the form of amplitude-frequency. Through the observation of the spectrum, the diagnosis of the data can be realized. The main feature extraction methods based on frequency domain are cepstrum analysis, power spectrum analysis and holo-spectrum analysis.

Cepstrum is an effective tool to detect periodic components in complex maps, and it has been broadly utilized for fault monitoring and diagnosis. For high-speed and large rotating machinery, its rotation condition is complex, especially when there are faults consisting of misalignment, gear defects, oil whirl, and so on, the vibration will become more complex. If there is a complex periodic structure on the spectrum, the recognition ability can be enhanced by analyzing it with cepstrum.

With a view to improving the production efficiency, product quality and reduce the processing costs of the workpiece, spindles and machine tools must work in the best environment with as little error as possible. The detection and diagnosis of spindle unbalance (UB), misaligned (MA), bent shaft (BS) or rolling bearing has always been considered as a difficult problem in rotary machining monitoring. David Ibarra-Zarate et al. [33] proposed a method based on cepstrum pre-whitening analysis. Figure 11 shows the signal extraction and selection process of this method.

Rotor unbalance often leads to rubbing in aero-engine, and it is very difficult to diagnose the friction position. Chen et al. [34, 35] used cepstrum to analyze the acceleration signal of the casing and successfully identified the blade-casing rubbing fault. The aero-engine rotor test-bed is shown in Figure 12. Due to the non-stationarity of dynamics and vibration of variable speed machinery, its condition is difficult to be monitored and diagnosed. Li et al. [36] combined order cepstrum with radial basis function (RBF) artificial neural network and applied it to gear fault detection. This method combine computational order tracking, cepstrum analysis and artificial neural network, which is effective for gear fault diagnosis.

The power spectrum (PSD) is the square of the Fourier transform (FT) amplitude of the signal, indicating the distribution of vibration power. Liang et al. [37] used the power spectrum to analyze faults of induction motors. When the faults show rich sideband and harmonic characteristics, this method has better identification ability. Jacek Urbanek et al. [38] put forward a method to extract the vibration signal components of rolling bearings which is named averaged instantaneous power spectrum (IPS). The flow chart for calculation of the averaged IPS is shown in Figure 13. Isa Yesilyurt et al. [39] proposed a method of smoothed instantaneous power spectrum, which is applied to the monitoring and analysis of gear faults.

Feature extraction based on holo-spectrum analysis is more and more widely used in rotor fault diagnosis. The traditional Fourier spectrum (FFT) has limitations in showing the whirling properties of the rotor fault (forward/reverse whirl). Patel et al. [40] used holo-spectrum to extract the whirling properties of the rotor to judge the fault of the rotor.

Although the frequency domain analysis can truly reflect the nature of the vibration signal, it can only analyze the stationary signal and can not reflect the change of the frequency component of the signal with time.

In view of the advantages and disadvantages of the first two methods, researchers have proposed joint time-frequency methods, which can characterize the change of non-stationary signal frequency with time. The methods based on joint time-frequency analysis include empirical mode decomposition (EMD), short-time Fourier transform (STFT), Hilbert-Huang transform (HHT) and wavelet transform (WT).

Based on EMD method, the signal can be decomposed into limited different intrinsic mode functions (IMFs), and the feature of the raw signal in different time domain is included in every IMF component [41]. Nguyen et al. [42] used EMD to process the original vibration signal of the bearing, so as to extract feature data, and apply it to deep learning network to diagnose the related faults. Wang et al. [43] put forward a fault diagnosis method for the rotating machinery. In this method, the fault signal is processed by ensemble EMD (EEMD), and the fault is identified by self-organizing map (SOM) neural network, as shown in Figure 14. Based on the combination of EMD-ICA and Hilbert envelope demodulation, Xie et al. [44] proposed a fault information extraction method to extract fault signal of turboprop engine reducer.

Since the Fourier transform (FT) is aimed at the stationary signal, the STFT has become an important method to study the non-stationary signal. Based on STFT, He et al. [45] proposed a time-frequency feature extraction method to extract the vibration signal of helicopter rotor, and the fault detection scheme based on this method is shown in Figure 15. Jin et al. [46] analyzed the vibration signal of internal combustion engine. N. Harish Chandra et al. [47] used STFT to process the run-up data of rotor bearings, so as to diagnose the fault of shaft misalignment and rotor stator friction.

The basic process of processing non-stationary signals with HHT is to process the raw signal to get IMFs by using the EMD method, and then perform Hilbert transform on every IMF to obtain the corresponding Hilbert spectrum. Li et al. [48] identified rotor cracks based on HHT.

