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Robust Control of a Two-Axis Piezoelectric Nano-Positioning Stage
Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011
Robust Control of a Two-Axis Piezoelectric Nano-Positioning Stage Fu-Cheng Wang*, Yan-Chen Tsai*, Chin-Hui Hsieh*, Lian-Sheng Chen*, Chung-Huang Yu** *Mechanical Engineering Department, National Taiwan University, Taipei 10617, Taiwan (Tel:+886233662680, e-mail:[email protected], [email protected] , [email protected], [email protected]) **Department of Physical Therapy and Assistive Technology, National Yang-Ming University, Taiwan (e-mail: [email protected]) Abstract: This paper applies robust control to a two-axis piezoelectric nano-positioning stage. As technology develops, the precision requirement for positioning platforms is becoming increasingly stringent. Since a traditional mechanical transmission structure cannot achieve high precision, a piezoelectric actuator is usually applied to drive the mechanism because of its high resolution, high accuracy, and large driving force. However, the non-linear dynamic characteristics of piezoelectric materials, such as hysteresis, might degrade system performance. Therefore, in the present study, we model a piezoelectric stage as a linear system, and regard the nonlinear factors as system uncertainties. We then apply robust control strategies to guarantee system stability and performance. Lastly, the designed controllers are implemented for experimental verification. The results demonstrate the effectiveness of these robust controllers. Keywords: piezoelectric, PZT, identification, robust control, loop-shaping
1. INTRODUCTION
2. SYSTEM DESCRIPTION AND IDENTIFICATION
As technology progresses, precise positioning requirements are becoming increasingly stringent. Frequently, traditional mechanical transmission structures might not meet these requirements due to friction and backlash. Therefore, piezoelectric materials are normally applied to drive position platforms because of their advantages such as high resolution, high accuracy, and large driving force. In piezoelectric materials, both direct and inverse piezoelectric effects are possible. The former occurs when the material produces an electric field in response to an applied force. The latter happens when the material undergoes a deformation in response to an applied electrical field. Using the inverse effect, piezoelectric actuators can drive positioning stages with a high degree of precision. However, non-linear characteristics of piezoelectric materials, such as hysteresis, might degrade the system performance. In order to reduce the non-linear effects, the Maxwell Slip model (Choi et al., 1997, 1999, 2002) and the Bouc-Wen model (Wen, 1976, Kuo and Low, 1995) have been proposed. However, these high-order models are difficult to construct and are not suitable for control design. Therefore, in this paper, we identify a piezoelectric stage as a linear model and regard the non-linear effects as system uncertainties and disturbances. We then apply robust control strategies to design suitable robust controllers to achieve system stability and performance.
Fig. 1 shows a two-axis PZT stage that consists of three layers: the top layer is a loading platform, the middle moves along the Y-axis, and the bottom moves along the X-axis. We installed ceramic piezoelectric actuators in the grooves of the stage, and stage movement is controlled by regulating the input voltages of the actuators. To measure stage movement, we used Mercury 5800 (MicroE Systems, 2010), which has a resolution of 1.22nm and digital outputs. The system structure is shown in Fig. 2, where Labview and PXI-8108 (National Instruments, 2010) were used for data acquisition. Because the voltage output of PXI-8180 is limited to 10V, a voltage amplifier, E-663, was also applied to drive the actuators.
This paper is arranged as follows: Section 2 introduces the PZT stage and obtains its transfer functions by identification techniques. Section 3 introduces robust control algorithms and designs robust controllers for the PZT stage. In Section 4, the designed robust controllers are implemented for experimental verification. Lastly, we draw conclusions in Section 5. 978-3-902661-93-7/11/$20.00 © 2011 IFAC
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Fig. 1 A two-axis piezoelectric stage. 10.3182/20110828-6-IT-1002.00741
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
repeated the experiments six times at each axis and obtained the following models of the PZT stage: G1X
-2.389 108 s4 + 1.278 105 s3 + 0.03251 s 2 + 73.78 s + 2472 , (1) s 4 + 2433 s3 + 3.698 106 s2 + 2.35 109 s + 6.383 1010
G X2
-1.413 108 s 4 + 5.818 10 6 s3 + 0.03211 s2 + 25.48 s + 134.8 , (2) s4 + 1487 s3 + 2.053 106 s 2 + 7.64 108 s + 2.257 109
G X3
-1.399 10 8 s4 + 5.959 106 s3 + 0.032 s2 + 24.04 s + 128.1 , s4 + 1478 s 3 + 2 106 s2 + 7.205 108 s + 2.242 109
(3)
G X4
-1.582 108 s4 + 7.753 10 6 s3 + 0.03348 s 2 + 28.48 s + 264.6 s4 + 1650 s3 + 2.184 106 s 2 + 8.627 108 s + 5.716 109
,(4)
G 5X
-1.438 10 8 s 4 + 6.343 106 s3 + 0.03259 s 2 + 24.4 s + 104.6 , (5) s 4 + 1517 s3 + 2.027 106 s 2 + 7.305 108 s + 1.519 109
G X6
-1.491 10 8 s 4 + 6.831 106 s3 + 0.03312 s2 + 25.66 s + 165.9 . s4 + 1570 3 + 2.082 106 s2 + 7.721 108 s + 3.176 109
(6)
GY1
-1.307 10 8 s 3 + 4.768 107 s 2 + 0.05036 s + 1.921 , s 3 + 1745 s 2 + 1.662 106s + 4.636 107
(7)
GY2
-1.578 108 s3 + 3.797 106 s2 + 0.05444 s + 2.199 , s 3 + 1971 s 2 + 1.791 106 s + 5.398 107
(8)
GY3
-1.316 10 8 s3 + 8.008 10 7 s 2 + 0.05011 s + 1.892 , s 3 + 1749 s 2 + 1.654 106 s + 4.572 107
(9)
1000 500
GY4
-1.317 108 s 3 + 7.774 107 s2 + 0.05017 s + 1.869 , s3 + 1750 s 2 + 1.655 106 s + 4.511 107
(10)
GY5
-1.318 108 s3 + 7.882 107 s2 + 0.05022 s + 1.887 , s3 + 1753 s 2 + 1.658 106 s + 4.567 107
(11)
GY6
-1.321 10 8 s 3 + 9.443 10 7 s2 + 0.05002 s + 1.89 . s 3 + 1751 s 2 + 1.652 106 s + 4.587 107
(12)
Fig. 2 Hardware control structure. At first, we tested the open-loop characteristics of the piezoelectric actuators. Given an input voltage, we measured the output displacement and drew the phase plot in Fig. 3, where the hysteretic effects of the actuators can be observed. 1500
0 -500 -1000 -1500 -40
-20
0 voltage(V)
20
Note that the orders of the transfer functions were automatically assigned by MATLAB for best fittings.
