Proceedings of the European Control Conference 2007 Kos, Greece, July 2-5, 2007
ThC10.3
Robust Control of Electrodynamic Shaker with Disturbance-Force Compensator Yasuhiro Uchiyama and Masayuki Fujita Abstract— This paper presents a disturbance-force compensator for an electrodynamic shaker. The characteristic of a shaking system is considered to be nonlinear and variable because of the influence of the test piece. In order to compensate for this problem, the influence of the disturbance force needs to be suppressed. It is assumed that the disturbance force is measured in the form of a controlled variable, and the controller is designed using µ–synthesis by considering the uncertainty of the shaker. Finally, the experiment using the electrodynamic shaker is executed. A good performance is achieved for the variation in the characteristic of the controlled plant.
Acceleration Rubber spring pickup Test piece
Armature Laser displacement sensor
I. INTRODUCTION
Fig. 1.
Recently, shaking systems that can more accurately replicate an actual situation have been widely employed for vibration testing. For instance, shaking systems are used in the automotive industry, civil engineering fields, and so on. An electrodynamic shaker has some worthwhile features such as good linearity and wide frequency response. The shaking system needs to be controlled for not only exercising stable control but also better replication of the given reference waveform. However, if the interaction between the shaker and the test piece cannot be ignored, the control of the shaking system becomes difficult. Further, if the characteristic of a controlled plant is nonlinear because of the influence of the test piece, the control becomes difficult in a similar manner. With regard to electrodynamic shakers, their influence is generally prominent in comparison with electrohydraulic shakers. There have been some instances [1], [2] in which electrohydraulic shakers have been successfully employed. In our previous studies [3], [4], a good performance was achieved. However, it has been assumed that the test piece is invariant and the perturbation of the controlled plant is small. If the characteristic of the specimen is variable because of influences such as damage during vibration testing, the plant perturbation assumes more significance. Because of this problem, the system stability can be compensated, although the control performance cannot satisfy the test specifications. In the case of multiple plants, a variant controller has been successfully employed, for recent examples [5], [6]. For employing this method in electrodynamic shakers, it is necessary to include a variant specimen model in the controlled plant; however, the identification of this model is very difficult. Therefore, a method in which the identification of the variant plant is unnecessary is needed. This requirement Y. Uchiyama is with IMV CORPORATION, Osaka 555-0011, JAPAN
[email protected] M. Fujita is with the Department of Mechanical and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, JAPAN
ISBN: 978-960-89028-5-5
Weight (970[g])
Electrodynamic shaker with a specimen (VEO-30).
can be satisfied by using a real-time compensator for the reaction force of the specimen in the electrohydraulic shaker [7]. However, in this control method, uncertainties are not explicitly considered. In this study, in order to compensate for the problem of the variant plant, let us attempt to suppress the influence of the disturbance force. By employing an assumption for measuring the disturbance force, the controller is designed using µ–synthesis. However, since the measurement of the disturbance force is difficult in actual experiment, the disturbance force is then estimated using displacement, acceleration, and current responses; in order to reduce noise and obtain an accurate value, these three variables have been employed. In this paper, the disturbance-force compensator for electrodynamic shakers (Fig. 1) is presented. Particularly for designing and evaluating the proposed controller, a singleaxis compact shaker has been employed. The rest of the paper is organized as follows. In Section 2, the problem is described. A mathematical model and an uncertainty weighting function are introduced in Section 3. A feedback controller is designed using µ-synthesis is discussed in Section 4. Section 5 provides some simulation and experimental results; further, force estimation is described in this section. Finally, the paper is summarized in Section 6. II. CONTROL PROBLEM The characteristic of the controlled plant is nonlinear and variable because of the influence of the test piece. Consequently, its control becomes difficult. This problem occurs because of the fact that the disturbance force added to the shaking system is induced by the interaction between the shaker and the test piece. If the specimen exhibits a resonant characteristic, the influence of this problem assumes significance. Further, it is considered that other forces such as friction are included in this disturbance force.
5219
ThC10.3
Kl
Cl
Armature ms
Fig. 2.
xs
L1
I1
E
R1
Ec
Cd
Schematic diagram of the shaker model and equivalent circuit.
