1m ago

1 Views

0 Downloads

1.51 MB

16 Pages

Tags:

Transcription

Robust feedforward-feedback control of a nonlinear andoscillating 2-DOF piezocantilever.Micky Rakotondrabe, Kanty Rabenorosoa, Joël Agnus, Nicolas ChailletTo cite this version:Micky Rakotondrabe, Kanty Rabenorosoa, Joël Agnus, Nicolas Chaillet. Robust feedforward-feedbackcontrol of a nonlinear and oscillating 2-DOF piezocantilever. IEEE Transactions on AutomationScience and Engineering, Institute of Electrical and Electronics Engineers, 2011, 8 (3), pp.506-519. 10.1109/TASE.2010.2099218 . hal-00635680 HAL Id: 0635680Submitted on 25 Oct 2011HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.

IEEE TRANSACTIONS ON AUTOMATION, SCIENCE AND ENGINEERING1Robust Feedforward-Feedback Control of aNonlinear and Oscillating 2-DOF PiezocantileverMicky Rakotondrabe, member, IEEE, Kanty Rabenorosoa, Joël Agnus and Nicolas Chaillet, member, IEEEAbstract— Many tasks related to the micro/nano-world require,not only high performances like submicrometric accuracy, butalso a high dexterity. Such performances are only obtained usingmicromanipulators and microrobots with multiple degrees offreedom (DOF). Unfortunately, these multi-DOF systems usuallypresent a strong coupling between the different axis making themvery hard to control.The aim of this work is the modeling and control ofa 2-DOF piezoelectric cantilever dedicated to microassembly/micromanipulation tasks. In addition to the coupling betweenthe two axis, the piezocantilever is very oscillating and stronglynonlinear (hysteresis and creep). In the proposed approach,the nonlinearity and vibration are first compensated thanks tothe feedforward technique. Afterwards, we derive a decoupledmodel in order to synthesize a linear robust H controller. Theexperimental results show the efficiency of the proposed approachand their convenience to the micromanipulation/microassemblyrequirements.Note to Practitioners— The main motivation of thisarticle is the need of both high performances and highdexterity in micromanipulation and microassembly tasks.In such a case, not only a submicrometric accuracyand stability are needed, but also numerous degrees offreedom. For that, in the literature, there exist piezoelectricbased structures with 2 or more DOF. Unfortunately, thecoupling between its axis, the nonlinearities (hysteresisand creep) and the structure vibration make them verydifficult to control and therefore make performances lost. Aclassical feedback controller can be employed but when thenonlinearities and vibration become strong, it is impossibleto synthesize a linear controller. In this paper, we show thatthe combination of feedforward techniques, to minimizethe nonlinearities and vibration, and feedback techniquesmakes possible to reach the high performances requiredin micromanipulation/microassembly. We notice that theproposed approach can also be applied to other nonlinear,oscillating and multi-DOF systems, such as piezotubes.Index Terms— Piezoelectric cantilever, 2 degrees of freedom,coupling, nonlinear, feedforward control, robust control, microassembly/micromanipulation.FEMTO-ST Institute, UMR CNRS 6174 - UFC / ENSMM / UTBMAutomatic Control and Micro-Mechatronic Systems depart., AS2M24, rue Alain Savary25000 Besançon - ton}@femto-st.frPaper type: regular paper. First submission: October 5th , 2009.I. I NTRODUCTIONHe design and development of microrobots, micromanipulators and microsystems in general used to workin the micro/nano-world (such as for micromanipulation andmicroassembly) are very different to that of classical systems.At this scale, the systems should have accuracy and resolutionthat are better than one micron. For instance, fixing a microlensat the tip of an optical fiber with 1µm of relative positioningerror or 0.4µrad of orientation error may cause a loss of 50%of the light flux [1]. In fact, to reach such high performances,the microrobots, micromanipulators and microsystems aredeveloped with smart materials instead of classical motors(example DC motors) and hinges. Smart materials minimizesthe mechanical clearances which induce the loss of accuracy.Furthermore, they allow less compact design than hingesbecause it is possible to design a microsystem with one bulkmaterial. Among the very commonly used smart materials arepiezoceramics. Their recognition is due to the high resolution,the short response time and the high force density that theyoffer.Beyond the accuracy and resolution, the systems used formicroassembly and micromanipulation need to be dexterous.Indeed, when producing hybrid and complex microstructuresand MEMS, the used micromanipulators, microrobots andmicrosystems should be able to perform complex trajectory orshould have a complex workspace. For that, they should havehigher number of degrees of freedom (DOF). For instance,[2] developped a 2-DOF piezoelectric stick-slip microrobotable to perform angular and linear motion, [3] [4] proposed3-DOF microrobots for x, y, θ motions, [5] proposed a 4-DOFpiezoelectric microgripper and [6] proposes a 6-DOF dexterous microhandling system. In fact, it has been demonstratedthat high DOF-number microassembly/micromanipulation systems offer more possibility for complex, hybrid assembledmicrostructures and MEMS than one-DOF systems [7]- [10].Among the commonly used microsystems and micromanipulators, piezoelectric microgrippers (called piezogrippers) areespecially adapted for microassembly and micromanipulationbecause of their ability to perform pick and place tasks withsubmicrometric resolution, and the possibility to control themanipulation force [11]. Most of existing microgrippers aremade of two single-DOF piezocantilevers with only an inplane positioning capability [12]- [15]. To fulfill the dexterityrequirement, our previous work [5] proposed a high dexteroustwo-fingered piezogripper (Fig. 1). With its 4 articular DOF,it is able to orient micro-objects or to pick-transport andplace them in and out of plane. In-plane means positioningT

