Positioning Control for Piezo Scanner using Multirate Perfect Inverse Model Based Iterative Learning Control

Transcription

1 21 IEEE/ASME International Conference on Advanced Intelligent Mechatronics Montréal, Canada, Jul 6-9, 21 Positioning Control for Piezo Scanner using Multirate Perfect Inverse Model Based Iterative Learning Control Takauki Shiraishi and Hiroshi Fujimoto Abstract Recentl, in high precision positioning field, the improvement of the positioning accurac is required for the development of the next generation technolog like nanotechnolog. In man industr application machines, we can often see the repetitive operation [?]. For the repetitive position command of the positioning machine, the investigation for the suppression of the repetitive tracking using iterative learning control (ILC) is increasing. For above reasons, the purpose of this paper is improvement of the positioning accurac in the high-speed repetitive position command. The plant is the Piezo-scanner of the atomic force microscope which is not onl used for measurement device but nano-manipulation. the requirement of positioning accurac of the AFM is nano-scale. In ILC, the inverse sstem of the plant is often used as the learning filter. This design of the learning filter using the inverse sstem is most important part in ILC sstem design. Thus, there are man methods of the inverse sstem design. In this paper, the ILC using the perfect inverse sstem of the plant which based on the perfect tracking control is proposed. Its proposed inverse sstem of the plant is the perfect inverse sstem of the plant, and the all poles of the proposed inverse sstem of the plant can be located on the origin in discrete-time domain. Thus, the proposed inverse plant has good characteristics for ILC sstem. In this paper, the effectiveness of the proposed method is shown b the some simulations and experimental results. I. INTRODUCTION The positioning accurac required of a precision positioning device was getting on improving, and required scale has changed from the microscale to nanoscale. In this paper, the piezo scanner of the atomic force microscope (AFM) is positioned at high speed on nano-scale. Usuall, AFM is used as a device which measures nanoscale surface topograph. However, it is utilized also for the nanoscale manipulation using the probe not onl as the measuring device. AFM in this paper has the XYZ scanner of the piezo-electric element. In measurement of surface topograph, X scanner is repetition operation. High accurac is demanded in order that the positioning accurac of the piezo scanner ma affect the measurement accurac of AFM directl. As pioneering investigation of the fast positioning control of XY scanner, the hardware of the piezo scanner is improved [2] or there are Zero Phase Error Traking Control (ZPETC) and Zero Magni- The authors would like to note that part of this research is supported b the Grant-in-Aid for Young Scientists (A) from the MEXT of Japan ( , 26868). T. Shiraishi is with Department of Electrical and Computer Engineering, Yokohama National Universit, 79-5 Tikiwadai, Yokohama, Japan shiraishi@hfl.dnj.nu.ac.jp H. Fujimoto is with Department of Graduate School of Frontier Science, Universit of Toko, Kashiwanoha, kashiwa-shi, Chiba, Jjapan. fujimoto@k.u-toko.ac.jp> tude Error Traking Control (ZMETC) which are the controlsstem designs which used the inverse sstem [3][4][5]. Iterative learning Control (ILC) is known b one of the was of suppressing the tracking to repetitive trajector [6][7]. ILC is used for positioning of the machine tool and the semiconductor photolithograph machine, and is the effective method in the field of fast positioning. This paper applies ILC to the piezo scanner of AFM, and aims at fast highl precise positioning in nanoscale. In ILC, using the inverse sstem of the plant for the learning filter is recommended, and the stabilit and repressed difference appears with the construction of the learning filter. Moreover, in actual iteration learning control, since the control sstem becomes unstable easil, the learning gain is set as the small value and the method of suppressing the graduall is used. This paper aims at suppression in the short time b appling the stable multi rate inverse sstem derived b Perfect Tracking Control (PTC) [8] to the learning