40
1500
10 voltage(V)
50 nm/s, Y-axis
position(nm)
1000 500
10 rX(t)
5
voltage(V)
position(nm)
50 nm/s, X-axis
0 -5 -10
0
5
10 time(s)
15
0 -5 -10
20
rY (t)
5
0
5
10 time(s)
15
20
0
(a)X-axis inputs
-500
(b)Y-axis inputs
2000
0 voltage(V)
20
40
0
-2000
Fig. 3 The hysteresis effects.
position(nm)
-20
2000 X(t)
0
5
10 time(s)
15
3540
position(nm)
2000
In order to achieve precise positioning, we consider a closedloop control structure for the stage. First, we generated chirp input voltage signals to drive the actuators and then measured the output displacement signals, as shown in Fig. 4. Using subspace system identification methods (Moor and Overschee, 1991, 1994), we can obtain the system’s transfer functions by a MATLAB command n4sid. Because the cross-coupling terms are relatively small and can be neglected, the MIMO system can be simplified as two SISO sub-systems. We
0
0
5
10 time(s)
15
20
2000 Y(t)
0
-2000
X(t)
-2000
20
position(nm)
-1500 -40
position(nm)
-1000
0
5
10 time(s)
15
(a) Outputs to rX
20
Y(t) 0
-2000
0
5
10 time(s)
15
(b) Outputs to rY
Fig. 4 Identification signals.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
3. ROBUST CONTROLLER DESIGN In this section, we introduce robust control algorithms and design robust controllers for the PZT system. The following Small Gain Theorem is normally considered for systems with uncertainties (Zhou et al., 1996):
1 / if and only if Z
1/ if and only if Z
M
G
r
Suppose Z RH and let 0 , the system of Fig. 5 is well posed and internally stable for all ( s ) RH with: (a)
N
N
z1
z2
M 1
; (b)
, where Z
is the
H norm of system Z. (a) The close-loop block diagram
w1
e1
N z1 z 2
Z
e
M
K 1 1 I ( I G0 K ) M
w2
2
Fig. 5 Small Gain Theorem. (b) Representation of the small-gain theorem
Suppose a nominal plant G0 has a normalized left coprime factorization of G0 ( s ) M ( s ) 1 N ( s ) , where M , N RH and M M * N N * I . Assume a perturbed system G can be expressed as: G ( M M ) 1 ( N N ) ,
with [ N
M ]
(13)
, and M , N RH . Consider a
closed-loop system with the controller K of Fig. 6(a), which can be rearranged as in Fig. 6(b) when r=0. Using Small-Gain Theorem, the closed-loop system is internally stable for all perturbations with N M , if and only if :
K 1 1 I ( I G0 K ) M
K ( I G0 K ) 1 I I
G0
From equations (1–12), the transfer functions of the X-axis or Y-axis varied at each experiment. Hence, we apply the ideas of gap metric (Georgiou and Smith, 1990) to select the nominal plant for controller design. Because the coprime factorization of a transfer function is not unique, the gap between G0 and G is defined as: The minimum perturbation of
N M changes G0 to G , denoted as δ (G0 , G ) .
G0 arg min max δ (G0 , Gi ), Gi . G0
which
Gi
(16)
Thus, we selected the nominal plants of the X-axis and Y-axis as follows: -1.491 10 8 s 4 + 6.831 106 s3 + 0.03312 s2 + 25.66 s + 165.9 , s 4 + 1570 3 + 2.082 106 s 2 + 7.721 108 s + 3.176 109 -1.316 10 8 s 3 + 8.008 10 7 s2 + 0.05011 s + 1.892 GY . s 3 + 1749 s 2 + 1.654 106 s + 4.572 107 GX
1
G0 ]
That is, the nominal plant G0 is chosen by:
. (14)
Thus, we can define the stability margin (Doyle and Zhou, 1998) as: K b(G0 , K ) ( I G0 K ) 1[ I I
Fig. 6 System structures.