Wavelet transform (WT) inherits and develops the concept of localization of STFT, and overcomes the disadvantage that the window length of STFT does no longer trade with frequency. The filtering characteristics of wavelet analysis in time-frequency domain are usually utilized to represent different frequency ranges of signals at different levels. Li et al. [49] proposed an adaptive harmonic window function based on wavelet transform, and the periodic signal and rotor axis trajectory can be extracted from strong noise. Fu et al. [50] utilized the improved harmonic wavelet transform to extract and analyze the vibration signals of the axis orbit, and obtained a high-resolution time-frequency picture. Muralidharan et al. [51] used continuous WT to extract fault features of centrifugal pumps, and input the feature signal into the classifier for classification. F. Al-Badour et al. [52] used the combination of continuous wavelet and wavelet packet transform to effectively extract feature signals and locate faults. Based on the integration of improved wavelet packet transform (IWPT), distance assessment technique and support vector machines (SVMs), Hu et al. [53] proposed a new fault diagnosis technique which can be utilized to the rotating machinery to extract significant frequency band features from the original vibration signal. Shan et al. [54] put forward a method to process the vibration signal of high-speed motorized spindle in actual running state. This method can be used to separate and extract different signal features of the motorized spindle from the strong noise by wavelet transform, and then analyze the frequency spectrum of the motorized spindle.

With the progress of computer technology, fault diagnosis technology is constantly developing. The feature extraction based on a single method is no longer adapt to complex and changeable working conditions. Therefore, the combination of multiple feature extraction methods is the development direction of fault diagnosis.

The artificial intelligence technique applied to feature extraction and fault diagnosis mainly include neural network, deep learning and other intelligent algorithms. Lobato et al. [55] proposed a diagnosis method which integrates stationary and non-stationary signal processing techniques, multi-attribute selection and classification by machine learning. According to the vibration data in the actual flight test, Hu et al. [56] of Boeing Company in the United States used artificial neural network (ANN) to detect the unbalanced state of aircraft engine. Zhu et al. [57] put forward a fault diagnosis method of convolution neural network based on symmetrical lattice images, which converts multiple vibration signals into symmetrized dot pattern (SDP) images. Then the convolutional neural network (CNN) is used to identify the SDP figure features in different vibration states. Yan et al. [58] of Harbin Institute of Technology came up with a deep belief network ((DBN)) based on multi-source heterogeneous information fusion for special diagnosis and diagnosis of rotor unbalanced faults. The experimental results show that under the appropriate DBN parameters, the accuracy of feature extraction and fault classification of rotor unbalanced faults reaches 100%. Figure 16 shows the fault diagnosis process of this method. Shao et al. [59] proposed a deep learning method based on deep belief network (DBN), which is utilized to learn features from the frequency distribution of vibration signals and to identify fault states. Figure 17 shows the DBN-based troubleshooting steps. Chen et al. [60] established a vibration fault diagnosis model based on the combination of cuckoo search algorithm with memory (CSM) and BP neural network. The faults such as rotor unbalance and rotor misalignment diagnosed by this method are consistent with the actual fault types. Li et al [61] of the University of Alberta in Canada came up with an adaptive tracking technique based on sinusoidal synthesis, which uses the sinusoidal characteristics of vibration signals to transform a nonlinear problem into a time-domain linear adaptive problem based on state space. Lu et al. [62] put forward a genetic algorithm based on dynamic search strategy, which showed good feature extraction and selection ability in two kinds of fault experiments of rotor unbalanced vibration and bearing damage. Deng et al. [63] presented an improved classification and regression tree (CART) algorithm, namely D-CART algorithm, which can quickly locate the rotor system fault and identify the fault type on the premise of ensuring the accuracy.

In summary, with the development of computer technology, feature extraction and fault diagnosis technology has developed rapidly from traditional signal processing technology to intelligent direction, which lays a foundation for the application of SR Technique. However, the feature extraction technique based on neural network, deep learning and other intelligent algorithms is still in the stage of development, and a series of bottlenecks need to be solved, such as the training difficulty and fitting ability of the model, the real-time performance of signal analysis and extraction, and so on. These are also the urgent problems that need to be solved in the future.

The core of this method is to acquire the influence coefficient matrix, which is used to express the linear relationship between the vibration response and imbalance on several specific planes of the rotor. While the influence coefficient matrix is known, the initial imbalance of the rotor can be located by dividing the vibration response matrix and the influence coefficient matrix [69, 70].