filter. Moreover, the adequate controller is designed from the convergence condition of ILC. Finall, simulation and the experiment show the effectiveness of the proposal method. II. IDENTIFICATION OF THE PLANT Fig. 1 shows the simplified structure of the plant which hardware is the commercial AFM (JSPM-52, JEOL Ltd,). The position of Z-scanner can measure b the photo diode which detects the reflected laser beam from cantilever tip. The frequenc characteristics of the plant Plant P[z] and the nominal plant [z] are shown in Fig.?? which is obtained b the servo analzer. The transfer function of the continuous-time domain and the discrete-time domain nominal plants (s), [z] are represented b eq.(1) and (2). Here, [z] is discretized with the zero-order hold at the Y Z X Cantilever Sample X Laser Z Y XYZ scanner A B C D Photo diode X Fig. 1. (t) u(t) D/A PZT Driver PZT Driver PZT Driver A/D D/A D/A Control input based imaging u[i] DSP 3D - image Displa Cfb Feedback Controller X scan Y scan Piezo-scanner of AFMs. e[i] [i] r[i] /1/$ IEEE 993

2 3 2 P[z] [z] L[z] ILC Gain[dB] 1 memor Q[z] memor Frequenc[Hz] 2 1 P[z] [z] r C fb[z] u fb u ff u P(s) plant Phase[deg] Fig. 3. Control sstem with iterative learning control. Fig Frequenc[Hz] Frequenc responses of the plant and the nominal Plant. the product of the matrices written in the form of eq.(7) becomes commutative. In this section, the stabilit of ILC is examined using the form of eq.(5) (7). B. Without ILC sampling time T s = 4[µs]. It is assume that the iterative reference trajector r which (s) = s s s (1) period is T ite. Then, the output and the tracking are j and e i = r j which is represented b [z] = (z )(z ) z 3.644z 2 (2) z.353 From eq.(2), the discretized nominal plant [z] has the unstable-zero A. Introduction of ILC III. CONTROL SYSTEM DESIGN At first, the signal used b ILC of this paper is explained. The following discrete time sstem which is discretized at the sampling time T s, linearit time invariant sstem. j [k] = G[q]u j [k] (3) where k shows k = kt s and j shows the iterative number (j = 1, 2, ). The transfer function G is the stable sstem. q is the time shift opetator. For example, q(u j [k]) is expresses q(u j [k]) = u j [k + 1]. Here, if the itetarive period T ite is T ite = NT s, the output j and thecontrol input u j at the iterative number j is represented b j = [ j [], j [1], j [N] ] T (4) u j = [ u j [], u j [1], u j [N] ] T. (5) where j and u j are the vector. Then, it is assume that the transfer function G[q] is causal sstem as follows. G[q] = g q + g 1 q 2 + g 2 q 3 + (6) Here, g k is the impulse response of G[z] at time t = kt s (k =, 1, ). Thus, transter function G can represent b g g 1 g G = (7) g N g N g From avobe equation, the relationship eq.(3) is shown b j = Gu j. where G is toeplitz and triangle matrix. Thus, e j = r Pu j = r PC fb e j = (I + PC fb ) r = S fb r. (8) where I, P, C fb, S fb are the unit matrix, the plant, the feedback controller, and the sensitivit function, respectivel. Thus, the tracking e i in iterative operation is onl determined b the sensitivit function S fb. C. Iterative learning control sstem There are some methods of ILC. In this paper, the control sstem adopted Previous Current Ccle Learning (PCCL) which is combined Current Ccle Learning and Previous Ccle Learning. Thus, PCCL has both characteristic [1], also PCCL is the one of the most populer tpe of ILC. The equation of data renewal of PCCL is represented b u j+1 = Q(u j + Le j ) + C fb e j+1. (9) where Q is the lowpass filter which adjusts the stabilit of the sstem. L is the learning filter which is the inverse sstem of the plant, usuall. The control block diagram is shown in Fig. 3. From eq. (9), the control input at the iteration number j +1 uses the signal of the one period before which is stored b the FIFO memor. The tracking of eq.(9) is represented b e j+1 = Q(I PL)S fb e j + (I Q)e 1. (1) The stabilit condition which is derived b eq.(1) is represented b Q(I PL)S fb < 1. (11) Hence, if the infinit norm Q(I PL)S fb is smaller than 1, the tracking will be converged. Thus, if Q and L can design as Q = I, L = P, (12) 994