IEEE TRANSACTIONS ON AUTOMATION, SCIENCE AND ENGINEERINGin the y-axis while out of plane in the z-axis (Fig. 1-a).In particular, when the 4-DOF microgripper is particularlycombined with external high range linear or rotary systems,the whole micromanipulation/microassembly system becomesitself dexterous [16]. For instance, such micromanipulationsystem allowed the manipulation of watch screws or thealignment of beam splitters for microspectrometers [7] [17].In fact, each finger that composes the 4-DOF microgripper isa 2-DOF piezocantilever that is able to move independentlyin the two orthogonal directions. Notwithstanding, when applying a reference deflection along y-axis (resp. z-axis) to thepiezocantilever, an undesirable deflection is obtained in the zaxis (resp. y-axis), making a loss of accuracy. This is due tomechanical imperfection of the structure and particularly tothe misalignment of the electrodes. In addition to these, thepresence of a manipulated object may also cause a couplingbetween the two axis. Beyond the coupling, the piezocantilever also presents nonlinearities (hysteresis and creep) andvibrations. While the hysteresis and creep also makes losethe accuracy, vibrations generate undesirable overshoots thatmay destroy the manipulated micro-objects or the actuatorsthemselves. At the end, despite the high dexterity, generalperformances are lost due to the coupling, nonlinearities andvibrations. It is obvious that the piezocantilever has to becontrolled. This control is at low-level and can only be veryuseful to ameliorate the efficiency of the whole (teleoperated orautomated) micromanipulation/microassembly systems. Thisis why several projects concerning the development of highperformances micromanipulation/microassembly stations include the low-level control and the performances improvementof each actuators these last years [18] [19].piezocantileverssupportmicro-objectzyx(a)10 mmPiezocantilever2In the literature, the control of 1-DOF piezocantilevers,including AFM-tubes working on one axis, is at its cruisingspeed. In open-loop control, both hysteresis and creep werecompensated by nonlinear compensators in cascade [20][21] [22]. Additionaly to the nonlinearity compensation, thevibration was minimized by using dynamic inversion method[23] [24] or inverse multiplicative approach [25]. Openloop control is convenient for sensor-less piezocantilevers,reducing the cost of the whole automated system. However,once external disturbances appear or model uncertainties become large, open-loop control techniques fail and closedloop methods should be used. Different closed-loop controllaws were therefore applied and have proved their efficiencyin the micro/nano-positioning: integral based control [26],state feedback technique [27], adaptive [28] [29] and robusttechniques [30] [31]. The above techniques applied for 1DOF can not be directly extended for 2-DOF piezocantilevers:coupling between axis should be delicately considered. In fact,if the coupling is badly characterized and modeled, the closedloop system may be unstable. In [32], a robust techniquewas proposed to control a nonlinear 2-DOF nanopositioner.It takes into account the coupling and the nonlinearities.However, when these nonlinearities and the coupling becomevery large and when the vibration is very badly damped,the technique fails. It is then necessary to propose a newapproach that permits to control strongly coupled, hysteretic,creeped and oscillating bi-variable piezocantilevers. Such atechnique can be used, not only for multi-DOF piezogrippers dedicated to micromanipulation/microassembly but alsofor AFM-piezotubes used as a scanner working on two orthree axis [33]. The object of this paper is to propose anew technique in that issue. First, we apply a feedforwardcompensation in order to minimize the effect of the hysteresisand vibration. Afterwards, a model taking into account thecoupling and residual nonlinearities is proposed. Finally, arobust feedback control law is synthesized in order to reachthe expected performances.The paper is organized as follows. In section-II, we presentthe functioning of the 2-DOF piezocantilever that will becontrolled. The feedforward technique for compensating hysteresis and vibration is presented in section-III. In sectionIV, we model the obtained system in order to further permitthe synthesis of a linear feedback control law. Section-V isdedicated to the synthesis of a robust H controller to rejectcoupling and uncertainty effects and to reach the specifiedperformances. Finally, conclusions end the paper.electrodeszpackagingythe twofingers(b)xpair oftoolsFig. 1. (a) a piezogripper with 2-DOF piezocantilevers. (b) the microgripperdevelopped in [5].II. P RESENTATION OF THE 2-DOF PIEZOCANTILEVERA. The piezocantilever working principleIn this section, we present the 2-DOF piezocantilever thatwill be controlled. Two of this piezocantilever compose a 4DOF piezogripper (Fig. 1-b) that is able to grip, orient andposition micro-objects along y, z and around x axis [5].The 2-DOF piezocantilever is made up of two piezolayerswith 4 local electrodes at its surfaces and one middle electrodefor ground (Fig. 2-a). It can be assimilated to a cantileverbeam clamped at one end. The two DOF are obtained with ajudicious application of voltages on the electrodes.

