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1 Actuators 2013, 2, 1-18; doi: /act Article OPEN ACCESS actuators ISSN State Sace Sstem Identification of 3-Degree-of-Freedom (DOF) Piezo-Actuator-Driven Stages with Unknown Configuration Yu Cao * and Xiongbiao Chen Deartment of Mechanical Engineering, Universit of Saskatchewan/57 Camus Drive, Saskatoon, SK, S7N 5A9, Canada; bc719@mail.usask.ca * Author to whom corresondence should be addressed; uc150@mail.usask.ca; Tel.: Received: 31 Januar 2013; in revised form: 28 Februar 2013 / Acceted: 4 March 2013 / Published: 8 March 2013 Abstract: Due to their fast resonse, high accurac and non-friction force, iezo-actuators have been widel emloed in multile degree-of-freedom (DOF) stages for various nano-ositioning alications. The use of fleible hinges in these iezo-actuator-driven stages allows the elimination of the influence of friction and backlash clearance, as observed in other configurations; meanwhile it also causes more comlicated stage erformance in terms of dnamics and the cross-couling effect between different aes. Based on the sstem identification technique, this aer resents the develoment of a model for the 3-DOF iezo-actuator-driven stages with unknown configuration, with its arameters estimated from the Hankel matri b means of the maimum a osteriori (MAP) online estimation. Eeriments were carried out on a commerciall-available iezo-actuator-driven stage to verif the effectiveness of the develoed model, as comared to other methods. The results show that the develoed model is able to redict the stage erformance with imroved accurac, while the model arameters can be well udated online b using the MAP estimation. These caabilities allow investigation of the comlicated stage erformance and also rovide a starting oint from which the mode-based control scheme can be established for imroved erformance. Kewords: cross-couling; dnamics; Hankel matri; state sace model; sstem identification

2 Actuators 2013, Introduction Piezo-actuator-driven stages have the advantages of fast resonse, high recision and generation of large forces. As such, the have been widel alied in semiconductors, biomedical science, roduction manufacturing and other devices that require nano-ositioning and maniulation [1 5]. With the ingenious design of fleible hinges, friction and backlash clearance can be eliminated, leading to imroved erformance. Meanwhile, the use of fleible hinges also caused a more comlicated stage. Modeling and control for one degree-of-freedom (DOF) iezo-actuator-driven stages have drawn considerable attention in the literature [6 10]. Due to the cross-couling effect between different aes, the methods develoed for 1-DOF iezo-actuator-driven stages ma not be readil etended to multile-ais ones [11], the research of which is still in its earl stage. In [12], a three-inut-three-outut state sace model was develoed for a 3-DOF micro-stage, along with the method for arameter identification; and b eeriments, it was shown that the develoed model was able to redict the erformance of the micro-stage with accetable accurac. An auto-regressive eogenous (ARX) model was develoed in [13] to describe the dnamic erformance of a biaial iezo-stage, and the model was then integrated in a feedforward comensator for recision tracking control with eerimental verification. However, the cross-couling between the two aes, which might have a negative effect on the erformance of the controller, was not considered in the ARX model. In [14], a fourth order linear transfer function was identified for a iezoelectric stage, where the cross-couling effect was neglected. On this basis, a chir signal was alied to each of the aes indeendentl, and with the measurement oututs, the arameters in each transfer function were estimated b using the sstem identification technique. In [15], the dnamic equations were combined with the Bouc-Wen model for each iezoelectric actuator to describe the erformance of a lane-te 3-DOF recision ositioning table or stage. The arameters of the model were otimized based on the real-coded genetic algorithm (RGA) method. From the numerical simulations and eerimental results, the 3-DOF cross-couling effect was reduced b the roosed control method, and good contour tracking erformance was obtained, due to successful identification of the dnamic models. A straightforward modeling method for multi-dof iezo-actuator-driven stages can be based on the internal configuration b means of hsics laws, as mentioned above. However, such details with regard to the internal structure are often not rovided b the manufactures. Therefore, sstem identification for multi-dof iezo-actuator-driven stages with unknown configuration is alwas required for the model develoment. In [16] and [17], modeling of a commerciall available 3-DOF iezo-actuator-driven stage was formulated as a single-inut-single-outut nonlinear regression roblem, with the cross-couling effect ignored. B emloing the online least squares suort vector machine and relevance vector machine, the model arameters were udated, once the subsequent measurement became available. The develoed model was alied to the inverse-model-based feedforward control scheme combined with roortional-integral-derivative (PID) regulator, and the erformance of the iezo-actuator-driven stage being controlled was imroved. An alternative method to imrove the erformance of multi-dof iezo-actuator-driven stage is the use of a robust linear controller, such as the sliding mode controller [18], in which the nonlinear effects are regarded as disturbance and then rejected b the robust controller. As such, a linear state sace model for the multi-dof iezo-actuator-driven stage is alwas desired. To meet this need, in this aer, we reort

