H∞ active control of frequency-varying disturbances in a main engine on the floating raft vibration isolation system:

Chunsheng Song; Yao Xiao; Chuanchao Yu; Wei Xu; Jinguang Zhang

doi:10.1177/1461348417725944

H∞ active control of frequency-varying disturbances in a main engine on the floating raft vibration isolation system:

Song, Chunsheng; Xiao, Yao; Yu, Chuanchao; Xu, Wei; Zhang, Jinguang 2017-08-21 00:00:00 Reducing the vibration of marine power machinery can improve warships’ capabilities of concealment and reconnais- sance. Being one of the most effective means to reduce mechanical vibrations, the active vibration control technology can overcome the poor effect in low frequency of traditional passive vibration isolation. As the vibrations arising from operation of marine power machinery are actually the frequency-varying disturbances, the H control method is adopted to suppress frequency-varying disturbances. The H control method can solve the stability problems caused by the uncertainty of the model and reshape the frequency response function of the closed loop system. Two-input two- output continuous transfer function models were identified by using the system identification method and are validated in frequency domain of which all values of best fit exceeds 89%. The method of selecting the weighting functions on the mixed sensitivity problem is studied. Besides, the H controller is designed for a multiple input multiple output (MIMO) system to suppress the single-frequency-varying disturbance. The numerical simulation results show that the magnitudes of the error signals are reduced by more than 50%, and the amplitudes of the dominant frequencies are attenuated by more than 10 dB. Finally, the single excitation source dual-channel control experiments are conducted on the floating raft isolation system. The experiment results reveal that the root mean square values of the error signals under control have fallen by more 74% than that without control, and the amplitudes of the error signals in the dominant frequencies are attenuated above 13 dB. The experiment results and the numerical simulation results are basically in line, indicating a good vibration isolation effect. Keywords Active control, frequency-varying disturbance, system identification, H control Introduction The mechanical noise generated by operation of the marine power machinery is the main noise source for warships’ underwater radiation, seriously aﬀecting warships’ stealth operational performance. By inheriting the ﬂoating raft’s properties of ‘‘mass eﬀect,’’ ‘‘tuning eﬀect,’’ and ‘‘mixed neutralized eﬀect,’’ as well as the advan- tages of the active and passive vibration isolation technology, the active ﬂoating raft vibration isolation technology is an important technical means to reduce mechanical vibrations for warships. Although extensive researches on the active vibration control methods have been carried out at home and abroad, the majority of those methods which are designed to suppress mechanical vibrations of the power equip- ment belong to the adaptive feed-forward control methods, such as the adaptive feed-forward control method School of Mechanical and Electronic Engineering, Wuhan University of Technology, Wuhan, China Hubei Maglev Engineering Technology Research Center, Wuhan University of Technology, Wuhan, China Institute of Advanced material manufacturing equipment and technology, Wuhan University of Technology, Wuhan, China Wuhan Guide Infrared Co., Ltd, Wuhan, China Corresponding author: Jinguang Zhang, School of Mechanical & Electronic Engineering, Wuhan University of Technology, Wuhan 430070, Hubei, China. Email: jgzhang@whut.edu.cn Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www.creativecommons.org/licenses/by/ 4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). 200 Journal of Low Frequency Noise, Vibration and Active Control 37(2) based on least mean square (LMS) algorithm, which is widely used owing to its low computational complexity and good convergence, the adaptive feed-forward control method based on neural networks. As a highly non- linear system, neural network has a very strong self-learning ability and can be combined with other algorithms to form a new controller that can achieve an uncertain or unknown system of eﬀective control. However, this requires a lot of parameters and greatly aﬀects the acceptability of the control eﬀect. Harmonic control method and the adaptive feed-forward control method based on recursive least square algorithm, which has large steady-state error although it converges faster than LMS algorithm. These control algorithms utilize the reference signals and the error signals to adjust the controller parameters according to corresponding updating formulas in the control process, before calculating and outputting a control signal to achieve the purpose of active vibration control. However, these control methods are subject to a common assumption that the disturbance signal is periodic disturbance by default, whose frequency is ﬁxed. In practice, the speed of marine engine is not constant, but ﬂuctuating within a certain range, which leads to frequency ﬂuctuation of the vibration signal within a certain range. The existing active vibration control methods which are designed to suppress the periodic disturbances work well, but further researches are required to verify if they do well in frequency-varying disturbances. When the reference signal frequency component of the system is complex and has perturbation, the adaptive feed-forward control method cannot obtain obvious control eﬀect. Many active control methods are adopted to suppress frequency-varying disturbances abroad. Ballesteros and Bohn designed a discrete-time H optimal gain-scheduling controller to reject harmonic disturbances with time-varying and known frequencies, which was experimentally validated on an active vibration control test bench. Experimental results showed that the controller could suppress a disturbance consisting of six independent har- monics with frequencies that varied within 20 Hz. In terms of the harmonic disturbance with frequency ﬂuctuation, Duarte et al. measured the frequencies of time-varying harmonic disturbances directly from the DC motors, designed a corresponding linear parameter varying (LPV) discrete-time controller, and ﬁnally carried out experi- ments on the lightweight elastic structure. Karimi and Emedi designed a H gain-scheduled controller for rejec- tion of time-ﬂuctuating narrow-band disturbances and experiments were conducted on the active vibration control system. Airimitoaie and Landau combined the adaptive feed-forward control with feedback control for attenu- ation of multiple unknown time-varying narrowband disturbances, and real-time experimental results were pro- vided. Landau et al. used nondirect adaptive control method to reduce the harmonic disturbances with frequency ﬂuctuation and compared with the direct adaptive algorithm based on endometrial principle. The two algorithms were ﬁnally veriﬁed by experiment. For the narrowband disturbance rejection problem of helicopter tail, Connolly et al. compared the LQG control algorithm with the H control algorithm, and the experimental results showed better performance of the H control method. Orivuori et al. proposed an adaptive LQ feedback control method with frequency estimator to eliminate frequency-varying disturbances on the diesel engine, and three inputs three outputs active vibration control experiments were conducted on a ﬂoating raft system. The majority of active control methods for purpose of suppressing frequency-varying disturbances belong to 8,12 the feedback control methods, which are mainly applied in structural vibration control ﬁeld and power machin- 7,13 ery’s vibration control ﬁeld. In the structural vibration control ﬁeld, the variation range of disturbance signal’s frequency is mostly above 5 Hz, and the control methods generally comprise frequency estimator. Thus, these methods use LPV technology to update controller parameters according to the estimated frequency in real time, thereby suppressing the frequency-varying disturbances. As the variation range of disturbance signal’s frequency produced by the diesel engine in a warship is small, which is probably less than 1 Hz, such disturbances can be suppressed eﬀectively with conventional feedback control methods which require no frequency estimator. LQ 7 13 control method, LQR control method, and H control method belong to the feedback control method, which can suppress disturbances whose frequency varies in a small range. To design a controller, these feedback control methods require an appropriate mathematical model of the controlled object, which should be able to accurately describe the dynamic characteristics of the controlled object. However, due to the nonlinearity of system, modeling error, and parameter perturbation, the mathematical model is usually slightly diﬀerent from the actual controlled object, which may lead to overﬂow instability for a feedback control method. As the H control method can describe the diﬀerence between the model and the actual controlled object through additive uncertainty and multiplicative uncertainty, the controller based on the H control method is robust enough to eﬀectively solve the problem of overﬂow instability. Meanwhile, the H control method can reshape frequency response function of the closed loop system by shaping design method based on the mixed sensitivity problem, and the designed closed-loop system can eﬀectively suppress frequency-varying disturbances. Thus, the H control method is utilized to suppress frequency-varying disturb- ances generated from operation of power machinery. Song et al. 201 Modeling of control channel for the floating raft vibration isolation system Introduction of the floating raft vibration isolation platform The active ﬂoating raft vibration isolation system is composed of two parts, namely mechanical device and electronic control device. As shown in Figure 1, the mechanical device is mainly made up of excitation motors, upper isolators, ﬂoating raft, lower isolators, displacement restrictors, electromagnetic actuators, supporting frames, and base. Four excitation motors are ﬁxed above the ﬂoating raft as sources of (a) Motor Upper Isolator 1#Acceleration Floating Raft Lower Isolator Displacement 1#Actuator Restrictor Actuator Supporting Frame Base (b) 4#Actuator 4#Acceleration sensor 1#Actuator 1#Acceleration sensor Figure 1. Mechanical device of the floating raft vibration isolation system. (a) Front view of the floating raft vibration isolation system and (b) top view of the floating raft vibration isolation system. 