Research Article Iterative Learning Control of Hysteresis in Piezoelectric Actuators

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1 Mathematical Problems in Engineering, Article ID 85676, 6 pages Research Article Iterative Learning Control of Hysteresis in Piezoelectric Actuators Guilin Zhang, 1 Chengjin Zhang, and Chaoyang Wang 3 1 College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 6659, China School of Mechanical, Electrical and Information Engineering, Shandong University at Weihai, Weihai 69, China 3 College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 6659, China Correspondence should be addressed to Chengjin Zhang; cjzhang@sdu.edu.cn Received April 1; Accepted 8 May 1; Published 5 May 1 Academic Editor: Qingsong Xu Copyright 1 Guilin Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We develop convergence criteria of an iterative learning control on the whole desired trajectory to obtain the hysteresiscompensating feedforward input in hysteretic systems. In the analysis, the Prandtl-Ishlinskii model is utilized to capture the nonlinear behavior in piezoelectric actuators. Finally, we apply the control algorithm to an experimental piezoelectric actuator and conclude that the tracking error is reduced to.15% of the total displacement, which is approximately the noise level of the sensor measurement. 1. Introduction Piezoelectric actuators (PEAs) have been widely used in nanopositioning systems due to their fast response and nanometer scale resolution [1 3]. However, the hysteresis existing in PEAs can greatly limit system performance [, 5]. Control of hysteretic system is an important area of control system research and a challenging problem [6 9]. Research on feedback and model-based feedforward control has been studied to achieve relatively high-precision positioning [1 13]. Iterativemethodscanbeusedtoimprovethepositioning performance if the positioning application is repetitive. Therefore, many researchers study the iterative and adaptive control methods to minimize the adverse effect of hysteresis [1 18]. The main challenge in iterative approaches for hysteretic systems is to assure convergence of the iterative algorithm. Leang and Devasia divide a general desired trajectory into some monotonicity partition [15, 16]. Afterwards, they prove the convergence of iterative learning control (ILC) algorithm oneachsinglebranch.inthispaper,westudythedesignof (ILC) algorithm to compensate for hysteresis-caused error in PEAs. The main contribution of our work is proving convergence of ILC algorithm on whole tracking trajectory. The remainder of this paper is organized as follows. First, we state the problem in the next section. Afterwards, we briefly review the Prandtl-Ishlinskii model in the context of thisworkandproveconvergenceoftheilcalgorithmwe designed. Finally, we implement the ILC algorithm on experimental stage and show our experimental results and conclusions.. Problem Statement Consider a hysteretic system of the following form: y (t) =H[V (t)], (1) where V(t) R is the input, y(t) R is the output, and H denotes the hysteresis function R R. For a given desired trajectory y d (t) defined on the finite time interval t [,T], theobjectiveistofindaninputv d (t) by way of the following iterative learning control (ILC) algorithm: V k+1 (t) = V k (t) +γe k (t), () where e k (t) = y d (t) y k (t), γ is a constant (to be determined), and V k (t) and y k (t) are the input and output at the kth iteration, respectively. Figure 1 depicts the block diagram of the ILC algorithm. The goal of the ILC algorithm is to generate