IEEE TRANSACTIONS ON AUTOMATION, SCIENCE AND ENGINEERING3Fig. 2-b pictures the functioning of the piezocantilever withcross section views. The structure at rest is presented with solidarea and the deformed actuator with dashed lines. P indicatesthe polarization of the piezoelectric material. It is remindedthat when the electric field (so the applied voltage) is in thesame direction than the polarization, the piezolayer contracts.In the first configuration (Fig. 2-c), the four electrodes are setto the same voltage VZ 0. The upper layer expands along xwhile the lower layer contracts, leading to a deflection along z.In the second situation (Fig. 2-d), the applied voltages on twoadjacent and two opposite electrodes are VY and VY (VY 0). So, while the left part of the piezocantilever expands, theright one contracts leading to a deflection along y axis. Finally,in the last configuration (Fig. 2-e), the electrodes are set atvoltages VZ VY and VZ VY , which yields a deflexionin y and z directions. The 2-DOF of the piezocantileverare very interesting for designing piezogrippers with highdexterity. Unfortunately, when a displacement along one axisis desired, a residual displacement along the orthogonal axisappears. This coupling is mainly due to the misalignementof the four electrodes and to the interference between theapplied electrical fields. Furthermore, hysteresis and creepnonlinearities as well as vibrations characterize the behaviorof the piezocantilever.B. The experimental setupFig. 3 shows the experimental setup. The piezocantilever,made up of PZT-layers, has the following total dimensions:15mm 2mm 0.5mm. Two optical sensors, from Keyence(LK2420), with 10nm of resolution and 0.1µm of accuracyare used to measure displacements at the tip of the piezocantilever along y and z axis. We use computer and DSpaceboard materials to acquire measurements and to provide control signals. These real-time materials work with a samplingfrequency of 5kHz which is high enough compared to thebandwidth of the system to be controlled. The control signalsthat they supply are amplified by a home-made high voltageamplifier having two independent lines. It can supply up to 200V at its output. The Matlab-Simulink TM software is usedto manage the data and signals.D-SPace boardlampczY-Z sensors2 lines amplifieryy-axis sensorx upportelectrodescross section viewPVz2-dof piezocantilever(c)PelectrodesVyVzpiezolayersVzVzVz VyVz-VyZFig. 3. Experimental setup based on one piezocantilever and two opticalsensors.z-Vyy(d)III. F EEDFORWARD CONTROL(e)-Vyz-axis sensorVyYVz-Vy Vz VyFig. 2. (a) presentation of the 2-DOF piezocantilever. (b) cross section viewof the piezocantilever. (c) achievement of z axis motion. (d) achievement ofy axis motion. (e) achievement of both y and z motions.In this section, we characterize and compensate the hysteresis and vibration of the piezocantilever. This compensation isnecessary in order to linearize the system and to attenuate thevibration and therefore to further make easy the synthesis ofa linear controller.The piezocantilever can be considered as a bi-variablesystem where inputs are voltages VY and VZ while outputs aredisplacements Y and Z (Fig. 4). As advised by previous works[23] [25], the nonlinearities and the vibration analysis can be