202 Journal of Low Frequency Noise, Vibration and Active Control 37(2) Acquire input signal Acquire acceleration signal Power Low Pass Amplifier Filter Data Acquisition Instrument Conditioning with Signal Source amplifier Floating Raft Acceleration IPC Isolation System transducer Figure 2. Schematic of model identification for the floating raft vibration isolation system. disturbance, which can produce frequency-varying disturbance when operating. The ﬂoating raft is connected with the base by four lower rubber isolators, and four electromagnetic actuators and the base are rigidly connected together. The electric control device consists of acceleration sensors, conditioning ampliﬁer, low- pass ﬁlter, and data acquisition instrument with signal source, power ampliﬁers, and dSPACE1103 system. The acceleration signals collected by the acceleration sensors go through the conditioning ampliﬁer and low- pass ﬁlter, and then output voltage signals which are directly proportional to the acceleration signals and collected by A/D ports of the dSPACE1103 system to output control signals through D/A ports. After the control signals are enlarged by power ampliﬁers to drive the electromagnetic actuator, the electromagnetic actuators output corresponding control forces to oﬀset the frequency-varying disturbances generated by the motors. Model identification of control channel for the floating raft vibration isolation system The ﬂoating raft vibration isolation system is a typical nonlinear and strong coupling system, whose internal mechanism is not clear. Thus, it is not suitable for analysis modeling. Floating raft vibration isolation system generally has multiple units and can produce diﬀerent perturbation and disturbance frequency. As the unit and the unit, and the unit and the raft are mutually coupled and interacted, ﬂoating raft isolation system is a multi- disturbance source and multidirectional vibration isolation system. In a ﬂoating raft vibration isolation system, as multiple units need installing on the same raft, which will result in a complex structure of the raft, the raft cannot be simply considered as a rigid body. The system identiﬁcation method only calls for input and output data of the controlled object with no need to understand the internal mechanism of the controlled object. In addition, this method can also accurately describe the dynamic characteristics of the controlled object, so it is used to establish the mathematical model of control channel for the ﬂoating raft vibration isolation system. The H control method requires a mathematical model of the controlled object, the function tfest() of System Identiﬁcation Toolbox on MATLAB can be called to identify an optimal discrete transfer function according to the frequency response of the controlled object. The structural model of the function is output error model, and the identiﬁcation algorithm is the prediction error method. Figure 2 is the schematic of model identiﬁcation for the ﬂoating raft vibration isolation system. The input signal is white noise signal whose frequency range is 0.1–200 Hz and eﬀective value of voltage is 1.5 V. The white noise signal generated from the signal source goes through the low-pass ﬁlter and the power ampliﬁer, and then drives the actuator on the ﬂoating raft vibration isolation system. The output signal acquired by the acceleration sensor is transmitted to data acquisition instrument after conditioning, ampliﬁcation, and ﬁltering; the acquired input signal is white noise signal which is ﬁltered. The sampling time is 12.8 s and the sampling frequency is 2560 Hz. In Figure 1, the 4#actuator and the 4# sensor are placed diagonally relative to the 1#actuator and the 1# sensor, and the corresponding power ampliﬁer’s numbers are, respectively, 1# and 4#. G11 represents the transfer function from 1# ampliﬁer to the 1# sensor, G14 represents the transfer function from 1# ampliﬁer to the 4# sensor, G41 represents the transfer function from 4# ampliﬁer to the 1# sensor, and G44 represents the transfer function from 4# ampliﬁer to the 4# sensor. The frequency characteristics of the four control channel are shown in Figure 3 from which it can be seen that the G11 and G44 are similar, and G14 and G41 are almost identical. Since we have Song et al. 203 20 20 -20 G14 -20 G11 G41 G44 -40 -40 -60 -60 -80 10 20 30 40 50 60 70 10 20 30 40 50 60 70 -200 -400 -500 -600 -1000 -800 -1500 -1000 -2000 0 20 40 60 10 20 30 40 50 60 70 Frequency (Hz) Frequency (Hz) Figure 3. Frequency characteristic of the control channels. chosen the same type of power ampliﬁer and actuator in the system, we have the same frequency response function in theory. And the structure of the ﬂoating raft isolation system is symmetrical; the arrangement of the motor is also symmetrical. Therefore, according to the symmetry, the frequency response function of each channel is theoretically the same, the transfer function G11 and G44, G14 and G41 should theoretically be approximately identical. The theoretical results can be well validated by the experimental data. The function d2c () in MATLAB can be called to transform the identiﬁed discrete transfer functions into the desired continuous transfer functions. The G11, G14, G41, and G44 control channel models’ orders are 10, and the identiﬁed models’ goodness of ﬁtness with the experimental data in the frequency domain is, respectively, 92.07, 91.74, 92.57, and 86.96%. The four control channel models’ poles are located in the left side of the complex plane, which indicates stable models performance. As the identiﬁed models are processed by model veriﬁcation, they can accurately describe the dynamic characteristics of the controlled object. As shown in Figure 3, the degree of curve ﬁtting is generally high, but at low frequencies, especially 0–15 Hz frequency is relatively low degree of ﬁt, which is mainly due to two reasons. First, the frequency response of the power ampliﬁer and actuator is relatively poor in the low frequency, and second, there is a certain standing wave eﬀect in passive vibration isolator. The wider the frequency characteristic analysis of the controlled object, the richer the dynamic characteristics of the controlled object, which requires a higher order of mathematical model to accurately describe the dynamic characteristics of the controlled object, however, higher order of mathematical model makes it more diﬃcult to design and analyze the control system. According to the frequency range of the disturbance signal and the design performance requirements of the control system, this paper chooses to analyze the frequency characteristics of the controlled object in the frequency range of 0–60 Hz. Because the vibration that needs to be suppressed is the vibration of the ship’s power machine, the main frequency components of the vibration signal are in the range of 10–50 Hz, so the result of the system model identiﬁcation meets the actual requirements. The identiﬁed model also needs to be modeled to determine whether it can accurately describe the dynamic characteristics of the controlled object. The model veriﬁcation mainly examines the ﬁtting degree of the model. The ﬁtting degree of the model means the ﬁtting degree between the model data and the experimental data. It shows the ﬁtting degree between the model frequency response and experimental frequency response of the four control channel in Figure 4. It can be seen from Figure 4, the ﬁtting degree in the frequency range of 15–65 Hz is very high, and the frequency response error of each channel is within 0.3. Phase (°) Am plitude (dB) Phase(°) Amplitude (dB) 204 Journal of Low Frequency Noise, Vibration and Active Control 37(2) 20 20 (a) (b) 0 0 -20 -20 Experimental frequency response -40 -40 Model frequency response Experimental frequency response -60 -60 Model frequency response -80 -80 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 0.4 0.35 Frequency response error 0.35 Frequency response error 0.3 0.3 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80 Frequcney (Hz) (c) (d) -10 -20 -20 -30 Experimental frequency response Experimental frequency response -40 -40 Model frequency response Model frequency response -50 -60 -60 -70 -80 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 0.4 0.35 Frequency response error Frequency response error 0.35 0.3 0.3 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80 Frequcney (Hz) Frequency (Hz) Figure 4. Model and experimental data fitting degree of each control channel. (a) Control channel G11, (b) control channel G14, (c) control channel G41, and (d) control channel G44. The frequency response is the evaluation index of the system identiﬁcation toolbox. The order and the ﬁtting degree of each control channel model are shown in Table 1. In Table 1, the order of each control channel model is 10 orders, which can meet the requirements of the model on the order, and the ﬁtting degree is more than 85%, which indicates a high degree of ﬁt between the model and the experimental data. The model can accurately describe dynamic characteristics of the controlled object in frequency range of 15–65 Hz. Amplitude(dB) Amplitude(dB) Amplitude(dB) Amplitude(dB) Amplitude(dB) Amplitude(dB) Amplitude(dB) Amplitude(dB) Song et al. 205 Table 1. Fitting degree of frequency response and order number. Control channel Order number Fitting degree of frequency response (%) G11 10 92.07 G14 10 91.74 G41 10 92.57 G44 10 86.96 The expressions of the continuous transfer functions for each control channel are shown in Appendix 1. e u K(s) G(s) Figure 5. Block diagram of feedback control for suppressing frequency-varying disturbance. H control theory of the floating raft vibration isolation system The feedback control method can eﬀectively suppress the frequency-varying disturbance, and the block diagram of feedback control is shown in Figure 5. The error signal e produced by the superposition of the disturbance signal d and the desired output signal y is detected and sent to the controller K(s); the controller calculates and outputs a control signal u according to the control law, and the control signal u goes through the control channel G (s) before generating the desired output signal y; the desired output signal and the disturbance signal cancel each other out to eliminate the frequency-varying disturbance. In case there is external disturbance force but without actuator control, the sensor measures the error signal e that is equal to the disturbance signal d. When the actuator controls but external disturbance force is absent, the sensor measures the error signal e that is equal to the desired output signal y. Uncertainty of the model and its description In the ﬂoating raft isolation system, due to the vibration isolators’ nonlinearity, the raft’s ﬂexibility, the parameter perturbation, unmodeled dynamics in high frequency, and the system identiﬁcation error, it is diﬃcult to establish a precise mathematical model. The diﬀerence between the mathematical model and the actual controlled object must be taken into consideration when designing the control system, so that the designed controller based on an error model can still guarantee the actual closed-loop system’s stability, achieving the desired designing goals. The diﬀerence between the mathematical model and the controlled object is called as model uncertainty, and the common description methods for model uncertainty are addition uncertainty description and multiplication uncer- tainty description, respectively, as shown in equations (1) and (2) GðsÞ GðsÞ¼ ðsÞ W ðsÞð1Þ GðsÞ GðsÞ ¼ ðsÞ W ðsÞð2Þ GðsÞ where GðsÞ is the actual controlled object, GðsÞ is the nominal model, ðsÞ is the uncertainty after normalized, W ðsÞ represents the addition uncertainty of model, and W ðsÞ represents the multiplication uncertainty of model. 