3 the tracking e j+1 can suppress to zero at the iteration number j = 2 (e 2 = O). From above reason, the learning filter L is designed as the inverse sstem of the plant, usuall. However, in actual sstems, there is the modeling between the plant and the nominal plant. Here, it is assume that P = (I + ) and L = Pn. Then, the tracking is represented b e j+1 = Q S fb e j + (I Q)e 1. (13) Generall, the nominal plant is designed give priorit from the low frequenc domain and correspond with the characteristic of the plant. Therefore, the modeling is the inclination which becomes large in the high frequenc domain. Moreover, in the positioning control sstem, the stead-state deviation becomes zero, and the feedback band is extended as much as possible, giving the adequate stabilit margin. As the result, the infinite norm of the sensitivit function becomes larger than 1 from the integration theorem of Bodo in the high frequenc domain. Therefore, Q filter has the desirable low pass characteristic, in order to suppress the gain of the high frequenc domain of S fb to less than 1. However, since the tracking also increases as the high frequenc domain is cut so that clearl from the second term of the right-hand side of the eq. (13), it is not desirable to enlarge the order of Q filter. Then, the construction of PCCL is considered. First, the renewal tpe of data of the e. (9) is transformed such as the following equation. u j+1 = Q(u j + P n (r j ) + C fb e j+1 = P n Qr + C fb e j+1 + Q(u j P n j ).(14) From eq.(eq:pccl1), Pn Qr is the feedforwar input which is generated b the reference, C fb e j+1 is the control input which is generated b the feedback controller, u j j is the control input as the estimated input disturbance at the one period before. Thus, it becomes clear from the control structure of PCCL to suppress the disturbance and the modeling b the periodic disturbance observer, and also to converge quickl b the feedforward input using the inverse sstem of the plant. P n D. Feedback controller The feedback controller C fb (s) which is PI compensator is represented b C fb (s) = s (15) s In implementation, C fb (s) is discretize b tustin at the sampling time T s. E. Notch filter To suppress the peak of the sensitivit function, the notch filter is inserted after the C fb. In ILC, the infinit norm of the control sstem have to keep less than 1 for the stable. The designable controller are the learning filter L, Q filter, and FB controller C fb. Although designing all three controllers collectivel is also considered, the engineer wants to give the phsical meaning of each controller. If it thinks from the eq. (1), the modeling will cause gain buildup from the first term of the right-hand side that L inserts the notch in the inverse model. QItismorenaturaltoinsertin or C fb. Here, when the notch characteristic is given to Q, it has influence on the second term of the right-hand side. It is better to include the notch characteristic in the feedback controller. The notch filter which negates the resonance peak is designed. The notch frequenc is 5.65 [khz]. s s C notch (s) = s s (16) F. Design of learning filter It explained using the inverse sstem of the plant for the learning filter. However, in the relative order, in the continuous time domain, the unstable-zero generates the 3rd more than plant b discretization. Therefore, the stable inverse sstem cannot be composed [11]. This paper compares the case where ZPETC, ZMETC, and PTC which compose the inverse sstem from the discrete domain are used for ILC. 1) Learning filter based on ZPETC: The inverse sstem of the plant derived b ZPETC[5] realizes the zero phase characteristic in all the frequencies below nquist frequenc, when the future value is given to the input. The discretized plant is represented b P[z] = B[z] A[z] = B s[z]b u [z]. (17) A[z] The characteristic polnomial A[z] is the stable. where B [z ] is the s-th order monic polnomial equation which include the unstable or limit of the stable zeros, and B + [z ] is the (m v)th order monic polnomial equation which include the stable zeros. B u [z] = b un z n + b u(n) z n + + b u (18) where the B u [z] is rewrited as B f u[z] = b u z n + b u1 z n + + b un. (19) Then, ZPETC based inverted model of the plant is obtained b G zpet [z] = z q A[z]Bu[z] f B s [z](b u [1]) 2 (2) Fig. 4 shows the bode diagram of the nominal plant [z] and G zpet[z]. Although the phase characteristic corresponds, the has arisen in the gain characteristic as nquist frequenc is approached. 2) Learning filter based on ZMETC: The inverse sstem of the plant derived b ZMETC[3][4] realizes the zero magnitude characteristic in all the frequencies below nquist frequenc. Then, ZMETC based inverted model of the plant is obtained b z q A[z] G zmet [z] = B s [z]bu[z] f (21) Fig. 5 shows the bode diagram of the nominal plant [z] and G zmet[z]. Although the gain characteristic corresponds, the has arisen in the phase characteristic as nquist frequenc is approached. 995

