Robust Acceleration Control of Electrodynamic Shaker Using µ Synthesis

Transcription

1 Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 5 Seville, Spain, Deceber -5, 5 WeIC8. Robust Acceleration Control of Electrodynaic Shaker Using µ Synthesis Yasuhiro Uchiyaa, Masakazu Mukai, and Masayuki Fujita Abstract This paper presents an acceleration control of an electrodynaic shaker that cannot eploy the iteration control ethod freely. There are considerable points in the design phase of the acceleration controller. The transfer functions using the acceleration controlled variable have zero at the origin and two crossover frequencies. It is not easy to design the acceleration controller by using the conventional design ethod directly. The controller is designed by using µ synthesis based on these points. Further, a two degrees of freedo controller is designed to iprove the transient response. Finally, the experient using an actual electrodynaic shaker is carried out. A coparison between the proposed controller and the controller derived fro the iteration control ethod is shown. I. INTRODUCTION Recently, ulti-axis shaking tables which ore accurately can replicate an actual situation are being widely used. For instance, the shaker is used in autootive industry, civil and architectural engineering, etc. An electrohydraulic shaker was generally used for a large-scale ulti-axis shaking table. An electrodynaic shaker has soe good features such as good linearity and a wide-frequency response. A ore accurate vibration test can be realized when the ulti-axis shaking table is coprised by electrodynaic shakers. Further, by having the feature of the wide-frequency response, acceleration is generally used to control the electrodynaic shaker. Then, the iportance of the acceleration control is ephasized. The controller of the ulti-axis shaker is required to provide not only stable control but also a good replication of the given reference wavefor as a response wavefor to the oveent associated with the shaking table. Conventional controllers eploy the open-loop ethod by using iterative copensation through repetitive excitations. Since the daage for a specien is accuulated at each iteration before the execution of the desired excitation, the application of the iterative copensation is undesirable. Furtherore, in the case that the test piece is easy to break, the iteration control cannot be applied. Thus, a new ethod without iteration control recently attracts attention. On the other hand, an experiental ethod, like the siulation of soilstructure interaction effects [], in which an actual vibration test and a coputer siulation are cobined, recently tends to increase. In this ethod, it is necessary to control a Y. Uchiyaa is with IMV CORPORATION, Osaka 555-, JAPAN uchiyaa@iv.co.jp M. Mukai is with the Departent of Electrical and Electronic Systes Engineering, Graduate School of Inforation Science and Electrical Engineering, Kyushu University, Fukuoka 8-858, JAPAN M. Fujita is with the Departent of Mechanical and Control Engineering, Tokyo Institute of Technology, Tokyo 5-855, JAPAN Fig.. Electrodynaic shaker with a specien (VEO-3). shaker in real tie without delaying the coputer siulation. However, the conventional controller of an electrodynaic shaker is based on a feedforward control using an FFT and can update only the drive signal per frae tie of the FFT. Hence, it is difficult to eploy the conventional controller in this ethod. In the exaples of the control applications for the electrodynaic shaker, an adaptive inverse control has been successfully applied for the shock testing [], and the ipleentation of the current and acceleration controllers can achieve good rando vibration control [3]. However, since the test applications are different fro this case, the approaches are not appropriate in this case. In contrast, there are soe instances with the electrohydraulic shaker. For exaple, the MCS (Minial Control Synthesis) ethod, one of the odel reference adaptive control ethods, has successfully been applied [4], the application of H control has extended the frequency response of shaking table [5], and the copensator by an adaptive filter has iproved the control results, the real-tie copensator of a reaction force has cancelled out its influence [6]. In our previous studies [7], [8], a good perforance is achieved for the control of the electrodynaic shaking syste. Since the design is for general case and there are abundant exaples of the control application, the displaceent is used as the controlled variable in [7], [8]. However, the vibration test cannot be executed at higher frequency band which is the advantage of the electrodynaic shaker and the prospective application using the displaceent control is restricted. In this paper, an acceleration control of the electrodynaic shaker (Fig. ) is presented. The final purpose of this study is to construct the controller of ulti-axis shaking syste. However, in this paper, due to ainly design and evaluate /5/$. 5 IEEE 67

