Robust Control with a Disturbance-force Compensator and Experimental Study for an Electrodynamic Shaker Yasuhiro Uchiyama, Daisuke Yonemura and Masayuki Fujita
Abstract This paper presents a robust double loop controller for an electrodynamic shaker, an inner loop of which includes a disturbance-force compensator. The characteristic of a shaking system is considered to be nonlinear with friction and be influenced by a resonant test piece. The control of the shaking system then becomes difficult. In order to compensate for this problem, the disturbance-force compensator is employed. Furthermore, because the reference signal is replicated well, a displacement feedback controller is introduced in an outer loop. Each controller is designed using μ –synthesis. As open-loop characteristics, a stability of the control system with the disturbance-force compensator is confirmed. Finally, the experiment using the electrodynamic shaker is executed, and proposed control system achieves a good performance.
1 Introduction Because durability of some piece is confirmed, shakers are used in vibration test of specimen. The specimen is test piece, and it is exposed to vibration environment under actual field conditions in the vibration test. Recently, shaking systems that can more accurately replicate an actual situation have been widely employed for vibration testing. An electrodynamic shaker has some worthwhile features such as good linearity and wide frequency response. The shaking systems are popularly used in the automotive industry. For other instance, they are used in the civil and architectural engineering fields, as shown in Fig. 1. The shaking system needs to be controlled for not only stable control but also better replication of the given referYasuhiro Uchiyama IMV CORPORATION, Osaka, JAPAN, e-mail:
[email protected] Daisuke Yonemura and Masayuki Fujita Department of Mechanical and Control Engineering, Tokyo Institute of Technology, Tokyo, JAPAN
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Y. Uchiyama, D. Yonemura and M. Fujita
Weight
X shaker
Steel-frame structure (Specimen)
Shaking Table
Fig. 1 Multi-axis shaking system using the electrodynamic shakers (DS-50000).
ence 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, the control becomes difficult in a similar manner. In applications in regard to vibration control, there are a lot of successful cases, like recent examples [1, 4]. In regard to the realization of above problems, there have been some instances [3, 5, 6] in which electrohydraulic shakers have been successfully employed. In our previous studies [7], 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. Therefore, if the plant perturbation becomes more significance, the control performance cannot satisfy the test specifications. For this control problem in the case of the electrohydraulic shaker, [3] apply the control with the disturbance observer-based compensation for the reaction force in specimen, and this method obtains good results. However, in this control method, uncertainties are not explicitly considered. In our study [8], let us similarly attempt to suppress the influence of the disturbance-force. In order to achieve the control performance and stability for uncertainties, the control loop is designed using μ – synthesis. The proposed method can be compensated not only the influence of the test piece but also the difference between the nominal model and the actual plant. Furthermore, in this study, a double loop controller with the disturbance-force compensator is constructed to achieve replication of the reference signal for the above problem situation. In an inner loop, the disturbance-force compensator suppresses the influence of the specimen and the nonlinear characteristics. In an outer loop, a feedback controller is designed to replicate the reference. Because the influence of the specimen and nonlinear characteristic is compensated in the inner loop, it is considered that the proposed control structure obtain better control stability and performance than a single loop controller. In this paper, because of mainly evaluating the proposed controller in experiments, a single-axis compact shaker is employed. The rest of the paper is organized as follows. A mathematical model and an uncertainty weighting function are introduced in Section 2. A feedback controller
Robust Control with a Disturbance-force Compensator
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Table 1 Parameters of the electrodynamic shaker and its perturbed regions. Symbol
Parameter
Perturbed region
R1 L1 B l Ga K1 C1 ms xs us I1 df
Resistance of drive coil Inductance of drive coil Magnetic flux density Length of drive coil Amplifier gain Suspension stiffness Suspension damping Armature mass Displacement of armature Input voltage to amplifier Current of drive coil Disturbance force
-10 – 10% -15 – 15% -10 – 10% -5 – 5% -30 – 30% -20 – 20% -10 – 10%
Disturbance force df Armature ms K1
xs Fs = BlI1
C1
Suspension
I1
Gaus R1
L1 Ec (=Blx. s)
Drive coil
Fig. 2 Schematic diagram of the shaker model and the equivalent circuit.
is designed using μ -synthesis is discussed in Section 3. Section 4 provides some simulation and experimental results. Finally, the paper is summarized in Section 5.
2 ELECTRODYNAMIC SHAKER MODEL 2.1 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. A simple shaker model and a linear equivalent circuit of the drive coil can be shown as Fig. 2. The dynamic equations of the shaker model can be obtained as BlI1 = ms x¨s +C1 x˙s + K1 xs + d f Ga us = R1 I1 + L1 I˙1 + Bl x˙s .
(1) (2)
The transfer function from the input voltage us to the displacement of armature xs is defined as Gs . In addition, the characteristic of an antialiasing filter and the
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AC coupling of the amplifier is added to Gs ; the resulting model is defined as the nominal model Gs .