3 Actuators 2013, 2 3 the model develoment based on the black bo sstem identification of for 3-DOF iezo-actuator-driven stages with unknown configuration. Secificall, a linear discrete state sace model, (k+1) = A(k) + Bu(k) and (k) = C(k) + Du(k) (A, B, C and D are sstem matrices), is adoted and alied to describe the dnamics of the iezo-actuator-driven stage. To identif the arameters of the state sace model, methods have been reorted in the literature [19 25]. In [22], a modified frequenc domain subsace identification algorithm was develoed based on the revious work. The ower sectrum estimates was strongl consistent when the measurements were corruted b bounded random noise. In [23], the numerical algorithms for the subsace state sace sstem identification (N4SID) method was combined with the multivariable outut-error state sace (MOESP) method for imroved erformance. The state sace model was obtained in [24] b identifing the Markov arameters (a kind of matri imulse resonse) that were indirectl calculated from an identified auto-regressive model or transfer function. In [25], the sstem matrices in the state sace model were derived through singular value decomosition (SVD) of the Hankel matri, which was directl identified from a Hankel-Toelitz model using the least squares method. The arameters are time-invariant, and thus, the model cannot be alied if the erformance of iezo-actuator-driven stage changes with the environmental condition, such as the temerature. To develo a state sace model with udating arameters, the SVD of the Hankel matri is strategicall combined with maimum a osteriori (MAP) online estimation in this stud. The arameters can be udated as new observations become available. Furthermore, MAP estimation utilizes rior information regarding the arameters and the measurement errors. Inclusion of osteriori arameter information can have the beneficial effect of reducing the variances of arameter estimators. To verif the effectiveness of the state sace model identified b using the MAP online estimation, eeriments were carried out on a commerciall available iezo-actuator-driven stage. The estimation errors obtained from the Hankel matri using online estimation were comared to those reorted in [25], for the illustration of the roosed method effectiveness. 2. Sstem Identification for a 3-DOF Piezo-Actuator-Driven Stage with Unknown Configuration In this section, it is assumed that the configuration or the internal structure of 3-DOF iezo-actuator-driven stages is unknown. Also, it is assumed that the stage is regarded as a linear multile-inut and multile-outut (MIMO) sstem b ignoring the nonlinearit, which is reasonable, as illustrated in the eeriments resented later in this aer. To reresent the linear dnamics and cross-couling effect of the stage, the simlified Hankel-Toelitz model is adoted and emloed in the resent stud, in which the Hankel matri is to be identified b imlementing the MAP online estimation method Simlified Hankel-Toelitz Model For a linear MIMO sstem, the discrete state sace reresentation is given b: ( k 1) A( k) Bu( k) w( k) ( k) C( k) Du( k) v( k) (1)