2 Iaginary Acc. controlled syste HPF General controlled syste 3 3 Real Fig.. Nyquist diagra of general control and acceleration control in lower frequency. the acceleration controller, a single-axis copact shaker is used. There are considerable points in the design phase of the acceleration controller. The transfer functions using the acceleration controlled variable have zero at the origin and two crossover frequencies. The controller is designed by using µ synthesis based on these points. The experient using an actual electrodynaic shaker is carried out to illustrate the effect of the proposed controller. Then a coparison between the proposed controller and the controller derived fro the iteration control ethod is shown based on these experiental results. II. THE CONTROL PROBLEM In acceleration control, a few attention should be paid to the following points: ) The controlled syste has zero at s =. ) There exist two crossover frequencies. 3) The response in lower frequency is influenced by noise. In the ite ), to satisfy the assuptions of the standard H control proble [9], it is eant that a weighting function and a constitution of the generalized plant is constrained. Nyquist diagra of two systes in lower frequency is shown in Fig.. In the general controlled syste, the locus of the open-loop characteristic starts fro the point greater than on the real axis, and the syste has one crossover frequency. However, in the case of the acceleration controlled syste to which is added the high pass filter characteristic like the AC coupling, the locus is drawn as shown in Fig., and it is found that the syste has two crossover frequencies. Therefore, it is needed that the stability argins at lower and higher crossover frequency are considered as the ite ). Generally, the noise in higher frequency band is cared when a controller is designed. Since the gain of the acceleration controlled syste is sall in lower frequency, the proble of the ite 3) is occurred easily. Thus, the noise in lower frequency band should be cared, too. III. ELECTRODYNAMIC SHAKER MODEL It is difficult to express the precise characteristic of an actual plant by using a atheatical odel, and soe uncertainties are generally present. In this section, a noinal odel of an electrodynaic shaker and an uncertainty weighting function are introduced. Fig. 3. Kd Cd Fs xa E R I L Ec M L R I Scheatic diagra of the arature and the equivalent circuit. A. Noinal odel The plant to be controlled is an electrodynaic shaker depicted in Fig.. The electrodynaic shaker is based on the principle that an electrodynaic force is generated in proportion to an electric current applied to the coil existing in the agnetic field. The siplified odel of the electrodynaic shaker can be shown as a vibration syste with a single degree of freedo. Assuing that the agnetic flux density is constant, a drive coil can be shown as a linear equivalent circuit. The scheatic odel of the shaker and the equivalent circuit are depicted in Fig. 3. The force F s and the reverse electrootive force E c can be represented as F s = BlI () E c = Blẋ a, () where x a denotes the displaceent of the arature, I denotes the current of the drive coil, l denotes the length of the drive coil, and B denotes the agnetic flux density. Fro Fig. 3, the following (3) and (4) can be obtained E = R I L I E c MI (3) =R I L I MI, (4) where M denotes a relative inductance, I denotes the current of the copper ring, L, denote an inductance, R, denote a resistance, and E denotes the input voltage to the drive coil. The dynaic equation of the arature can be obtained as F s = ẍ s C d ẋ a K d x a, (5) where K d denotes the stiffness coefficient of the suspension, and C d denotes the daping coefficient of the suspension. Considering the voltage u a as input and the acceleration ẍ a as output, the characteristic of the electrodynaic shaker can be represented as the following state-space for ẋ = A a x B a u a, y a = C a x a, x =[x a ẋ a I I ], A a = K d C d Bl BlL L L M RL MR L L M L L M B a = [ C a = [ K d BlM L L M MR L L M LR L L M ] T G al L L M GaM L L M, C d Bl ], where G a denotes the aplifier gain., (6) 67