2.2 Modeling Uncertainty In order to add the nonlinear characteristic of the shaker, friction is generated by increasing the clamping pressure of the suspension system. The nonlinear characteristic for the input voltage level can be then confirmed in frequency characteristics. The nominal parameter and the perturbation range are obtained from the characteristics measured by various input levels. The each perturbed range is set as shown in Table 1. However, because the influence of the friction, which is identified with the transient characteristic, is introduced as the nonlinear characteristic in the frequency response, it is difficult that the controller is designed to maintain stability against these perturbations perfectly. Furthermore, a test piece, which has resonant characteristic, is considered. 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. Therefore, an additional uncertainty for the influence of the specimen and the parameter perturbations of the shaker are considered. By using these perturbations, 16 perturbed models can be obtained. Here, the magnitude of the weighting function Wdd is selected to cover all the perturbations as follows: √ 0.04(s2 + 1500 2s + 15002 ) √ . (3) Wdd = s2 + 150 2s + 1502
3 CONTROLLER DESIGN 3.1 Control Objectives 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 uncertainties which are considered above section. In this study, a double loop control is considered because of achieving these objectives. Because the influence of the specimen and the nonlinear characteristics are suppressed, the disturbance-force compensator is employed in the inner loop. On the other hand, the outer loop controller is designed in order to obtain the performance wherein the reference signal is replicated well. The each robust controller is designed with μ –synthesis, which has many applications like [2]. The design of the inner loop controller is described in the reference [8], in which the same shaker was used. In particular, when the feedback controller in the outer loop is used, the compensation in the crossover frequency of the outer loop is care-
Robust Control with a Disturbance-force Compensator
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0
Gain [dB]
−20 −40 −60 −80 0.01
0.1
1.0 10 Frequency [Hz]
100
1000
Fig. 3 Frequency characteristic of the disturbance-force compensator K f . Fig. 4 Feedback structure of displacement control.
rd +
-
ed
Δd
Wdd
Wsd Kd
Gs
+
+
xs
fully considered. The crossover frequency exists at 10 Hz neighborhood, and the disturbance-force compensator K f in the reference is also designed to improve the control performance in this frequency band. Therefore, the same controller is employed in the inner loop, and the frequency characteristic is shown in Fig. 3.
3.2 Design of Outer Loop Controller A displacement controller in the outer loop is considered. Let us consider the feedback structure shown in Fig. 4, where rd and ed denote the reference and the control error respectively; the block Kd , the feedback controller. A weighting function Wsd for the replication performance of the reference signal 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, Wsd is now chosen as Wsd =
10 · 0.5 · 2π 1.0 · 2π 5.0 · 2π s2 . · · · 2 s + 0.5 · 2π s + 1.0 · 2π s + 5.0 · 2π s + 1.78s + 1.58
(4)
Further, the characteristic of a high pass filter includes Wsd to deal with the AC coupling of the amplifier. 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 is constructed to treat the control objectives of the μ –synthesis framework. Here, the block structure of the uncertainty Δ is defined as
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rd +
-
ed
Kd
xs
+
Gs
+ Kf
d^f
I1
..
xs
GE
Fig. 5 Control system with the disturbance-force compensator.
Δ := diag (Δd , Δsd ) , Δd ∈ C1×1 , Δsd ∈ C1×1 ,
(5)
where C denotes the set of all complex numbers, |Δd | ≤ 1, |Δsd | ≤ 1, and Δsd is a fictitious uncertainty block for considering robust performance. The D-K iteration procedure is employed, and the controller which satisfies the objective is obtained after 3 iterations. The degree of this controller has been reduced from 20 states to 12 states.
4 CONTROL RESULTS 4.1 Control Stability in Simulation The control system with the disturbance-force compensator is shown in Fig. 5, where GE denotes the disturbance-force estimator; dˆf , the estimated disturbance force. The disturbance force is estimated using the displacement and acceleration responses of the armature and the current response of the drive coil. An open-loop characteristic is calculated in the simulation, and the stability of the system is confirmed. The case of the control system without the disturbance-force compensator is considered, and its effect is confirmed. The simulation results with the inner loop controller are shown in Fig. 6, where the solid line denotes the result with the nominal model; the dashed lines denote the results of using perturbed models in Section 2. The simulation results without it are similarly shown in Fig. 7. The same displacement feedback controller is employed in each result. Since the stability margin is satisfied at crossover frequency, it appears that the both control system are stable. However, comparing the results with and without the inner loop, the results with the perturbed models are consistent with the nominal result in the frequency band from 1 Hz to 40 Hz in the case of using the proposed control system. Because the disturbance-force compensator is suppressed the influence of the specimen and perturbed parameters, the good results are obtained in the neighborhood of the crossover frequency. Therefore, the phase margin of 55 degrees is obtained in the case of using the double loop control structure, but the phase margin without the inner loop is very small. As the results, the control system with the disturbance-force compensator is more stable than that without it.