2 3 Mixed sensitivity problem When designing the control system, it is necessary to not only consider the robust stability requirement of the closed-loop control system due to the diﬀerences between the mathematical model and the actual controlled object, but also take into account the performance requirement of the control system. The mixed sensitivity 206 Journal of Low Frequency Noise, Vibration and Active Control 37(2) problem is a typical designing problem for the H1 controller, which can transform both requirements into the optimization problem of the H norm when designing a control system. A typical block diagram of the mixed sensitivity problem is shown in Figure 5. Where, G is the nominal model, K is the controller to be designed, W , W , and W are three diﬀerent weighting functions; w is the input signal; z , z , and z are the weighted 1 2 3 1 2 3 outputs; y is the error signal measured by sensor; u is the control signal output from the controller. The area surrounded by the dashed box is augmented controlled object, including the controlled object G and three weighting functions. The mixed sensitivity problem can be understood as selecting three appropriate weighting functions and solving a controller K to make the system meet the robust stability and performance requirements, namely the controller K enables the establishment of formula (3) W S W =ð1 þ GKÞ 1 1 W R ¼ W K=ð1 þ GKÞ 1 ð3Þ 2 2 W T W GK=ð1 þ GKÞ 3 3 1 1 S is the sensitivity function, T is the complementary sensitivity function, R is the controller sensitivity function. Where,kk W S 1 represents the performance requirements,kk W R 1&kk W T 1 both represent the 1 2 3 1 1 1 robust stability requirements. Selection of the weighting function The solution of mixed sensitivity problem needs to be converted into a standard H control problem. Doyle et al. oﬀers a classic DGKF which is also called as the Algebraic Riccati equation RIC method to solve the standard H control problem and a central controller is designed by solving Riccati equations in this method. In order to solve mixed sensitivity problem requires three appropriate weighting functions W , W , and W should be 1 2 3 selected in advance, because the quality of these three weighting functions can directly aﬀect the designed con- troller’s capacity to meet the designing requirements. Currently, there have yet not been mature rules to select the weighting function, and cut-and-try method is mostly used. When designing the control system, ﬁrst select the weighting function W and W , followed by selection of the weighting function W . The weighting function W 2 3 1 2 and W are related to model uncertainties, including the absolute error and the relative error between the model response and the experimental frequency response. The weighting function W should be selected according to the frequency characteristics of external interference. In order to suppress the frequency-varying disturbance, whose dominant frequency ranges from 1 Hz, the amplitude of the performance-related sensitivity function S in dominant frequency must be less than 1. For the single-input single-output system, the system performance requirement can be described by equation (4) kk W S 1 ) Sð jwÞ W ð jwÞ ð4Þ 1 1 Equation (4) shows that as the sensitivity function S’s frequency characteristic Sð jwÞ must be located below the W ð jwÞ , the amplitude of W ð jwÞ in dominant frequency must be larger than 1. It can make the W ð jwÞ 1 1 1 present a peak shape in dominant frequency, thus the Sð jwÞ and the W ð jwÞ manifest themselves as a null in dominant frequency, which indicates that the entire feedback control system can highly suppress the disturbance signal containing the dominant frequency component. If only a dominant frequency is to be suppressed, W can be selected as a single peak shape with the peak value located at the dominant frequency, as shown in formula (5) and (6) W ¼ k ð5Þ 2 2 s þ 2ws þ w pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ w ¼ w 1 ð6Þ In equations (5) and (6), w is the dominant frequency which needs suppressing; k and are regarded as adjustment parameters when selecting W . 1 Song et al. 207 For a MIMO control system, if there is only a frequency-varying disturbances containing a dominant frequency component that needs suppressing, the selection method for weighting functions is as the same as the SISO control system, except that those three weighting functions need to become a corresponding diagonal matrix. H controller designed for a MIMO control system The W is chosen to describe the model uncertainty and its value is determined by the absolute error between the model frequency response and the experimental frequency response. The four identiﬁed channels’ model frequency responses and experimental frequency responses are limited from 0 to 60 Hz, and the absolute error are all less than 0.2. In order not to increase the order numbers for controllers and to obtain a larger robust stability, the optional weighting function W is assigned as 0.1 and W as 0. 2 3 The H1 controller is designed with the following steps: (I) Adjust the parameters and and tentatively give a weighting function W . By relying on the transfer function of the control channel and the weighting functions W , W , and W , call the hinfsyn () function in the 1 2 3 MATLAB software to solve the H1 controller K. If there is no solution to the mixed sensitivity problem, reduce the parameter k appropriately. After k is reduced, if the mixing sensitivity problem remains unsolved, which may be due to the excessive weighting function W , reduce the amplitude of W and reexecute step (I). 2 2 (II) Verify whether the closed-loop control system is inherently stabile, namely to check whether the designed controller and closed-loop control system are stable. If the control system is not inherently stable, proceed from step I and modify the selection parameters of the weighting function W appropriately. If the system is inherently stable, proceed to the next step. (III) Check Bode Plots regarding the sensitivity function of the closed-loop control system to determine whether the system meets the performance requirements. If not, go back to step (I); if yes, end the controller design. The weighting function W is selected according to formula (5), and the W is determined by the dominant 1 2 frequency of disturbance signal. Assume that a disturbance whose dominant frequency varies close to 30 Hz is to be suppressed and the variation range of frequency is no more than 1 Hz. w can be assigned to 2 30 ¼ 188:4 rad=s. According to formula (5) and (6), by conducting a series of spreadsheets, and in order to ensure the stability for the controller K and the closed-loop control system under stable conditions, the weighting functions eventually are identiﬁed as "# s þ0:1885sþ35530 W ¼ s þ0:1885sþ35530 ð7Þ 0:10 W ¼ 00:1 W ¼ 0 The mathematical model of the control channel is a two-input two-output transfer function matrix made up of four channels. According to the mathematical model G (s) of the control channel, the weighting functions W , W , 1 2 and W , and the hinfsyn () function in MATLAB can be called to solve the robust controller K. The controller K is a two-input two-output state-space equation, and the dimension of the matrix A is 44. Due to too large controller order, it is not conducive to the control algorithm, and thus the controller order should be reduced. The bstmr () function in MTALB can be used to complete order reduction. Under the condition that there is little change of the dynamic characteristics of the controller K, the dimension of the matrix A is reduced to 4. The state-space expression of the controller after reducing order follows as formula (8) 2 3 0:05745774583 187:8103102 16:00317480 8:547472727 6 7 187:5005506 0:1353067232 8:501991657 16:05663768 6 7 A ¼ ð8Þ 6 7 4 5 15:53041937 8:236529397 0:08355357455 188:4515155 8:196244109 15:44411324 186:8724015 0:06379850336 208 Journal of Low Frequency Noise, Vibration and Active Control 37(2) 2 3 2:724889168 0:8050106367 6 7 1:590320803 0:4766626217 6 7 B ¼ 6 7 4 5 1:085861943 3:082330315 0:2714032574 0:7586295242 0:5600031142 2:861390715 0:3605602097 1:693404853 C ¼ 1:494554152 0:3491156794 1:716977793 2:405479480 D ¼ where A represents the system state matrix, B represents the control input matrix, C represents the system output matrix, and D represents the input and output correlation matrix. With the same method above, the weighing functions and the controller K can be designed when the disturbance whose dominant frequency varies close to 35 Hz. The weighing functions and the controller K are deﬁned as "# s þ0:2199sþ48360 W ¼ s þ0:2199sþ48360 ð9Þ 0:10 W ¼ 00:1 W ¼ 0 2 3 0:1140915794 218:7348982 16:01016303 17:27993062 6 7 6 218:5730602 1086776480 17:26447051 16:00866239 7 6 7 A ¼ 6 7 15:89387633 17:22096667 0:1291900427 218:7462161 4 5 17:19691173 15:89185783 218:5563591 0:080873009323 2 3 2:335157528 0:03278774785 6 7 6 2:208785663 0:5466691529 7 6 7 B ¼ ð10Þ 6 7 0:43824719078 2:675072276 4 5 0:3265658986 0:3694493762 "# 0:30425 3:1997 0:17276 0:53507 C ¼ 0:031683 0:15135 1:93023 2:5354 "# D ¼ Upon examination, the designed control system meets the internal stability condition, and the shaping design results based on the mixed sensitivity problem is shown as Figure 6. As shown in Figure 7, there are two curves for the singular value of the sensitivity function S in the closed-loop control system, and the two curves are below the 1 1 singular value of the function W , and the singular values of the sensitivity function S and the function W at 1 1 the frequency 30 Hz nearby present concave shapes, which indicates that the closed-loop control system can eﬀectively suppress frequency-varying disturbance with dominant frequency of 30 Hz. In Figure 8, we can con- clude that control system can eﬀectively suppress frequency-varying disturbance with dominant frequency of 35 Hz in the same way. The simulation block diagram for a MIMO feedback control system is shown in Figure 9. Sweep signal is used to simulate frequency-varying disturbance, and the sweep signal 1 and the sweep signal 2 are, respectively, used as the disturbance signal 1 and the disturbance signal 2, of which the duration are both 5 s. To distinguish two disturbance signals, one of sweep signals goes through a gain of 0.8 narrow links. The simulation results are shown in Figure 10, where Figure 10(a) is the time-domain diagram and self-power spectrum for the error signal 1 before and after control, and Figure 10(b) is the time-domain diagram and power Song et al. 209 Augmented controlled object Z1 w + Z2 Z3 G W uy Figure 6. Block diagram of the mixed sensitivity problem. σ (S) - 1 σ (W ) 60 1 -20 -40 10 20 30 40 50 60 70 80 Frequency (Hz) Figure 7. Shaping design result for 30 Hz frequency-varying disturbance. (S) -1 (W ) -20 -40 10 20 30 40 50 60 70 80 Frequency (Hz) Figure 8. Shaping design result for 35 Hz frequency-varying disturbance. Singular Values ( dB) Singular Values (dB) 210 Journal of Low Frequency Noise, Vibration and Active Control 37(2) Figure 9. Robust control for a MIMO control system. MIMO: multiple input multiple output. Figure 10. Time-domain diagram and self-power spectrums of the error signals of 30 Hz. spectrum for the error signal 2 before and after control. As it can be seen, the magnitudes of two error signals are reduced by more than 50%, and the amplitudes of the error signal 1 and the error signal 2 in 30 Hz have fallen by about 15 dB. This indicates that the control system can eﬀectively suppress frequency-varying disturbance with dominant frequency of 30 Hz. In order to verify universality of the H controller designing method, another controller is designed for frequency-varying disturbances with dominant frequency of 35 Hz, which is shown in Figure 11. The simulation results show that when the disturbance signal’s frequency varies around 35 Hz, the dominant frequency’s magni- tudes of the error signal 1 and the error signal 2 fall by about 10 dB. It shows that the designed system can eﬀectively suppress frequency-varying disturbance whose dominant frequency varies around 30 and 35 Hz. Song et al. 211 Figure 11. Time-domain diagram and self-power spectrums of the error signals of 35 Hz. 1# Power Amplifier 1# Actuator U1 E1 1# Acceleration dSPACE1103 4# Acceleration E4 Conditioning Low pass filter Amplifiers U4 