2.7 Inverse Model Control
3.1.3 Controller Formulation
( )
dB for all , 0
1 ( ) L
L j M P
L j
ω ≤ ω ≥
+ ω (3.11)
The ML contour on the NC therefore forms a boundary that must not be vio- lated by a plot of the open-loop transmission, L(s) = G(s) ⋅ P(s).
2. Tracking specifi cation for the reference input
L U h
( )
( ) ( ) for all , 0,
1 ( )
T j L j T j P
L j
ω ≤ + ωω ≤ ω ω∈⎡⎣ ω ⎤⎦ (3.12)
where
ω = ω + ζ ω + ⋅ ω +
L 2 2
l l l la
( ) 1
( / 2 / 1) ( / 1)
T j
s s s
ω = ω +
ω + ζ ω +
ua
U 2 2
u u u
/ 1
( )
( ( 2 / 1)
T j S
s s
The parameters ζl, ωl, ωla, ζu, ωu, and ωua are selected to meet desired time- domain specifi cations for the step input such as settling time, rising time, etc.
The subscripts, u and l, denote the upper and lower bounds, respectively. The additional pole (−ωua) descends the lower tracking bound, and the additional zero (−ωua) raises the upper tracking bound in the high-frequency range.
3. Specifi cation for plant-input disturbance rejection
h
( )
for all , 0,
( ) m
Y j P
D jω ≤ αω ω∈⎡⎣ ω ⎤⎦ (3.13) where αm is a constant that limits the output for the step plant-input disturbance.
–1.0 –0.5
–1.5
–300 –150 0 150 300
0.0 0.5 1.0 1.5
Displacement(mm)
Applied voltage (V)
FIGURE 3.3 Hysteresis nonlinear behavior of the smart structure. (From Choi, S.B. et al., ASME J. Dyn. Syst. Meas. Control, 121, 27, 1999. With permission.)
These three specifi cations generate robust bounds on Lo(jω), which is the nomi- nal loop transmission, at selected frequencies, and the bounds are plotted on the NC.
The synthesized Lo(jω) must lie on or just above the bound at each frequency to satisfy the required performance. The compensator has the following form:
2 2
2 2
( / 1) ( / 2 / 1)
( )
( / 1) ( / 2 / 1)
q zr zr zr
q r
k
i pj pj pj
i j
s z s s
G s
s s p s s
+ ω + ζ ω +
=
∏
+∏
ω + ζ ω +∏ ∏
(3.14)where i, j, k, q, and r are arbitrary natural numbers. The properly designed com- pensator guarantees that the variation in |L(jω)/(1 + L(jω) )| is less than or equal to
|TU(jω)| − |TL(jω)|.
A prefi lter is not required in the disturbance rejection problem. The designed compensator, however, guarantees only the variation in |L(jω)/(1 + L(jω) )| less than or equal to that allowed. Therefore, it is necessary that a prefi lter be designed to ensure robust performance in tracking problem. The system can be ranged in dB by use of the prefi lter so that the control ratio can be increased or decreased within the tracking bounds given by Equation 3.12. This implies that the control ratio of the closed-loop transfer function can be altered by the prefi lter. Because the required frequency response is characterized by the reference input, the prefi lter needs to be designed so that the closed-loop transfer function satisfi es the specifi ed tracking bounds. The peak value of the closed-loop Bode plot should be less than 0 dB in order to remove the overshoot of the step response. And, for good tracking perfor- mance of the sinusoidal input, the closed-loop transfer function should be close to 0 dB. The form of the prefi lter can be expressed by
+ ω + ζ ω +
=
∏
+∏
ω + ζ ω +∏ ∏
2 2
2 2
( / 1) ( / 2 / 1)
( )
( / 1) ( / 2 / 1)
q zr zr zr
q r
i pj pj pj
i j
s z s s
F s
s p s s
(3.15)
It is noted that to realize the compensator (3.14) and the prefi lter (3.15) utilizing the microcomputer, the discretized transfer functions and the equivalent difference equations should be obtained [15].
3.1.3.2 Application to the Structure
In this section, the fi rst and second fl exible modes are considered in the design of the QFT controller. The number of fl exible modes is determined from the investiga- tion of the system responses before and after employing the controller. Through the computer simulation, the open-loop responses of the nominal system are observed in time domain by investigating the response effect of each fl exible mode (tested up to the 5th mode), and after employing the controller, the closed-loop responses due to residual modes (3rd, 4th, and 5th modes) are observed by investigating the excita- tion magnitude, which is an indicator of the control spillover effect. The controlled response due to the residual modes is small enough to be negligible. To identify the system parameters such as ωi and ζi, the frequency response test of the smart
structure, whose properties are listed in Table 3.1, is undertaken. The measured natu- ral frequencies are as follows: ω1 = 36 Hz, ω2 = 122.7 Hz with a tip mass of 1.252 g and ω1 = 30 Hz, ω2 = 121.6 Hz with a tip mass of 1.730 g. The anti-resonance fre- quency varies from 71.5 to 80 Hz due to the tip mass variation. The fi rst-mode damp- ing ratio, ζ1, varies from 0.030 to 0.044, the second-mode damping ratio, ζ2, from 0.09 to 0.11, and the anti-resonance damping ratio, ζ3, from 0.050 to 0.059. On the other hand, in order to quantify the structured uncertain bound of K, the relationship between the displacement of the structure and the applied voltage is experimentally obtained as shown in Figure 3.3. It is clear that the hysteresis nonlinearity increases as the applied voltage increases. From this result, it is distilled that the plant gain, K, varies from 0.2 to 0.8 mm/100 V.