2 Mathematical Problems in Engineering k (t) H[ ](t) y k (t) + y d (t) Lemma 3. Let the operator F:C[,T] C[,T]be incremental and strictly increasing with constants k 1,k as defined by Definition 1.Then, k+1 (t) + + ILC e k (t) (k V 1 (t) F[V 1 ] (t)) (k V (t) F[V ] (t)) (k k 1 ) V 1 (t) V (t) (7) Figure 1: The ILC scheme. a feedforward control input V k (t) V d (t) in the norm sense, where sup V (t). (3) t [,T] In this paper, C[, T] isusedtodenotethespaceofcontinuous functions on t [,T],andC m [, T] denotes the space of continuous monotone functions on t [,T]. Definition 1 (incremental strictly increasing operators). An operator F:C[,T] C[,T]is called incremental and strictly increasing, if, for V 1, V C[,T], considering the ordering V 1 (t) V (t) for all t [,T], there exist constants k 1,k >such that k 1 (V 1 (t) V (t)) F[V 1 ] (t) F[V ] (t) () k (V 1 (t) V (t)), for any t [,T]. Lemma. Let the operator F:C[,T] C[,T]be incremental and strictly increasing. If <γ 1/k and V (t) V d (t) in () for all t [,T],thenV k (t) V d (t) for any t [, T]. Proof. We use the mathematical induction to prove this assertion. First, we prove V d (t) V 1 (t) for any t [,T]. Consider V d (t) V 1 (t) = V d (t) V (t) γ(y d (t) y (t)) = V d (t) V (t) γ(f[v d ] (t) F[V ] (t)) V d (t) V (t) γk (V d (t) V (t)) =(1 γk )(V d (t) V (t)), where 1 γk.wecanobtainv d (t) V 1 (t). Then,we assume V d (t) V k (t),forallt [,T],and V d (t) V k +1(t) = V d (t) V k (t) γ(y d (t) y k (t)) = V d (t) V k (t) γ(f[v d ] (t) F[V k ] (t)) (6) V d (t) V k (t) γk (V d (t) V k (t)) =(1 γk )(V d (t) V k (t)); we also get V d (t) V k+1 (t) ; therefore, the assertion holds. (5) for all V 1 (t), V (t) C[, T]. Proof. See [19]. Theorem. Consider a system of the form y(t) = F[V](t).Let the operator F:C[,T] C[,T]beincrementaland strictly increasing. If the constant <γ 1/k and V, V d C[,T], V (t) V d (t) for all t [,T], then the iterative control algorithm converges; that is, V d (t) V k (t) ρk V d (t) V (t), (8) where ρ<1and V k (t) V d (t) uniformly in t [,T]as k. Proof. ToprovetheconvergenceoftheILCalgorithm,we show contraction of the input (). Subtracting V d (t) from both sides of () andsubstitutingf[v](t) inplaceofy(t),weget V d (t) V k+1 (t) = V d (t) V k (t) γ(y d (t) y k (t)) = V d (t) V k (t) γ(f[v d ] (t) F[V k ] (t)), for all t [,T].Takingthefunction norm of (9), we get V d(t) V k+1 (t) (9) = V d (t) V k (t) γ(f[v d ] (t) F[V k ] (t)). (1) By Lemma, weobtainv d (t) V k (t) for all t [,T].Since the operator F is incrementally strictly increasing with constants k 1 k and <γ 1/k,thenbyLemma 3 we obtain V d(t) V k+1 (t) (1 γk 1 ) V d (t) V k (t) ρ V d (t) V k (t), (11) where ρ=1 γk 1 <1.Because<ρ<1,(11) is a contraction. By induction, we obtain that V d (t) V k (t) ρk V d (t) V (t) ; () therefore, the sequence V d (t) V d (t) as k ; that is, V k (t) converges to V d (t) uniformly in t [,T].Proof is completed.

3 Mathematical Problems in Engineering 3 3. The Prandtl-Ishlinskii Hysteresis Model w The Prandtl-Ishlinskii (PI) model can be used to capture the rate-independent hysteresis nonlinearity in piezoelectric actuators. In this section, the PI model is presented. The PI model utilizes the play or stop operators and a density function to characterize the hysteresis behavior. The hysteresis play operator is illustrated in Figure, while its detailed formulations have been presented in []. For a given input V(t) C m [, T], the play operator F r [V](t) with threshold r is defined by r r> F r [V;ψ]() =f r (V (),ψ), F r [V;ψ](t) =f r (V (t),f r [V;ψ](t i )), (13) with for t i <t t i+1, i N, f r (V,w) = max (V r,min (V +r,w)), (1) where ψ is initial value of the operator F r,and=t <t 1 < < t N = T is a partition of [, T], suchthattheinput function V is monotone on each subinterval [t i,t i+1 ] Property of Play Operator Lemma 5. For any r,theoperatorf r can be extended uniquely to a Lipschitz continuous operator F r :C[,T] R C[, T].AnditholdsforallV 1, V C[,T], for all initial values ψ 1,ψ R,andforallt [,T].Consider F r [V 1 ;ψ 1 ] (t) F r [V ;ψ ] (t) max ( sup V 1 (τ) V (τ), ψ 1 ψ ) (15) τ [,t] F r [V 1 ;ψ 1 ] (t) F r [V ;ψ ] (t), if V 1 V, ψ 1 ψ. (16) Proof. See [] Section.3. Lemma 6. If V 1 (t) V (t),forallt [,T],andinitialvalue ψ 1 =ψ =, a Lipschitz continuous operator F r :C[,T] R C[,T] is incremental and strictly increasing; that is, there exist constants k 1,k >such that k 1 (V 1 (t) V (t)) F r [V 1 ](t) F r [V ](t) k (V 1 (t) V (t)). Proof. Consider F r [V 1 ;ψ 1 ] (t) F r [V ;ψ ] (t) =F r [V 1 ] (t) F r [V ] (t) =f r (V 1 (t),f r [V 1 ] (t 1)) f r (V (t),f r [V ] (t 1)). (17) Figure : Play operator. Because F r [V 1 ](t) F r [V ](t) and ψ 1 =ψ =,(15)canbe simplified as F r [V 1 ] (t) F r [V ] (t) sup (V 1 (τ) V (τ)). τ [,t] (19) Let α(t) > and sup τ [,t] (V 1 (τ) V (τ)) = α(t)(v 1 (t) V (t)). We get F r [V 1 ] (t) F r [V ] (t) sup (α (τ)) (V 1 (t) V (t)). τ [,t] () Let k = sup τ [,t] (α(τ)).from(18)and(), we obtain k 1 (V 1 (t) V (t)) F r [V 1 ] (t) F r [V ] (t) therefore, the assertion holds. k (V 1 (t) V (t)); (1) 3.. Property of the Prandtl-Ishlinskii Model. The PI model assumes that the output y(t) of a piezoelectric actuator is a weighted superposition of basic F r [V]. Theoutputy(t) is written as y (t) =H[V] (t) = F r [V] (t) p (r) dr, () where p(r) is a given density function, satisfying p(r) with p(r)dr <. Lemma 7. If V 1 (t) V (t), forallt [,T], and all initial values ψ 1 = ψ in all operators F r,thenpioperatorh : C[, T] C[, T] is incremental and strictly increasing; that is, there exist constants λ 1,λ >such that Since V 1 (t) V (t), forallt [,T],andinitialvalueψ 1 = ψ =,thenf r [V 1 ]() F r [V ]().Byinduction,F r [V 1 ](t) F r [V ](t). There must exist a constant k 1 >such that F r [V 1 ] (t) F r [V ] (t) k 1 (V 1 (t) V (t)). (18) λ 1 (V 1 (t) V (t)) H[V 1 ] (t) H[V ] (t) for any t C[,T]. λ (V 1 (t) V (t)) (3)