4# Power Amplifier 4# Actuator Figure 12. Schematic diagram of the MIMO feedback control system. MIMO: multiple input multiple output. H control experiments for a MIMO control system The diagram of MIMO feedback control system is shown in Figure 12, the system is a two-input two-output feedback control system. The voltage signal proportional to the acceleration signal from the low-pass ﬁlter can be regarded as error signals, as shown in E1 and E4 of Figure 11. E1 represents the voltage signal from the condi- tioning and ﬁltered 1# acceleration signal, and E4 represents the voltage signal from the conditioning and ﬁltered 4# acceleration signal. The ﬁlter output port connects with the A/D port of the dSPACE1103 system’s wiring panel CLP1103. The control board DS1103 is a kind of controller hardware, depending on the designed controller to calculate the control amount. The calculation for controller adopts ﬁxed-step calculation method, of which the 212 Journal of Low Frequency Noise, Vibration and Active Control 37(2) 0.6 0.6 E4 E1 0.4 0.4 0.2 0.2 0 0 -0.2 -0.2 -0.4 -0.4 Without control Robust control Without control Robust control 0 2 4 6 8 10 0 2 4 6 8 10 1.5 2 U1 U4 0.5 0 0 -0.5 -1 -1 -1.5 -2 0 2 4 6 8 10 0 2 4 6 8 10 Time (s) Time (s) Figure 13. Time-domain diagram of error signals and control signals when the speed of motor is 1800 r/min. (a) 0 (b) 0 Without control Without control -10 -10 Robust control Robust control -20 -20 -30 -30 -40 -40 -50 -50 -60 -60 -70 -70 -80 -80 -90 -100 -90 0 20 40 60 80 0 20 40 60 80 Frequency(Hz) Frequency(Hz) Figure 14. Self-power spectrum of the error signals of 30 Hz. step is 0.0001 and the solver is ode8. After crossing D/A converter, the control amount outputs two control signals, namely U1 and U4 as shown in Figure 11. The amplitudes of the two control signals are conﬁned to 5 V. The control signal U1 goes across 1# power ampliﬁer, and then drives l# electromagnetic actuator to output control force; after going across 4# power ampliﬁer, the control signal U4 drives 4# electromagnetic actuator to output control force. On the coordinative role of the two actuators, the vibration from frequency ﬂuctuation disturbance generated by the motor above the 1#actuator can be inhibited. In the experiments, two diﬀerent controllers are designed to suppress the disturbances with dominant frequency of 30 and 35 Hz. When the frequency is 30 Hz, we choose the speed of the motor above 1# actuator 1800 r/min, when the frequency is 35 Hz, we choose the speed of the motor above 1# actuator 2100 r/min. When the speed of the motor above 1# actuator is 1800 r/min, the active vibration control experiment is conducted according to Figure 11, and experimental results are shown in Figures 13 and 14, respectively. Without control, the two control signals are 0, and the amplitudes of the error signals are approximately 0.5 and 0.3 m/s . After the robust control, two control signals gradually increase and eventually approach a steady Amplitude(dB) Voltage (V) Acceleration(m/s ) Amplitude(dB) Acceleration(m/s ) Voltage (V) Song et al. 213 E1 E4 0.5 0.5 -0.5 -0.5 Without control Robust control Without control Robust control -1 -1 0 2 4 6 8 10 0 2 4 6 8 10 Frequency (Hz) U1 U4 -1 -1 -2 -2 0 2 4 6 8 10 0 2 4 6 8 10 Time (s) Time (s) Figure 15. Time-domain diagram of error signals and control signals when the speed of motor is 2100 r/min. 0 0 (a) (b) Without control Without control -10 Robust control Robust control -20 -20 -30 -40 -40 -50 -60 -60 -80 -70 -80 -100 -90 -100 -120 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 Frequency (Hz) Frequency (Hz) Figure 16. Self-power spectrum of the error signals of 35 Hz. state; the two error signals achieve fast convergence, and their amplitudes are greatly reduced. In the interception map, the signals are used as error signals without control from 0 to 5 s and the signals are used as error signals with robust control from 5 to 11 s. The RMS values of error signal E1 before and after the control are, respectively, 0.3689 and 0.0823, so the RMS value is decreased 77.68%. The RMS values of error signal E4 before and after the controls are, respectively, 0.2597 and 0.0659, which indicates a 74.64% decrease in the RMS value. Figure 14(a) and (b), respectively, represent the self-power spectrum of the error signals E1 and E4 before and after the control, where the amplitudes of the frequency components in the range of about 30 Hz within the error signals E1 and E4 are greatly reduced by 13.91 and 13.16 dB, respectively, while other frequency components show no signiﬁcant changes. When the speed of the motor above 1# actuator is 2100 r/min, experimental results are shown in Figures 15 and 16, respectively. After the robust control, two control signals gradually increase and eventually approach a steady state; the two error signals achieve fast convergence, and their amplitudes are greatly reduced. In the interception map, the signals are used as error signals without control from 0 to 4 s and the signals are used as error signals with robust control from 4 to 10 s. The RMS values of error signal E1 before and after the control are, Amplitude(dB) Acceleration(m/s ) Voltage(V) Amplitude (dB) Acceleration(m/s ) Voltage(V) 214 Journal of Low Frequency Noise, Vibration and Active Control 37(2) respectively, 0.6258 and 0.0614, so the RMS value is decreased 90.19%. The RMS values of error signal E4 before and after the control are, respectively, were 0.3369 and 0.0335, which indicates a 90.06% decrease in the RMS value. Figure 16(a) and (b), respectively, represents the self-power spectrum of the error signals E1 and E4 before and after the control, where the amplitudes of the frequency components in the range of about 30 Hz within the error signals E1 and E4 are greatly reduced by 22.48 and 33.88 dB respectively, while other frequency components show no signiﬁcant changes. Conclusion The system identiﬁcation method is adopted to establish two-input and two-output continuous transfer function model of the control channel for the ﬂoating raft vibration isolation system, and the model can accurately describe the dynamic characteristics of the controlled object. The H controller is designed with the shaping design method based on mixed sensitivity problem and the selection method of weighting function is described. The H controller is designed for a MIMO system to suppress single frequency-varying disturbance. Simulation analysis is conducted against the designed controller, and the results show that the amplitudes of the dominant frequencies are atte- nuated more than 10 dB before and after the robust control. An actual ﬂoating raft vibration isolation system is built for the single excitation source multichannel control experiments. Each experiment achieves good vibration isolation eﬀect, the RMS values of error signals drop by more than 74% before and after the control, and the amplitudes of the dominant frequency have fallen more than 13 dB. Experimental results show that the controller can eﬀectively suppress frequency-varying disturbance. Declaration of conflicting interests The author(s) declared no potential conﬂicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) disclosed receipt of the following ﬁnancial support for the research, authorship, and/or publication of this article: This study is supported by National Natural Science Funds of China (51205296, 51275368). References 1. Wei C, Wang Y, Chen S, et al. Research on the control of electromagnetic suspension vibration isolator based on block normalized LMS algorithm. J Vib Shock 2012; 31: 100–103. 2. Yang T, Li X, Zhu M, et al. Experimental investigation of active vibration control for diesel engine generators in marine applications. J Vib Eng 2013; 2: 160–168. 3. Zhang Z and Guan D. Feedback control of Rijke-type thermoacoustic oscillations transient growth. J Low Freq Noise Vib Active Control 2015; 34: 219–232. 4. An F, Sun H and Li X. Adaptive active control of periodic vibration using maglev actuators. J Sound Vib 2012; 331: 1971–1984. 5. Li S and Yu D. Simulation study of active control of vibro-acoustic response by sound pressure feedback using modal pole assignment strategy. J Low Freq Noise Vib Active Control 2016; 35: 167–186. 6. Daley S, Zazas I and Hatonen J. Harmonic control of a ‘smart spring’ machinery vibration isolation system. Proc IMechE, Part M: J Engineering for the Maritime Environment 2008; 222: 109–119. 7. Orivuori J, Zazas I and Daley S. Active control of frequency varying disturbances in a diesel engine. Control Eng Pract 2012; 20: 1206–1219. 8. Ballesteros P and Bohn C. Active vibration control for harmonic disturbances with time-fluctuating frequencies through LPV gain scheduling. In: Proceedings of the 2011 Chinese control and decision conference (CCDC), Sichuan, China, 23–25 May 2011, pp.728–733. 9. Duarte F, Ballesteros P and Shu X. An LPV discrete-time controller for the rejection of harmonic time-fluctuating disturbances in a lightweight flexible structure. In: Proceedings of the American control conference, Washington DC, USA, 17–19 June 2013, pp.4092–4097. 10. Karimi A and Emedi Z. H gain-scheduled controller design for rejection of time-fluctuating narrow-band disturbances applied to a benchmark problem. Eur J Control 2013; 19: 279–288. 11. Airimitoaie TB and Landau ID. Indirect adaptive attenuation of multiple narrow-band disturbances applied to active vibration control. IEEE Trans Control Syst Technol 2014; 22: 761–769. 12. Landau ID, Airimitoaie TB and Silva AC. Adaptive attenuation of unknown and time-fluctuating narrow band and broadband disturbances. Int J Adapt Control Signal Process 2015; 29: 1367–1390. Song et al. 215 13. Connolly AJ, Green M, Chicharo JF, et al. The design of LQG and H controllers for use in active vibration control and narrow band disturbance rejection. In: Proceedings of the 34th IEEE conference on decision and control, New Orleans, LA, USA, 13–15 December 1995, pp.2982–2987. IEEE. 14. Balas MJ. Feedback control of flexible systems. IEEE Trans Autom Control 1978; 23: 673–679. 15. Zhou K and Doyle JC. Essentials of robust control. Upper Saddle River, 1984, p.38. 16. Ballesteros P and Bohn C. Active rejection of harmonic disturbances with nonstationary harmonically related frequencies using varying-sampling-time LPV control. Solid State Phenomena, 2016, p.248. 17. Doyle JC, Glover K, Khargonekar PP, et al. State-space solutions to standard H2 and H1 control problems. IEEE Trans Autom Control 1989; 8: 831–847. Appendix 1 2 2 15:976ðs 60:93Þðs þ 76:76s þ 21930Þðs 399:6s þ 104100Þ G ðsÞ¼ 2 2 2 ðs þ 17:45s þ 12330Þðs þ 17:39s þ 29680Þðs þ 43:07s þ 82190Þ 2 2 ðs þ 125:9s þ 74800Þðs 90:76s þ 185700Þ 2 2 ðs þ 60:05s þ 105100Þðs þ 23:93s þ 151800Þ 2 2 5:8297ðs 1316Þðs 64:14s þ 12880Þðs þ 132:6s þ 49340Þ G ðsÞ¼ 2 2 2 ðs þ 19:65s þ 14040Þðs þ 19:52s þ 29150Þðs þ 37:18s þ 89980Þ 2 2 ðs 237:6s þ 95770Þðs 63:68s þ 167300Þ 2 2 ðs þ 80:69s þ 120600Þðs þ 25:05s þ 146500Þ 2 2 9:7756ðsþ640:4Þðs 61:88s þ 12330Þðs þ 161:2s þ 627200Þ G ðsÞ¼ 2 2 2 ðs þ 15:68s þ 14140Þðs þ 20:54s þ 28290Þðs þ 44:91s þ 89690Þ 2 2 ðs 235:5s þ 85160Þðs 75:66s þ 166200Þ 2 2 ðs þ 49:7s þ 110100Þðs þ 20:93s þ 146900Þ 2 2 14:248ðs 107:5Þðs þ 104:1s þ 21920Þðs þ112:7s þ 63420Þ G ðsÞ¼ 2 2 2 ðs þ 19:84s þ 11450Þðs þ 15:97s þ 29870Þðs þ 48:55s þ 78030Þ 2 2 ðs 484:8s þ 124700Þðs 118:8s þ 192400Þ 2 2 ðs þ 56:47s þ 105300Þðs þ 19:31s þ 151600Þ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png "Journal of Low Frequency Noise, Vibration and Active Control" SAGE http://www.deepdyve.com/lp/sage/h-active-control-of-frequency-varying-disturbances-in-a-main-engine-on-0Y4DDMskXg