4 Gain [db] Phase [deg] Gain [db] Phase [deg] G zpet frequenc [Hz] G zpet frequenc [Hz] Fig. 4. G zmet Bode diagram of and G zpet [z] frequenc [Hz] G zmet frequenc [Hz] Fig. 5. Bode diagram of and G zmet [z]. 3) Learning filter based on PTC: Perfect Tracking Control (PTC) [8] is the control method which can achieve zero tracking on the sample point to the nominal plant. The state equation which realized the controlable form is represented b ẋ(t) = A c x(t) + b c e(t), (22) where x(t) is the is the state variable. Then, the plant is discretized. The relation of the control period is T u = T and T r = nt u, respectiver. The state equation discretized b sampling-period T u is Here x[k] shows x(kt u ). x [k + 1] = A k x[k] + b k e[k] (23) A k = e A ct u, b k = Tu e A cτ b c dτ (24) Since the order of the plant is n, the n sample future of the eq. (23) is considered. It is considered as timet = it r = kt u.. x [i + 1] = A i x[i] + B i e[i] (25) = A n k, B i = [A n k b k,,a k b k,b k ] A i the control input is obtained from eq.(25) as follows. u ptc [i] = B i (I zr A i )e[i + 1] (26) Here, zr is the shift operator of sampling time T r. As for matrix B i, the holomorph will be guaranteed if the plant is the controllable. Moreover, all the poles of the inverse sstem are at the origin of z plane. B the eq.(26), since the state variable was necessar, the state variable undetectable in this paper was derived b central difference. For example, ė[k] of deflection was derived b the following equation. ė[k] = G. Design of Q filter Q filter is represented b [k + 1] [k 1] 2T u (27) ( z ) Nq z Q[z] = (28) 4 The filter is the zero phase low pass filter, and cut-off frequenc is not small according to the increase in N q. First, it is Fig. 6 about S fb [z] in case the plant and the nominal plant are eqs. (1) (2), and the frequenc response of the Q filter is shown. As Fig. 4 and 5 showed, both of G zpet [z] and G zmet [z] should express the plant with sufficient accurac, but Fig. 6 the big difference is seen in S fb [z]. In the case where G zpet [z] is used, below in [db], the gain has become, and has brought the desirable result. On the other hand, when G zpet [z] is used, the modeling serves as the big gain in the high region. It can check that the modeling is greatl concerned with stabilit from these results. Moreover, since the perfect inverse sstem can be composed from PTC, it can expect that the gain is falling rather than these two methods. Then, the design to the actual plant is considered. First, the frequenc characteristics of sensitivit function S fb [z] and modeling are investigated. Since the modeling into consideration, high order fitting model hi (s) shown in the following equation is prepared. hi (s) = s s s s s s s s s s (29) The frequenc response at the time of making the plant into the eq. (29) is shown in Fig. 7. Although it is considered b Fig. 6 that Nq = 1 is enough as the Q filter, if it Fig. 7 Sees, the gain of the high region is rising greatl under the effect of the modeling. Since it is considered that there is the bigger modeling than the actuall result, it can be guessed that the Q filter about Nq = 2 is needed. After designing the Q filter and the learning filter, memor read timing is determined according to each learning filter. 996