Robust Control with a Disturbance-force Compensator
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Gain [dB]
40 20 0 −20 0.01
0.1
1 Frequency [Hz]
10
100
0.1
1 Frequency [Hz]
10
100
Phase [deg]
180 90 0 −90 −180 0.01
Fig. 6 Open-loop characteristics with the disturbance-force compensator in the inner loop (solid line: with the nominal model, dashed line: with the perturbed models).
Gain [dB]
40 20 0 −20 0.01
0.1
1 Frequency [Hz]
10
100
0.1
1 Frequency [Hz]
10
100
Phase [deg]
180 90 0 −90 −180 0.01
Fig. 7 Open-loop characteristics without the disturbance-force compensator (solid line: with the nominal model, dashed line: with the perturbed models).
4.2 Experimental Results The control performance is confirmed by an experiment. To implement the controller with a processing board, the controller is discretized via the Tustin transform at the sampling frequency of 5120 Hz. The electrodynamic shaker with setting the resonant specimen, which is described in Fig. 8, is employed. The test piece is constructed with a 970 g weight and rubber springs. A first order resonant frequency of the specimen is selected with the adjustment of the number of the springs, either 11 Hz or 13 Hz is set. Each frequency is close to the crossover frequency.
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Y. Uchiyama, D. Yonemura and M. Fujita Acceleration Rubber spring pickup Test piece
Weight (970[g]) Armature
Fig. 8 Experimental singleaxis compact electrodynamic shaker with the resonant specimen (VEO-30).
Laser displacement sensor
Disp. [mm]
0.3 0.15 0 −0.15 −0.3 0
1
2
3 4 Time [s]
5
6
Fig. 9 Experimental transient result without the disturbance-force compensator when the resonant frequency is set at 11 Hz (solid line: response signal, dashed line: reference signal).
This experiment is executed by an excitation using a seismic waveform data as the reference signal. The control performance is evaluated by the result of the transient response. For making the reference from the original data, the acceleration data is transformed to the displacement data. After that, its level is adjusted because of executing with this shaker, and the dominant frequency band is fitted to the assumed frequency band of the control. First, the control without the disturbance-force compensator is considered. The resonance frequency is set at 11 Hz. The control result is shown in Fig. 9, where the solid line denotes the response signal; the dashed line, the reference signal. As the result, this system attains stability regardless of the influence of the friction and resonant specimen. However, the response signal is especially inconsistent with the reference signal for a period from 1 s to 2 s. Furthermore, it is considered that the characteristic of the specimen varies, the resonance frequency is changed from 11 Hz to 13 Hz. The control result is similarly shown in Fig. 10. It is difficult to obtain exactly the uncertainty model for the transient influence of friction, and the phase margin becomes smaller due to setting higher resonant frequency, therefore the control system becomes unstable. Next, the proposed control result with the inner loop controller is similarly shown in Fig. 11. The same feedback controller as the above result is employed. Because the influence of the nonlinear characteristic is enlarged, the input level of the reference is set to double for the experiment in Fig. 9. The response signal is consistent
Robust Control with a Disturbance-force Compensator
9
Disp. [mm]
0.3 0.15 0 −0.15 −0.3 0
1
2
3 4 Time [s]
5
6
Fig. 10 Experimental transient result without the disturbance-force compensator when the resonant frequency is set at 13 Hz (solid line: response signal, dashed line: reference signal).
Fig. 12 Experimental transient result with the disturbance-force compensator when the resonant frequency is set at 13 Hz (solid line: response signal, dashed line: reference signal).
0.3 0 −0.3 −0.6 0
1
2
3 4 Time [s]
5
6
1
2
3 4 Time [s]
5
6
0.6
Disp. [mm]
Fig. 11 Experimental transient result with the disturbance-force compensator when the resonant frequency is set at 11 Hz (solid line: response signal, dashed line: reference signal).
Disp. [mm]
0.6
0.3 0 −0.3 −0.6 0
with the reference signal. Because the disturbance-force compensator in the inner loop suppresses the influence of the nonlinear characteristic and the specimen, the good performance is achieved. After that, it is also considered that the resonance frequency is changed from 11 Hz to 13 Hz. The experimental result is similarly shown in Fig. 12. The same input level is used at the above experiment. In this case, too, the proposed method yields the response signal which is consistent with the reference signal. 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 the actual plant.
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5 Conclusion In this paper, the robust double loop controller of the electrodynamic shaker has been presented. It has been considered that the controlled plant has the nonlinear characteristic and the resonant specimen is employed. Because these influences are compensated, the disturbance-force compensator has been employed in the inner loop. In addition, the outer loop controller has been designed in order to obtain the performance wherein the reference signal is replicated well. The each controller has been designed using μ –synthesis. As the open-loop characteristics, the control system with the disturbance-force compensator has been more stable than that without it. The experiment using the electrodynamic shaker has been executed, and proposed control system has obtained the good performance. These results support the conclusion that the proposed controller is especially effective in this case.
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