In this study, our purpose is to compensate for this problem by suppressing the influence of the disturbance force. For instance, a simple shaker model with a test piece exhibiting a resonant characteristic is considered, as shown in Fig. 2. The dynamic equations of the shaker model are
df = Cl (x˙ s − x˙ l ) + Kl (xs − xl ) ,
0 −20 10 Frequency [Hz]
100
10 Frequency [Hz]
100
180
Drive coil
¨s + Cd x˙ s + Kd xs + df F s = ms x 0 = ml x ¨ l − df
20
−40 1
Fs = BlI1 Phase [deg]
Kd
40
xl Gain [dB]
Test piece ml
(1) (2) (3)
where Fs denotes the electrodynamic force; xs and xl denote the displacements of the armature and the specimen, respectively; df denotes the disturbance force; Kd and Cd denote the stiffness and damping coefficient of the suspension system of the shaker, respectively; Kl and Cl denote the stiffness and damping coefficient of the specimen, respectively; and ms and ml denote the masses of the armature and the specimen, respectively. According to these equations, a control input for counteracting df is generated by the controller. This control input is added to Fs , and the influence of df is then suppressed. Further, as the disturbance force is measured in the form of a controlled variable, the need for introducing the characteristic that induces the disturbance force is eliminated from the controller design. III. ELECTRODYNAMIC SHAKER MODEL It is difficult to express the precise characteristic of an actual plant. A nominal model of the electrodynamic shaker and uncertainty weighting functions are introduced. A. Nominal model The electrodynamic shaker is based on the principle that an electrodynamic force is generated in proportion to a current applied to the drive coil existing in the magnetic field. The armature displacement and acceleration are measured using a laser displacement sensor and piezoelectric accelerometer, respectively. The current is measured using a current transducer. Assuming that the magnetic flux density is constant, a drive coil can be shown as a linear equivalent circuit in Fig. 2. The force Fs and the reverse electromotive force Ec can be represented as Fs = BlI1
(4)
Ec = Blx˙ s ,
(5)
90 0 −90 −180 1
Fig. 3. Bode diagram of the shaker system with the specimen (blue line: normal level, red line: double level, solid line: simulated result, dashed line: measured result).
where I1 denotes the current of the drive coil; l denotes the length of the drive coil; B denotes the magnetic flux density. From Fig. 2, the following (6) and (7) can be obtained E = R1 I1 + L1 I˙1 + Ec E = Ga us ,
(6) (7)
where E denotes the input voltage to the drive coil; Ga denotes an amplifier gain; us denotes the input voltage to the amplifier. As stated above, the mathematical model of the electrodynamic shaker obtained using (1)–(7). As shown in Fig. 1, the test piece, which is constructed with a 970 g weight and four rubber springs, is used. A firstorder resonant frequency of the specimen is approximately 13 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 order to add the nonlinear characteristic of the shaker, friction is generated by increasing the clamping pressure of the suspension system. For investigating the characteristic, a transfer function is measured by using input signals of white noise, the rms value of which differs correspondingly. The result of the transfer function from us to xs is shown in Fig. 3, where blue line denotes the result with the normal level; red line denotes the result setting the level to double from normal; solid line denotes the measured result; dashed line denotes the simulated result. The nonlinear characteristic for the input voltage level is existed in a static state. By comparing the measured value with the simulated value, the nominal values are determined. In this study, the characteristic which is measured using the input of the normal level is defined as the nominal model. B. Modeling uncertainty As shown in Fig. 4, the parameter perturbations are considered as additive uncertainties. The each perturbed range is set as shown in Table I. In the case of considering the equivalent circuit of the drive coil, the transfer function from the input voltage E to
5220
ThC10.3 ~
~
Gdrv
wg δg
0
B
Wd ∆d
wb δb df
Gd us
+
Ga
Fig. 4.