IEEE TRANSACTIONS ON AUTOMATION, SCIENCE AND ENGINEERINGdone independently, by choosing properly the frequencies ofthe input signals. Therefore, we first analyze and compensatethe hysteresis. Afterwards, we consider the vibration.4Y [ µm]21010YVyZ [ µm]3200Hh 10 1 2systemVzZ 20 4002040VY [V ](a)5Fig. 4. 20coupling VY Z 3 40 20Y [ µm]020402040VY [V ](b)Z [ µm]60Block-scheme representing the 2-DOF piezocantilever.40020 50 20A. Hysteresis compensationTo characterize the hysteresis, a sine input signal is appliedto the system and the resulting output signal is plotted versusthe input. The frequency of this input is chosen to be low inorder to avoid the effect of the dynamic part on the hysteresisshape. However, it should not be too low in order to avoid theeffect of the creep [25] [31]. In our case, we have two inputsignals VY and VZ . Following our previous characterizationwork [34], fY fZ 0.05Hz is a good choice. Concerningthe amplitude, the piezocantilever can be powered by a voltageup to 100V but our experiments will be limited to 40V (bothfor y and z axis), which corresponds to the expected range ofdisplacements.First, we apply a VY voltage while VZ is set to zero. Twoamplitudes VY 40V and VY 20V are used. As pictured inFig. 5-a, a strong hysteresis characterizes the VY Y transfer.hThis hysteresis is nearly equal to 17% ( H 100% 7.5µm42µm ).Furthermore, a residual displacement appears on the z axis(Fig. 5-b). This corresponds to the VY Z coupling.Now we set VY to zero and apply a sine voltage VZ .As pictured in Fig. 5-d, a strong hysteresis (16.67% 20µm120µm 100%) also characterizes the VZ Z transfer.Finally, a residual displacement appears on the y axis (Fig. 5c) corresponding to the VZ Y coupling. The asymmetry ofthe coupling curve is due to the imperfection of the mechanicaldesign of the 2-DOF piezocantilever.The two hysteresis being too strong, they make difficult thesynthesis of further feedback controller. It is then necessary tominimize these hysteresis. The principle used for that is thefeedforward compensation based on the inverse model. Thereare different hysteresis models and compensation techniquesfor smart materials: the Preisach [35], the Prandtl-Ishlinskii[36] [22] and the Bouc-Wen techniques [37]. We use thePrandtl-Ishlinski model (PI-model) because of the simplicityof its implementation and ease of obtaining a compensator orinverse model.1) Prandtl-Ishlinskii (PI) hysteresis modeling: in the PI approach, a hysteresis is modeled by the sum of many elementaryhysteresis operators, called play operators. Each play operator,denoted by γj (.), is characterized by a threshold rj and a 10 40coupling VZ Y 15 40 200VZ [V ](c)Fig. 5. 602040 40 20(d)0VZ [V ]Hysteresis characteristics of the 2-DOF system piezocantilever.weighting wj [38]. Thus, the relation between the input Vi(i {Y, Z}) and the output δ (δ {Y, Z}) is given by:δ nPhystγj (Vi (t))j 1 nPhystj 1ωj · max {Vi (t) rj , min {Vi (t) rj , δ(t )}}(1)where δ(t ) indicates the value of the output at precedenttime and nhyst the number of play operators. The identificationof the parameters rj and wj , well described in [25], is doneusing the maximum voltage input Vi 40V .2) Prandtl-Ishlinskii (PI) hysteresis compensation: to compensate a hysteresis that has been modeled with a PI-model,another PI hysteresis model (called hysteresis compensator orPI inverse model) is put in cascade with it. For the 2-DOFpiezocantilever, a compensator is put for each axis. In Fig. 6,YRH [µm] is the new input for the y axis while ZRH [µm] isfor the z axis, and where subscript RH means reference forthe hysteresis compensated system.YRHHystcomp-YYVysystemZ RHFig. 6.sators.Hystcomp-ZVzZBlock-scheme representing the system with the hysteresis compen-Like the PI direct model, each elementary operator of thePI inverse model is characterized by a threshold rk′ and a