4 Actuators 2013, 2 4 n n where A R n m, B R q n, C R q m and D R n 1 are sstem matrices, R 1 is the state, u R m is q 1 the inut, R n 1 is the outut, w R 1 and v R q reresent the ignored nonlinearit and uncertainties of the iezo-actuator-driven stage and m and q are the number of inuts and oututs, resectivel. B iteration, one has: k ( ) Ak ( ) Bu ( k) w( k) ( k) C ( k) D u ( k) w ( k 1) v ( k) for an { Z, 0}, where u and are defined as column vectors of the inut and outut data going stes towards the future, v and future, u( k) ( k) u( k 1) ( k 1) u ( k), ( k) u( k 1) ( k 1) (3) w are defined as column vectors of the noises and disturbance going stes towards the B is the controllabilit matri, sstem Markov arameters and C is the observabilit matri, (2) D is the Toelitz matri for the 1 2 nm T T 2 T 1 T T qn B A B A B AB B R, C C ( CA) ( CA ) ( CA ) R, D CB D C 0 0 q m q n D CAB CB D 0 R, 0 CA C 0 R, CA B CA B CA B D 0 CA CA C If m n 1 2 A A A I R nn, there eists an interaction matri M such that: (4) Substituting Equation (5) into Equation (2) ields: A C 0 M (5) ( k ) ( B MD) u( k) M( k) M w( k 1) M v( k) w( k) (6) Combining Equations (2) and (6) leads to the following equation, which is the so-called simlified Hankel-Toelitz model: where ( k) C ( k) Du( k) w( k 1) v( k) C ( B MD ) u ( k ) C M ( k ) D u ( k) ( k). (7) ( k) w ( k 1) v ( k) C M w ( k 1) C Mv ( k ) C w ( k ). Using the following denotations: one has Equation (7) rewritten as: Γ C( B MD), Φ CM, (8)

5 Actuators 2013, 2 5 u ( k ) ( k) Γ Φ D ( k ) ( k) u ( k) where ( k ) reresents the combined model noises and can be regarded as the model estimation error. Define: H 0 CB ΓΦ D (10) the square matri, H, can be estimated without knowing M. 0 Once H is identified, an adjacent 0 H 1 C A B can be calculated b using Equation (5) such that: Similarl, H1 C( MC) B ( CM) CB ΦH 0 (11) H M M M M Φ H ( ) i ( ) ( ) ( ) ( ) i i C A B C C C B C C C B 0 (12) Using H0, H1, as building blocks, a Hankel matri of an size can be constructed. For eamle: H H H H 0 1 (9) T n (13) 2.2. Reconstruction of the Sstem Matrices The Hankel matri is arranged with Markov arameters of increasing order going from left to right. Let the Hankel matrices be: n1 2 n1 1 CB CAB CA B CAB CA B CA B 2 n n1 2 H(0) CAB CA B CA B, H(1) CA B CA B CA B n2 n21 n1n 2 n21 n22 n1n21 CA B CA B CA B CA B CA B CA B where n1, n2 Z. Comaring Equation (14) with Equations (4), (12) and (13); H (0) and H (1) can then be etracted from H b rearrangement of its elements. The state sace matrices are reconstructed from the Hankel matri b emloing the following Lemma 1. Lemma 1: An s-th order state sace model can be reconstructed as: T 2 s s (1) ss A U V (14) H (15) where B is the first m columns of 2 T s V, C is the first q rows of 2 s U s and s min{ m( n 1), q( n 1)}. The s 1 2 matri U and s V are made u of s left and right singular vectors of s H (0), and the diagonal matri,, s is made u of s corresonding singular values of H (0) [24] MAP Online Estimation Equation (9) can be rewritten as, b ignoring ( k ): where 1 ( k) θ X ( k). (16)

6 Actuators 2013, 2 6 u ( k ) θ Γ Φ D, X ( k ). u ( k) B using the least squares method, θ is identified to be a time-invariant matri, which might not be able to accuratel describe the environment-deendent erformance of the iezo-actuator drive stage. In order to al the state sace model in the control of iezo-actuator-driven stage, the model arameters should be udated as new observation data is available. Therefore, MAP online estimation was emloed to identif the arameter matri in Equation (16) instead. The MAP online estimation method is used to udate the arameters as the new observation data oints becomes available, which is given b: T 1 θi1 θi Pi1Xi1σ i1e i1 (17) where X has the same definition as the one given in Equation (16), θ i is the value of identified arameters based on the first i grous of data, P i is the covariance of identified arameters from the first i grous of data, σ is the variance matri of measurement errors and i E is the estimation error of i the i-th grou of data. Integration of the rior information regarding the arameters and the information regarding the measurement errors can have the beneficial effect of reducing variances of arameter estimators. As a result, the arameter identification could be imroved. Since the Hankel-Toelitz model is a regression model given the zero initial condition, E was also i calculated b using the regression method as: E i 1 i i (18) where is the measurement outut of the iezo-actuator-driven stage and i is the estimation outut i of the iezo-actuator-driven stage calculated through i-1 iterations Model of the 3-DOF Piezo-Actuator-Driven Stage A three-inut-three-outut state sace model (1) is emloed for the 3-DOF iezo-actuator drive stage. B imlementing the singular value decomosition on the Hankel matri, which is estimated based on the Hankel-Toelitz model, as shown in Lemma 1, the sstem matrices of the state sace model can be derived. Since the 3-DOF iezo-actuator-driven stage is reviousl assumed to be linear, the model identification can be imlemented on each inut channel individuall. For eamle, when an inut signal is onl rovided in one channel ui ( i 1,2,3), the three-dimensional outut [ 1 2 3] T i i i, i can be obtained from the identified one-inut-three-outut model b aling the method mentioned above: i( k 1) Aii( k) Bu i i( k) wi( k) ( k) C ( k) Du ( k) v ( k) (19) i i i i i i 3 3 where Ai R 31, Bi R 3 3, Ci R 31 and Di R are sstem matrices of the one-inut-three-outut sstem. The states for all three channels in Equation (19) ma be stacked as:

7 Actuators 2013, 2 7 ( k 1) ( ) ( ) 1 A1 k 1 B1 u k 1 T ( k 1) A ( k) B u ( k) + w ( k) w ( k) w ( k) ( k 1) A ( k) B u ( k) According to the definition of the linear sstem, the outut can be eressed as the sum of i ( i 1,2,3), such that: 1( k) u1( k) C C C ( k) D D D u ( k) v ( k) v ( k) v ( k) ( k) u3( k) (20) (21) As such, the state sace model for the three-inut-three-outut sstem can be eressed as: where: ( k 1) A( k) Bu( k) w( k) ( k) C( k) Du( k) v( k), B diag B B B, C C C C, D D D D A diag A A A ( k) u3( k) w3( k) , ( k) u1( k) w1( k) ( k) ( k), u( k) u ( k), w( k) w ( k), vk ( ) v1( k) v2( k) v3( k). (22) 3. Results and Discussion To verif the effectiveness of the state sace model and the roosed identification method, eeriments were imlemented on a commerciall-available 3-DOF iezo-actuator-driven stage (P-558.TCD, Phsik Instrumente), as shown in Figure 1a. Driven b four iezoelectric actuators, the P558.TCD can generate linear dislacements in the vertical direction Z and rotation around two orthogonal horizontal aes R and R. Table 1 shows the motion range and resolution in each DOF. Table 1. Motion range and resolution in each degree-of-freedom (DOF) DOF Z R Motion range 50 μm ±250 μrad ±250 μrad Resolution 0.5 nm 50 nrad 50 nrad Figure 1. Eerimental settings on the iezo-actuator-driven stage: (a) icture and (b) schematic. R (a) (b)

8 Actuators 2013, 2 8 For dislacement measurements, three caacitive sensors built in the stage are emloed. All dislacements were measured with a samling interval of 2 ms in the resent stud. Both the actuators and the sensors in the stage were connected to a host comuter via a digital controller (E-761, Phsik Instrumente) and controlled through Labview, as shown in Figure 1b. As instructed b its manual, the iezo controller can drive the actuator with a maimum oerating frequenc of Hz if an inut voltage in the range of V is alied. During oeration, the motion of the four iezoelectric elements must be coordinated to reduce the internal forces generated due to the over actuation, which ma cause reduced stiffness and even break or damage the iezo-actuator-driven stage. This is realized b a user rogram interface rovided b the manufacturer, which is used to generate the voltage inut of each iezoelectric actuator from the user defined reference signal Linearit of the 3-DOF Piezo-Actuator-Driven Stage To eamine the linearit of the 3-DOF iezo-actuator-driven stage, a case stud was conducted rior to the sstem identification. In articular, a 1 Hz 1 μm sinusoidal reference signal with 1 μm offset, a 2 Hz 200 μm sinusoidal reference signal with 2 s time dela and a 100 μm ste reference signal with 3 s time dela were rovided to the Z, R and R channel, resectivel, and the corresonding oututs were measured. Then, the stage dislacement outut, as these three signals were alied simultaneousl, was measured. The criterion used for the linearit eamination is that if the outut with three inut signals equals or aroimatel equals the sum of the oututs when the signals is alied individuall, the 3-DOF iezo-actuator-driven stage is linear or can be aroimatel considered to be linear. Figure 2 shows the comarison between the two oututs mentioned above. It can be seen that the overlaed with each other, indicating that the stage can be aroimatel considered to be a linear sstem. Differences between the measured outut when the three inuts were rovided to the different channels simultaneousl and the sum of the oututs when the three inuts were rovided searatel eist. For eamle, in the R direction, the maimum difference is aroimatel 3 μm, which is onl 1.5% of the amlitude of the reference signal. This difference might be due to the nonlinearities of the 3-DOF iezo-actuator-driven stage, which is ignored in the model develoment resented in this aer. Figure 2. Linearit of the 3-DOF iezo-actuator-driven stage: (a c) comarison between the measured outut when the three inuts were rovided to the different channels simultaneousl and the sum of the oututs when the three inuts were rovided searatel; (d f) difference between these two oututs. (a) (b) (c)