4 Mathematical Problems in Engineering bit D/A Amplifier XE-51 Piezoelectric actuators Controller AD711 Plant bit A/D SGS sensor Figure 3: Structure of the experimental setup. Proof. Consider H[V 1 ] (t) H[V ] (t) = F r [V 1 ] (t) p (r) dr F r [V ] (t) p (r) dr = (F r [V 1 ] (t) F r [V ] (t))p(r) dr k (V 1 (t) V (t))p(r) dr =k p (r) dr (V 1 (t) V (t)) λ (V 1 (t) V (t)), for any t C[,T],whereλ =k can get () p(r)dr. Similarly,we H[V 1 ] (t) H[V ] (t) = (F r [V 1 ] (t) F r [V ] (t))p(r) dr k 1 p (r) dr (V 1 (t) V (t)) λ 1 (V 1 (t) V (t)), for any t C[,T],whereλ 1 = k 1 completed. (5) p(r)dr. Proofis Theorem 8. Consider a hysteretic system of the form y(t) = H[V](t). Let the PI hysteresis operator H satisfy the condition of Lemmas 6 and 7.Iftheconstant<γ 1/λ and V, V d C[, T] in (), V (t) V d (t) for all t [,T], then the iterative control algorithm converges; that is, V d (t) V k (t) ρk V d (t) V (t), (6) where <ρ<1and V k (t) V d (t) uniformly in t [,T]as k. Proof. The proof is identical to the proof of Theorem.. Experimental Results.1. Experimental Setup. TheILCalgorithmisappliedonan experimental piezoelectric actuator PST15/7/VS, which isapreloadedpztfrompiezomechanikingermany.the natural frequency of the actuator is khz. The actuator provides a maximum displacement of μm and includes an integrated high-resolution strain gauge position sensor (SGS). The excitation module comprises a voltage amplifier (HVPZT XE-51.B) with a fixed gain of 15, which provides the excitation voltage for the actuator. The AD711-EVA controller board equipped with -bit ADC and-bit DAC is utilized to generate and acquire input voltage and output displacement signals. The experimental data are acquired at a sampling frequencyof 1kHz. Figure 3 illustrates the structure of the experimental system... Experimental Result. We apply the ILC algorithm to track a sinusoidal trajectory y d (t) = 5 sin((1/π)t (π/)) + 7.The constant γ in ()ischosentobeγ =.5 and the initial input V (t) = 5sin((1/π)t (π/))+5. The experimental results are shown in Figure. Figure (a) showstheresultsoftheilcalgorithmtotrack the desired trajectory, and the feedforward input of the ILC algorithm is depicted in Figure (b). The maximum error at the kth iteration e k (t) is shown in Figure (c), and it demonstrated that convergence of ILC algorithm is achieved. Figure (d) shows the tracking error at the 3th iteration. The maximum error is approximately.15 μm, which is.15%of the total displacement range. 5. Conclusions In this paper, we designed an ILC algorithm to compensate for hysteresis-caused tracking error in piezoelectric actuators andprovedconvergenceofthisalgorithmonthewhole tracking trajectory. Experiments were carried out to verify the effectiveness of the ILC algorithm. The experimental results show that the tracking error can be reduced to the noise level of the sensor measurements. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