,
(15)
such that the closed-loop system is internally stable for all perturbations N M with if and only if b(G , K ) .
These give the system gaps as δ (GX , GX i )<0.016256 and δ (GY , GY i )<0.016411 .
In order to improve system performance, loop-shaping techniques (Glover and McFarlane, 1989, 1992) are usually applied. As shown in Fig. 7, we set weighting functions to
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Root Locus
shape the plant as Gs ( s ) W2 ( s )G( s )W1 ( s ) , and then design a standard H robust controller K for the shaped plant.
2000 1500
Finally, we implement the controller K W1 KW2 for the original plant G ( s) .
K
W1
G
Imaginary Axis
1000 500 0 -500 -1000
W2
-1500 -2000 -6000
Gs
-4000
-2000
0
2000
4000
6000
Real Axis
(a) Gx Root Locus 2000 1500
W2
K
W1
G
1000
Imaginary Axis
500
K
0 -500 -1000
Fig. 7 Loop-shaping design procedures.
-1500 -2000
We use root locus techniques to estimate the settling time of the system’s step responses. For example, the root loci of Gx is shown in Fig. 8(a). Therefore, we set the following weighting function WX
3 109 ( s 600) , s( s 250)
1010 ( s 400) , s ( s 300)
(18)
0
1000
2000
3000
4000
Real Axis
(b) Gs , X GX WX Fig. 8 Root loci of the X-axis.
(17)
with two extra poles at s 0 and s 250 , and an extra zero at s 600 , to adjust the root loci as Fig. 8(b). Hence, the estimated settling time is 0.001sec with a gain of 3 109 . Similarly, we choose the weighting function in the y-axis as WY
-1000
4. SIMULATIONS AND EXPERIMENTS In this section, we implement the designed robust controllers for simulation and experimental verification. Given step inputs, the simulation results are obtained using the SimulinkTM structure of Fig. 9, and the experimental results are obtained using the hardware settings of Fig. 2.
with two extra poles at s 0 and s 300 , and an extra zero at s 400 . Thus, the estimated settling time is 0.01s with a gain of 1 1010 . The corresponding controllers are designed as : 1.435 s5 2650 s 4 3.609 106 s3 1.936 109 s 2 3.141 1011 s 1.93 1012 , s 5 2263 s 4 3.174 106 s 3 1.982 109 s 2 4.376 1011 s 2.77 1012 1.491 s4 + 3100 s3 + 3.296 106s2 + 8.706 108 s + 2.738 1010 , s4 + 3041 s3 + 3.928 106 s2 + 1.225 109 s + 4.084 1010
K, X K ,Y
with the following stability margins:
Fig. 9 Simulink structure.
bmax (Gs , X , K , X ) 5717, bmax (Gs ,Y , K ,Y ) 0.5569.
These stability margins are much greater than the system perturbations, such that robust stability can be guaranteed. These controllers will be implemented for experimental verification in section 4.
The simulation and experimental results are compared in Fig. 10, and illustrated in Table 1. First, the experimental results match with the simulations. That is, the identified transfer functions have caught basic dynamics of the PZT system. Second, the experimental responses have oscillations of about 300Hz, which is resulted from the system’s noises. In the
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
future, we will adjust the weighting functions to suppress these oscillations. Third, the experimental responses tend to have a shorter settling time, which might be caused by extra damping effects in the physical systems. Commnad Exp. Simu.
200
Axis Commnad Exp. Simu.
200
X
150 position(nm)
150 position(nm)
Table 2 Statistic data of the ramp responses.
250
250
100
50
0
0
2
Y
-50 1.995
2.005 2.01 2.015 2.02 2.025 2.03 2.035 2.04 2.045 time (s)
2
(a) 200 nm, X-axis
400
position(nm)
300
200
200
500
1000
2000
5000
RMSE
0.0072
0.0101
0.0175
0.0308
0.0586
0.1193
MAE
6.21
5.595
6.645
9.03
11.79
22.71
RMSE
0.0066
0.0092
0.0157
0.0267
0.0499
0.1017
MAE
5.335
5.8
6.625
7.025
11.08
21.49
In addition, to test the effect of loads on the platform during operation, we place 340g and 1000g masses on the stage and again test the system performance. The results are shown in Fig. 11, which indicates that the performance seems to be irrelevant to the carrying loads.
Commnad Exp. Simu.
500
400
100
2.005 2.01 2.015 2.02 2.025 2.03 2.035 2.04 2.045 time (s)
(b) 200 nm, Y-axis
Commnad Exp. Simu.
500
error
100
50
-50 1.995
position(nm)
with more rapid movement and the y-axis tends to have smaller error than the x-axis.
300
0.18
200
0.16 X-axis Noload X-axis 340g
0.16
Y-axis Noload Y-axis 340g
0.14
X-axis 1kg
100
100
Y-axis 1kg
0.14
0.12
0.12 0.1
2
2.005 2.01 2.015 2.02 2.025 2.03 2.035 2.04 2.045 time (s)
2.005 2.01 2.015 2.02 2.025 2.03 2.035 2.04 2.045 time (s)
0.1
RMS
1.995
2
RMS
0
0 1.995
0.08
0.08 0.06
0.06
(c) 500 nm, X-axis
(d) 500 nm, Y-axis
0.04
0.04
0.02
0.02 0
1000
1000
Commnad Exp. Simu.