9 Actuators 2013, 2 9 Figure 2. Cont. (d) (e) (f) 3.2. Sstem Identification of the 3-DOF Piezo-Actuator-Driven Stage Figure 3 shows the flow chart of sstem identification. Since different signals alied in sstem identification ma lead to the difference in the model identified, the effects of aling the random signal and the chir signal in the arameter estimation were investigated in the signal selection in this stud. The two signals were comared, and the one with less model rediction error was emloed as the inut for order selection, in which state sace models with different orders were identified and comared. The one with less model rediction error was emloed as the model for the iezo-actuator-driven stage. Figure 3. Flow chart of black bo sstem identification. Random inut Chir inut Which one leads to better rediction? Initial order = 3 Order +1 Minimum error? No Comare with least square method in [22] Data verification For signal selection, a 20 μm reference chir signal with 20 μm offset and frequenc ranging from 1 to 100 Hz was rovided to channel 1 (reference Z channel), and the corresonding outut in each channel was measured. Based on the emirical knowledge of our revious stud on iezoelectric

10 Actuators 2013, 2 10 actuators, the order of the state sace model was originall set to be three, and θ in Equation (17) was 0 set to be a zero matri. Since the covariance of the arameters is unknown, P 0 is set to be a diagonal matri with big covariance designated in the diagonal elements. B aling online estimation with the identified Hankel matri, the sstem matrices of the state sace model (Equation (19), I = 1) were obtained. The estimation error varied, deending on the values of arameter in Equation (2). Figure 4 (a c) shows the estimated error versus the -value. It can be seen that if = 8, the estimation errors in all three outut directions aroached or reached their individual minimum values. Therefore, it is reasonable to set = 8, as the chir signal is rovided to channel 1. Figure 4. Estimation error changes with -value when reference inut was alied in channel (a) Z direction; (b) R direction; (c) R direction. (a) (b) (c) For other two channels, a 200 μrad reference chir signal with frequenc ranging from 1 to 100 Hz was alied. B emloing the aforementioned rocedure, was set to 25 and 27 for channel 2 and 3 resectivel. Table 2 shows the rediction error in each direction, as a 1 Hz sinusoidal reference inut was alied to the three channels, resectivel. The rediction errors are calculated in terms of the 2-norm of the error vector (defined as the difference between the measurement and the model rediction). It is seen that the diagonal rediction error is μm, μrad and μrad in the Z, R and R direction, resectivel, which is 0.49%, 2.43% and 2.16% of the desired movement in the individual direction. Table 2. Model rediction error if chir inuts were alied. Direction 1 Hz 20 μm sinusoidal inuts with 20 μm offset in channel 1 1 Hz 200 μrad sinusoidal inuts in channel 2 1 Hz 200 μrad sinusoidal inuts in channel 3 Z (μm) R (μrad) R (ΜRAD)