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References (20)

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Active Rejection of Harmonic Disturbances with Nonstationary Harmonically Related Frequencies Using Varying-Sampling-Time LPV Control
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Active vibration control for harmonic disturbances with time-varying frequencies through LPV gain scheduling
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Feedback Control of Rijke-Type Thermoacoustic Oscillations Transient Growth
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State-space solutions to standard H2 and H∞ control problems
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State-space solutions to standard H/sub 2/ and H/sub infinity / control problems
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Juha Orivuori, I. Zazas, S. Daley (2012)
Active control of frequency varying disturbances in a diesel engine
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A. Connolly, M. Green, J. Chicharo, R. Bitmead (1995)
The design of LQG and H/sub /spl infin// controllers for use in active vibration control and narrow band disturbance rejection
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Abstract

Reducing the vibration of marine power machinery can improve warships’ capabilities of concealment and reconnais- sance. Being one of the most effective means to reduce mechanical vibrations, the active vibration control technology can overcome the poor effect in low frequency of traditional passive vibration isolation. As the vibrations arising from operation of marine power machinery are actually the frequency-varying disturbances, the H control method is adopted to suppress frequency-varying disturbances. The H control method can solve the stability problems caused by the uncertainty of the model and reshape the frequency response function of the closed loop system. Two-input two- output continuous transfer function models were identified by using the system identification method and are validated in frequency domain of which all values of best fit exceeds 89%. The method of selecting the weighting functions on the mixed sensitivity problem is studied. Besides, the H controller is designed for a multiple input multiple output (MIMO) system to suppress the single-frequency-varying disturbance. The numerical simulation results show that the magnitudes of the error signals are reduced by more than 50%, and the amplitudes of the dominant frequencies are attenuated by more than 10 dB. Finally, the single excitation source dual-channel control experiments are conducted on the floating raft isolation system. The experiment results reveal that the root mean square values of the error signals under control have fallen by more 74% than that without control, and the amplitudes of the error signals in the dominant frequencies are attenuated above 13 dB. The experiment results and the numerical simulation results are basically in line, indicating a good vibration isolation effect. Keywords Active control, frequency-varying disturbance, system identification, H control Introduction The mechanical noise generated by operation of the marine power machinery is the main noise source for warships’ underwater radiation, seriously aﬀecting warships’ stealth operational performance. By inheriting the ﬂoating raft’s properties of ‘‘mass eﬀect,’’ ‘‘tuning eﬀect,’’ and ‘‘mixed neutralized eﬀect,’’ as well as the advan- tages of the active and passive vibration isolation technology, the active ﬂoating raft vibration isolation technology is an important technical means to reduce mechanical vibrations for warships. Although extensive researches on the active vibration control methods have been carried out at home and abroad, the majority of those methods which are designed to suppress mechanical vibrations of the power equip- ment belong to the adaptive feed-forward control methods, such as the adaptive feed-forward control method School of Mechanical and Electronic Engineering, Wuhan University of Technology, Wuhan, China Hubei Maglev Engineering Technology Research Center, Wuhan University of Technology, Wuhan, China Institute of Advanced material manufacturing equipment and technology, Wuhan University of Technology, Wuhan, China Wuhan Guide Infrared Co., Ltd, Wuhan, China Corresponding author: Jinguang Zhang, School of Mechanical & Electronic Engineering, Wuhan University of Technology, Wuhan 430070, Hubei, China. Email: jgzhang@whut.edu.cn Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www.creativecommons.org/licenses/by/ 4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). 200 Journal of Low Frequency Noise, Vibration and Active Control 37(2) based on least mean square (LMS) algorithm, which is widely used owing to its low computational complexity and good convergence, the adaptive feed-forward control method based on neural networks. As a highly non- linear system, neural network has a very strong self-learning ability and can be combined with other algorithms to form a new controller that can achieve an uncertain or unknown system of eﬀective control. However, this requires a lot of parameters and greatly aﬀects the acceptability of the control eﬀect. Harmonic control method and the adaptive feed-forward control method based on recursive least square algorithm, which has large steady-state error although it converges faster than LMS algorithm. These control algorithms utilize the reference signals and the error signals to adjust the controller parameters according to corresponding updating formulas in the control process, before calculating and outputting a control signal to achieve the purpose of active vibration control. However, these control methods are subject to a common assumption that the disturbance signal is periodic disturbance by default, whose frequency is ﬁxed. In practice, the speed of marine engine is not constant, but ﬂuctuating within a certain range, which leads to frequency ﬂuctuation of the vibration signal within a certain range. The existing active vibration control methods which are designed to suppress the periodic disturbances work well, but further researches are required to verify if they do well in frequency-varying disturbances. When the reference signal frequency component of the system is complex and has perturbation, the adaptive feed-forward control method cannot obtain obvious control eﬀect. Many active control methods are adopted to suppress frequency-varying disturbances abroad. Ballesteros and Bohn designed a discrete-time H optimal gain-scheduling controller to reject harmonic disturbances with time-varying and known frequencies, which was experimentally validated on an active vibration control test bench. Experimental results showed that the controller could suppress a disturbance consisting of six independent har- monics with frequencies that varied within 20 Hz. In terms of the harmonic disturbance with frequency ﬂuctuation, Duarte et al. measured the frequencies of time-varying harmonic disturbances directly from the DC motors, designed a corresponding linear parameter varying (LPV) discrete-time controller, and ﬁnally carried out experi- ments on the lightweight elastic structure. Karimi and Emedi designed a H gain-scheduled controller for rejec- tion of time-ﬂuctuating narrow-band disturbances and experiments were conducted on the active vibration control system. Airimitoaie and Landau combined the adaptive feed-forward control with feedback control for attenu- ation of multiple unknown time-varying narrowband disturbances, and real-time experimental results were pro- vided. Landau et al. used nondirect adaptive control method to reduce the harmonic disturbances with frequency ﬂuctuation and compared with the direct adaptive algorithm based on endometrial principle. The two algorithms were ﬁnally veriﬁed by experiment. For the narrowband disturbance rejection problem of helicopter tail, Connolly et al. compared the LQG control algorithm with the H control algorithm, and the experimental results showed better performance of the H control method. Orivuori et al. proposed an adaptive LQ feedback control method with frequency estimator to eliminate frequency-varying disturbances on the diesel engine, and three inputs three outputs active vibration control experiments were conducted on a ﬂoating raft system. The majority of active control methods for purpose of suppressing frequency-varying disturbances belong to 8,12 the feedback control methods, which are mainly applied in structural vibration control ﬁeld and power machin- 7,13 ery’s vibration control ﬁeld. In the structural vibration control ﬁeld, the variation range of disturbance signal’s frequency is mostly above 5 Hz, and the control methods generally comprise frequency estimator. Thus, these methods use LPV technology to update controller parameters according to the estimated frequency in real time, thereby suppressing the frequency-varying disturbances. As the variation range of disturbance signal’s frequency produced by the diesel engine in a warship is small, which is probably less than 1 Hz, such disturbances can be suppressed eﬀectively with conventional feedback control methods which require no frequency estimator. LQ 7 13 control method, LQR control method, and H control method belong to the feedback control method, which can suppress disturbances whose frequency varies in a small range. To design a controller, these feedback control methods require an appropriate mathematical model of the controlled object, which should be able to accurately describe the dynamic characteristics of the controlled object. However, due to the nonlinearity of system, modeling error, and parameter perturbation, the mathematical model is usually slightly diﬀerent from the actual controlled object, which may lead to overﬂow instability for a feedback control method. As the H control method can describe the diﬀerence between the model and the actual controlled object through additive uncertainty and multiplicative uncertainty, the controller based on the H control method is robust enough to eﬀectively solve the problem of overﬂow instability. Meanwhile, the H control method can reshape frequency response function of the closed loop system by shaping design method based on the mixed sensitivity problem, and the designed closed-loop system can eﬀectively suppress frequency-varying disturbances. Thus, the H control method is utilized to suppress frequency-varying disturb- ances generated from operation of power machinery. Song et al. 201 Modeling of control channel for the floating raft vibration isolation system Introduction of the floating raft vibration isolation platform The active ﬂoating raft vibration isolation system is composed of two parts, namely mechanical device and electronic control device. As shown in Figure 1, the mechanical device is mainly made up of excitation motors, upper isolators, ﬂoating raft, lower isolators, displacement restrictors, electromagnetic actuators, supporting frames, and base. Four excitation motors are ﬁxed above the ﬂoating raft as sources of (a) Motor Upper Isolator 1#Acceleration Floating Raft Lower Isolator Displacement 1#Actuator Restrictor Actuator Supporting Frame Base (b) 4#Actuator 4#Acceleration sensor 1#Actuator 1#Acceleration sensor Figure 1. Mechanical device of the floating raft vibration isolation system. (a) Front view of the floating raft vibration isolation system and (b) top view of the floating raft vibration isolation system. 