900
700 position(nm)
position(nm)
600 500 400
7000
8000
9000 10000
0
0
1000
2000
3000
4000 5000 6000 speed (nm/s)
7000
8000
9000 10000
(b) RMSE, Y-axis
Fig. 11 Error to different loads.
500 400
Finally, we test the system’s ability to track three trajectories: a square, a circle, and a star, as shown in Fig. 12 (a), (c), and (e). The corresponding time trajectories of the x-axis and yaxis are shown in Fig. 12 (b), (d), and (f), which test the ability of the stage to track straight lines, sinusoidal signals, and simultaneous ramps, respectively. This analysis is illustrated in Table 3, which shows that the errors are smallest for the square, and largest for the circle. It is because the average moving speeds are 100nm/s for the square, 1257nm/s for the circle, and 200nm/s for the star. That is, the RMSEs and MAEs become bigger when the moving faster.
100
100
0
0
1.995
2
2.005 2.01 2.015 2.02 2.025 2.03 2.035 2.04 2.045 time (s)
(e) 1000 nm, X-axis
2
2.005 2.01 2.015 2.02 2.025 2.03 2.035 2.04 2.045 time (s)
(f) 1000 nm, Y-axis
Fig. 10 Step responses. Table 1 Statistic data of step responses.
axis
4000 5000 6000 speed (nm/s)
600
200
200
Y-
3000
300
300
axis
2000
(a) RMSE, X-axis
800
700
X-
1000
Commnad Exp. Simu.
900
800
1.995
0
200nm
500nm
1000nm
Settling
Experiment
0.01
0.009
0.009
time(s)
Simulation
0.015
0.015
0.015
Overshoot
Experiment
3.25
2.66
3.1
X-axis
Y-axis
X-axis
Y-axis
(%)
Simulation
3.8
3.8
3.8
RMSE
RMSE
MAE
MAE
Settling
Experiment
0.012
0.011
0.011
(nm)
(nm)
(nm)
(nm)
time(s)
Simulation
0.015
0.015
0.015
square
0.0111
0.01
3.545
3.355
Overshoot
Experiment
2.95
3.76
2.8
(%)
Simulation
2.3
2.3
2.3
circle
0.0315
0.0265
7.056
6.039
star
0.0118
0.0093
4.71
4.364
Table 3 Tracking errors.
The responses to ramp inputs are also tested, and the root mean square error (RMSE) and the maximum absolute error (MAE) are illustrated in Table 2. The system errors increase 3543
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
REFERENCES 250
250 Command Ex perim ent
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(a) Square tracking
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(b) Time trajectories for (a). 50
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Command Ex perim ent
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Command Experiment
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(e) Star tracking
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(f) Time trajectories for (e).
Fig. 12 Geometric tracking tests.
5. CONCLUDING REMARKS This paper has demonstrated H robust control for a piezoelectric nano-positioning stage. We first identified the stage as linear transfer functions, and regarded the system nonlinearities as model perturbations and disturbances. Then we designed robust controllers for the PZT system. The simulation and experimental results showed the effectiveness of the designed robust controllers in improving stability and performance of the piezoelectric stage. For future work, we can modify the weighting functions to suppress the system’s noises, as illustrated in (Wang et al., 2011). Furthermore, reduced-order robust control and robust PID control techniques can also be applied to simplify the controllers for system miniaturization, as shown in (Wang et al., 2009, 2010).
Choi, G.H., Choi, G.S., and Kim, H.S. (1997). A study on position control of piezoelectric actuators. IEEE International Symposium on Industrial Electronics, 851–855. Choi, G.H., Choi, G.S., and Oh, J.H., (1999). Repetitive tracking control of a coarse-fine actuato. Proceedings of the IEEE International Conference on Advanced Intelligent Mechatronics, 19–23. Choi G.H., Choi, G.S., and Lim, Y.A. (2002). Tracking position control of piezoelectric actuators for periodic reference inputs. Mechatronics, 12(5), 669–684. Doyle, J.C. and Zhou, K. (1998). Essentials of robust control, Prentice Hall, Upper Saddle River, New Jersey. Georgiou, T.T. and Smith, M.C. (1990). Optimal robustness in the gap metric. IEEE Transactions on Automic Control, 35 (6), 673–686. Guo, W. and Low, T.S. (1995). Modeling of a Three-Layer Piezoelectric Bimorph Beam with Hysteresis. Journal of Microelectromechanical Systems, 4(4), 230–237. Glover, K. and McFarlane, D.C. (1989). Robust controller design using normalized coprime factor plant descriptions, Lecture Notes in Control and Information Sciences, Springer–Verlag, London. Glover, K. and McFarlane, D.C. (1992). A loop shaping design procedure using synthesis, IEEE Transactions on Automatic Control, 37 (6), 759–769. MicroE Systems. (2010) Available online: http://www.microesys.com/encoderselectorchart/ National Instruments. (2010) Available online: http://sine.ni.com/nips/cds/view/p/lang/zht/nid/206087 Moor, B.D. and Overschee, P.V. (1991). Subspace algorithms for the stochastic identification problem. 30th IEEE Conference on Decision and Control, 1321-1326. Moor, B.D. and Overschee, P.V. (1994). N4SID: subspace algorithms for the identification of combined deterministic-stochastic system. Autimatica, 30 (1), 75–93. Wang, F.C. and Chen,H.T.(2009). Design and Implementation of Fixed-Order Robust Controllers for a Proton Exchange Membrane Fuel Cell System, International Journal of Hydrogen Energy, 34 (6), 2705-2717. Wang, F.C. and Ko, C.C. (2010). Multivariable Robust PID Control for a PEMFC System, International Journal of Hydrogen Energy, 35 (19), 10437–10445. Wang, F.C., Tsai, Y.C. and Hsieh, C.H. (2011). Design and implementation of fixed-order robust controllers for a precision piezoelectric stage. Submitted to CDC-ECC 2011. Wen, Y.K. (1976). Method for random vibration of hysteretic systems. Journal of the Engineering Mechanics Division-ASCE, 102 (2), 249–263. Zhou, K., Doyle, J.C., and Glover, K. (1996) Robust and Optimal Control. Prentice Hall, Upper Saddle River, New Jersey.