11 Actuators 2013, 2 11 Similar to the use of the chir signal, 40 μm and 200 μrad reference random signals were also alied to each channel, resectivel. The order of each sub-model was chosen to be three, and was set to nine, 14 and 13 for the three channels, resectivel. The same 1 Hz sinusoidal inuts were rovided to difference channels, and the outut was measured and comared with the model rediction. Table 3 illustrates the model rediction error. In contrast to the chir signal, it can be concluded that the model rediction errors is much bigger when random signals are used in the model identification. For eamle, when a 1 Hz 200 μrad sinusoidal reference inut was rovided to channel 2, the model rediction error in the R direction reached μrad b using the random inuts, which is over 10-times larger than that derived b using the chir signal. As a result, a chir signal was emloed as the reference inut for model identification below. Table 3. Model rediction error if random inuts were alied. Direction Z (μm) R (μrad) 1 Hz 20 μm sinusoidal inuts with 20 μm offset in channel 1 R (ΜRAD) Hz 200 μrad sinusoidal inuts in channel Hz 200 μrad sinusoidal inuts in channel To determine the order of the state sace model, the arameter identification, as described reviousl, was reeated with varing values of n (Equation (1)) in each channel. Tables 4 6 show the estimation errors in each channel. Parameter was chosen to have different values for varing orders based on the method mentioned above. It can be concluded that if the chir signal was used in channel 1, the estimation error in the Z direction reached its minimum value of μm with the order of the sub-model being si or seven. For the R direction, the otimal choice was to set n = 7. Therefore, the sub-model for channel 1 was considered to be a seventh order state sace sstem. The sstem matrices were determined as given in Equation (23). Using a similar rocedure, the orders of the sub-model for the other two channels were both chosen to be four, and the sstem matrices were determined, as shown in Equations (24) and (25) A , B C , D (23)

12 Actuators 2013, 2 12 A C , B, , D2 033 (24) A3, B3, C , D (25) Table 4. Estimation error from the chir inuts in channel 1. Order Z (μm) Estimation error R (ΜRAD) R (ΜRAD) Table 5. Estimation error from the chir inuts in channel 2. Order Z (μm) Estimation error R (μrad) R (μrad)

13 Actuators 2013, 2 13 Table 6. Estimation error from the chir inuts in channel 3. Estimation error Order Z (μm) R (μrad) R (μrad) Model Verification To illustrate the effectiveness of the MAP online estimation method, 1, 5 and 10 Hz sinusoidal reference inuts were rovided to different channels, resectivel. For comarison, the estimation method introduced in [25] was imlemented as well. The arameter was defined as 21, four and seven, resectivel, for the three inut channels. Tables 7 and 8 show the rediction error in each direction based on the different identification methods. The rediction errors were calculated in terms of the 2-norm of the error vector, illustrating that the rediction error increases with the frequenc. Table 7. Estimation error b aling the online estimation method. Inut Channel Z (μm) R (μrad) R (ΜRAD) 1 Hz 10 μm Hz 200 μrad Hz 200 μrad Hz 10 μm Hz 200 μrad Hz 200 μrad Hz 10 μm Hz 200 μrad Hz 200 μrad In contrast to the identification method introduced in [25], the use of osteriori arameter information in MAP online estimation leads to better estimations on the Hankel matri. For eamle, the estimation errors for the 5 Hz, 10 μm sinusoidal inuts to channel 1 were μm, μrad and in the Z, R and R directions, resectivel. These results are 7.3%, 40.6% and 24%, resectivel, of those derived using the identification method introduced in [25].

14 Actuators 2013, 2 14 Table 8. Estimation error b aling the identification method introduced in [25]. Inut Channel Z (μm) R (μrad) R (ΜRAD) 1 Hz 10 μm Hz 200 μrad Hz 200 μrad Hz 10 μm Hz 200 μrad Hz 200 μrad Hz 10 μm Hz 200 μrad Hz 200 μrad Figure 5 shows the outut in each direction as a result of a 10 μm 10 Hz sinusoidal reference inut with 10 μm offset in the Z direction comared with the model rediction. It can be clearl seen that the identified state sace model is able to describe the couling effect between each ale. Figure 5. Comarison of eerimental results and model rediction under 10 μm 10 Hz sinusoidal inut in channel 1: (a) Z direction; (b) R direction; (c) R direction. (a) 3.4. Case Stud with Combined Inuts (b) (c) To verif the identified linear state sace model with combined inuts, three eeriments were imlemented. In the first eeriment, the reference inuts simultaneousl alied to the three channels are a 1 Hz and 20 μm sinusoidal reference with a 20 μm offset, a 2 Hz and 200 μrad sinusoidal reference with a time dela of two seconds and a 100 μrad ste inut with a time dela of three seconds. The oututs in the three directions were measured, and the redicted oututs were obtained according to the identified state sace model of Equations (23 25), resectivel. In the second eeriment, a 1 Hz and 1 μm sinusoidal inut with 1 μm offset and a 2 Hz, 0.5 μrad sinusoidal reference inut with a 2 s time dela were rovided to the iezo-actuator-driven stage with the third channel ket zero. The oututs were measured and comared to the redicted oututs. To validate the model in high frequenc, the inut of 1 Hz and 2 μm sinusoid in channel 1 was relaced with a 10 Hz 40 μm one in the third eeriment. Also, the corresonding oututs in the three directions were