IEEE TRANSACTIONS ON AUTOMATION, SCIENCE AND ENGINEERING5weighting gain wk′ . They are computed from the parametersrj and wj of the direct model. We have [36]:rk′ andw1′ kPj 1wl · (rk rj ); k 1 · · · nhyst(2)1w1wk′ w1 kPwjj 2 w! kk 1P· w1 wjj 2!; k 2 · · · nhyst(3)3) Experimental results: first, the PI hysteresis model,described by (Eq. 1), is identified. For a trade-off on accuracyand complexity, we choose nhyst 15. In Fig. 7, theexperimental curves and the simulation are plotted. As seenin Fig. 7-b, the identified model for the VZ Z transferwell fits to the real (experimental) hysteresis. However, thereis a residual error between the model VY Y (Fig. 7-a)and the corresponding real hysteresis. This is due to the factthat the real hysteresis is non-symmetrical while the employedmodel is symmetrical. An adapted model can be used but itscompensator has a high complexity for implementation [39].In addition, the aim is to reduce the hysteresis in order tofacilitate the feedback synthesis and the residual error can beconsidered as uncertainty. This uncertainty will be taken intoaccount during the feedback control design.17% (Fig. ?-a) was reduced into 8.75%. This residual hysteresis is due to the asymmetry of the real hysteresis, alreadycommented above. Fig. 8-b presents the coupling YRH Z.It is shown that this coupling stays unchanged.After that, a sine reference ZRH with amplitude 60µmis applied while YRH is set to zero. As pictured in Fig. 8d, the hysteresis which was initially 16.67%(see Fig. ?-d)is completely removed. However, the coupling ZRH Ybecomes slightly larger (compare Fig. ?-c and Fig. 8-c). Thiscoupling will be considered as a disturbance to be removedduring the modeling.20Y [ µm]Z [ µm]3210100 1 10 2 20 20(a)0 1001020coupling YRH Z 3 20YRH [ µm] 10Y [ µm]600102050100YRH [ µm](b)Z [ µm]40 520 100 20 15Y [ µm]20: experimental result(c): model simulation10 40coupling Z RH Y 20 1000 50050Z RH [ µm]100 60 100(d) 500Z RH [ µm]Fig. 8. Experimental results of the 2-DOF piezocantilever with the hysteresiscompensator. 10 20 40 30 20 10010203040VY [V ](a)B. Vibration compensationZ [ µm]60: experimental result40: model simulation200 20 40 60 40 30(b)Fig. 7. 20 10010203040VZ [V ]Experimental results and model simulation of the hysteresis.The hysteresis compensator has been computed using(Eq. 3), implemented in the Matlab-Simulink TM software andtested. The experimental process is performed as follows.First, a sine input reference YRH with amplitude 20µmis applied while ZRH is set to zero. Fig. 8-a presents theoutput Y . It appears that the hysteresis which was initiallyNow, let us analyze the step responses of the new systemrepresented by the scheme in Fig. 6. For that, we first apply astep YRH 20µm, ZRH being equal to zero. The response ofY is plotted in Fig. 9-a. Then, we apply a step ZRH 60µm,YRH being equal to zero. The response of Z is plotted inFig. 9-d. It appears that the structure is more oscillating andhas a badly damped vibration in the y axis than in z axis. Theovershoots are 77% and 4.8% in y and z axis respectively. Theradcorresponding resonant frequencies are 5400 rads and 2670 s .Finally, the couplings YRH Z and ZRH Y are picturedin Fig. 9-b and Fig. 9-c respectively.Similarly to the hysteresis phenomenon, it is hard to synthesis a feedback controller when the vibration characterizingpiezocantilevers is too badly damped [34]. This is whywe propose to minimize the vibration along y axis. Fig. 10gives the block-scheme of the 2-DOF piezocantilever withthe previous hysteresis compensators and the new vibration