+
1 Ld s +Rd
+
+ I1
B
+
+
+
Gain [dB]
~
Ga
-
l
−40
−60 0.1
Schematic diagram of additive uncertainties. TABLE I P ERTURBED PARAMETERS . Symbol R1 L1 B Ga
−20
1.0
10 Frequency [Hz]
100
1000
Fig. 5. Uncertainty weighting Wd and additive perturbations of the drive coil (solid line: Wd , dotted line: parameter perturbation).
Perturbed region -10 – 10% -15 – 15% -10 – 10% -5 – 5%
Wul
Wuh
df Kf
the current I1 is represented as 1/ (Ld s + Rd ). It is difficult that this characteristic is described with an accurate model, and the perturbation is then considered as an unstructured additive uncertainty. Frequency responses of these additive perturbations in the drive coil are plotted in Fig. 5. Here, the magnitude of the uncertainty weighting function Wd is chosen to cover all the model perturbations as follows: 0.11(s + 500 · 2π) . (8) s + 100 · 2π Further, it is assumed that uncertainties which cannot be considered exist, and hence the magnitude of the weighting function is enlarged in higher frequency band. Since the assumptions of the simplified H∞ control problem [8] is satisfied, additive perturbation is used and Wd is proper. The perturbation of the amplifier gain and the magnetic flux density are represented as Wd =
˜ a = Ga + wg δg , δg ∈ [−1, 1] G ˜ = B + wb δb , δb ∈ [−1, 1] , B
(9) (10)
˜ a and B ˜ denote the actual parameters; wg and wb where G denote weighting coefficients. It is noted that the parameter perturbations of ms , Kd , Cd are not considered, because it is considered that the disturbance force can be used in the design procedure.
us
A. Control objectives The following items are set as control objectives. • Stabilize the system even when an uncertainty exists. • Maintain the performance wherein the disturbance force is suppressed well. A controller is designed to maintain a robust stability against the uncertainty model. Moreover, it is desired that a controller maintains a good robust performance despite the uncertainty which is considered above section. The robust
Ga Fig. 6.
~
Gdrv
~
B
+ l
ef
Ws
Feedback structure of displacement control.
controller for this system is then designed with µ–synthesis, which has many applications like [9]. Let us consider the feedback structure shown in Fig. 6, where the block Kf denotes the feedback controller. In this design section, due to consider a simple control construction, it is assumed that the disturbance force is directly measured. However, because that is difficult in actual situation, an estimator is then employed and is described in the next section. B. Choice of weighting functions The weighting function Ws for the suppression performance of the disturbance force is considered. For improving the performance, the gain of the control frequency band is required to be enlarged within obtaining robust stability. According to the above points, Ws is now chosen as Ws =
IV. C ONTROLLER D ESIGN The design of the controller of an electrodynamic shaker is carried out using MATLAB.
~
100 · 2π 200 · 2π 3.5 · 20 · 2π · · s + 20 · 2π s + 100 · 2π s + 200 · 2π s s · . · s + 2.0 · 2π s + 0.4 · 2π
(11)
The frequency response of Ws is shown in Fig. 7. Practically, measuring signals are influenced as a result of noise. Since the signal-to-noise ratio of the displacement response signal is low at high frequency, the magnitude of the controller is required to be made small at the high frequency band. A weighting function Wuh is then chosen as Wuh = 400 ·
s + 2.5 · 2π . s + 4000 · 2π
(12)
On the other hand, since the signal-to-noise ratio of the acceleration response signal is low at low frequency, the magnitude of the controller is required to be made small at
5221
ThC10.3 20
∆ Gain [dB]
0
Wul
−20
Wuh −40 −60 0.1
Fig. 7.
wg Wd 1.0
10 Frequency [Hz]
100
1000
wb Gd
Weighting function Ws for the control error.
-
P
100 80 Gain [dB]
Ws
+
Kf
60
Generalized plant P
Fig. 9.
40 20 1.2
−20 0.01
0.1
1.0 10 Frequency [Hz]
100
Magnitude [−]
0 1000
Fig. 8. Weighting function Wuh and Wul for the control input (solid line: Wul , chained line: Wuh ).
1 0.8 0.6 0.4 0.1
1
10
100
Frequency [Hz]
the low frequency band. A weighting function Wul is then chosen as √ s2 + 2.5 · 2π 2s + (2.5 · 2π)2 √ . (13) Wul = 0.9 · s2 + 0.002 · 2π 2s + (0.002 · 2π)2
The frequency responses of Wuh and Wul are shown in Fig. 8. It is noted that Wuh and Wul are separated in order to consider the simple adjustment of weighting functions.
Fig. 10. Max singuler value and µ plot of 2nd D-K iteration (solid line: max singuler value, dashed line: µ upper bound).
The robust performance condition is equivalent to the following structured singular value µ. sup µ∆ (Fl (P, Kf )(jω)) < 1
(17)
ω∈R
C. µ–synthesis The design objective for stability and control performance is formalized as requirement for a closed-loop transfer function with weighting functions. Therefore, the generalized plant P , which is shown in Fig. 9, is constructed to treat the control objectives of the µ–synthesis framework. Here, the block structure of the uncertainty ∆ is defined as ∆ := diag (δg , δb , ∆d , ∆p ) , δg ∈ C 1×1 , (14) δb ∈ C 1×1 , ∆d ∈ C 1×1 , ∆p ∈ C 1×3 ,
where |δg | ≤ 1, |δb | ≤ 1, |∆d | ≤ 1, ∆p ∞ ≤ 1, and ∆p is a fictitious uncertainty block for considering robust performance. Next, it is considered that P is partitioned as P11 P12 P = . (15) P21 P22 From Fig. 9, the linear fractional transportation on P by Kf is defined as follows: Fl (P, Kf ) := P11 + P12 Kf (I − P22 Kf )−1 P21 .
(16)
Since the controller satisfies this condition, the D-K iteration procedure is employed. The controller which satisfies (17) is obtained after 2 iteration as shown in Fig. 10. The degree of this controller has been reduced from 29 states to 16 states. V. CONTROL RESULT The control performance is confirmed by a simulation and an experiment. The controller is discretized via the Tustin transform at the sampling frequency of 5120 Hz. A. Estimation of disturbance force It is difficult that the disturbance force is directly measured. In this study, the disturbance force is estimated using the displacement and acceleration responses of the armature and the current response of the drive coil. Due to reduce the influence of noise and obtain the accurate estimate values, the order of the estimator is needed to be low. Therefore, the three variables are employed. The schematic diagram for estimating the disturbance force is depicted in Fig. 11. As
5222
ThC10.3 df Ga Ld s +Rd
Bl
1 ms
+
.. xs
+
1 s
1 s
xs
Gain [dB]
us
-
+
I1
Cd Kd
d^f
10 Frequency [Hz]
100
10 Frequency [Hz]
100
180 Phase [deg]
Estimator Fig. 11.
30 20 10 0 −10 −20 −30 1
Schematic diagram of the disturbance estimator.
shown in this figure, the estimated disturbance force dˆf is given as follows, Cd ˆ (18) x ¨s , df = BlI1 − Kd xs − ms + s + ωp
Fig. 12. Simulation results with the resonant specimen (solid line: with control, dashed line: without control, chained line: the nominal model of the shaker).
The control performance is confirmed with the experiment, which is executed using the shaker showed in Fig. 1. The performance with the proposed controller is evaluated using
Gain [dB]
30 20 10 0 −10 −20 −30 1
10 Frequency [Hz]
100
10 Frequency [Hz]
100
Phase [deg]
180 90 0 −90 −180 1
B. Simulation results
C. Experimental results
0 −90 −180 1
where ωp denotes the adjustment parameter to keep the gain low in a low frequency band. It is noted that the parameter perturbations of ms , Kd , Cd cannot be considered in the robust design and the stability analysis is certainly incomplete. However, in order to apply this proposed method to the electrodynamic shaker experimentally, the influence of these perturbations is confirmed by the actual excitation experiment. In contrast, because this estimate construction is employed, it is considered that these value mismatch are included in the estimated disturbance force. Then, in the case of compensating the estimated disturbance force, this difference can be small.