3 Phase [deg] Fig. 4. Bode diagra of the shaker syste (solid line: siulated value, dashed line: easured result). TABLE I PERTURBED PARAMETERS. Sybol Perturbation region R -4 % L -3 3% B - % K d -5 5% C d - % - % R - % L -3 3% In Fig., a specien constructed by a 97 g weight and rubber springs are located on an arature. The firstorder resonant frequency of the specien is approxiately 3 Hz. Due to the influence of the resonance, a peak notch appeared in the transfer characteristic of the shaker in the neighborhood of the resonant frequency. In this paper, the influence of the specien is approxiated by the following (7). G e = s ζ ω s ω s ζ ω s ω, (7) where ω =π.8, ζ =.5, ω =π 4, ζ =.4. Although the peak notch characteristic at Hz and 55 Hz can be represented by using the for (7), these characteristics are considered as an uncertainty. Add a high pass filter that has the cutoff frequency.5 Hz as the AC coupling. The bode diagra of the odel is shown in Fig. 4. In Fig. 4, the solid line denotes the siulated value and the dashed line denotes the easured result. In addition, the characteristic of an antialiasing filter and the AC coupling of the aplifier are added to the transfer function (6) and (7). Then the resulting odel is defined as the noinal odel G s. B. Modeling uncertainty In this study, the paraeter perturbations of the shaker and the influence of the peak notch characteristic which is neglected are considered as an unstructured additive perturbation. To estiate the quantities of additive odel Fig. 5. Uncertainty weighting W d and additive perturbations (solid line: W d, dotted line: paraeter perturbation, dashed line: influence of the peak notch characteristic). ~ Gs Wd d Ws r Kb Gs es - Fig. 6. Feedback structure. perturbations, the differences between the noinal transfer function and the perturbed transfer functions are calculated when only one paraeter is changed and the others are fixed. The paraeter perturbations are shown in Table I. According to the perturbation, 6 perturbed odels can be obtained. In addition, the influence of the peak notch characteristic can also be obtained by the sae ethod. Frequency responses of these additive perturbations are plotted in Fig. 5. Here, the agnitude of the uncertainty weighting function W d is chosen to cover all the odel perturbations as follows: W d =.3(s 6) s 4 ya s 7 s. (8) Also, it is assued that uncertainties which cannot be considered exist, and hence the agnitude of the weighting function is enlarged in the lower and higher frequency band. Since an additive perturbation is used and W d is proper, it is noted that the assuptions of the standard H control proble is satisfied. IV. CONTROLLER DESIGN The design of the controller of an electrodynaic shaker is carried out using MATLAB. A. Control objectives Let us consider the feedback structure shown in Fig. 6. In Fig. 6, r and e s denote the reference and the control error. The block K b represents the controller. The following ites are set as control objectives. Stabilize the syste even when an uncertainty exists. Maintain the perforance wherein the reference signal is replicated well. 67

4 Fig. 7. Weighting function W s and W u. (solid line: W s, chained line: W u) u w Gs - Kb Wu Wd Ws P y z Fig. 8. Generalized plant P A controller is designed to aintain a robust stability against the uncertainty odel. Moreover, it is desired that a controller aintains a good perforance despite this uncertainty. Thus, the controller for this syste is designed by µ synthesis, which has any applications like []. A DOF controller is fabricated to iprove the transient response. B. Choice of weights and µ synthesis A weighting function W s for the replication perforance of the reference signal is considered. It is needed that the gain of the control frequency band is enlarged to iprove the control perforance in the frequency band. In the acceleration control, it is noticed that the characteristic like a high pass filter is included, because this plant has zero at s =. According to the above points, W s is now chosen as α π W s = s π 6 π s 6 π π s π s s 4 π s s 5 π s s π, (9) where, α is the adjustent paraeter. α can be enlarged as long as the uncertainty is satisfied. Here, the adjustent paraeter α is set at.. The frequency response of the perforance weighting function W s is shown by the solid line in Fig. 7. Practically, easuring acceleration ay be influenced by noise ore strongly. Hence the effect of the noise should be considered in acceleration control. Particularly, it is desired to ake the agnitude of the controller low at lower frequency in this paper case. To