202 Journal of Low Frequency Noise, Vibration and Active Control 37(2) Acquire input signal Acquire acceleration signal Power Low Pass Amplifier Filter Data Acquisition Instrument Conditioning with Signal Source amplifier Floating Raft Acceleration IPC Isolation System transducer Figure 2. Schematic of model identification for the floating raft vibration isolation system. disturbance, which can produce frequency-varying disturbance when operating. The ﬂoating raft is connected with the base by four lower rubber isolators, and four electromagnetic actuators and the base are rigidly connected together. The electric control device consists of acceleration sensors, conditioning ampliﬁer, low- pass ﬁlter, and data acquisition instrument with signal source, power ampliﬁers, and dSPACE1103 system. The acceleration signals collected by the acceleration sensors go through the conditioning ampliﬁer and low- pass ﬁlter, and then output voltage signals which are directly proportional to the acceleration signals and collected by A/D ports of the dSPACE1103 system to output control signals through D/A ports. After the control signals are enlarged by power ampliﬁers to drive the electromagnetic actuator, the electromagnetic actuators output corresponding control forces to oﬀset the frequency-varying disturbances generated by the motors. Model identification of control channel for the floating raft vibration isolation system The ﬂoating raft vibration isolation system is a typical nonlinear and strong coupling system, whose internal mechanism is not clear. Thus, it is not suitable for analysis modeling. Floating raft vibration isolation system generally has multiple units and can produce diﬀerent perturbation and disturbance frequency. As the unit and the unit, and the unit and the raft are mutually coupled and interacted, ﬂoating raft isolation system is a multi- disturbance source and multidirectional vibration isolation system. In a ﬂoating raft vibration isolation system, as multiple units need installing on the same raft, which will result in a complex structure of the raft, the raft cannot be simply considered as a rigid body. The system identiﬁcation method only calls for input and output data of the controlled object with no need to understand the internal mechanism of the controlled object. In addition, this method can also accurately describe the dynamic characteristics of the controlled object, so it is used to establish the mathematical model of control channel for the ﬂoating raft vibration isolation system. The H control method requires a mathematical model of the controlled object, the function tfest() of System Identiﬁcation Toolbox on MATLAB can be called to identify an optimal discrete transfer function according to the frequency response of the controlled object. The structural model of the function is output error model, and the identiﬁcation algorithm is the prediction error method. Figure 2 is the schematic of model identiﬁcation for the ﬂoating raft vibration isolation system. The input signal is white noise signal whose frequency range is 0.1–200 Hz and eﬀective value of voltage is 1.5 V. The white noise signal generated from the signal source goes through the low-pass ﬁlter and the power ampliﬁer, and then drives the actuator on the ﬂoating raft vibration isolation system. The output signal acquired by the acceleration sensor is transmitted to data acquisition instrument after conditioning, ampliﬁcation, and ﬁltering; the acquired input signal is white noise signal which is ﬁltered. The sampling time is 12.8 s and the sampling frequency is 2560 Hz. In Figure 1, the 4#actuator and the 4# sensor are placed diagonally relative to the 1#actuator and the 1# sensor, and the corresponding power ampliﬁer’s numbers are, respectively, 1# and 4#. G11 represents the transfer function from 1# ampliﬁer to the 1# sensor, G14 represents the transfer function from 1# ampliﬁer to the 4# sensor, G41 represents the transfer function from 4# ampliﬁer to the 1# sensor, and G44 represents the transfer function from 4# ampliﬁer to the 4# sensor. The frequency characteristics of the four control channel are shown in Figure 3 from which it can be seen that the G11 and G44 are similar, and G14 and G41 are almost identical. Since we have Song et al. 203 20 20 -20 G14 -20 G11 G41 G44 -40 -40 -60 -60 -80 10 20 30 40 50 60 70 10 20 30 40 50 60 70 -200 -400 -500 -600 -1000 -800 -1500 -1000 -2000 0 20 40 60 10 20 30 40 50 60 70 Frequency (Hz) Frequency (Hz) Figure 3. Frequency characteristic of the control channels. chosen the same type of power ampliﬁer and actuator in the system, we have the same frequency response function in theory. And the structure of the ﬂoating raft isolation system is symmetrical; the arrangement of the motor is also symmetrical. Therefore, according to the symmetry, the frequency response function of each channel is theoretically the same, the transfer function G11 and G44, G14 and G41 should theoretically be approximately identical. The theoretical results can be well validated by the experimental data. The function d2c () in MATLAB can be called to transform the identiﬁed discrete transfer functions into the desired continuous transfer functions. The G11, G14, G41, and G44 control channel models’ orders are 10, and the identiﬁed models’ goodness of ﬁtness with the experimental data in the frequency domain is, respectively, 92.07, 91.74, 92.57, and 86.96%. The four control channel models’ poles are located in the left side of the complex plane, which indicates stable models performance. As the identiﬁed models are processed by model veriﬁcation, they can accurately describe the dynamic characteristics of the controlled object. As shown in Figure 3, the degree of curve ﬁtting is generally high, but at low frequencies, especially 0–15 Hz frequency is relatively low degree of ﬁt, which is mainly due to two reasons. First, the frequency response of the power ampliﬁer and actuator is relatively poor in the low frequency, and second, there is a certain standing wave eﬀect in passive vibration isolator. The wider the frequency characteristic analysis of the controlled object, the richer the dynamic characteristics of the controlled object, which requires a higher order of mathematical model to accurately describe the dynamic characteristics of the controlled object, however, higher order of mathematical model makes it more diﬃcult to design and analyze the control system. According to the frequency range of the disturbance signal and the design performance requirements of the control system, this paper chooses to analyze the frequency characteristics of the controlled object in the frequency range of 0–60 Hz. Because the vibration that needs to be suppressed is the vibration of the ship’s power machine, the main frequency components of the vibration signal are in the range of 10–50 Hz, so the result of the system model identiﬁcation meets the actual requirements. The identiﬁed model also needs to be modeled to determine whether it can accurately describe the dynamic characteristics of the controlled object. The model veriﬁcation mainly examines the ﬁtting degree of the model. The ﬁtting degree of the model means the ﬁtting degree between the model data and the experimental data. It shows the ﬁtting degree between the model frequency response and experimental frequency response of the four control channel in Figure 4. It can be seen from Figure 4, the ﬁtting degree in the frequency range of 15–65 Hz is very high, and the frequency response error of each channel is within 0.3. Phase (°) Am plitude (dB) Phase(°) Amplitude (dB) 204 Journal of Low Frequency Noise, Vibration and Active Control 37(2) 20 20 (a) (b) 0 0 -20 -20 Experimental frequency response -40 -40 Model frequency response Experimental frequency response -60 -60 Model frequency response -80 -80 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 0.4 0.35 Frequency response error 0.35 Frequency response error 0.3 0.3 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80 Frequcney (Hz) (c) (d) -10 -20 -20 -30 Experimental frequency response Experimental frequency response -40 -40 Model frequency response Model frequency response -50 -60 -60 -70 -80 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 0.4 0.35 Frequency response error Frequency response error 0.35 0.3 0.3 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80 Frequcney (Hz) Frequency (Hz) Figure 4. Model and experimental data fitting degree of each control channel. (a) Control channel G11, (b) control channel G14, (c) control channel G41, and (d) control channel G44. The frequency response is the evaluation index of the system identiﬁcation toolbox. The order and the ﬁtting degree of each control channel model are shown in Table 1. In Table 1, the order of each control channel model is 10 orders, which can meet the requirements of the model on the order, and the ﬁtting degree is more than 85%, which indicates a high degree of ﬁt between the model and the experimental data. The model can accurately describe dynamic characteristics of the controlled object in frequency range of 15–65 Hz. Amplitude(dB) Amplitude(dB) Amplitude(dB) Amplitude(dB) Amplitude(dB) Amplitude(dB) Amplitude(dB) Amplitude(dB) Song et al. 205 Table 1. Fitting degree of frequency response and order number. Control channel Order number Fitting degree of frequency response (%) G11 10 92.07 G14 10 91.74 G41 10 92.57 G44 10 86.96 The expressions of the continuous transfer functions for each control channel are shown in Appendix 1. e u K(s) G(s) Figure 5. Block diagram of feedback control for suppressing frequency-varying disturbance. H control theory of the floating raft vibration isolation system The feedback control method can eﬀectively suppress the frequency-varying disturbance, and the block diagram of feedback control is shown in Figure 5. The error signal e produced by the superposition of the disturbance signal d and the desired output signal y is detected and sent to the controller K(s); the controller calculates and outputs a control signal u according to the control law, and the control signal u goes through the control channel G (s) before generating the desired output signal y; the desired output signal and the disturbance signal cancel each other out to eliminate the frequency-varying disturbance. In case there is external disturbance force but without actuator control, the sensor measures the error signal e that is equal to the disturbance signal d. When the actuator controls but external disturbance force is absent, the sensor measures the error signal e that is equal to the desired output signal y. Uncertainty of the model and its description In the ﬂoating raft isolation system, due to the vibration isolators’ nonlinearity, the raft’s ﬂexibility, the parameter perturbation, unmodeled dynamics in high frequency, and the system identiﬁcation error, it is diﬃcult to establish a precise mathematical model. The diﬀerence between the mathematical model and the actual controlled object must be taken into consideration when designing the control system, so that the designed controller based on an error model can still guarantee the actual closed-loop system’s stability, achieving the desired designing goals. The diﬀerence between the mathematical model and the controlled object is called as model uncertainty, and the common description methods for model uncertainty are addition uncertainty description and multiplication uncer- tainty description, respectively, as shown in equations (1) and (2) GðsÞ GðsÞ¼ ðsÞ W ðsÞð1Þ GðsÞ GðsÞ ¼ ðsÞ W ðsÞð2Þ GðsÞ where GðsÞ is the actual controlled object, GðsÞ is the nominal model, ðsÞ is the uncertainty after normalized, W ðsÞ represents the addition uncertainty of model, and W ðsÞ represents the multiplication uncertainty of model. 