The control performance of the proposed method is verified with a simulation. The model of the electrodynamic shaker using the resonant specimen, which is described in Section 3, is employed as the controlled plant. The frequency transfer function from the input voltage us to the armature displacement xs is used, and the results with and without the proposed controller are compared with the nominal model of the shaker. It is noted that the characteristic of the test piece is eliminated from the nominal model. The controller is required to suppress the influence of the specimen and decrease the difference between the transfer characteristic and the characteristic of the nominal model. The simulation results of the transfer function are shown in Fig. 12, where the solid line denotes the result with control; the dashed line denotes the result without control; the chained line denotes the characteristic of the nominal model. By comparing both results, the proposed method is achieved to suppress the influence of the disturbance force. The proposed method yields the characteristic which is consistent with the nominal characteristic.
90
Fig. 13. Experimental results with the normal execution level (solid line: with control, dashed line: without control, chained line: the nominal model of the shaker).
the transfer function from the input voltage to the armature displacement in a manner similar to that of the simulation result. The transfer function is obtained in random wave excitation, of which the frequency band is from 1 to 100 Hz. Moreover, the experiment is executed changing the input level of the random waveform to investigate the influence of the nonlinear characteristic. The experimental results of the transfer functions are shown in Fig. 13, where the solid line denotes the result with control; the dashed line denotes the result without control; the chained line denotes the characteristic of the nominal model. This nominal model eliminates the characteristic of the specimen. As the result of the dashed line in Fig. 13, the influence of the resonant specimen can be confirmed as the peak notch characteristic in the neighborhood of the resonance frequency, which is approximately 13 Hz. Comparing the results with and without control, the proposed method is achieved to suppress the influence of the test piece. Furthermore, since the measured transfer function with
5223
30 20 10 0 −10 −20 −30 1
Gain [dB]
Gain [dB]
ThC10.3
10 Frequency [Hz]
100
90 0 −90 −180 1
10 Frequency [Hz]
100
10 Frequency [Hz]
100
180 Phase [deg]
Phase [deg]
180
30 20 10 0 −10 −20 −30 1
10 Frequency [Hz]
90 0 −90 −180 1
100
Fig. 14. Experimental results setting the execution level to double from the experiment of Fig. 13 (solid line: with control, dashed line: without control, chained line: the nominal model of the shaker).
Fig. 15. Experimental results changing the characteristic of the specimen (solid line: with control, dashed line: without control, chained line: the nominal model of the shaker).
control is consistent with the nominal model, the proposed method yields good result. After that, the input level of the random waveform is set to double for the above experiment. The transfer function is changed as shown in Fig. 3. The experimental results using the same controller are shown in Fig. 14. Because of changing the input level, the transfer function without control is inconsistent with the characteristic of the nominal level. That is especially significant at the low frequency band. However, as the result with the proposed controller, the measured transfer characteristic is consistent with the nominal characteristic. It figures that this good result is obtained in order to compensate for the estimated disturbance force in which the difference between the nominal model and the actual plant is included. Further, it is considered that the characteristic of the specimen is varied, and one of four rubber springs is then dismounted. The experimental results with the normal execution level are shown in Fig. 15. As shown in this figure, the resonance frequency changes from 13 Hz to 11 Hz. In this case, the proposed method yields the characteristic which is consistent with the nominal characteristic, too. These results support the conclusion that the proposed method can be compensated not only the influence of the test piece but also the difference between the nominal model and actual plant.
of the controlled plant. These results support the conclusion that the proposed controller is especially useful in this case. In the feature, let us construct the double loop controller, and the proposed controller is employed as the inner loop. If the specimen is broken during vibration testing and the characteristic of the controlled plant is varied, it will appear that the outer loop controller can achieve a good replication of the reference.
VI. CONCLUSION In this paper, the disturbance-force compensator of the electrodynamic shaker was presented. It was considered that the controlled plant was the nonlinear characteristic and the resonant specimen was employed. The control of the shaking system then became difficult. In order to compensate for this problem, the influence of the disturbance force was suppressed. It was assumed that the disturbance force was measured in the form of the controlled variable, and the controller was designed using µ–synthesis. The experiment using the electrodynamic shaker was executed, and good performance was achieved for the variation of the characteristic
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