keep the controller gain low at lower frequency, the weighting function W u is chosen as s.5 π W u =. s. π. () The frequency response of W u is shown by the chained line in Fig. 7. First define a block structure as := { diag ( d, perf ), d C, perf C }, () where d, perf, and perf is a fictitious uncertainty block for considering robust perforance. Next, it is considered that P is partitioned as [ ] P P P =. () P P Fig. 9. Frequancy response of the controller K b (solid line: W u is set as (), dashed line: W u =.). Fro Fig. 8, the linear fractional transportation on P by K b is defined as follows: F l (P, K b ):=P P K b (I P K b ) P. (3) The robust perforance condition is equivalent to the following structured singular value µ. sup µ (F l (P, K b )(jω)) < (4) ω R Since the controller satisfies this condition, the D-K iteration procedure is eployed. The controller which satisfies (4) is obtained after 4 iterations, and the degree of this controller has been reduced fro 9 states to 7 states. Fig. 9 shows the frequency response of the controller K b. Also, assuing that the restriction of the control input is eased, the other controller is designed with setting W u which is set as., and the characteristic is shown by the dashed line in Fig. 9. Coparing to both controllers, the effect of () can be seen. C. Two degrees of freedo design To iprove the syste response, here a two degrees of freedo controller shown in Fig. will be eployed. In Fig., the block F d denotes the reference odel. In order to obtain the precise replication in the frequency band in which an accurate control perforance is desired, an appropriate characteristic is selected as F d. It is noticed that G s F d should be proper. Furtherore, since the agnitude of G s F d is not enlarged too uch, it is also desired that the agnitude of the reference odel F d is sall at lower 673

5 6 4 r Fig.. us F d /G ~ s G s F d K b es - ya Two degrees of freedo structure... Fig.. Frequancy response of G s F d. frequency in the acceleration feedback. Then using both high-pass and low-pass filters, F d is chosen as s 3 ( ) 3 F d = s 3 π (π) (π) 3. s 7 π (5) The frequency response of G s F d is shown in Fig.. The gain of G s F d is contained up to 4 db. V. EXPERIMENTAL RESULT The control perforance is confired by an experient. The experient is executed by the syste described in the section. To ipleent the controller with the processing board, the controller is discretized via the Tustin transfor at the sapling frequency of 5 Hz. A. Static perforance The frequency characteristic is easured to evaluate the control perforance in the static case. In this experient, a rando wavefor is inputted to the controller, and the transfer function of the closed-loop characteristic is easured by the FFT analysis. The easured transfer function is copared to the siulation result. The results of the closed-loop characteristic are shown in Fig.. The DOF controller shown in Fig. is used in this experient. Since the noinal odel differs fro the actual plant, the experiental result cannot yield a siilar result with the siulation at high frequency band over 9 Hz and low frequency band up to Hz. However, the good result is obtained in the controlled frequency band. It appears that the design specification is satisfied in the static case. B. Transient perforance This experient is executed by an excitation using a easured wavefor data as the reference signal. The control perforance is evaluated by the result of the transient response during the excitation. The original wavefor of this Phase [deg] Fig.. Bode diagra of the closed-loop characteristic (solid line: experiental result, dashed line: siulated value) Acc. [/s ] Acc. [/s ] Fig. 3. signal. 3 3 Error Peak error :.4 [/s ] Rs error : 5.4 [%] Tie [s] Control results of the conventional controller with the first drive reference is easured at the oving vehicle. For aking the reference fro the original, the level of the original adjusted to be able to excitation with this shaker and the doinant frequency band are fitted to the assued frequency band of the control. To copare a conventional ethod with the proposal ethod, an open-loop control with the iterative excitation is also used in the experient. The iteration process of this conventional ethod is suarized as follows: ) The first drive is calculated fro the easured transfer function. ) The excitation is executed, and an error signal is obtained. 3) The next drive signal is coputed fro the previous drive and the error signal. 4) Go back to ). First, the conventional control is considered. the control results of using first drive signal are shown in Fig. 3 and those after the 4th iteration are shown in Fig. 4. The upper figure represents the reference signal which is denoted by 674