15 Actuators 2013, 2 15 measured and comared to the oututs redicted b the identified state sace model. The comarison is shown in Figure 6, from which it can be concluded that the model is able to describe the erformance (both dnamics and cross-couling effect) of the 3-DOF iezo-actuator-driven stage. Figure 6. Comarison of eerimental results and model rediction from combined inuts to all three channels in the first eeriment (a c); in the second eeriments (d f) and in the third eeriments (g i). (a) (b) (c) (d) (e) (f) (g) (h) (i)

16 Actuators 2013, Conclusions A straightforward modeling method for multi-dof iezo-actuator-driven stages is based on the internal configuration b means of hsics laws, as reorted in the literature [12 15]. However, such details with regard to the internal structure are often not rovided b the manufactures. Therefore, sstem identification for multi-dof iezo-actuator-driven stages with unknown configuration is alwas required for the model develoment. The contribution of this aer is the develoment of a black bo model used to describe the dnamics of 3-DOF iezo-actuator-driven stages with unknown hsical configuration, which allows the investigation of the comle sstem erformance with unknown hsical configuration b means of the linear state sace model. B combining the MAP online estimation methods, the Hankel matri of the state sace model was identified and the model arameters were udated as new observations were available. To show the effectiveness of the roosed estimation method, model verification eeriments were carried out on the iezo-actuator-driven stage, and the oututs obtained were comared to the redictions of the state sace model identified using the method introduced in [25]. From the model verification results, it was shown that the linear state sace model can redict the dnamic erformance of a iezo-actuator-driven stage with imroved accurac. B making use of the osteriori arameter information, the MAP online estimation method erforms better in the model identification than the least squares method. Moreover, the identified arameters are udated online as new and more data becomes available. The develoed model and arameter identification methods rovide a starting oint from which to adativel comensate for the dnamics and cross-couling effects of the iezo-actuator-driven stage b means of the mode-based control scheme. Acknowledgments Suort for the resent stud from the China Scholarshi Council (CSC) and the Natural Sciences and Engineering Research Council (NSERC) of Canada is acknowledged. References 1. Ouang, P.R.; Tjitorodjo, R.C. Micro-motion devices technolog: The state of arts review. Int. J. Adv. Manuf. Tech. 2008, 38, Devasia, S.; Eleftheriou, E. A surve of control issue in nano-ositioning. IEEE Trans. Contr. Sst. Technol. 2007, 15, Ge, P.; Jouaneh, M. Tracking control of a iezoelectric actuator. IEEE Trans. Contr. Sst. Technol. 1996, 14, Leang, K.K.; Devasia, S. Design of hsteresis comensating iterative control for iezo-ositioners: alication on atomic force microscoes. Mechatronics 2006, 16, Lin, C.J.; Yang, S.R. Precise ositioning of iezo-actuated stages using hsteresis observer based control. Mechatronics 2006, 16, Chen, X.B.; Zhang, Q.S.; Kang, D.; Zhang, W.J. On the dnamics of iezoelectric ositioning sstems. Rev. Sci. Instrum. 2008, 79,

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18 Actuators 2013, Lim, R.K.; Phan, M.Q.; Longman, R.W. State-Sace Sstem Identification with Identified Hankel Matri; Technical Reort No. 3045; Princeton Universit: Princeton, NJ, USA, Setember b the authors; licensee MDPI, Basel, Switzerland. This article is an oen access article distributed under the terms and conditions of the Creative Commons Attribution license (htt://creativecommons.org/licenses/b/3.0/).