Goodman et al. [71] made a comprehensive discussion on the influence coefficient method in 1964 and put forward its general expression for the first time. In addition, they introduced the least square method into the influence coefficient method to calculate the final correction mass of multi-speed and multi-plane balance of rotating machinery. Kang et al. [70] obtained better balance accuracy by reducing the number of equilibrium conditions. Lei et al. [72] put forward an improved influence coefficient method for anisotropic rotor bearing systems, which uses the amplitude and phase of the forward precession instead of the traditional unbalanced response, and can obtain the similar result with the conventional method. Considering the randomly distributed residual unbalance and the residual unbalance of the balance disk, Ranjan et al. [73] proposed a general influence coefficient method which is suitable for the active magnetic bearing rotor system. It can ensure the safe operation of unbalance rotors at high speed by use of active magnetic bearings. Fujisawa et al. [74] studied a multi-plane multi-speed balance method, which uses the least square method of influence coefficient and takes into account the vibration characteristics of multi-span rotors to calculate the correction amount.

For the complex and changeable high-speed rotor system, the fast and accurate calculation of imbalance is regarded as an important factor affecting the balance effect. Kang et al. [75] researched the dynamic balance of uneven rotor-bearing system. On the basis of the motion equation of asymmetric rotor, the influence coefficient matrix expression of asymmetric rotor was derived by way of the use of complex coordinate illustration and finite element method. Moon et al. [76] proposed to use the reference influence coefficient method of the reference model for active balancing, and tested the method on a machine tool. The balance experimental device is shown in Figure 18. By using this method, the vibration is reduced to 54% near the critical speed, the reduction effect in other parts is more than 80%, and the average reduction effect is 70% of the initial unbalance response. According to the actual conditions of dynamic balancing in the field, Untaroiu et al. [77] put forward a general optimization, that is, to minimize the maximum residual vibration of dynamic balancing with influence coefficient. The technique can be easily used in all rotor balancing methods that using correction amounts, and even to active balancing. When the two-plane influence coefficient method is used in dynamic balancing, it does not need an excessive amount of statistics of the rotor system, and it can gather and process data automatically. Xu et al. [78] used the two-plane influence coefficient method to balance the rigid rotor, and the vibration reduction rate of the rigid rotor can reach more than 85% after one dynamic balance correction.

For the turbomachinery with complex deformation, multiple constraints and less measurement information, Bin et al. [69] put forward a high-speed balancing method for N+1 supported turbomachinery without trial weight, which can improve the balancing efficiency and accuracy of the rotor. Based on the multi-plane influence coefficient, this method can determine the relationship between the synchronous vibration and the unbalance excitation. It is based on the prediction of the steady-state response of the finite element method to calculate the multi-plane influence coefficients at several rotational speeds. The experimental device and imbalance location results are shown in Figure 19. Through the experimental verification, the bearing vibration is reduced by an average of 25% and the maximum is reduced by 50% at 2700 r/min.

The method is to utilize the modal characteristics of the unbalance vibration of the rotor, decompose the unbalance according to each order of modals, correct the imbalance respectively, and finally restrain the unbalance vibration of the rotor. Compared with the influence coefficient method, this method is more sensitive to higher vibration modals.

Han et al. [79] studied a new type of generalized modal balance of isotropic rotor systems. In this method, the unbalance modal response is obtained by analyzing the modes of non-isotropic rotor systems. It was simulated to verify its theoretical results and effectiveness. Because the low pressure rotor of large steam turbine is difficult to be balanced by the traditional influence coefficient method, based on modal analysis, Bin et al. [80] put forward a high speed balancing method of large flexible rotor, so as to realize the dynamic balance of this kind of large special flexible rotor. Quartarone et al. [81] improved the modal balance method and put forward a method to identify the balance mass group of high-speed rotors, which allows one critical speed to be balanced selectively at a time and minimizes the impact on the unbalanced response in the rest of the speed range. This method is used to balance the turbine molecular pump and its vibration is significantly reduced.

In view of the disadvantage that the traditional spectrum analysis only pays attention to the amplitude-frequency relationship while neglecting the phase information, the holographic spectrum theory is put forward to overcome the shortcomings of the former two methods by means of dual-sensor measurement [82].

Liu et al. [83] proposed Low-Speed Holo-Balancing method to balance a flexible rotor without test runs at high speed. The variation of the first two modal components of the rotor during the start and stop phase is analyzed by using the decomposition of the holographic spectrum, and the first two low-mode components can be offset together when the speed is lower than the first critical speed. Based on FFT spectrum and information fusion, the holographic spectrum technique was used to the dynamic balancing of flexible rotor systems, and holographic balancing method was proposed [84]. Combined with computer simulation and genetic algorithm, this method simplifies the dynamic balancing process and improves the speed of unbalance and phase recognition, and its effectiveness was validated by balancing of 300 MW turbo-generator units shown in Figure 20.