5 Mathematical Problems in Engineering 5 5 Displacement (μm) Input voltage (V) Time (s) Time (s) 1 k= k=8 k= Desired k= k=8 k= k=16 (a) Displacement versus time (b) Input versus time y d (t) y k (t) (μm) 1 5 Error (μm) Iteration number, k. 6 Time (s) 8 1 (c) e k (t) versus k (d) e 3 (t) versus time Figure : Experiment results. Acknowledgment This work is supported by the National Natural Science Foundation of China (no. 6117). References [1] S. Devasia, E. Eleftheriou, and S. O. R. Moheimani, A survey of control issues in nanopositioning, IEEE Transactions on Control Systems Technology,vol.15,no.5,pp.8 83,7. [] M. Al Janaideh, S. Rakheja, and C.-Y. Su, An analytical generalized prandtl-ishlinskii model inversion for hysteresis compensation in micropositioning control, IEEE/ASME Transactions on Mechatronics, vol. 16, no., pp. 73 7, 11. [3] Q. Xu and P.-K. Wong, Hysteresis modeling and compensation of a piezostage using least squares support vector machines, Mechatronics, vol. 1, no. 7, pp , 11. [] M. Al Janaideh and P. Krejci, Inverse rate-dependent prandtlishlinskii model for feed forward compensation of hysteresis in a piezomicropositioning actuator, IEEE/ASME Transactions on Mechatronics, vol. 18, pp , 13. [5]Y.Qin,Y.Tian,D.Zhang,B.Shirinzadeh,andS.Fatikow, A novel direct inverse modeling approach for hysteresis compensation of piezoelectric actuator in feedforward applications, IEEE/ASME Transactions on Mechatronics,vol.18,no.3,pp , 13. [6] X. Tan and J. S. Baras, Modeling and control of hysteresis in magnetostrictive actuators, Automatica, vol., no. 9, pp ,. [7] H. C. Liaw, B. Shirinzadeh, and J. Smith, Enhanced sliding mode motion tracking control of piezoelectric actuators, Sensors and Actuators A: Physical,vol.138,no.1,pp.19,7. [8] J.-X. Xu and K. Abidi, Discrete-time output integral slidingmode control for a piezomotor-driven linear motion stage, IEEE Transactions on Industrial Electronics, vol.55,no.11,pp , 8. [9] H.-J. Shieh and C.-H. Hsu, An adaptive approximator-based backstepping control approach for piezoactuator-driven stages, IEEE Transactions on Industrial Electronics, vol.55,no.,pp , 8. [1] S. Bashash and N. Jalili, Robust multiple frequency trajectory tracking control of piezoelectrically driven micro/nanopositioning systems, IEEE Transactions on Control Systems Technology,vol.15,no.5,pp ,7. [11] P. Krejci and K. Kuhnen, Inverse control of systems with hysteresis and creep, IEE Proceedings: Control Theory and Applications,vol.18,no.3,pp ,1. [] L. Sun, C. Ru, W. Rong, L. Chen, and M. Kong, Tracking control of piezoelectric actuator based on a new mathematical model, Micromechanics and Microengineering, vol. 1,no.11,pp.139 1,.

6 6 Mathematical Problems in Engineering [13] P. Ge and M. Jouaneh, Tracking control of a piezoceramic actuator, IEEE Transactions on Control Systems Technology, vol., pp.9 16,1996. [1] X. Tan and J. S. Baras, Adaptive identification and control of hysteresis in smart materials, IEEE Transactions on Automatic Control,vol.5,no.6,pp ,5. [15] K.K.LeangandS.Devasia, Iterativefeedforwardcompensation of hysteresis in piezo positioners, in Proceedings of the nd IEEE Conference on Decision and Control,vol.3,pp , December 3. [16] K. K. Leang and S. Devasia, Design of hysteresis-compensating iterative learning control for piezo-positioners: application to atomic force microscopes, Mechatronics, vol. 16, no. 3-, pp , 6. [17] R.V.Iyer,X.Tan,andP.S.Krishnaprasad, Approximateinversion of the preisach hysteresis operator with application to control of smart actuators, IEEE Transactions on Automatic Control, vol. 5, no. 6, pp , 5. [18] Y. Wu and Q. Zou, Iterative control approach to compensate for both the hysteresis and the dynamics effects of piezo actuators, IEEE Transactions on Control Systems Technology,vol.15,no.5, pp.936 9,7. [19] R. Venkataraman and P. S. Krishnaprasad, Approximate inversion of hysteresis: theory and numerical results, in Proceedings ofthe39thieeeconfernceondecisionandcontrol, pp. 8 5, December. [] M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer,NewYork,NY,USA,1996.

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