2 3 Mixed sensitivity problem When designing the control system, it is necessary to not only consider the robust stability requirement of the closed-loop control system due to the diﬀerences between the mathematical model and the actual controlled object, but also take into account the performance requirement of the control system. The mixed sensitivity 206 Journal of Low Frequency Noise, Vibration and Active Control 37(2) problem is a typical designing problem for the H1 controller, which can transform both requirements into the optimization problem of the H norm when designing a control system. A typical block diagram of the mixed sensitivity problem is shown in Figure 5. Where, G is the nominal model, K is the controller to be designed, W , W , and W are three diﬀerent weighting functions; w is the input signal; z , z , and z are the weighted 1 2 3 1 2 3 outputs; y is the error signal measured by sensor; u is the control signal output from the controller. The area surrounded by the dashed box is augmented controlled object, including the controlled object G and three weighting functions. The mixed sensitivity problem can be understood as selecting three appropriate weighting functions and solving a controller K to make the system meet the robust stability and performance requirements, namely the controller K enables the establishment of formula (3) W S W =ð1 þ GKÞ 1 1 W R ¼ W K=ð1 þ GKÞ 1 ð3Þ 2 2 W T W GK=ð1 þ GKÞ 3 3 1 1 S is the sensitivity function, T is the complementary sensitivity function, R is the controller sensitivity function. Where,kk W S 1 represents the performance requirements,kk W R 1&kk W T 1 both represent the 1 2 3 1 1 1 robust stability requirements. Selection of the weighting function The solution of mixed sensitivity problem needs to be converted into a standard H control problem. Doyle et al. oﬀers a classic DGKF which is also called as the Algebraic Riccati equation RIC method to solve the standard H control problem and a central controller is designed by solving Riccati equations in this method. In order to solve mixed sensitivity problem requires three appropriate weighting functions W , W , and W should be 1 2 3 selected in advance, because the quality of these three weighting functions can directly aﬀect the designed con- troller’s capacity to meet the designing requirements. Currently, there have yet not been mature rules to select the weighting function, and cut-and-try method is mostly used. When designing the control system, ﬁrst select the weighting function W and W , followed by selection of the weighting function W . The weighting function W 2 3 1 2 and W are related to model uncertainties, including the absolute error and the relative error between the model response and the experimental frequency response. The weighting function W should be selected according to the frequency characteristics of external interference. In order to suppress the frequency-varying disturbance, whose dominant frequency ranges from 1 Hz, the amplitude of the performance-related sensitivity function S in dominant frequency must be less than 1. For the single-input single-output system, the system performance requirement can be described by equation (4) kk W S 1 ) Sð jwÞ W ð jwÞ ð4Þ 1 1 Equation (4) shows that as the sensitivity function S’s frequency characteristic Sð jwÞ must be located below the W ð jwÞ , the amplitude of W ð jwÞ in dominant frequency must be larger than 1. It can make the W ð jwÞ 1 1 1 present a peak shape in dominant frequency, thus the Sð jwÞ and the W ð jwÞ manifest themselves as a null in dominant frequency, which indicates that the entire feedback control system can highly suppress the disturbance signal containing the dominant frequency component. If only a dominant frequency is to be suppressed, W can be selected as a single peak shape with the peak value located at the dominant frequency, as shown in formula (5) and (6) W ¼ k ð5Þ 2 2 s þ 2ws þ w pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ w ¼ w 1 ð6Þ In equations (5) and (6), w is the dominant frequency which needs suppressing; k and are regarded as adjustment parameters when selecting W . 1 Song et al. 207 For a MIMO control system, if there is only a frequency-varying disturbances containing a dominant frequency component that needs suppressing, the selection method for weighting functions is as the same as the SISO control system, except that those three weighting functions need to become a corresponding diagonal matrix. H controller designed for a MIMO control system The W is chosen to describe the model uncertainty and its value is determined by the absolute error between the model frequency response and the experimental frequency response. The four identiﬁed channels’ model frequency responses and experimental frequency responses are limited from 0 to 60 Hz, and the absolute error are all less than 0.2. In order not to increase the order numbers for controllers and to obtain a larger robust stability, the optional weighting function W is assigned as 0.1 and W as 0. 2 3 The H1 controller is designed with the following steps: (I) Adjust the parameters and and tentatively give a weighting function W . By relying on the transfer function of the control channel and the weighting functions W , W , and W , call the hinfsyn () function in the 1 2 3 MATLAB software to solve the H1 controller K. If there is no solution to the mixed sensitivity problem, reduce the parameter k appropriately. After k is reduced, if the mixing sensitivity problem remains unsolved, which may be due to the excessive weighting function W , reduce the amplitude of W and reexecute step (I). 2 2 (II) Verify whether the closed-loop control system is inherently stabile, namely to check whether the designed controller and closed-loop control system are stable. If the control system is not inherently stable, proceed from step I and modify the selection parameters of the weighting function W appropriately. If the system is inherently stable, proceed to the next step. (III) Check Bode Plots regarding the sensitivity function of the closed-loop control system to determine whether the system meets the performance requirements. If not, go back to step (I); if yes, end the controller design. The weighting function W is selected according to formula (5), and the W is determined by the dominant 1 2 frequency of disturbance signal. Assume that a disturbance whose dominant frequency varies close to 30 Hz is to be suppressed and the variation range of frequency is no more than 1 Hz. w can be assigned to 2 30 ¼ 188:4 rad=s. According to formula (5) and (6), by conducting a series of spreadsheets, and in order to ensure the stability for the controller K and the closed-loop control system under stable conditions, the weighting functions eventually are identiﬁed as "# s þ0:1885sþ35530 W ¼ s þ0:1885sþ35530 ð7Þ 0:10 W ¼ 00:1 W ¼ 0 The mathematical model of the control channel is a two-input two-output transfer function matrix made up of four channels. According to the mathematical model G (s) of the control channel, the weighting functions W , W , 1 2 and W , and the hinfsyn () function in MATLAB can be called to solve the robust controller K. The controller K is a two-input two-output state-space equation, and the dimension of the matrix A is 44. Due to too large controller order, it is not conducive to the control algorithm, and thus the controller order should be reduced. The bstmr () function in MTALB can be used to complete order reduction. Under the condition that there is little change of the dynamic characteristics of the controller K, the dimension of the matrix A is reduced to 4. The state-space expression of the controller after reducing order follows as formula (8) 2 3 0:05745774583 187:8103102 16:00317480 8:547472727 6 7 187:5005506 0:1353067232 8:501991657 16:05663768 6 7 A ¼ ð8Þ 6 7 4 5 15:53041937 8:236529397 0:08355357455 188:4515155 8:196244109 15:44411324 186:8724015 0:06379850336 208 Journal of Low Frequency Noise, Vibration and Active Control 37(2) 2 3 2:724889168 0:8050106367 6 7 1:590320803 0:4766626217 6 7 B ¼ 6 7 4 5 1:085861943 3:082330315 0:2714032574 0:7586295242 0:5600031142 2:861390715 0:3605602097 1:693404853 C ¼ 1:494554152 0:3491156794 1:716977793 2:405479480 D ¼ where A represents the system state matrix, B represents the control input matrix, C represents the system output matrix, and D represents the input and output correlation matrix. With the same method above, the weighing functions and the controller K can be designed when the disturbance whose dominant frequency varies close to 35 Hz. The weighing functions and the controller K are deﬁned as "# s þ0:2199sþ48360 W ¼ s þ0:2199sþ48360 ð9Þ 0:10 W ¼ 00:1 W ¼ 0 2 3 0:1140915794 218:7348982 16:01016303 17:27993062 6 7 6 218:5730602 1086776480 17:26447051 16:00866239 7 6 7 A ¼ 6 7 15:89387633 17:22096667 0:1291900427 218:7462161 4 5 17:19691173 15:89185783 218:5563591 0:080873009323 2 3 2:335157528 0:03278774785 6 7 6 2:208785663 0:5466691529 7 6 7 B ¼ ð10Þ 6 7 0:43824719078 2:675072276 4 5 0:3265658986 0:3694493762 "# 0:30425 3:1997 0:17276 0:53507 C ¼ 0:031683 0:15135 1:93023 2:5354 "# D ¼ Upon examination, the designed control system meets the internal stability condition, and the shaping design results based on the mixed sensitivity problem is shown as Figure 6. As shown in Figure 7, there are two curves for the singular value of the sensitivity function S in the closed-loop control system, and the two curves are below the 1 1 singular value of the function W , and the singular values of the sensitivity function S and the function W at 1 1 the frequency 30 Hz nearby present concave shapes, which indicates that the closed-loop control system can eﬀectively suppress frequency-varying disturbance with dominant frequency of 30 Hz. In Figure 8, we can con- clude that control system can eﬀectively suppress frequency-varying disturbance with dominant frequency of 35 Hz in the same way. The simulation block diagram for a MIMO feedback control system is shown in Figure 9. Sweep signal is used to simulate frequency-varying disturbance, and the sweep signal 1 and the sweep signal 2 are, respectively, used as the disturbance signal 1 and the disturbance signal 2, of which the duration are both 5 s. To distinguish two disturbance signals, one of sweep signals goes through a gain of 0.8 narrow links. The simulation results are shown in Figure 10, where Figure 10(a) is the time-domain diagram and self-power spectrum for the error signal 1 before and after control, and Figure 