In summary, the imbalance location method is mainly based on the above three methods, and other methods are improved and optimized on the basis of them. Whether the position of the unbalance and the size of the unbalance can be accurately discriminated is the key to the implementation of the follow-up adjustment method to restrain the vibration. Therefore, the rapid positioning of unbalanced vibration and high-precision calculation of unbalance are important research content in the future.

The dynamic performance of pole assignment system is principally determined by the distribution of poles on the foundation plane. Once the state area expression or transfer operate of the system is understood, the pole of the control system is affected to the selected position by adding the state feedback controller [85]. For asymmetric systems, Ariyatanapol et al. [86] projected the pole-delay methodology of the time-delay half exploitation single-input state feedback management, which may accurately piece the specified closed-loop points while not dynamic different poles. Based on H∞ controller and feedforward compensation controller. Real-time online force automatic compensation was implemented by Gao et al. [87]. Combined with the pole placement theory, Liu et al. [88] proposed an improved hybrid H2/H∞ control technique and achieved the synchronous management of attitude stabilization and vibration suppression for versatile satellite. Fan et al. [89] projected an adaptive observer-based state-feedback management technique for stabilization of the integrated fault designation and mechanism fault-tolerant management systems subject to regulate input constraints. As shown in Figure 21, the initial feasible solution of nonlinear optimization is obtained by pole placement method to improve system performance.

Optimal management theory was shaped and developed below the impetus of space technology. For a controllable dynamic system or motion method, the optimum control theme is pointed out from all control schemes, so once the motion of the system is transferred from an initial state to a specific target state, its performance index price is perfect. The rotor imbalance will cause synchronous vibration of magnetic levitation centrifugal compressor system. As shown in Figure 22, a synchronous vibration management methodology associate in nursing optimum notch filter was planned by Peng et al. [90], which stabilized the synchronous vibration force in the full speed range. Based on the classical optimal control theory, Tombul et al. [91] simplified the nonlinear control problem to linear control, and obtained the nonlinear optimal control law through the sequence of linear time-varying systems. When the shaft is rotating at a constant angular velocity, this method enables fast and robust control of unbalanced vibrations.

Rosyid et al. [92] applied the optimal control method to the control of the rotor-bearing-support system with a large degree of freedom. Two pairs of different weighting matrices were used to control the unbalance vibration of low-speed and high-speed rotors respectively through a linear quadratic regulator. A model prophetic best management methodology for magnetically suspended regulator is bestowed by Zhang et al. [93]. The optimum controller is obtained by means that of iterating a Riccati distinction equation, and stability of the management theme is investigated through faux algebraical Riccati technique. Simulations and experiments show that the controller will effectively suppress the cone-shaped vortex K.E. caused by the gyro effect. As shown in Figure 23, Yang et al. [94] established an optimal control model for controlling the vibration of a rotor system of a tilting pad gas bearing using piezoelectric actuators, proportional differential compensation and gap sensors. The optimal control theory was used to obtain the optimal control rules between the actuator voltage and the dynamic displacement of the rotor, and the results proved that the optimal control system could effectively suppress the unbalanced response of the system.

For the optional control method, the determination and selection of the optimal control objective function is complicated. With the event of intelligent technology, intelligent algorithmic program are wide employed in the regulation system. Carvalho et al. [95] introduced the fuzzy algorithm to improve the traditional influence coefficient method, and the robustness of vibration control was improved through fuzzy processing of rotor vibration response. Xu et al. [96] proposed a fuzzy self-tuning single neuron PID control method, and the schematic diagram is shown in Figure 24. Compared with ancient inflammatory disease and single somatic cell inflammatory disease management ways, it has several characteristics, such as fast response time, small overshoot, few oscillations, strong robustness, and good stability. Oke [97] applied evolution algorithm to reduce the residual imbalance of the rotor system. The results show that the algorithm is superior in optimizing time and effectively reducing vibration values. Yang [98] considered the discrete distribution of the unbalance mass and phase, and projected a hybrid genetic algorithmic program with neighborhood search. The influence of the difference between theoretical calculation and the actual addition of the balance weight was avoided, and the balance accuracy and speed was improved. Yao et al. [99] improved the PID algorithm with fuzzy control, as shown in Figure 25, and the minimum mean square filter was connected to the fuzzy control to compensate the unbalanced quality. Simulations show that the improved fuzzy PID formula performs higher than ancient PID management in overshoot management, and also the transient time of fuzzy PID is considerably shortened.

The selection and design of automatic balancing control strategy for complex rotor system need to be determined according to actual working conditions and specific actuators. Simple algorithm, good working stability, high balance accuracy and high balance efficiency are important characteristics for evaluating regulation methods. In the future, the development trend of automatic balance control methods is to combine the reliability of traditional control methods with the high efficiency of modern intelligent control methods to deal with the self-recovery regulation control of unbalanced vibration under different working conditions.