IEEE TRANSACTIONS ON AUTOMATION, SCIENCE AND ENGINEERING35Y [ µm]Z [ µm]8306constitutes the vibration compensator and which is convolvedwith the input reference. For instance, if the reference inputis a step, the control signal is a staircase with two steps.625204152first impulse response10050010203040(a)1050coupling YRH Z 20t[ms ]Y [ µm](b)702040A160A280second impulse responset[ms ]Z [ µm]6085060404TTdt [s]30220010coupling Z RH Y 20100200300(c)Fig. 9.400t[ms ]00(a)50100150200impulset[ms ](d)A1Vibration characteristics of the 2-DOF piezocantilever.referenceinputcompensator. In this, the new references inputs are YRV andZRV ZRH . Subsrcipt RV means reference for the vibrationcompensated system.To compensate a dynamic part, an inverse dynamic modelcan be used [23] [24]. This technique necessitates a bistability condition (direct and inverse models stable) on themodel. However, such a condition is not always guaranteedfor real systems. Another technique to minimize or cancelvibration is input shaping techniques. Input shaping techniquesavoid the dynamic inversion and have a simplicity of implementation [25]. There are different kinds of input shapingmethods but the one presented here is the Zero Vibration(ZV) method which has the particularity to be simple forcomputation [40].YRVVibr YRHcomp-YHystcomp-YZ RVZ RHHystcomp-ZYVysystemVzZFig. 10. Block-scheme representing the system with the hysteresis compensators and the vibration compensator.1) The ZV input shaping technique: When an impulse isapplied to an oscillating system, a vibration appears. Let ωn bethe natural frequency and ξ the damping ratio. When a second2·π ,impulse is applied at time Td T /2, with T 2ωn ·1 ξthe vibration caused by the second impulse can cancel theone caused by the first impulse (Fig. 11-a) if the amplitudesof both are judiciously chosen. For any reference input, theprevious sequence of impulses, also called shaper, is convolvedwith it. Fig. 11-b shows the bloc-diagram of the shaper, whichcontrolsignal oscillating outputsystemA2delayTdshaper (vibration compensator)(b)Fig. 11. Principle of the ZV input shaping technique to minimize vibration.If the identified parameters ωn or ξ are quite differentfrom the real parameters, a residual vibration will remainafter compensation. In fact, the vibration caused by the firstimpulse will not be exactly cancelled by that of the secondone. Therefore, if the overshoots of the two vibrations arevery high, the resulting interfered signal may also have a highovershoot. To avoid such a problem, it is advised to use morethan two impulses in the shaper. In that case, each impulseamplitude and the corresponding vibration are small. So, theresulting interfered signal will have a lower overshoot if any.Let an oscillating system be modeled by the followingsecond order model:δ δRK1ωn 2· s2 (4)2·ξωn·s 1where K is the static gain.If Ai and ti are the amplitudes of the impulses and theirapplication times (delays), the shaper is computed using thefollowing expressions [40]:

IEEE TRANSACTIONS ON AUTOMATION, SCIENCE AND ENGINEERING7Y [ µm] A1 a1(1 β)m 1 a2 A 2(1 β)m 1 Ai : . . amAm (1 β)m 1 ti : ξ.π 30 tm (m 1) · Td(5) Y)1510with vibration compensator(YRV Y )5 0where β e 1 ξ2 , m is the number of impulses in theshaper, ai indicates the ith monomial of the polynomial fromm 1(1 β). We have a1 1 and am β m 1 .2) Experimental result: first, we identify the transferYRH Y . We obtain: K 0.93, ωn 6101rad/s andξ 0.02. Fig. 12 shows that the identified model reasonablyfits to the experimental result.Y [ µm]: experimental result: model simulation30(YRH20 t1 0t2 T d.without vibration compensator2505101520253035t[ms ]Fig. 13.Experimental step responses of the piezocantilever in Y -axis:comparison results of with and without compensator.After that, a harmonic analysis is performed. For that, a sineinput signal is applied and the corresponding output magnitudeis plotted. Fig. 14 presents the comparison of the results withand without compensator. It shows that the initial resonancepeak is highly damped when we apply the input shapingmethod.25Y20 magnitude [dB] of2015without vibration compensator(YRH Y )101551005 505101520253035t[ms ]Fig. 12. Step response of the piezocantilever in Y -axis: experimental resultand model simulation. 10with vibration compensator(YRV Y ) 15 2010 110 010 110 210 310 4Fig. 14. Experimental harmonic responses of the piezocantilever in Y -axis:comparison results of with and without compensator.Then, different shapers, characterized by different numbersof impulses, were computed and implemented accordingly toFig. 10. Higher the number of impulses is, lower is the overshoot of the obtained output Y , when a step reference inputYRV is applied. However, the complexity of the implementedcompensator increases versus the number of impulses. It istherefore unecessary to have a shaper with a high numberof impulses. In our case, when the number of impulse ismore than 4, the overshoot stops decreasing drastically. So,we propose to finally choose a shaper with four impulses.The first experiment concerns the step response on Y . Inorder to compare the results with and without compensator,both results are plotted as in Fig. 13. The figure clearly showsthat the overshoot which is 72.22% without compensator ishighly reduced when using the 4-impulse compensator.C. Scheme of the new systemIn the previous sub-sections, we have compensated thehysteresis of two axes and the vibration of the y axis usingthe feedforward techniques. The new system to be modeledand controlled with feedback is now a bi-variable systemwith smaller hysteresis and vibration, but still with a strongcoupling. It has inputs YRV and ZRV while outputs are Yand Z (Fig. 15). The next section will be dedicated to themodeling of this new system.

IEEE TRANSACTIONS ON AUTOMATION, SCIENCE AND ENGINEERINGYRVVibr YRHcomp-YHystcomp-Y8YVysystemZ RVZ RHHystcomp-ZZVzFig. 15. Block-scheme of the new system to be modeled and controlled withfeedback technique.IV. M ODELINGIn this section, we first characterize the system. Afterwards,we propose to model the system by a decoupling techniquefor y and z axis. The two decoupled models are advantageousrelative to one multivariable model because we handle simpler models and therefore synthesize simple controllers. Theidentification part and the scheme of the nominal model usedfor the controllers design end the section.of ZRV are applied: 0µm, 30µm and 60µm. The responseon Y , plotted in Fig. 17-a, shows the residual hysteresis inYRV Y . The effect of the constant ZRV on the (threehysteresis) curves is that they are slightly angled and shifted.A sine input ZRV with amplitude 60µm is now applied. Threeconstant values are given to YRV : 0µm, 10µm and 20µm. Theresponse of Z, plotted in Fig. 17-d shows the linearity of thedirect transfer ZRV Z. The applied constant YRV affectsthe three linear curves by a slight angle.Fig. 17-b presents the coupling transfer YRV Z. Itconfirms that the effect of YRV on Z is negligible. Indeed,we have: Z YRV 0, ZRV(6)Fig. 17-c presents the coupling transfer ZRV Y , whichcan b

compensated by nonlinear compensators in cascade [20] [21] [22]. Additionaly to the nonlinearity compensation, the vibration was minimized by using dynamic inversion method [23] [24] or inverse multiplicative approach [25]. Open-loop control is convenient for sensor-less piezocantilevers, reducing the cost of the whole automated system. However,

Related Documents: