Time-Optimal Motion Control of Piezoelectric Actuator: STM Application

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1 Time-Optimal Motion Control of Piezoelectric Actuator: STM Application Yongai Xu, Peter H. Mecl Abstract This paper exaes the problem of time-optimal motion control in the context of Scanning Tunneling Microscopy (STM) application an hysteresis compensation inherent to PZT actuators. The problem was ivie into velocity tracing problem an point-to-point motion problem, an the time-optimal input profile was obtaine using the comman shaping technique of constraine least-square optimization. The optimal scanning velocity given a specific scanning istance an the effect of sampling rate have also been stuie. To further improve the tracing performance, the piezoelectric hysteresis nonlinearity was moele using the classical Preisach Moel, an explicitly compensate using the propose continuous numerical inversion algorithm. goes bac to the starting point in the x axis while the y axis is avance to the next ban as fast as possible [], as shown in Fig.. The motion control problem is accoringly ivie into a velocity tracing problem (control over the active-scan region) an a point-to-point motion problem (control over the transition region). In this paper a time-optimal input profile is esigne using the comman shaping technique of constraine leastsquare optimization, which was first introuce by M. C. Reynols an P. Mecl [8]. It enables the STM to accurately scan along the specifie trajectory at a faster rate without exciting its vibrational moe. I. INTRODUCTION anotechnology is currently a very active research area N with a broa range of applications in materials science, computing, chemical an biological etection, an rug treatment. Scanning Tunneling Microscopy (STM) is a wiely use tool to manipulate materials at the atomic level. By its nature, however, the probe/actuator system of a STM is inherently flexible an the inuce vibration is the major limiting factor in achievable scanning precision an spee. Many researchers have aresse this problem in various ways. S. Devasia an co-worers []-[3] aopte a feeforwar approach an propose a moel-base inversion metho to achieve the imum input energy. S. Salapaa et al. [4] an G. Schitter et al. [5] iniviually evelope a controller base on H technique. X. Tan an J. S. Baras [6] propose a robust control framewor for smart actuators by combining inverse control with the l robust control theory. C. J. Li et al. [7] esigne a nonlinear piezo-actuator controller using a self-tuning regulator base on learning parameter estimation. A typical scanning trajectory of STM consists of two parts: an active-scan region in which the probe scans at a precise velocity; an a transition region in which the probe Manuscript receive September 22, Yongai Xu is with the School of Mechanical Engineering, Purue University, West Lafayette, IN USA (phone: ; fax: ; xuy@ purue.eu). Peter H. Mecl is with the School of Mechanical Engineering, Purue University, West Lafayette, IN USA. Fig.. Scanning trajectory (from []) Another major concern in STM application is hysteresis inherent to the piezoelectric actuator. Hysteresis is an inputoutput nonlinearity which, if left uncompensate, will egrae the system s overall performance. There are many ways to moel hysteresis, but the most popular one is the classical Preisach moel an its erivatives [9][0]. Though the Preisach moel oes not provie any physical insight into the problem, it can successfully prouce behaviors very similar to physical systems. A STM system can be moele as the composition of a hysteresis nonlinear component H an a linear ynamic component G as shown in Fig. 2. Tao an Kootovic [] have shown that accurate position control is achievable if an inverse operator H can be foun such that H an H cancel each other, thus ing the controller to be esigne base on the linear ynamics of the plant. Many hysteresis inversion methos have been propose.

2 D. Hughes an J. T. Wen [2] utilize the monotonic nature of the first-orer reversal curves (both the first-orer escening curves an ascening curves) to invert the hysteresis function. R. Venataraman an P. S. Krishnaprasa [3] propose an inversion algorithm base on the contraction mapping principle by exploiting the properties of Lipschitz continuity an incrementally strict increase of the Preisach operator uner some mil assumptions. X. Tan et al. [4] evelope Closest Match Algorithm for the inversion irectly base on a iscretize hysteresis moel. plant u r In this paper a continuous numerical inversion algorithm (CNIA) is propose base on the classical Preisach Moel an guarantee to converge. The spee of convergence can be manipulate by choosing an appropriate stopping threshol. Simulations serve to emonstrate its effectiveness in attenuating the hysteresis nonlinearity. II. TIME-OPTIMAL FEEDFORWARD DESIGN A. Time-optimal comman shaping For now we focus on the x-axis movement because the actuator oesn t move in the y-axis uring the active P H P u u c y H G P 2 Fig. 2. Plant moel P 3 t t 2 P 2 P 3 t t 2 Fig. 3. Scanning reference trajectory. P 4 P 4 scanning perio an therefore the y-axis movement oesn t affect the scanning accuracy. The x-axis reference trajectory (position reference an velocity reference, respectively) shown in Fig. 3 is ivie into two parts. The first part PP 3 is basically a velocity tracing problem. The scanning uration Tscan = t2 t is etere by the specific scanning istance an velocity, thus t is the optimization goal. By setting the tracing tolerance on PP 2 to infinity an the tracing tolerance on PP 2 3 to the esire value that meets the velocity tracing precision, the system is mae to reach the esire velocity as fast as possible an then settle own to a nearly constant velocity for the uration of T scan. The secon part PP 3 4 is essentially a point-to-point motion problem without any tracing constraints, which enables the system to return to the starting point in the x-axis as fast as possible. This way the shape comman input will be time optimal. The comman shaping technique for tracing [8] is use to solve the optimization problem an can be formulate on the iscrete-time state space representation as follows. Given the system x( + ) = Ax( ) + Bu( ) () y ( ) = Cx( ) we have x( ) = A x + ΦU ( ) (2) where 2 Φ = [ A B A B B], an U( ) = [ u u() u( )] T. The system output shoul satisfy the tracing tolerance: y ( ) = Cx( ) = CA x + CΦU( ) (3) y ( ) y( ) e ( ) If y ( ) y( ) > 0, (3) becomes CΦU( ) y ( ) + e ( ) CA x (4) Denote R = CΦ, f = y ( ) + e ( ) CA x, (4) can be rewritten as RU (5) f For y ( ) y( ) < 0, we have a similar expression RU (6) where f = y ( ) + e ( ) +CA x. 2 From (5) (6) we set up the tracing constraint for the -th sampling instant as R U ( ) f 2 R f2 f (7)

3 Also the actuator has a saturation limit: u ( ) F (8) By incorporating the tracing constraint an actuator limit at each point on the trajectory in a least-square programg scheme, a time-optimal input profile can be obtaine by increasing until a solution satisfying the constraints results. The optimization problem is formulate as {( ) ( )} T 2 x x x x ( ) ( ), U ( ) (9) subject to (7), (8) There are many stanar pacages for solving linear least-square programg problems such as MATLAB. Compare to other optimization schemes, the largest avantage of this metho is that tracing can be irectly specifie in the solution scheme, thus avoiing manipulating weighting factors that have little physical meaning. The linear component G of STM moel we use to o the simulation is a 4 th -orer transfer function aopte from [], which contains a pair of non-imum phase (NMP) zeros at 643 Hz an two vibration moes with natural frequencies of 24.9 Hz an Hz, respectively. 2 2 K( s 2ζ zωzs+ ωz ) Gs () = s + 2ζ ωs+ ω ( i i i ) i= Fig. 5. Position an velocity outputs to the time-optimal comman TABLE I STM MODEL PARAMETERS Damping ratio Natural frequency (Hz) NMP zero st moe n moe K The resulting comman profile is shown in Fig. 4. This profile oesn t have the symmetry shown in L. Y. Pao s wor [5] an at the en of the maneuver the input oesn t go bac to zero because the STM moel oesn t have a rigi boy moe. The position an velocity outputs for 4 Fig. 4. Time-optimal comman profile for v = 0 µm/s Fig. 6. Time optimality comparison between ifferent scanning velocities three cycles of the shape time-optimal comman are shown in Fig. 5. B. Time optimality uner ifferent velocities The time optimality of the solution using ifferent scanning velocities is investigate an the results are shown in Fig. 6 for three scanning istances of 20, 4 an 8 µm, respectively. It is observe that, for a given scanning istance, a higher scanning velocity oes not necessarily result in a shorter cycle time because the time spent on the transition to reach the esire velocity may offset the time gaine uring the active scanning perio T scan. It is also

4 Fig. 7. The effect of sampling frequency on reucing the velocity tracing error notice that the optimal velocity v, for which cycle time is imize, ecreases when the scanning istance is reuce, for example, v for 20, 4 an 8 µm are.5,.2 4 an.0 0 µm/s, respectively. The effect of the sampling rate on reucing velocity tracing error is also exae. The comparison of tracing errors in three cases (the sampling time of 0.2, 0. an 0.05 ms, respectively) is shown in Fig. 7, which is a zoom-in of the scanning portion (the esire scanning velocity is 4 0 µm/s ). It can be observe that higher sampling frequency yiels a feeforwar profile with smaller velocity tracing error between two consecutive sampling instants. III. CONTINUOUS NUMERICAL INVERSION OF HYSTERESIS NONLINEARITY Lie the Closest Match Algorithm (CMA) evelope by X. Tan et al. [4], our propose CNIA searches for the numerical approximation of the inverse hysteresis instea of trying to fin the exact inversion. This significantly reuces the mathematical complexity of the inversion problem. The ifference is that X. Tan s wor was base on a iscrete moel of hysteresis while ours is essentially base on a continuous Preisach moel. The avantage from this ifference can be illustrate with the analogy of numerical approximation of curves. CMA is lie curve-fitting using a staircase profile (iscrete approximation). The only way to reuce the approximation error is to refine the staircase, which means more experimental ata have to be collecte for the moel ientification, thus tremenously increasing the computational buren (they use least-square estimation). CNIA is lie approximation using linear interpolation (hence we call it continuous ), therefore much less ientification ata are neee without sacrificing the curve-fitting accuracy. Moreover, if the linear interpolation is replace by higherorer polynomial fitting, which is easy to o, a further reuction in the approximation error can be expecte. The search metho of CNIA yiels a control input u * that satisfies * * Γˆ ( u ; ψ ) f ε, u u, u () [ ] where ˆΓ is the Preisach operator of the ientifie hysteresis moel, ψ is the input history, f is the esire hysteresis output an ε > 0 is the e inversion error. The algorithm is outline as follows with an example of searching on a escening curve as illustrate in Fig. 8. We assume that the initial conitions of u, ψ an the corresponing hysteresis ouput f =Γ ˆ ( u; ψ ) are given. The following iteration steps are for the -th sampling instant ( ). The notation ( n u ) ( ) represents the n-th iteration value of u ( ). Without confusion, we rop the inex an only use u to simplify the notation. * u represents the returne value for u ( ) that satisfies (). In aition to u *, the algorithm also returns the * upate input history ψ for the next sampling instant. Continuous Numerical Inversion Algorithm: Step : Let u = u( ), ψ ε 2 Fig. 8. Example of CNIA = ψ( ), an f = f ( ), as Point 0 in Fig. 8. Set iteration step n = 0. If f = f, go to Step 5. Step 2: Initialize the search step u > 0. If f > f, it means that the esire hysteresis output is going own () along a escening curve. Then let u = u u (this gives Point in Fig. 8). Otherwise, if f < f, it means that the esire hysteresis output is going up f

5 () along an ascening curve. Then let u = u + u. An now n =. Step 3: Saturation an Convergence chec: Chec if u u u. If not, let ( n u ) = u if u < u ; let u = u if u > u. ( n) ( n) ( n ) Calculate the hysteresis output f ˆ ( u ; ψ ) at the same time upate the input history to f f ε =Γ ; ψ. If (as Point 3 in Fig. 8), the convergence has been reache. Go to step 5. Otherwise (as Point an 2 in Fig 8) continue to next step. Step 4: Moify the searching point: For the case of f being on a escening curve: If f f > 0, eep the current u as it is. Let ( ) ( ) n = n+, n n u = u u (this gives Point 2 in Fig. 8). Go to step 3. If f f < 0, it means that the last increment of u is too large an the n-th test point has sippe over to the other sie of f. Then reuce the increment by u = u/2 an moify the n-th test point to ( n ) u = u + u (this gives Point 3 in Fig. 8). Go to step 3. For the case of f being on an ascening curve: If f f < 0, eep the current u as it is. Let ( ) ( ) n = n+, n n u = u + u. Go to step 3. If f f > 0, it means that the last increment of u is too large an the n-th test point has sippe over to the other sie of f. Then reuce the increment by u = u/2 an moify the n-th test point to ( n ) u = u u. Go to step 3. * * Step 5: Let u = u, ψ = ψ. Exit. As each reversal curve (either ascening or escening) of the hysteresis is monotonic, the propose search algorithm is guarantee to converge. The number of iterations neee epens on the choice of the stopping threshol ε. Since the algorithm is implemente in a igital controller where A/D converters are use, the value of ε shoul not be smaller than the quantization error. CNIA outline above uses the variant-step search algorithm, but other avance search algorithms can be reaily aopte. The simulation result of open loop hysteresis compensation is shown in Fig. 9 an 0 which serve to emonstrate the effectiveness of CNIA in attenuating hysteresis nonlinearity. The notation of u, u r, u c in the figures is efine in Fig. 2. It is observe that the Fig. 9. Hysteresis compensation an compensation error Fig. 0. Linearization of the hysteresis nonlinearity compensation error u r u c ε where the threshol ε is set to 0.2% of the input range in this example. Fig. 0 shows that the relationship between the reference input u r an the virtual input u c to the linear component G of the plant with hysteresis nonlinearity is almost linear when the compensation of CNIA is applie. IV. CLOSED-LOOP IMPLEMENTATION Although the time-optimal comman is an open loop signal, it can be easily incorporate into a close-loop framewor as illustrate in Fig.. If the reference moel G 0 is ientical to the actual plant G an H cancels H perfectly, the tracing error e = 0 an the resulting closeloop response will be ientical to the original open loop response. Otherwise the feebac controller C will moify the actual control input u to compensate for moeling errors. The only requirement is that the imum able actuator limit for the time-optimal comman be less than the actual actuator limit so that some actuator effort is available for the feebac control.