6 Acc. [/s ] Acc. [/s ] 3 3 Error Peak error :.5 [/s ] Rs error : 6.4 [%] Tie [s] Fig. 4. Control results of the conventional controller after the 4th iteration. Acc. [/s ] Acc. [/s ] 3 3 Error Peak error :.56 [/s ] Rs error : 9.7 [%] Tie [s] Fig. 5. Control results of the DOF controller. the dashed line and the response signal which is denoted by the solid line. The lower figure shows the error signal. Because of the influence of the nonlinear perturbation of the syste, it is difficult for the control using first drive signal to yield good results and the tracking error is enlarged, as shown in Fig. 3. In Fig. 4, since the drive signal is updated by the 4th iterative copensation, the wavefor replication can yield a good perforance as copared to Fig. 3. Next, the proposed ethod is considered, the control result is shown in Fig. 5. It is shown that the response wavefor yields siilar results with the reference. Coparing with the results of Fig. 4, the tracking perforance at the first excitation of the DOF controller obtains good results as well as the results after the 4th iteration of the conventional ethod. The power spectru density of this control results is shown in Fig. 6. As the frequency response, the response spectru yields siilar results with the reference, and it appears that the DOF controller yield a good perforance in the desired frequency band. Also, the rs error (= rs value of the error wavefor / rs value of the reference wavefor %) and the peak error are written in the control results respectively. It is found that both errors of the DOF controller are nearly equal to the results of Fig. 4. These results support the conclusion that this ethod is PSD [(/s ) /Hz] Fig. 6. Power spectru density of the DOF control results. useful. VI. CONCLUSIONS In this paper, the acceleration control of the electrodynaic shaker was presented. The case in which the conventional ethod using iterative copensation cannot be eployed was considered. The proposed controller was designed by using µ synthesis based on the reark points. Further, a two degrees of freedo controller was also designed to iprove the transient response. The experient using an actual electrodynaic shaker was shown. Then a coparison between the proposed controller and the conventional controller was shown. These results support the conclusion that the proposed acceleration controller is especially useful in the case. Although it is needed to take into account an influence of elastic odes on the shaking table in the controlled frequency band, it appears that the application of this ethod to a ulti-axis shaking table is available. REFERENCES [] K. Konagai and R. Ahsan, Siulation of nonlinear soil-structure interaction on a shaking table, Journal of Earthquake Engineering, vol. 6, no., pp. 3 5,. [] A. M. Karshenas, M. W. Dunnigan, and B. W. Willias, Adaptive inverse control algorith for shock testing, IEE Proceedings of Control Theory and Applications, vol. 47, no. 3, pp ,. [3] C. M. Liaw, W. C. Yu, and T. H. Chen, Rando vibration test control of inverter-fed electrodynaic shaker, IEEE Transactions on Industrial Electronics, vol. 49, no. 3, pp ,. [4] D. P. Stoten and E. Goez, Recent application results of adaptive control on ulti-axis shaking tables, in Proc. of the 6th SECED International Conference, Seisic Design Practice into the Next Century, 998, pp [5] A. Maekawa, C. Yasuda, and T. Yaashita, Application of H control to a 3-D shaking table (in Japanese), Transactions of the SICE of Japan, vol. 9, pp. 94 3, 993. [6] Y. Dozono, T. Horiuchi, T. Konno, and H. Katsuata, Shaking-table control by real-tie copensation of the reaction force caused by a nonlinear specien, Transactions of the ASME Journal of Pressure Vessel Technology, vol. 6, no., pp. 7, 4. [7] Y. Uchiyaa and M. Fujita, Application of two-degree-of-freedo control to electro-dynaic shaker using adaptive filter based on H filter, in Proc. of the 7th European Control Conference, Cabridge, UK, Sept. 3. [8] Y. Uchiyaa, M. Mukai, and M. Fujita, Robust control of ulti-axis shaking syste using µ-synthesis, in Proc. of the 6th IFAC World Congress on Autoatic Control, Prague, Czech, July 5. [9] K. Zhou and J. C. Doyle, Essentials of Robust Control. Upper Saddle River, NJ: Prentice Hall, 998. [] M. Fujita, T. Naerikawa, F. Matsuura, and K. Uchida, µ-synthesis of an electroagnetic suspension syste, IEEE Transactions on Autoatic Control, vol. 4, no. 3, pp ,