5 Gain [db] Gain [db] δ zpet S fb 3 δ S zmet fb Q Q frequenc [Hz] Fig Frequenc responses of Q[z], S fb [z], zpet. δ zpet S fb 3 δ S zmet fb Q Q frequenc [Hz] Fig. 7. Frequenc responses of Q[z], S fb [z], zpet. IV. SIMULATION AND ENPERIMENTAL RESULT A. Simulation Triangular wave trajector is used for actuation of the direction of X with the piezo scanner of AFM. The reason for using the triangular wave is to keep the constant speed during the X scan. The frequenc of the triangular wave was made into 1 [khz]. The simulation result at then [ this ] is shown in Fig. 9. It is the result in (a) ZMETC, (b) ZPETC, and (c) PTC, respectivel. It is the case where the plant and the nominal plant are made into the eq.(1) (2) (considerable in Fig. 6). Moreover, the Q filter is not used in order to clarif the difference in the learning filter. It is the tracking for six periods, respectivel. Since 1 period serves as the answer of onl FB, its following nature is bad, but in (b) ZPETC and (c) PTC, the tracking can be powerfull suppressed after 2 period, respectivel. (b) Although the result with most sufficient ZPETC was brought, since it asked for the state variable (ė,ë) undetectable in PTC b central difference, such the result was brought. It can check that the analses in Fig. 6 are right. B. Experimental result The triangular wave trajector of 1 [khz] of the same conditions as simulation was given. The amplitude is 8. It is [nm]. An experimental result is shown in Fig. 1 and 11. Respectivel, the are the reference track, and the position and deflection. Deflection piles up the experimental result and average value of 1 periods. Convergence is dramaticall fast even when which learning filter is used. However, since the order of the Q filter had become in N q = 1 oscillation in G zpet [z], it was considered as N q = 2. Under this effect, deflection is large the little in G zpet [z]. V. CONCLUSION In this paper, ILC was applied to the piezo scanner and fast positioning was carried out. Moreover, ZPETC, ZMETC, and PTC were used for the learning filter of ILC, and comparison analses were carried out. It analzed about the stabilit of ILC and the adequate Q filter was designed from the stabilit condition. From simulation and the experimental result, it was checked that the learning filter based on ZPETC is effective. Although the learning filter based on PTC was considered to be the effective method, the effect b central difference was checked. Since the learning filter based on ZMETC has buildup of the gain in the high region, it is necessar to make the number of section of the Q filter increase as compared with the two previous methods. Fast operation of 1 [khz] was achieved not onl in simulation but in the experiment. ACKNOWLEDGEMENT The authors would like to note that part of this research is supported b the Grant-in-Aid for Young Scientists (A) from the MEXT of Japan ( , 26868). REFERENCES [1] M. Sitti, Atomic force microscope probe based controlled pushing for nanotribological characterization, IEEE-ASME T. Mech., vol. 9, pp , 24. [2] N.Kodera, H.Yamashita, and T.Ando : Rev. Sci.Instrum., 76,5378(25) [3] J. A. Butterworth, L. Y. Pao, and D. Y. Abramovitch. The effect of nonminimum-phase zero locations on the performance of feedforward model-inverse control techniques in discrete-time sstems., In American Control Conference, 28, pp , 28. [4] J. A. Butterworth, L. Y. Pao, D. Y. Abramovitch, Architectures for Tracking Control in Atomic Force Microscopes, in Proc. IFAC 17th World Congress, pp , 28. [5] M.Tomizuka: Zero Phase Error Tracking Algorithm for Digital Control, Trans. ASME, Jounal of Dnamic Sstems, Measurement, and Control, Vol.19, pp (1987) [6] K. L. Moore, Iterative Learining Control for Deterministic Sstems Springer, pp.17, [7] H.-S. Ahn, Y. Chen, and K. L. Moore. Iterative learning control: Brief surve and categorization. Sstems, Man, and Cbernetics, Part C: Applications and Reviews, IEEE Transactions on, 37(6): , 27. [8] H. Fujimoto, Y. Hori and A. Kawamura: Perfect Tracking Control Method Based on Multirate Feedforward Control, T. SICE, pp (27) (in Japanese) [9] B. T. Fine, S. Mishra, M. Tomizuka, Model Inverse Based Iterative Learning Control Using Finite Impulse Response Approximations, Proc. 29 American Control Conference, pp , 29. [1] J. X. Xu, S. K. Panda, and T. H. Lee, Realitime Iterative Learning Control, Springer, pp.7-13, 29. [11] K. J. Åström and J. Sternb : Zeros of sampled stetem, Automatica, vol. 2, no.1, pp (1984). 997

6 ref ref ref ref, ref, ref, (a) ZMETC based ILC Fig. 8. Simulation results (tracking ) (a) ZMETC based ILC Fig. 9. Simulation results (tracking ). 6 ref 6 ref 6 ref position[nm] position[nm] position[nm] (a) ZMETC based ILC Fig. 1. Experimental results (tracking ) [nm] [nm] [nm] (a) ZMETC ibased ILC Fig. 11. Experimental results (Reference and Output). 998