10(b) is the time-domain diagram and power Song et al. 209 Augmented controlled object Z1 w + Z2 Z3 G W uy Figure 6. Block diagram of the mixed sensitivity problem. σ (S) - 1 σ (W ) 60 1 -20 -40 10 20 30 40 50 60 70 80 Frequency (Hz) Figure 7. Shaping design result for 30 Hz frequency-varying disturbance. (S) -1 (W ) -20 -40 10 20 30 40 50 60 70 80 Frequency (Hz) Figure 8. Shaping design result for 35 Hz frequency-varying disturbance. Singular Values ( dB) Singular Values (dB) 210 Journal of Low Frequency Noise, Vibration and Active Control 37(2) Figure 9. Robust control for a MIMO control system. MIMO: multiple input multiple output. Figure 10. Time-domain diagram and self-power spectrums of the error signals of 30 Hz. spectrum for the error signal 2 before and after control. As it can be seen, the magnitudes of two error signals are reduced by more than 50%, and the amplitudes of the error signal 1 and the error signal 2 in 30 Hz have fallen by about 15 dB. This indicates that the control system can eﬀectively suppress frequency-varying disturbance with dominant frequency of 30 Hz. In order to verify universality of the H controller designing method, another controller is designed for frequency-varying disturbances with dominant frequency of 35 Hz, which is shown in Figure 11. The simulation results show that when the disturbance signal’s frequency varies around 35 Hz, the dominant frequency’s magni- tudes of the error signal 1 and the error signal 2 fall by about 10 dB. It shows that the designed system can eﬀectively suppress frequency-varying disturbance whose dominant frequency varies around 30 and 35 Hz. Song et al. 211 Figure 11. Time-domain diagram and self-power spectrums of the error signals of 35 Hz. 1# Power Amplifier 1# Actuator U1 E1 1# Acceleration dSPACE1103 4# Acceleration E4 Conditioning Low pass filter Amplifiers U4 4# Power Amplifier 4# Actuator Figure 12. Schematic diagram of the MIMO feedback control system. MIMO: multiple input multiple output. H control experiments for a MIMO control system The diagram of MIMO feedback control system is shown in Figure 12, the system is a two-input two-output feedback control system. The voltage signal proportional to the acceleration signal from the low-pass ﬁlter can be regarded as error signals, as shown in E1 and E4 of Figure 11. E1 represents the voltage signal from the condi- tioning and ﬁltered 1# acceleration signal, and E4 represents the voltage signal from the conditioning and ﬁltered 4# acceleration signal. The ﬁlter output port connects with the A/D port of the dSPACE1103 system’s wiring panel CLP1103. The control board DS1103 is a kind of controller hardware, depending on the designed controller to calculate the control amount. The calculation for controller adopts ﬁxed-step calculation method, of which the 212 Journal of Low Frequency Noise, Vibration and Active Control 37(2) 0.6 0.6 E4 E1 0.4 0.4 0.2 0.2 0 0 -0.2 -0.2 -0.4 -0.4 Without control Robust control Without control Robust control 0 2 4 6 8 10 0 2 4 6 8 10 1.5 2 U1 U4 0.5 0 0 -0.5 -1 -1 -1.5 -2 0 2 4 6 8 10 0 2 4 6 8 10 Time (s) Time (s) Figure 13. Time-domain diagram of error signals and control signals when the speed of motor is 1800 r/min. (a) 0 (b) 0 Without control Without control -10 -10 Robust control Robust control -20 -20 -30 -30 -40 -40 -50 -50 -60 -60 -70 -70 -80 -80 -90 -100 -90 0 20 40 60 80 0 20 40 60 80 Frequency(Hz) Frequency(Hz) Figure 14. Self-power spectrum of the error signals of 30 Hz. step is 0.0001 and the solver is ode8. After crossing D/A converter, the control amount outputs two control signals, namely U1 and U4 as shown in Figure 11. The amplitudes of the two control signals are conﬁned to 5 V. The control signal U1 goes across 1# power ampliﬁer, and then drives l# electromagnetic actuator to output control force; after going across 4# power ampliﬁer, the control signal U4 drives 4# electromagnetic actuator to output control force. On the coordinative role of the two actuators, the vibration from frequency ﬂuctuation disturbance generated by the motor above the 1#actuator can be inhibited. In the experiments, two diﬀerent controllers are designed to suppress the disturbances with dominant frequency of 30 and 35 Hz. When the frequency is 30 Hz, we choose the speed of the motor above 1# actuator 1800 r/min, when the frequency is 35 Hz, we choose the speed of the motor above 1# actuator 2100 r/min. When the speed of the motor above 1# actuator is 1800 r/min, the active vibration control experiment is conducted according to Figure 11, and experimental results are shown in Figures 13 and 14, respectively. Without control, the two control signals are 0, and the amplitudes of the error signals are approximately 0.5 and 0.3 m/s . After the robust control, two control signals gradually increase and eventually approach a steady Amplitude(dB) Voltage (V) Acceleration(m/s ) Amplitude(dB) Acceleration(m/s ) Voltage (V) Song et al. 213 E1 E4 0.5 0.5 -0.5 -0.5 Without control Robust control Without control Robust control -1 -1 0 2 4 6 8 10 0 2 4 6 8 10 Frequency (Hz) U1 U4 -1 -1 -2 -2 0 2 4 6 8 10 0 2 4 6 8 10 Time (s) Time (s) Figure 15. Time-domain diagram of error signals and control signals when the speed of motor is 2100 r/min. 0 0 (a) (b) Without control Without control -10 Robust control Robust control -20 -20 -30 -40 -40 -50 -60 -60 -80 -70 -80 -100 -90 -100 -120 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 Frequency (Hz) Frequency (Hz) Figure 16. Self-power spectrum of the error signals of 35 Hz. state; the two error signals achieve fast convergence, and their amplitudes are greatly reduced. In the interception map, the signals are used as error signals without control from 0 to 5 s and the signals are used as error signals with robust control from 5 to 11 s. The RMS values of error signal E1 before and after the control are, respectively, 0.3689 and 0.0823, so the RMS value is decreased 77.68%. The RMS values of error signal E4 before and after the controls are, respectively, 0.2597 and 0.0659, which indicates a 74.64% decrease in the RMS value. Figure 14(a) and (b), respectively, represent the self-power spectrum of the error signals E1 and E4 before and after the control, where the amplitudes of the frequency components in the range of about 30 Hz within the error signals E1 and E4 are greatly reduced by 13.91 and 13.16 dB, respectively, while other frequency components show no signiﬁcant changes. When the speed of the motor above 1# actuator is 2100 r/min, experimental results are shown in Figures 15 and 16, respectively. After the robust control, two control signals gradually increase and eventually approach a steady state; the two error signals achieve fast convergence, and their amplitudes are greatly reduced. In the interception map, the signals are used as error signals without control from 0 to 4 s and the signals are used as error signals with robust control from 4 to 10 s. The RMS values of error signal E1 before and after the control are, Amplitude(dB) Acceleration(m/s ) Voltage(V) Amplitude (dB) Acceleration(m/s ) Voltage(V) 214 Journal of Low Frequency Noise, Vibration and Active Control 37(2) respectively, 0.6258 and 0.0614, so the RMS value is decreased 90.19%. The RMS values of error signal E4 before and after the control are, respectively, were 0.3369 and 0.0335, which indicates a 90.06% decrease in the RMS value. Figure 16(a) and (b), respectively, represents the self-power spectrum of the error signals E1 and E4 before and after the control, where the amplitudes of the frequency components in the range of about 30 Hz within the error signals E1 and E4 are greatly reduced by 22.48 and 33.88 dB respectively, while other frequency components show no signiﬁcant changes. Conclusion The system identiﬁcation method is adopted to establish two-input and two-output continuous transfer function model of the control channel for the ﬂoating raft vibration isolation system, and the model can accurately describe the dynamic characteristics of the controlled object. The H controller is designed with the shaping design method based on mixed sensitivity problem and the selection method of weighting function is described. The H controller is designed for a MIMO system to suppress single frequency-varying disturbance. Simulation analysis is conducted against the designed controller, and the results show that the amplitudes of the dominant frequencies are atte- nuated more than 10 dB before and after the robust control. An actual ﬂoating raft vibration isolation system is built for the single excitation source multichannel control experiments. Each experiment achieves good vibration isolation eﬀect, the RMS values of error signals drop by more than 74% before and after the control, and the amplitudes of the dominant frequency have fallen more than 13 dB. Experimental results show that the controller can eﬀectively suppress frequency-varying disturbance. Declaration of conflicting interests The author(s) declared no potential conﬂicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) disclosed receipt of the following ﬁnancial support for the research, authorship, and/or publication of this article: This study is supported by National Natural Science Funds of China (51205296, 51275368). References 1. Wei C, Wang Y, Chen S, et al. Research on the control of electromagnetic suspension vibration isolator based on block normalized LMS algorithm. J Vib Shock 2012; 31: 100–103. 2. Yang T, Li X, Zhu M, et al. 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Appendix 1 2 2 15:976ðs 60:93Þðs þ 76:76s þ 21930Þðs 399:6s þ 104100Þ G ðsÞ¼ 2 2 2 ðs þ 17:45s þ 12330Þðs þ 17:39s þ 29680Þðs þ 43:07s þ 82190Þ 2 2 ðs þ 125:9s þ 74800Þðs 90:76s þ 185700Þ 2 2 ðs þ 60:05s þ 105100Þðs þ 23:93s þ 151800Þ 2 2 5:8297ðs 1316Þðs 64:14s þ 12880Þðs þ 132:6s þ 49340Þ G ðsÞ¼ 2 2 2 ðs þ 19:65s þ 14040Þðs þ 19:52s þ 29150Þðs þ 37:18s þ 89980Þ 2 2 ðs 237:6s þ 95770Þðs 63:68s þ 167300Þ 2 2 ðs þ 80:69s þ 120600Þðs þ 25:05s þ 146500Þ 2 2 9:7756ðsþ640:4Þðs 61:88s þ 12330Þðs þ 161:2s þ 627200Þ G ðsÞ¼ 2 2 2 ðs þ 15:68s þ 14140Þðs þ 20:54s þ 28290Þðs þ 44:91s þ 89690Þ 2 2 ðs 235:5s þ 85160Þðs 75:66s þ 166200Þ 2 2 ðs þ 49:7s þ 110100Þðs þ 20:93s þ 146900Þ 2 2 14:248ðs 107:5Þðs þ 104:1s þ 21920Þðs þ112:7s þ 63420Þ G ðsÞ¼ 2 2 2 ðs þ 19:84s þ 11450Þðs þ 15:97s þ 29870Þðs þ 48:55s þ 78030Þ 2 2 ðs 484:8s þ 124700Þðs 118:8s þ 192400Þ 2 2 ðs þ 56:47s þ 105300Þðs þ 19:31s þ 151600Þ

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"Journal of Low Frequency Noise, Vibration and Active Control" – SAGE

Published: Aug 21, 2017

Keywords: Active control; frequency-varying disturbance; system identification; H∞ control

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H∞ active control of frequency-varying disturbances in a main engine on the floating raft vibration isolation system:

H∞ active control of frequency-varying disturbances in a main engine on the floating raft vibration isolation system:

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H∞ active control of frequency-varying disturbances in a main engine on the floating raft vibration isolation system:

H∞ active control of frequency-varying disturbances in a main engine on the floating raft vibration isolation system:

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