Now, considering the fi rst two fl exible modes to be controlled, the plant transfer function (3.10) becomes
ω + ζ ω +
= ω + ζ ω + ⋅ ω + ζ ω +
2 2
3 3 3
2 2 2 2
1 1 1 2 2 2
/ 2 / 1
( ) ( / 2 / 1) ( / 2 / 1)
s s
P s K
s s s s (3.16)
The nominal plant is adopted as that with the tip mass of 1.730 g and the plant gain of 0.2 mm/100 V. Three perturbed plants are considered as follows: perturbed plant with variable tip mass, perturbed plant with hysteresis nonlinearity of the struc- ture, and perturbed plant with variable tip mass as well as hysteresis nonlinearity.
Figure 3.4 presents the nominal plant and three perturbed plants. It is evident that the perturbed systems have different natural frequencies and magnitudes from the nominal one.
The imposed robust tracking bounds for the plant uncertainties are as follows:
ML = 3 dB, ωh = 350 rad/s, ωl = 198.4 rad/s, ωla = 198.4 rad/s, ωu = 233.4 rad/s, ωua = 233.4 rad/s, αm = 0.5, ζl = 2, and ζu = 1. The lower and upper tracking bounds are predetermined so that the settling time of the system for the step input has the value between 0.025 and 0.075 s, and the overshoot does not exist. The loop shaping process is performed so that the loop transmission satisfi es stability and performance bounds on the NC in the low frequencies and subsequently in the higher frequencies.
As a fi rst step, an integrator should be included in the initial compensator to remove TABLE 3.1
Dimensional and Mechanical Properties of the Smart Structure
Young’s Modulus Thickness Density Width Length
Composite beam (glass/epoxy)
20 GPa 0.60 mm 1865 kg/m3 25 mm 12 cm
Piezoceramic (PZT)
64 GPa 0.80 mm 7700 kg/m3 24 mm 6 cm
Piezoceramic strain constant −300 × 10−12 (m/m)/(V/m)
Tip mass 1.252–1.730 g
Sensing position (from fi xed) 100 mm
the steady-state error for the step reference input of the system. With this integrator, the nominal loop transmission keeps a phase angle of 90° up to the frequency of 110 rad/s. From the Nichols plot of the nominal loop transmission, it is clear that gain increment is necessary because the plot is located much lower from the bounds in the low frequencies. Furthermore, it is fi gured out that more than one zero is needed to compensate the phase lag over the frequency of 110 rad/s. The increased gain is 55 dB (=562.34) and the added zero is the fi rst complex-zero pair of the compensa- tor. A complex-pole pair is also added at a frequency that is higher than that of the fi rst complex-zero pair in order to maintain the compensator to be proper. Now, the nominal loop transmission intrudes the top of the U-contour after the frequency of 220 rad/s. Thus, another complex-zero pair is added and adjusted at a frequency higher than that of the fi rst complex-pole pair. As a last process, the second complex- pole pair should be included in the compensator to maintain a proper compensator and then it is adjusted until the plot of the nominal loop transmission sweeps and just touches the U-contour. Finally, the designed QFT compensator is composed of an integrator, two complex-pole pairs, and two complex-zero pairs without any simple real pole and zero as follows:
+ ⋅ ⋅ + ⋅ + ⋅ ⋅ +
= ⋅ + ⋅ ⋅ + ⋅ + ⋅ ⋅ +
2 2 2 2
2 2 2 2
( /143 2 0.4 /143 1) /770 2 0.45 /770 1) ( ) 562.34
( /520 2 0.4 /520 1) ( /3800 2 0.6 /3800 1)
s s s s
G s s s s s s
(3.17) The plot of the nominal loop transmission is shown in Figure 3.5. Since the QFT design process is carried out at a number of discrete frequencies, these frequen- cies are denoted in the fi gure. It is noted that the plot at each chosen frequency satisfi es the specifi ed bound, that is, Lo(jω) does not violate the U-contour and any points of Lo( jω) are on or above the performance bound curve for the frequencies.
–500 50 100 150 200
–25 0 25 50
Nominal plant
Perturbed plant with variable tip mass Perturbed plant with hysteresis nonlinearity Perturbed plant with variable tip mass and hysteresis nonlinearity
Magnitude(dB)
Frequency (Hz)
FIGURE 3.4 Nominal and perturbed plants. (From Choi, S.B. et al., ASME J. Dyn. Syst.
Meas. Control, 121, 27, 1999. With permission.)
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.