2.7 Inverse Model Control
3.1.4 Control Results
The Bode plots of the loop transmission are presented in Figure 3.6a. The Bode plots of the closed-loop transfer function in Figure 3.6b transgress the proposed tracking bounds. The peak magnitude is greater than 0 dB and overshoot is expected for the step input. Thus, the prefi lter to improve time responses for the reference input is synthesized with two simple poles and one simple zero as follows:
= +
+ ⋅ +
/100 1 ( ) ( /70 1) ( /210 1) F s s
s s (3.18)
The control ratio including the prefi lter is drawn in Figure 3.6c. It is evident that the peak magnitude is reduced and hence the Bode plots do not transgress the tracking bounds. The iterative process to design the compensator (3.17) and the prefi lter (3.18) is summarized in Figure 3.7.
converter and the high-voltage amplifi er with a gain of 1000. The MetraByte’s DAS-20 I/O board has 12 bit resolution for both the D/A and the A/D conversion with a range of ±10 V. The sampling rate is chosen as 1600 Hz, which is enough to take account of the fi rst two fl exible modes. A low-pass digital fi lter is employed in the sensing process of the displacement to avoid observation spillover due to the residual modes.
Figure 3.9 shows the simulated and measured forced-vibration control responses of the smart structure excited by the fi rst-mode natural frequency. In this case, only the compensator is used without the prefi lter. It is clearly observed that in spite of the varying tip mass the imposed vibration is promptly rejected with relatively small control input voltage. The designed compensator is very effective in the rejection of the plant-input disturbance. And there exists an excellent agreement between the simulation and experimental results.
75 50
0
Magnitude(dB)
–25 –50
Nominal Perturbed
1 10 100
Frequency (rad/s)
Frequency (rad/s) (a)
Nominal Perturbed Tracking bounds
Nominal Perturbed Tracking bounds
Magnitude(dB)
Frequency (rad/s) (b)
(c)
1,000 10,000
1 10 100 1,000 10,000
1 10 100 1,000 10,000
25
75 50
0 –25 –50 25
Magnitude(dB)
75 50
0 –25 –50 25
FIGURE 3.6 Bode plots of the smart structure. (a) Loop transmission, (b) closed-loop trans- fer function, and (c) control ratio for reference input. (From Choi, S.B. et al., ASME J. Dyn.
Syst. Meas. Control, 121, 27, 1999. With permission.)
The simulated and measured step responses are presented in Figure 3.10. The open- loop step responses clearly exhibit unwanted vibration. Furthermore, it is also observed from the measured open-loop step responses that the hysteresis nonlinear behavior causes the displacement to be biased with respect to the original place, that is, zero dis- placement after removing the step input voltage. It is seen from the closed-loop control responses that the prefi lter smoothes the control input voltage and reduces the overshoot of the step response, as expected from the frequency responses in Figure 3.6c. There remains neither steady-state error nor undesirable chattering. It is noted from the mea- sured control input history that the additional voltage is applied to the piezoceramic actu- ator after 1.5 s so as to recover the initial equilibrium position of the smart structure.
Prefilter
Adjustment of the controller gain
Design of pole or zero
Yes Yes
No
No Selection of nominal plant and discrete
frequencies ω1< ω < ··· < ωm i=1(ω1)
sk G(s)= 1 Determination of the control specifications (stability, tracking, and disturbance rejection)
On or above bounds ω = ωi ?
i=i+1
i=m ?
Implementation Discretization of the
and the prefilter Determination of k as in
FIGURE 3.7 Flowchart of the QFT control design.
Figure 3.11 presents the measured 1 Hz sinusoidal tracking responses up to 10,000 cycles with an amplitude of 1 mm centered at 0.5 and 0 mm to investigate the hyster- esis behavior of the smart structure. The reference inputs (R) are chosen as 0.5 + 0.5 sin(2πt) mm for the biased trajectory and 0.5 sin(2πt) mm for the symmetric trajec- tory. A voltage of −275 × [0.5 + 0.6 sin(2πt)] V is applied for the open-loop tracking of the sinusoidal reference input centered at 0.5 mm, and −320 × [0.5 sin(2πt)] V for the reference input centered at 0 mm. The open-loop response of the biased sinu- soidal trajectory follows the reference path to a certain extent. But the magnifi ed open-loop error plot is biased negatively. In addition, the plot is shifted downward as the cycle number increases. This is arisen from the hysteresis behavior of the piezo- ceramic actuator, which is mainly caused by warm-up and follow-up polarization of the actuator. On the contrary, the open-loop response of the symmetric sinusoidal trajectory tells that the continuous change of the dipole of the piezoceramic actua- tor cannot achieve proper sinusoidal tracking of the smart structure. The open-loop error plot is also of negative bias, but the shift of the error plot exists upward with the increment of the cycle number. The maximum error in the open-loop sinusoi- dal tracking is 0.183 mm for the biased, and 0.177 mm for the symmetric case. The controlled responses of the sinusoidal tracking have a maximum error of 0.033 mm.
These results are quite self-explanatory justifying that the QFT controller imple- mented with the prefi lter provides robust and accurate tracking control performance against plant uncertainties such as hysteresis nonlinearity of the smart structure.
Microcomputer C:\
PZT
Composite beam D/A
Tip mass Noncontacting proximitor
(Bently Nevada 7200)
High-voltage amplifier (Trek 609A-3)
Power supply (HPS-303D) Dynamic signal
analyzer (HP35665A) A/D
D/A
FIGURE 3.8 Experimental apparatus for vibration and position tracking control.