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Robust Control of a Two-Axis Piezoelectric Nano-Positioning Stage Fu-Cheng Wang*, Yan-Chen Tsai*, Chin-Hui Hsieh*, Lian-Sheng Chen*, Chung-Huang Yu** *Mechanical Engineering Department, National Taiwan University, Taipei 10617, Taiwan (Tel:+886233662680, e-mail:[email protected], [email protected] , [email protected], [email protected]) **Department of Physical Therapy and Assistive Technology, National Yang-Ming University, Taiwan (e-mail: [email protected]) Abstract: This paper applies robust control to a two-axis piezoelectric nano-positioning stage. As technology develops, the precision requirement for positioning platforms is becoming increasingly stringent. Since a traditional mechanical transmission structure cannot achieve high precision, a piezoelectric actuator is usually applied to drive the mechanism because of its high resolution, high accuracy, and large driving force. However, the non-linear dynamic characteristics of piezoelectric materials, such as hysteresis, might degrade system performance. Therefore, in the present study, we model a piezoelectric stage as a linear system, and regard the nonlinear factors as system uncertainties. We then apply robust control strategies to guarantee system stability and performance. Lastly, the designed controllers are implemented for experimental verification. The results demonstrate the effectiveness of these robust controllers. Keywords: piezoelectric, PZT, identification, robust control, loop-shaping
1. INTRODUCTION
2. SYSTEM DESCRIPTION AND IDENTIFICATION
As technology progresses, precise positioning requirements are becoming increasingly stringent. Frequently, traditional mechanical transmission structures might not meet these requirements due to friction and backlash. Therefore, piezoelectric materials are normally applied to drive position platforms because of their advantages such as high resolution, high accuracy, and large driving force. In piezoelectric materials, both direct and inverse piezoelectric effects are possible. The former occurs when the material produces an electric field in response to an applied force. The latter happens when the material undergoes a deformation in response to an applied electrical field. Using the inverse effect, piezoelectric actuators can drive positioning stages with a high degree of precision. However, non-linear characteristics of piezoelectric materials, such as hysteresis, might degrade the system performance. In order to reduce the non-linear effects, the Maxwell Slip model (Choi et al., 1997, 1999, 2002) and the Bouc-Wen model (Wen, 1976, Kuo and Low, 1995) have been proposed. However, these high-order models are difficult to construct and are not suitable for control design. Therefore, in this paper, we identify a piezoelectric stage as a linear model and regard the non-linear effects as system uncertainties and disturbances. We then apply robust control strategies to design suitable robust controllers to achieve system stability and performance.
Fig. 1 shows a two-axis PZT stage that consists of three layers: the top layer is a loading platform, the middle moves along the Y-axis, and the bottom moves along the X-axis. We installed ceramic piezoelectric actuators in the grooves of the stage, and stage movement is controlled by regulating the input voltages of the actuators. To measure stage movement, we used Mercury 5800 (MicroE Systems, 2010), which has a resolution of 1.22nm and digital outputs. The system structure is shown in Fig. 2, where Labview and PXI-8108 (National Instruments, 2010) were used for data acquisition. Because the voltage output of PXI-8180 is limited to 10V, a voltage amplifier, E-663, was also applied to drive the actuators.
This paper is arranged as follows: Section 2 introduces the PZT stage and obtains its transfer functions by identification techniques. Section 3 introduces robust control algorithms and designs robust controllers for the PZT stage. In Section 4, the designed robust controllers are implemented for experimental verification. Lastly, we draw conclusions in Section 5. 978-3-902661-93-7/11/$20.00 © 2011 IFAC
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Fig. 1 A two-axis piezoelectric stage. 10.3182/20110828-6-IT-1002.00741
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
repeated the experiments six times at each axis and obtained the following models of the PZT stage: G1X
-2.389 108 s4 + 1.278 105 s3 + 0.03251 s 2 + 73.78 s + 2472 , (1) s 4 + 2433 s3 + 3.698 106 s2 + 2.35 109 s + 6.383 1010
G X2
-1.413 108 s 4 + 5.818 10 6 s3 + 0.03211 s2 + 25.48 s + 134.8 , (2) s4 + 1487 s3 + 2.053 106 s 2 + 7.64 108 s + 2.257 109
G X3
-1.399 10 8 s4 + 5.959 106 s3 + 0.032 s2 + 24.04 s + 128.1 , s4 + 1478 s 3 + 2 106 s2 + 7.205 108 s + 2.242 109
(3)
G X4
-1.582 108 s4 + 7.753 10 6 s3 + 0.03348 s 2 + 28.48 s + 264.6 s4 + 1650 s3 + 2.184 106 s 2 + 8.627 108 s + 5.716 109
,(4)
G 5X
-1.438 10 8 s 4 + 6.343 106 s3 + 0.03259 s 2 + 24.4 s + 104.6 , (5) s 4 + 1517 s3 + 2.027 106 s 2 + 7.305 108 s + 1.519 109
G X6
-1.491 10 8 s 4 + 6.831 106 s3 + 0.03312 s2 + 25.66 s + 165.9 . s4 + 1570 3 + 2.082 106 s2 + 7.721 108 s + 3.176 109
(6)
GY1
-1.307 10 8 s 3 + 4.768 107 s 2 + 0.05036 s + 1.921 , s 3 + 1745 s 2 + 1.662 106s + 4.636 107
(7)
GY2
-1.578 108 s3 + 3.797 106 s2 + 0.05444 s + 2.199 , s 3 + 1971 s 2 + 1.791 106 s + 5.398 107
(8)
GY3
-1.316 10 8 s3 + 8.008 10 7 s 2 + 0.05011 s + 1.892 , s 3 + 1749 s 2 + 1.654 106 s + 4.572 107
(9)
1000 500
GY4
-1.317 108 s 3 + 7.774 107 s2 + 0.05017 s + 1.869 , s3 + 1750 s 2 + 1.655 106 s + 4.511 107
(10)
GY5
-1.318 108 s3 + 7.882 107 s2 + 0.05022 s + 1.887 , s3 + 1753 s 2 + 1.658 106 s + 4.567 107
(11)
GY6
-1.321 10 8 s 3 + 9.443 10 7 s2 + 0.05002 s + 1.89 . s 3 + 1751 s 2 + 1.652 106 s + 4.587 107
(12)
Fig. 2 Hardware control structure. At first, we tested the open-loop characteristics of the piezoelectric actuators. Given an input voltage, we measured the output displacement and drew the phase plot in Fig. 3, where the hysteretic effects of the actuators can be observed. 1500
0 -500 -1000 -1500 -40
-20
0 voltage(V)
20
Note that the orders of the transfer functions were automatically assigned by MATLAB for best fittings.
40
1500
10 voltage(V)
50 nm/s, Y-axis
position(nm)
1000 500
10 rX(t)
5
voltage(V)
position(nm)
50 nm/s, X-axis
0 -5 -10
0
5
10 time(s)
15
0 -5 -10
20
rY (t)
5
0
5
10 time(s)
15
20
0
(a)X-axis inputs
-500
(b)Y-axis inputs
2000
0 voltage(V)
20
40
0
-2000
Fig. 3 The hysteresis effects.
position(nm)
-20
2000 X(t)
0
5
10 time(s)
15
3540
position(nm)
2000
In order to achieve precise positioning, we consider a closedloop control structure for the stage. First, we generated chirp input voltage signals to drive the actuators and then measured the output displacement signals, as shown in Fig. 4. Using subspace system identification methods (Moor and Overschee, 1991, 1994), we can obtain the system’s transfer functions by a MATLAB command n4sid. Because the cross-coupling terms are relatively small and can be neglected, the MIMO system can be simplified as two SISO sub-systems. We
0
0
5
10 time(s)
15
20
2000 Y(t)
0
-2000
X(t)
-2000
20
position(nm)
-1500 -40
position(nm)
-1000
0
5
10 time(s)
15
(a) Outputs to rX
20
Y(t) 0
-2000
0
5
10 time(s)
15
(b) Outputs to rY
Fig. 4 Identification signals.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
3. ROBUST CONTROLLER DESIGN In this section, we introduce robust control algorithms and design robust controllers for the PZT system. The following Small Gain Theorem is normally considered for systems with uncertainties (Zhou et al., 1996):
1 / if and only if Z
1/ if and only if Z
M
G
r
Suppose Z RH and let 0 , the system of Fig. 5 is well posed and internally stable for all ( s ) RH with: (a)
N
N
z1
z2
M 1
; (b)
, where Z
is the
H norm of system Z. (a) The close-loop block diagram
w1
e1
N z1 z 2
Z
e
M
K 1 1 I ( I G0 K ) M
w2
2
Fig. 5 Small Gain Theorem. (b) Representation of the small-gain theorem
Suppose a nominal plant G0 has a normalized left coprime factorization of G0 ( s ) M ( s ) 1 N ( s ) , where M , N RH and M M * N N * I . Assume a perturbed system G can be expressed as: G ( M M ) 1 ( N N ) ,
with [ N
M ]
(13)
, and M , N RH . Consider a
closed-loop system with the controller K of Fig. 6(a), which can be rearranged as in Fig. 6(b) when r=0. Using Small-Gain Theorem, the closed-loop system is internally stable for all perturbations with N M , if and only if :
K 1 1 I ( I G0 K ) M
K ( I G0 K ) 1 I I
G0
From equations (1–12), the transfer functions of the X-axis or Y-axis varied at each experiment. Hence, we apply the ideas of gap metric (Georgiou and Smith, 1990) to select the nominal plant for controller design. Because the coprime factorization of a transfer function is not unique, the gap between G0 and G is defined as: The minimum perturbation of
N M changes G0 to G , denoted as δ (G0 , G ) .
G0 arg min max δ (G0 , Gi ), Gi . G0
which
Gi
(16)
Thus, we selected the nominal plants of the X-axis and Y-axis as follows: -1.491 10 8 s 4 + 6.831 106 s3 + 0.03312 s2 + 25.66 s + 165.9 , s 4 + 1570 3 + 2.082 106 s 2 + 7.721 108 s + 3.176 109 -1.316 10 8 s 3 + 8.008 10 7 s2 + 0.05011 s + 1.892 GY . s 3 + 1749 s 2 + 1.654 106 s + 4.572 107 GX
1
G0 ]
That is, the nominal plant G0 is chosen by:
. (14)
Thus, we can define the stability margin (Doyle and Zhou, 1998) as: K b(G0 , K ) ( I G0 K ) 1[ I I
Fig. 6 System structures.
,
(15)
such that the closed-loop system is internally stable for all perturbations N M with if and only if b(G , K ) .
These give the system gaps as δ (GX , GX i )<0.016256 and δ (GY , GY i )<0.016411 .
In order to improve system performance, loop-shaping techniques (Glover and McFarlane, 1989, 1992) are usually applied. As shown in Fig. 7, we set weighting functions to
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Root Locus
shape the plant as Gs ( s ) W2 ( s )G( s )W1 ( s ) , and then design a standard H robust controller K for the shaped plant.
2000 1500
Finally, we implement the controller K W1 KW2 for the original plant G ( s) .
K
W1
G
Imaginary Axis
1000 500 0 -500 -1000
W2
-1500 -2000 -6000
Gs
-4000
-2000
0
2000
4000
6000
Real Axis
(a) Gx Root Locus 2000 1500
W2
K
W1
G
1000
Imaginary Axis
500
K
0 -500 -1000
Fig. 7 Loop-shaping design procedures.
-1500 -2000
We use root locus techniques to estimate the settling time of the system’s step responses. For example, the root loci of Gx is shown in Fig. 8(a). Therefore, we set the following weighting function WX
3 109 ( s 600) , s( s 250)
1010 ( s 400) , s ( s 300)
(18)
0
1000
2000
3000
4000
Real Axis
(b) Gs , X GX WX Fig. 8 Root loci of the X-axis.
(17)
with two extra poles at s 0 and s 250 , and an extra zero at s 600 , to adjust the root loci as Fig. 8(b). Hence, the estimated settling time is 0.001sec with a gain of 3 109 . Similarly, we choose the weighting function in the y-axis as WY
-1000
4. SIMULATIONS AND EXPERIMENTS In this section, we implement the designed robust controllers for simulation and experimental verification. Given step inputs, the simulation results are obtained using the SimulinkTM structure of Fig. 9, and the experimental results are obtained using the hardware settings of Fig. 2.
with two extra poles at s 0 and s 300 , and an extra zero at s 400 . Thus, the estimated settling time is 0.01s with a gain of 1 1010 . The corresponding controllers are designed as : 1.435 s5 2650 s 4 3.609 106 s3 1.936 109 s 2 3.141 1011 s 1.93 1012 , s 5 2263 s 4 3.174 106 s 3 1.982 109 s 2 4.376 1011 s 2.77 1012 1.491 s4 + 3100 s3 + 3.296 106s2 + 8.706 108 s + 2.738 1010 , s4 + 3041 s3 + 3.928 106 s2 + 1.225 109 s + 4.084 1010
K, X K ,Y
with the following stability margins:
Fig. 9 Simulink structure.
bmax (Gs , X , K , X ) 5717, bmax (Gs ,Y , K ,Y ) 0.5569.
These stability margins are much greater than the system perturbations, such that robust stability can be guaranteed. These controllers will be implemented for experimental verification in section 4.
The simulation and experimental results are compared in Fig. 10, and illustrated in Table 1. First, the experimental results match with the simulations. That is, the identified transfer functions have caught basic dynamics of the PZT system. Second, the experimental responses have oscillations of about 300Hz, which is resulted from the system’s noises. In the
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
future, we will adjust the weighting functions to suppress these oscillations. Third, the experimental responses tend to have a shorter settling time, which might be caused by extra damping effects in the physical systems. Commnad Exp. Simu.
200
Axis Commnad Exp. Simu.
200
X
150 position(nm)
150 position(nm)
Table 2 Statistic data of the ramp responses.
250
250
100
50
0
0
2
Y
-50 1.995
2.005 2.01 2.015 2.02 2.025 2.03 2.035 2.04 2.045 time (s)
2
(a) 200 nm, X-axis
400
position(nm)
300
200
200
500
1000
2000
5000
RMSE
0.0072
0.0101
0.0175
0.0308
0.0586
0.1193
MAE
6.21
5.595
6.645
9.03
11.79
22.71
RMSE
0.0066
0.0092
0.0157
0.0267
0.0499
0.1017
MAE
5.335
5.8
6.625
7.025
11.08
21.49
In addition, to test the effect of loads on the platform during operation, we place 340g and 1000g masses on the stage and again test the system performance. The results are shown in Fig. 11, which indicates that the performance seems to be irrelevant to the carrying loads.
Commnad Exp. Simu.
500
400
100
2.005 2.01 2.015 2.02 2.025 2.03 2.035 2.04 2.045 time (s)
(b) 200 nm, Y-axis
Commnad Exp. Simu.
500
error
100
50
-50 1.995
position(nm)
with more rapid movement and the y-axis tends to have smaller error than the x-axis.
300
0.18
200
0.16 X-axis Noload X-axis 340g
0.16
Y-axis Noload Y-axis 340g
0.14
X-axis 1kg
100
100
Y-axis 1kg
0.14
0.12
0.12 0.1
2
2.005 2.01 2.015 2.02 2.025 2.03 2.035 2.04 2.045 time (s)
2.005 2.01 2.015 2.02 2.025 2.03 2.035 2.04 2.045 time (s)
0.1
RMS
1.995
2
RMS
0
0 1.995
0.08
0.08 0.06
0.06
(c) 500 nm, X-axis
(d) 500 nm, Y-axis
0.04
0.04
0.02
0.02 0
1000
1000
Commnad Exp. Simu.
900
700 position(nm)
position(nm)
600 500 400
7000
8000
9000 10000
0
0
1000
2000
3000
4000 5000 6000 speed (nm/s)
7000
8000
9000 10000
(b) RMSE, Y-axis
Fig. 11 Error to different loads.
500 400
Finally, we test the system’s ability to track three trajectories: a square, a circle, and a star, as shown in Fig. 12 (a), (c), and (e). The corresponding time trajectories of the x-axis and yaxis are shown in Fig. 12 (b), (d), and (f), which test the ability of the stage to track straight lines, sinusoidal signals, and simultaneous ramps, respectively. This analysis is illustrated in Table 3, which shows that the errors are smallest for the square, and largest for the circle. It is because the average moving speeds are 100nm/s for the square, 1257nm/s for the circle, and 200nm/s for the star. That is, the RMSEs and MAEs become bigger when the moving faster.
100
100
0
0
1.995
2
2.005 2.01 2.015 2.02 2.025 2.03 2.035 2.04 2.045 time (s)
(e) 1000 nm, X-axis
2
2.005 2.01 2.015 2.02 2.025 2.03 2.035 2.04 2.045 time (s)
(f) 1000 nm, Y-axis
Fig. 10 Step responses. Table 1 Statistic data of step responses.
axis
4000 5000 6000 speed (nm/s)
600
200
200
Y-
3000
300
300
axis
2000
(a) RMSE, X-axis
800
700
X-
1000
Commnad Exp. Simu.
900
800
1.995
0
200nm
500nm
1000nm
Settling
Experiment
0.01
0.009
0.009
time(s)
Simulation
0.015
0.015
0.015
Overshoot
Experiment
3.25
2.66
3.1
X-axis
Y-axis
X-axis
Y-axis
(%)
Simulation
3.8
3.8
3.8
RMSE
RMSE
MAE
MAE
Settling
Experiment
0.012
0.011
0.011
(nm)
(nm)
(nm)
(nm)
time(s)
Simulation
0.015
0.015
0.015
square
0.0111
0.01
3.545
3.355
Overshoot
Experiment
2.95
3.76
2.8
(%)
Simulation
2.3
2.3
2.3
circle
0.0315
0.0265
7.056
6.039
star
0.0118
0.0093
4.71
4.364
Table 3 Tracking errors.
The responses to ramp inputs are also tested, and the root mean square error (RMSE) and the maximum absolute error (MAE) are illustrated in Table 2. The system errors increase 3543
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
REFERENCES 250
250 Command Ex perim ent
Command Experiment
200
150 Y -a x is (n m )
200
100
50
Y -ax is(nm )
150
0
-50
0
1
2
3
4 X-ax is(nm )
5
6
7
8
100 250 Command Ex perim ent 200
50 Y -a x is (n m )
150
0
100
50
0
-50
-50 -50
0
50
100 X-axis(nm)
150
200
0
1
2
3
250
(a) Square tracking
4 X-ax is(nm )
5
6
7
8
(b) Time trajectories for (a). 50
300
Command Ex perim ent
0 -50
Command Experiment
Y -axis(n m )
200
-100 -150 -200 -250 -300 -350
100
-400
Y -ax is (n m )
-450 0
1
2
3
4 X-ax is(nm )
5
6
7
8
0 250 Command Ex perim ent
200 150
-100 Y-a xis(nm )
100
-200
50 0 -50 -100 -150 -200
-300 -500
-250
-400
-300
-200 -100 X-axis(nm)
0
0
1
2
3
100
(c) Circle tracking
4 X-ax is(nm )
5
6
7
8
(d) Time trajectories for (c). 500 Command Ex perim ent
450
400
Command Experiment
300 Y-a xis(nm )
400 350
200
100
Y -ax is(nm )
300
0
250
-100
0
2
4
6 X-ax is(nm )
8
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200 450 Command Ex perim ent
400
150
350 300 Y -axis(n m )
100 50
250 200 150 100
0
50 0
-50 -100
-50
0
100
200 X-axis(nm)
300
(e) Star tracking
400
500
0
2
4
6 X-ax is(nm )
8
10
12
(f) Time trajectories for (e).
Fig. 12 Geometric tracking tests.
5. CONCLUDING REMARKS This paper has demonstrated H robust control for a piezoelectric nano-positioning stage. We first identified the stage as linear transfer functions, and regarded the system nonlinearities as model perturbations and disturbances. Then we designed robust controllers for the PZT system. The simulation and experimental results showed the effectiveness of the designed robust controllers in improving stability and performance of the piezoelectric stage. For future work, we can modify the weighting functions to suppress the system’s noises, as illustrated in (Wang et al., 2011). Furthermore, reduced-order robust control and robust PID control techniques can also be applied to simplify the controllers for system miniaturization, as shown in (Wang et al., 2009, 2010).
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