6 ACKNOWLEDGMENT The author woul lie to than his fellow Ph.D. stuent Michael C. Reynols, who has by now grauate an become a faculty member at University of Aransas, Fort Smith, for his many constructive an insightful comments. u G + + e C + y H H+G r _ u r u u f Fig.. Close-loop implementation framewor Fig. 2. Close-loop implementation on the STM moel with the hysteresis nonlinearity Fig. 2 illustrates the close-loop response obtaine by incorporating the shape comman input, a 5 th -orer feebac controller by pole-placement, an the hysteresis compensation using CNIA whose convergence threshol is set to 2 mv. It can be observe that the uncertainty ue to the hysteresis inversion error has been taen care of by the feebac compensation, while in the open loop case, the tracing error ue to this uncertainty is as large as 0.2 µm. V. CONCLUSION This paper introuce a practical high-spee, highprecision motion control strategy for scanning of piezoactuators use in scanning tunneling microscopy. The timeoptimal feeforwar input was obtaine using the comman shaping technique of constraine least-square optimization, an the hysteresis nonlinearity in the plant was compensate using the propose continuous numerical inversion algorithm. Simulations were one an the high performance of the propose controller was emonstrate. Future wor inclues experimental test on an actual piezo-actuator an stuy of the combine feeforwar/feebac tracing strategy. REFERENCES [] H. Perez, Q. Zou, an S. Devasia, Design an control of optimal feeforwar trajectories for scanners: STM example, Proceeings of the American Control Conference, May 2002, pp [2] D. Croft, G. She, an S. Devasia, Creep, hysteresis, an vibration compensation for piezo-actuators: atomic force microscopy application, Proceeings of the American Control Conference, June 2000, pp [3] D. Croft, an S. Devasia, Vibration compensation for high spee scanning tunneling microscopy, Review of Scientific Instruments, Vol. 70, No. 2, December 999, pp [4] S. Salapaa, A. Sebastian, J.P. Clevelan, an M.V. Salapaa, Design, ientification an control of a fast nanopositioning evice, Proceeings of the American Control Conference, May 2002, pp [5] G. Schitter, P. Menol, H.F. Knapp, F. Allgoewer, an A. Stemmer, High performance feebac for fast scanning atomic force microscopy, Review of Scientific Instruments, Vol. 72, No. 8, August 200, pp [6] X. Tan, an J.S. Baras, A robust control framewor for smart actuators, Proceeings of the American Control Conference, June 2003, pp [7] C. J. Li, H.S.M. Beigi, S. Li, an J. Liang, Nonlinear piezoactuator control by learning self-tuning regulator, Transactions of the ASME, Vol. 5, December 993, pp [8] M.C. Reynols, an P. Mecl, The applica-tion of comman shaping to the tracing problem, Proceeings of the American Control Conference, June 2003, pp [9] I.D. Mayergoyz. Mathematical Moels of Hysteresis. Springer- Verlag, New Yor, 99. [0] I.D. Mayergoyz, an G. Frieman, Generalize Preisach moel of hysteresis, IEEE Transactions on Magnetics, Vol. 24, No., Jan. 988, pp [] G. Tao, an P.V. Kootovic. Aaptive control of systems with actuator an sensor nonlinearities. Wiley, 996. [2] D. Hughes, J.T. Wen, Preisach moeling an compensation for smart material hysteresis, in Active Materials an Smart Structures, Proceeings of SPIE, Vol. 2427, 994, pp [3] R. Venataraman, P. S. Krishnaprasa, Approximate inversion of hysteresis: theory an numerical results, Proceeings of the 39th IEEE Conference on Decision an Control, 2000, pp [4] X. Tan, R. Venataraman, an P. S. Krishnaprasa, Control of hysteresis: theory an experimental results, in Moeling, Signal Processing, an Control in Smart Structures, Proceeings of SPIE, Vol. 4326, 200, pp [5] L. Y. Pao, Minimum-time control characteristics of flexible structures, Journal of Guiance, Control, an Dynamics, Vol. 9, No., January-February 996, pp