T E C H N I C A L P A P E R
Robust control design for precision positioning of a generic
piezoelectric system with consideration of microscopic hysteresis
effects
Paul C.-P. Chao•Pen-Yen Liao• Meng-Yen Tsai•
Chin-Teng Lin
Received: 31 August 2010 / Accepted: 27 February 2011 / Published online: 18 February 2011 Ó Springer-Verlag 2011
Abstract This study performs precision positioning of a generic piezoelectric structure against hysteresis effects by finite elements, microscopic hysteresis cancellation and robust H? compensation. The designed control algorithm
is expected to be effective in enhancing servo performance of hard disk drives. The precision positioning is accom-plished by adding a polarization term into the linear constitutive equations of piezoelectric materials. This polarization term is then described by the well-known Preisach model. Applying basic principles of finite ele-ments and Hamilton’s thoery, the macroscopic governing equations of an arbitrary piezoelectric system in finite elements are obtained. Based on the macro-model, a con-troller consisting of two parts is designed to perform the precision positioning of a generic piezo-structure. The first part is responsible for direct hysteresis cancellation at the microscopic level, while the second one is a robust H?
controller to overcome inevitable cancellation errors. In this way, the control effort is then more effective than the conventional PI and double-lead controller without microscopic hysteresis cancellation. A simple piezoelectric structure of a bender-bimorph cantilever beam is consid-ered for designs and experimental validation. Based on
experimental results, the proposed control design is found effective to suppress hysteresis effects as opposed to con-ventional controllers.
1 Introduction
Smart materials have drawn much attention in recent years in areas of sensing and actuation. Among all materials, such as piezoelectric, electrostrictive, electrorheological materials, and shape memory alloys, etc., the piezoelectric materials are most used in variety of sensors and actuators due to simple construction, small size, high sensitivity and precision (Shaffer and Fried 1970; Lee 1979). Known applications include scanning tunneling microscopy, ultrasonic motors, resonators, accelerometers and hard disk drives. However, it is well noted that the actuation behaviors of piezoelectric structures have two major non-linearities, i.e., hysteresis and creep (Newcomb and Flinn
1982; Chen and Montgomery 1980), undermining the performance of the piezoelectric actuators. The maximum error due to the hysteresis effect can be as much as 10–15% of the actuated distance if the piezoelectric structure is excited in an open-loop fashion (Peng et al.2010).
For hard disk drives, the piezo-electric actuators have been designed and used for (a) precision positioning for track-following control (Mori et al.1991; Li and Horowitz
2002; Li et al.2003; Huang et al.2006; Oboe et al.1999; Tsai and Yen1999; Tokuyama et al.2001; Lou et al.2002) or (b) flying height control of the magnetic reading head (Gao and Swei 2000; Liu et al.2002; Suzuki et al.2003; Tagawa et al. 2003). For afore-objective (a), the piezo-actuator is usually part of a dual-stage piezo-actuator system, which consists of a ‘‘fine’’ microactuator (MA) realized by piezo-structures and a ‘‘coarse’’ voice coil motor (VCM). P. C.-P. Chao (&) M.-Y. Tsai C.-T. Lin
Department of Electrical Engineering,
National Chiao Tung University, Hsinchu 300, Taiwan e-mail: [email protected]
P. C.-P. Chao
Institute of Imaging and Biomedical Phonics, National Chiao Tung University, Tainan 711, Taiwan P.-Y. Liao
Department of Mechanical Engineering,
Chung-Yuan Christian University, Chung-Li 320, Taiwan DOI 10.1007/s00542-011-1250-x
For (b), the piezo-actuator could serves as a sole actuator or part of a dual-stage one. Part of reasons for the piezo-actuators being only used in micro-piezo-actuators is their inability to achieve long-range precision positioning due to hysteresis. It is evident in some of the previous works (Oboe et al. 1999; Tsai and Yen 1999; Tokuyama et al.
2001; Lou et al. 2002) that the hysteresis effects are still present as a critical factor in achieving required positioning precision. To tackle the hystereis, the control scheme synthesized in the present study aims to offer the potential capability of effectively counteracting the hysteresis effects of the microactuators, and also, extending positioning range of the piezo-MA to ease the burden of the VCM or even utilize the piezo-MA solely to perform the precision positioning.
Research works (Mayergoyz 1991; Ge and Jouaneh
1995; Freeman and Joshi 1995; Hughes and Wen 1995; Yu et al.2002) had been conducted to prescribe hysteresis effects of piezoelectric structures via casting the lumped input–output relation of piezoelectric device/system into a pre-chosen mathematical models, such as Preisach (Mayergoyz 1991; Preisach 1935; Hutton 2009), Bouc-Wen (Andronikou et al. 1983), Prandtl–Ishlinksii (Dong and Tan 2009; Janaideh et al. 2009) or domain wall (Chen and Wang 1998). These approaches of ‘‘macro-scopic’’ modeling on the hysteresis effects requires experimental identification on the parameters of the hys-teresis model adopted, whenever a new piezo-structure is considered. To simplify the modeling, a novel micro-scopic modeling technique on the hysteresis effects was proposed with the aim to incorporate the hysteresis effects in the level of microscopic scales (Simkovics et al.2000). This is accomplished by adding a polarization term to one of linear constitutive equations for piezoelectric materials. The polarization term is then described by the well-known hysteresis model, Preisach model. Applying basic princi-ples of finite elements and Hamilton’s theory, the mac-roscopic governing equations of an arbitrary piezoelectric system in finite elements can be obtained. Based on the model, a controller consisting of two parts is designed in this study to perform precision positioning of a generic piezo-structure. The first part is responsible for a direct cancellation of the hysteresis term at the microscopic level, while the second one is a robust H? controller
responsible to overcome the errors of the aforementioned hysteresis cancellation. In this way, the control effort is more effective than the conventional PI and double-lead controller (Kim et al. 2009) without microscopic hyster-esis cancellation. A typical piezoelectric structure of bender-bimorph cantilever beam is considered for designs and experimental validation. General closeness is found between the simulations and experimental data. Further-more, the designed controller is found effective in
experiments to suppress hysteresis effects and achieve excellent precision positioning.
The remainder of this paper is organized as follows. In section two, linearized piezoelectric constitutive equations are modified with an addition of the polarization prescribed by Preisach model. Section three proposes finite element modeling with an exemplary demonstration of a piezo-electric bimorph beam. Section four presents experimental studies to verify the effectiveness of the proposed modeling technique.
2 Modified constitutive equations
Based on the principle of virtual displacement, the internal mechanical stress T varied by the virtual strains dS in piezoelectric materials must be equal to the external forces R applied on the considered region X, yielding
Z
X
TijdSijdX¼ R: ð1Þ
On the other hand, the dielectric displacement D varied by the virtual electric field strength dE is equal to electric charges Q accumulated in the considered volume, yielding Z
X
DidEidX¼ Q: ð2Þ
The above equations (1, 2) are commonly assumed coupled by linear piezoelectric material relations
T¼ cES etE;
D¼ eS þ eSE; ð3Þ
where cEcontains elastic moduli, etand e are piezoelectric coefficients and eSare dielectric constants or permittivities. It is known that application of strong electric fields and large mechanical stress to piezoelectric materials leads to switchings of electric orientations of ferroelectric grains (dipoles) inside the materials. The phenomenon is known as ‘‘polarization,’’ as illustrated by Fig.1. While polari-zing, certain amount of energy is consumed to make possible re-orientation of grains, leading to the difference in the load-response relations with ascending and descen-ding loads. This is known as ferroelectric hysteresis. To reflect the effects of ferroelectric hysteresis, one can add an additional term ^aP proportional to dielectric polarization Pinto the second equation in Eq. 3, yielding
T¼ cES etE;
D¼ eS þ eSEþ ~aPðEÞ; ð4Þ
where ~a represents the electric displacement change due to unit polarization. With the modified constitutive equations
(4) in hand, the hysteresis effects due to varied input electric field E are in fact characterized in microscopic level as the changes in polarization P(E) in Eq.4.
3 Preisach model for polarization
The effects of ferroelectric hysteresis had shown high similarity with ferromagnetic material behavior. There-fore, the Preisach model (Mayergoyz 1991), which is commonly used to describe ferromagnetic hysteresis, is adopted herein to prescribe the polarization term P(E) in Eq.4. Besides Preisach, other microscopic models effec-tively describing the hysteresis can also be used herein to model P(E), like Prandtl–Ishlinksii model in Janaideh et al. (2009). Utilizing the Preisach modeling technique, each polarization term in P(E) in Eq.4 can be expressed as functions of the corresponding applied electric field E by the integral
PðtÞ ¼ ZZ
a b
l a; bð Þcab½E tð Þdadb; ð5Þ
where cabis the elementary operator as shown in Fig.2,
which is also a function depending on E. cabvaries from
zero to one, emulating an on–off to distinguish the change in input E either in ascending or descending for reflecting the hysteresis effect. The rule of the dependence of cabon
E follows the illustration in Fig.3, where a and b are maximum and minimum input values of E. l(a, b) in Eq. 5 is in fact a function of a and b, so-called Preisach function, which captures a variety of different hysteresis characteristics with given values of a and b, corre-sponding switching values of E between ascending and descending. For a certain piezoelectric material, if l(a, b) is known, the polarization due to hysteresis can be cal-culated directly by the integration in Eq. 5. However, since determination of the weighting function l(a, b) requires double differentiation which may amplify the errors in experimental data, another approach (Mayergoyz
1991) is used for numerical implementation of the Pre-isach function. Also, l(a, b) in Eq.5 could be in varied forms to model asymmetric hysteresis loops and saturated hysteresis output, as handled by Prandtl–Ishlinksii model in (Janaideh et al. 2009). The numerical method to cal-culate P(t) is stated as follows.
For the case of monotonically increasing input E(t) at the final state, P(t) is in fact a double integral of the weighting function l(a, b) on a region that is circumscribed by a set of interface links whose final segment is horizontal (see Fig.3a, c). Here, the integration is equivalent to the sum of trapezoids within this region S?(t). Thus, the gen-eral formula for estimating the piezoelectric expansion, when the input E(t) is monotonically increasing at the final state, is PðtÞ ¼X n k¼1 Fa0k;b0k1 F a0k;b 0 k þ F EðtÞ; b0n ; ð6Þ
where function F is a double integral of the weighting l(a, b) over the shaded region in the (b, a) plane as shown in Fig.3c. Similarly, for the case of monotonically decreasing input E(t) at the final state (see Fig.3b, d), since the integration is equivalent to the sum of the trapezoids within the region S?(t) in Fig.3d, the piezoelectric expansion is as following
Fig. 1 Illustration of polarization
PðtÞ ¼X n1 k¼1 Fða0 k;b 0 k1ÞFða0k;b 0 kÞþ½Fða 0 n;b 0 n1ÞFða0n;EðtÞÞ : ð7Þ With Eqs.6–7 in hand, the microscopic polarization P can be captured via prior identification for the dependence of polarization F on {a, b} based on macroscopic experimental data and using the technique of finite element modeling, in later sections. This identification procedure is realized by (1) sampling the (b, a) plane as in Fig.4, (2) experimentally recording the correspondence between sampled (a, b)’s and F’s from measurements, and finally (3) interpolating the values of F’s within sampled (a, b)’s using
Fða; bÞ ffi c0þ c1aþ c2bþ c3ab; ð8Þ
Fða; bÞ ffi c4þ c5aþ c6b; ð9Þ
for square and triangular areas, respectively, shown in Fig.4. With the above F identified; i.e., ci’s in Eqs.8,9are
determined, the polarization P(t) either in ascending or descending branches at final state, as shown in Eqs.6,7, respectively, can be computed. Note that the identification process is based on macroscopic experimental data and the
equations of motion built from finite element modeling in the next section.
4 Finite element modeling
A simple bimorph piezoelectric beam as depicted and photographed in Fig. 5is considered for demonstrating the S+ S -Limiting Triangle T0 Interface link β α = α β α ) ( 1 3 1=E t α + S S -Interface link α β ) , ( 1α β T Limiting Triangle T0 β α= ) ( 1 3 1=E t α S1 S2 S3 S+(t) α β α= β 0 α 2 α1 α 0 β β1 β2 ) ( 3 t E S1 S2 S3 S+(t) α β β α = 0 α 1 α 2 α 3 α 0 β β1β2 E3(t)
(a)
(b)
(c)
(d)
Fig. 3 aLinks with
monotonically increasing input voltage E3(t), b links with monotonically decreasing input E3(t), c the region S?(t) whose final input is monotonically increasing, d the region S?(t) whose final input is monotonically decreasing
afore-proposed microscopic modeling on hysteresis effects, which are seen as results of microscopic polarization. A subsequent system modeling based on finite elements is conducted to reflect hysteresis effects in macroscopic level. As shown in Fig.5a, the considered piezo-beam owns length L, width b, and a pair of piezoelectric patches extending from x = x1 to x2 in parallel polarizations.
The thicknesses of the beam and piezo-electric patches are 2h and hp, respectively. Assuming small slenderness
ratio, Euler beam theory is employed to model the canti-lever beam. The modeling process starts with a special case and alternative representation of the constitutive Eqs.4
as
r11 ¼ Epe11 h31D;
E3¼ h31e11þ b33Dþ ^aPðE3Þ;
ð10Þ where 1 and 3 refer to directions of x and z in Fig.5a, respectively, and ^a denotes the electric field change due to unit polarization. Also in Eqs.10, r11and e11are the stress
and strain in the x direction of piezoelectric elements; h31
and b33 are the piezoelectric and dielectric constants,
respectively; Ep is Young’s modulus of piezoelectric
material. Potential and kinetic energies are next formulated
for application of Hamilton’s equations to derive equations of motion. The proposed procedure of finite element modeling can easily be extended to arbitrary piezoelectric structures with hysteresis effects via establishing the polarization term P in Eqs. 10.
The finite element modeling for the piezo-beam starts with considering deformation for the ith element of the beam as shown in Fig.6, where R0 is the distance from
origin O of the global coordinate system to the origin O0of the local coordinate system. The displacements of the Euler beam in the x- and z-direction are, respectively,
u1ðx; tÞ ¼ uðx; tÞ zwxðx; tÞ; ð11aÞ
u2ðx; tÞ ¼ 0; ð11bÞ
u3ðx; tÞ ¼ wðx; tÞ; ð11cÞ
where u(x, t) and w(x, t) represent axial and transverse displacements, respectively; moreover the subscript x denotes the spatial derivative with respect to x. The position vector of an arbitrary point located on the neutral axis after deformation is
R x; z; tð Þ ¼ x þ u zwð xÞ i þ z þ wð Þ k: ð12Þ
The total kinetic energy of the piezoelectric bender-bimorph beam consisting of one beam and two piezoelec-tric patches as shown in Fig.5a is
T¼ Tbþ Tp ¼ Z Vb 1 2qbRt RtdVbþ Z Vp 1 2qpRt RtdVpjupþ Z Vp 1 2qpRt RtdVpjdown ¼1 2 ZL 0 qbAb u2tþ w2t dxþ Zx2 x1 qpAp u2tþ w2t dx ð13Þ
where subscripts t denotes time differentiation; qband qpare
the mass densities of the beam and piezoceramics, respectively. Also, the cross section areas for beam and piezo-patches are Ab ¼ 2hb and Ap¼ hpb, respectively; Vb
and Vpare volumes of beam and piezo-patches, respectively;
2h is the height of the beam; b is its width. The linear Lagrangian strains of Euler beam are
p h p h h 2 b + −V b A p A 1 x x2 b L +− +− x
z
(a)
(b)
Fig. 5 a Schematic diagram of the cantilever beam with two piezoelectric layers bounded on the top and bottom, where the input voltage V is applied between the intermediate electrode and the top/ bottom electrodes, b photograph of the piezoelectric beam considered for the experiment
i u Configuration Deformed z O 0 R le x Configuration body Rigid− 1 + i w i w X Z 1 + i u 1 + i x w ' O x
e11 ¼ ux zwxx; e12¼ 0; e13 ¼ e31 ¼ wx: ð14Þ
On the other hand, the potential energy is U¼ Ubþ Up; ¼1 2 Z Vb r11e11dVbþ 1 2 Z Vp r11e11þ DE3 ð ÞdVjupper þ1 2 Z Vp r11e11þ DE3 ð ÞdVjlower ¼1 2 ZL 0 ½EbAbu2xþEbIbw2xxdx þ1 2 Zx2 x1 ½EpApu2x EpApuxwxxð2h þ hpÞ þEpIpw2xx 2h31ApDux þh31ApDwxxð2h þ hpÞ þ Apb33D 2 A p^aPDdx; ð15Þ
where Iband Ipare the moments of inertia about the neutral
axis of the beam and piezoelectric layer, respectively. The virtual work done by the applied voltage V on the piezo-actuators can be expressed as
dW¼ 2 Zx2
x1
VdDbdx; ð16Þ
where the input voltage V is applied across the total thickness of the piezo-patches as shown in Fig.5a. Using Hamilton’s principle,
Zt2
t1
d Tð U þ WÞ dt ¼ 0; ð17Þ
one obtains the governing equations in axial interval [0, L] as follows, u: qbAbþ 2qpAp uttþ EbAbþ 2EpAp uxx 1 2EpApð2h þ hpÞwxxx¼ 0; ð17aÞ w: q bAbþ qpApwtt E bAbþ EpApwxxxx þ1 2EpApð2h þ hpÞuxxx¼ 0; ð17bÞ D: h31Apux 1 2h31Apwxx 2hþ hp Apb33D þ1 2Ap^aPþ Vbp ¼ 0: ð17cÞ
Note in the above equations that the piezoelectric-related terms, with subscript p, only exist in the interval [x1, x2], where the beam structure does not exist. Based on
Eq.17c, the electric displacement can be derived as
D¼Vbpþ h31Apux 1 2h31Apwxx 2hþ hp þ1 2Ap^aP Apb33 : ð18Þ The clamped-free boundary conditions at x = 0 and x = L of the piezo-beam in Fig.5a, respectively, are
uð0; tÞ ¼ 0; wð0; tÞ ¼ 0; wxð0; tÞ ¼ 0;
uxðL; tÞ ¼ 0; wxxðL; tÞ ¼ 0; wxxxðL; tÞ ¼ 0:
ð19Þ
With potential/kinetic energies and boundary condi-tions in hands, the finite element modeling can then be performed by expressing the energies (13,15) in terms of nodal coordinates and then derive the equations of motion. This process starts with substituting expression (18) into (13, 15) for eliminating the dependence of potential and kinetic energies on the electric displacement D. In the next step, as shown in Fig. 7, a set of (Ne? 1)
equally-spaced nodes along the axial, neutral axis of the beam, x, are selected to build Ne beam elements. Within
the axial range from x = x1 to x2, there are also nodes
associated with piezoelectric patches. Note that three degrees of freedom are associated with each node; i.e., {ui, wi, wx,i}. Assuming cubic interpolation functions
between degrees of freedom of adjacent nodes, one is able to express the dynamic states of an arbitrary axial location within the ith element based on the beam theorem, yielding qi¼ Ne i;u Tu iþ Nei;w Tw i ð20Þ where
qi u½ i wi wxi uiþ1 wiþ1 wxiþ1 T
;
ui¼ u½ i 0 0 uiþ1 0 0T; wi¼ 0½ wi wxi 0 wiþ1 wxiþ1 T ; Ne i;u¼ N½ u1 0 0 Nu2 0 0 T ; Ne i;w¼ 0 N½ w1 Nw2 0 Nw3 Nw4 T ; Nu1¼ 1 ^ x s; Nu2¼ ^ x s; Nw1¼ 1 s32^x 3 3^x2sþ s3 ; Nw2¼ 1 s3^x 3s 2^x2s2þ ^xs3 ; Nw3¼ 1 s32^x 3þ 3^x2s ; Nw4¼ 1 s3x^ 3sþ ^x2s2 :
Note in the above equations that s is the axial length of the ith beam element; ^x is the local axial coordinate; the superscript e for Nie’s indicates that these interpolations are
in local dimension. The potential and kinetic energies in
x1 x2 z Node (Ne+1) Node 1 Ne,p Elements Ne Elements x
Fig. 7 The elements and nodes of the piezo-beam from side view of Fig.5a
Eqs.13,15 are next re-represented as sum of those sub-energies corresponding to each element. Expressing these sub-energies in terms of nodal coordinates {ui, wi, wx,i}’s
based on Eq.20and assembling sub-energies, one is able to derive kinetic and potential energies in terms of newly-defined nodal coordinates. Based on the correspondence in the mass, damping and stiffness matrices between assembled energies and bulk energies in a standard dynamic system, one can derive
M €Qþ C _Qþ KQ ¼ F; ð21Þ where Q u½ 1 w1 wx;1 . . . uNeþ1 wNeþ1 wx;Neþ1 T M¼X Ne k¼1 Zs 0 qb AbNTk;uNk;uþ IbNTk;wxNk;wx þ AbNTk;wNk;w h i dx 8 < : 9 = ; þX Ne;p kp¼1 Zs 0 2qp ApNTkp;uNkp;uþ IpNTkp;wxNkp;wx þ ApNTkp;wNkp;w h i dx 8 < : 9 = ;; ð21aÞ K¼X Ne k¼1 Zs 0 Eb AbNTk;uxNk;uxþ IbN T k;wxxNk;wxx 8 < : 9 = ;dx þX Ne;p kp¼1 Zs 0 2Ep ApNTkp;uxNkp;uxþ IpN T kp;wxxNkp;wxx h i 8 < : h 2 31Aphþ hp 2 b33 NTk p;wxxNkp;wxxdx ) ; ð21bÞ F¼ X Ne;p kp¼1 Zs 0 h31 2bpVþ Ap^aP hþ hp 2b33 Nkp;wxx 8 < : 9 = ;dx; ð21cÞ where Ni’s are global versions of Ni
e
’s defined in Eqs.20; i.e., each column vector Ni is an extension of the
corre-sponding Nie to full system dimension with zeros at the
entries not related the ith element. In addition, kp is the
summation index for those elements of piezoelectric ele-ments, while Ne,p is the total number of piezoelectric
ele-ments. In the final step of finite element modeling, boundary conditions (19) at node 1 and (Ne? 1) are applied.
Fur-thermore, the nodes for beam and piezo-patches at the same axial position are assumed identical dynamical states for continuity, leading to the equations of motion in the dimensions of 3(Ne- 1). It is seen from Eqs.21,21cthat
the microscopic hysteresis effect due to polarization P is incorporated in the term of Ap^aP, acting as a modification on
the piezoelectric actuation by the applied voltage V in the level of macroscopic behaviors. This finding enables iden-tification of microscopic polarization P through macro-scopic experimental data of beam responses, as conducted in the following section.
5 Polarization identification
With the system equations of motion established in Eqs.21 and the piezoelectric force term in Eq.21c, the microscopic polarization ‘‘^aP’’ as a function of applied voltage V is identified based on macroscopic experiment data in this section. To this end, a series of pre-designated voltages are applied in a quasi-static fashion to the piezo-electric beam, with the aim not to trigger system dynamics. In this way, the system equations21is well approximated by
KQ¼ F; ð22Þ
where the piezoelectric force F can be carried out based on integration (21c), yielding F¼ cF Vþ 1 2hp^aP Vð Þ ð23Þ where cF is a 3(Ne- 1) column vector with constant
entries resulted from the integration. With Eqs.22and23, the identification on polarization ^aPðVÞ in Eq. (23) can be achieved by applying a series of applied voltage and recording piezo-beam reflections corresponding to the DOFs in Q. For maximum sensitivity, the beam tip vertical deflection; i.e., wx;Neþ1, is selected to collect data for
identifying ^aP Vð Þ in Eq.23.
A simple experimental piezo-cantilever beam system as shown in Fig.5is set up with the piezoceramic patches in model no. of APC-856 (American Piezo Ceramics 1775). The material properties and dimensions of these piezo-creamic patches, and brass are listed in Table 1. A set of voltages corresponding to intersections in Fig.8 are applied to the piezo-beam and their resulted tip vertical displacements are listed in Table2. Based on these data, the interpolation functions in Eqs.8, 9 can be approxi-mated well for those square and triangular areas in Fig.8; thus, identifying the polarization ^aPðVÞ in Eq.21c. Figure9 show resulted errors and corresponding error percentages in predicting polarization based on the iden-tified ^aPðVÞ, where it is seen from subfigure (a) that the errors at the intersections of grids are zeros since the
Table 1 Material properties and dimensions of APC-85
Brass Piezoceramics Young’s modulus E (N/m2) 11 9 1010 5.8 9 1010 Density q (kg/m3) 8,940 7,500 h31(N/Coulomb): b31 (V - m/Coulomb) -4.698 9 108 3.115 9 107 Total length L (mm) 50 50 Width b (mm) 20 20 Thickness t (mm) 0.36 0.24
corresponding predictions of polarization ^aPðVÞ are directly obtained from experiments for interpolations. They are free of interpolation errors. It is also seen from sub-figure (b) that the maximum value of error percentages in polarization is controlled under 3% except for the points near the origin. These errors are not of much concern since the hysteresis effects are not significant as subjected to smaller applied voltages.
6 Control design
A control scheme is synthesized in this section to perform precision positioning of a generic piezo-structure. The design process starts with transforming the system equa-tions (21) to those in modal coordinates (Rao1986). This is accomplished by introducing the modal transformation
Q¼ Ug; ð24Þ
where U is the modal transformation matrix; g are modal coordinates. The modal matrix U is a square matrix whose columns are eigenvectors of the system. Using the modal transformation (24) and performing subsequent standard manipulation, the equations of motion (21) can then be expressed as € giþ 2fixi_giþ x 2 igi¼ u T iF; for i¼ 1; 2; . . .; ð25Þ
where fiis the damping ratio and xiis the natural frequency
of the ith mode. To simplify the ensuing design, the modal equations (25) are truncated to the first n equations and, furthermore, represented in a state-space form as
_x¼ Ax þ B V þ1 2hp^aP Vð Þ ; y¼ Cx; ð26Þ where x¼ g½ 1g_1 gng_n; A¼ 0 1 x2 n 2fnxn . . . 0 1 x2 n 2fnxn 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 ; B¼ 0 uT 1F .. . 0 uT nF 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 ; ð27Þ and C is determined based on the system outputs to be controlled. If the controller is designed to perform preci-sion positioning of the piezo-beam tip, C¼ ^CoU^ where ^U
is responsible for truncated modal transformation and ^
Co¼ 0½ . . . 0 1 0:
It appears in system equations (26) that the control effort; i.e., the applied voltage V, generates a nonlinear term due to hysteresis polarization,1
2hp^aPðVÞ, in addition
to the linear term V. To overcome the hysteresis, a con-troller consisting of two parts is designed as
V ¼ ^V1
2hp^a ^PðVÞ; ð28Þ
where ^a ^PðVÞ cam be obtained from the previously established and identified Preisach model. Note that ^
a ^PðVÞ still deviates slightly from the realistic polarization ^
aPðVÞ, as shown in the last section. It is intended in this study that the second term in control law (28) is responsible for a direct cancellation of the actual hysteresis term, ^aPðVÞ at the microscopic level in Eq.10, while the first term would be designed via H?control theory to overcome the
Fig. 8 Discretized b–a plane for experimental identification
Table 2 Measured piezo-beam tip displacements at sampled inter-sections in Fig.8 60 (volt) 62.012 201.54 331.76 446.48 561.21 685.23 855.76 50 40.308 173.63 303.86 415.48 514.7 648.02 0 40 31.006 155.03 279.05 381.37 489.89 0 0 30 21.704 130.22 244.95 356.57 0 0 0 20 12.402 117.82 229.44 0 0 0 0 10 9.3018 105.42 0 0 0 0 0 0 0 0 0 0 0 0 (lm) 0 ai 0 10 20 30 40 50 60 (volt) bi
errors of microscopic hysteresis cancellation; i.e.,ðPðVÞ ^
PðVÞÞ. In this way, the control effort is then expected to be more effective and energy-economic than designing a direct control scheme without microscopic hysteresis cancellation. The control law in Eq.28 leads to a new form of system equations26as
_x¼ Ax þ B þ DBð Þ ^V; y¼ Cx; ð29Þ where DB¼ B 1 2hp ^ d ; ^d¼ ^a P V ð Þ ^P Vð Þ V^ ð29aÞ With formulation (29), the system uncertainty arising from modeling error of the identified Preisach model, ^d, can be treated as an additive uncertainty to the controlled system. The system plant G in the frequency domain is then a sum of nominal and uncertain parts, namely G0and
DG, respectively, yielding
G¼ G0þ DG; ð30Þ
where the transfer functions
G0¼ C sI Að Þ1B; ð31aÞ
DG¼ C sI Að Þ1DB: ð31bÞ
To apply a standard H? control design, DG is further
modeled by DG¼ WaDa, where Dj aj 1. The transfer
function of Wacan be derived based on measured modeling
error of the identified Preisach model, ^d, and computed DG via Eqs.29aand31b, respectively. The upper bounds of the modeling error DG over all frequencies can well be approximated by Wa¼ 3:0795 sð þ 975:2Þ s2þ 659s þ 1:593 106 sþ 7117 ð Þ sð2þ 62:17s þ 4:164 105Þ : ð32Þ With uncertainty modeled and identified, the system block diagram can be represented by Fig.10, where the exogenous inputs and controlled outputs are regulated by four weighting functions. e is the error signal; ^u¼ V is the controlled signal; n is the noise signal; u is the control input; r is the reference signal; G0is the nominal plant; K is the
controller; Weis used to reflect the requirements on control
objective; Wu does some restrictions on the control or
actuator signals; Wn reflect dynamic characteristics of
noises. A typical design process of H? control design is
initialized by determining the aforementioned weighting functions, which would affect the system sensitivity function, control sensitivity function and complementary sensitivity functions that are used to examine whether or not the originally-set performance specifications are satisfied. The determination of weighting functions for the considered piezo-beam as shown in Fig.5 is stated in the followings. First, to satisfy the common criterion for stability and performance requirement; i.e., Wk eSk1 1, where S is the
sensitivity function, Weis designed of the form
We¼
s=Msþ xb
sþ xbee
; ð33Þ
where Ms and xb are determined by required natural
frequency and damping ratio of the controlled system, Fig. 9 aError in predicted polarizations over the b–a domain, b error
percentage
denoted by xn and f, respectively. The determination follows xb xn= ffiffiffi 2 p ;X¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:5þ 0:5 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ 8f2 q r ;xmax ¼ Xxn; Ms:¼ Sk k1¼ Sðjxj maxÞj ¼ XpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX2þ 4f2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 X2Þ2þ 4f2X2 q :
Second, weighting Wu is designated such that (1/Wu)
successfully bound KSj j over low and high frequency. This is to reflect saturation limit of the piezo-actuator and its bandwidth. In this way, the weighting can be designed of the form
Wu¼
sþ xbc=Mu
eusþ xbc
; ð34Þ
where Muis designated as 60, which is the maximum input
voltage to the piezoelectric beam considered in this study, while the bandwidth frequency of the beam, xbc¼
1; 000ðrad/sÞ; is designated as the controller bandwidth. In the next step, a suitable eu is to be chosen to satisfy
WuKS
k k1 1 by the small gain theorem (Zhou and Doyle
1998), which is set as 0.05. Third, weighting Wnis used to
incorporate sensor noise that is relatively significant in the ranges of high frequencies. Therefore, Wnis determined as
a form of a high-pass filter as
We¼
s sþ xnc
; ð35Þ
where xnc is the cut-off frequency of the sensor. In this
study, where laser interferometers are used to measure the piezo-beam tip motion, simple experiments show that the cuff-off frequency is around 60 Hz. Having determined all transfer functions in the blocks in Fig.10, following the common procedure of H?control design, the last step is to
convert the system blocks in Fig.10into that in Fig.11via linear fractional transformation (LFT) (Zhou and Doyle
1998). Through some simple algebraic manipulations, one can obtain z ^ u e y 2 6 6 4 3 7 7 5¼MðsÞ p n u 2 6 4 3 7 5; whereMðsÞ¼ 0 0 Wa 0 0 Wu We 0 WeG0 I Wn G0 2 6 6 4 3 7 7 5: ð36Þ
With the expression of M(s) in hands, the next step is to design controller K(s) such that the exogenous input to
controlled output transfer function kTzxk1 (Tzx is the
complementary sensitivity function) is minimized. However, it is difficult to find an optimal H? controller
since it is both numerically and theoretically complicated. In practice, a so-called sub-optimal H1 controller (Zhou
and Doyle1998) suit well the control goal of minimizing Tzx
k k1 to some degree. Utilizing the solution procedure based on maximum entropy (Zhou and Doyle1998), a H?
controller of order 15 can be found with the input/output transfer function satisfying kTzxk10:9617. A
model-reduction is further performed to reduce the order of the controller to 7 for easy implementation, yielding
The designed H? controller in Eq.37 is ready to be
implemented to the piezo-beam system, as shown in Fig.5. Note that based on control design theory, the robustness of the controller should be examined after implementation via
WaKS
k k1 1; ð38Þ
for robust stability with additive uncertainty considered (Zhou K, Doyle JC. Essentials of Robust Control, Prentice Hall: New Jersey1998), and
WeS WuKS 1 1; ð39Þ
for robust performance. Figures 12 show comparison of frequency responses between {S, KS, KS} and inverses of
1=We; 1=Wu; 1=Wa
f g; respectively, where it can be seen that the criteria in Eqs.38, 39 are all satisfied with the designed controller in Eq.37.
Fig. 11 LFT representation of the system
K sð Þ ¼ 56:742s
6þ 1:1028e6s5þ 1:2854e9s4þ 1:8534e13s3þ 2:125e15s2þ 7:5876e18s þ 2:371e20
7 Experimental validation
A closed-loop experimental piezo-beam system is set up with the piezoceramic patches in model no. of APC-856 (American Piezo Ceramics 1775). Figure13 shows the experiment framework. Implementation of designed control algorithm is accomplished by the dSPACE module, which provides a 64-bit floating-point processor to minimize the computation errors associated with finite word length. Furthermore, the sampling rate of dSPACE is selected as 20 kHz, which is adequate to suppress the primary distur-bance below 400 Hz due to spindle runout, the secondary around 400–1,000 Hz due to disk flutter vibration (Li and Horowitz 2002; Li et al. 2003; Huang et al. 2006), and minimize negative effects of quantization. The output control signal is amplified by a power amplifier (HAS 4051) to provide enough voltage to derive the piezoelectric beam. The motion of beam tip is measured by a laser displacement sensor (MTI 250, MICROTRAK 7000). The sensor signal is fed back to the dSPACE module for computing control output. Note that the resolution of the laser displacement is ±0.1–0.2 lm. The material properties and dimensions of the piezocreamic patches, and brass are listed in Table1. It is pertinent to note at this point that while applied to hard disk drives with a fixed point processor, one can use available simulators off-line to emulate realistic digital computation environment; e.g., the Fixed Point Blockset, a feature of MATLAB/Simulink, to estimate errors and noise
due to quantization and finite word length, respectively. In this way, the scaling of the controller coefficients in Eq.37
and power/memory resources for computation in the fixed point processor can be optimized before implementation of control design.
In experiments, two types of commands—steps and sinusoids—are applied to examine controller performance. Figure14 shows simulated and experimental responses with the designed controller (28) to perform microscopic hysteresis cancellation and the H? controller in Eq.37to
compensate the error resulted from previous hysteresis cancellation, where it is seen that both are close to each Fig. 12 Validation of robust
performance and stability
other and the controller successfully performs positioning within 0.05 s for a long range of 100 lm. On the other hand, Fig.15show experimental step responses employing the designed H? controller and PI-double-lead
compen-sator for comparison. It is seen from this figure that the H?
controller appears more effective with efficient control
effort (not the large impulsive control effort required by PI-double-lead) as shown in subfigure (b) for achieving similar performance, as shown in subfigure (a), in terms of precision tracking at steady state and small settling time. Also, Fig.16a, b show the steady-state results for sinu-soidal commands in 1 Hz and 10 Hz, respectively. It is seen from subfigure (a) that the hysteresis cancellation with a H?compensator renders much smaller positioning errors
for 1 Hz command than 10 Hz. This is due to the fact that the Preisach model is originally identified based on a quasi-static process. The positioning errors would be magnified as the frequency of the sinusoidal command increases (Ben Mrad and Hu 2002).
It is pertinent to note at this point that the application of the proposed control scheme is not originally limited to hard disk drives; thus, the piezo-beam considered for experimental study as shown in Fig.5 is in larger size, which does not exhibit clear hysteresis effects until driven to 100 lm. With this size, the bandwidth of the considered piezo-actuator is lower (around 100 Hz) than that (above 10 kHz) of the much smaller piezoelectric micro-actuators designed and used for dual-stage control in hard disk drives (Mori et al. 1991; Li and Horowitz 2002; Li et al. 2003; Huang et al. 2006; Oboe et al.1999; Tsai and Yen1999; Tokuyama et al.2001; Lou et al.2002; Gao and Swei2000; Liu et al.2002; Suzuki et al.2003; Tagawa et al.2003). It is the main reason that the experimental sinusoidal com-mands adopted herein are primarily around 1 and 10 Hz and up to 100 lm positioning range to exhibit hysteresis. In addition, a displacement sensor with better resolution and lower noise than the one used herein is also needed to conduct experiments to mimic the environment of hard drives. However, even though with the above limitations on the experimental study, it is for sure that since the techniques of modeling and control design proposed in this study are generic and the sampling rate of the dSPACE microprocessor is up to 20 kHz, the proposed controller can be applied to varied sizes of piezo-actuators, including the micro-sized actuators documented in (Mori et al.1991; Li and Horowitz 2002; Li et al.2003; Huang et al.2006; Oboe et al. 1999; Tsai and Yen 1999; Tokuyama et al.
2001; Lou et al.2002; Gao and Swei2000; Liu et al.2002; Suzuki et al. 2003; Tagawa et al. 2003). Note that the sampling rate of 20 kHz is fast enough to suppress the primary disturbance below 400 Hz due to spindle runouts and the secondary around 400–1,000 Hz due to disk flutter vibration (Li and Horowitz 2002; Li et al. 2003; Huang et al. 2006). Therefore, the same level of performance presented in this section for the larger-sized actuator is also expected for the micro-sized ones in a production hard disk drive.
In addition to the positioning technique of the pro-posed hysteresis cancellation and an H? controller, a
Fig. 14 Simulated and experimental step responses with the micro-scopic hysteresis cancellation assisted by H?compensation
Fig. 15 Experimental step responses employing the designed H? controller and PI-double-lead compensator
conventional, widely-used-in-industry PI and double-lead controller (Lee 2005) is designed based on the nominal plant—without considering the hysteresis term in the sec-ond equation of Eq.4—and tested for comparison. The PI control is chosen to achieve zero steady-state error, while the double-lead to render necessary stability. Note that the aforementioned comparison aims mainly to show effec-tiveness of microscopic hysteresis cancellation, not deter-mining superiority between H? and PI-double-lead
controllers. Figure17 shows resulted experimental results for sinusoidal commands in 1 and 10 Hz. It is seen from
these figures that the hysteresis cancellation with a robust H? compensator renders a much better control
perfor-mance in terms of much smaller errors for 1 Hz sinusoidal command and moderately better performance for 10 Hz command, than the PI and double-lead controller. For a clearer view on hysteresis rejection, the afore-obtained steady state simulation and experimental results are re-represented in the voltage-displacement domain in Fig.18, as loops. It is seen from this figure that the experimental loop with 1 Hz sinusoidal command encloses a very small area and close to those simulated counterpart, Fig. 16 Simulated and experimental tracking results for the cases
with the hysteresis cancellation assisted by H? compensation for sinusoidal commands in frequencies of 1 and 10 Hz
Fig. 17 Simulated and experimental tracking results for the cases with the hysteresis cancellation assisted by H?and PI-double-lead compensations for sinusoidal commands in frequencies of 1 and 10 Hz
while the difference between experimental and simulated is magnified due to imprecision of hysteresis modeling as subjected to faster commands. Note that both simulated loops appear with almost no enclosed area, showing excellent theoretical performance of the hysteresis can-cellation with a robust H? compensator. Finally, Fig. 19
shows the experimental loops for three different cases with (1) the hysteresis cancellation equipped with an H?
con-troller for compensating the resulting error; (2) a conven-tional PI plus double-lead control without considering
hysteresis; (3) no feedback control (open-loop). It is seen from this figure that for those cases in 1 Hz, the controller (1) and (2) renders slightly smaller loop areas than (3), showing the basic effects of the feedback compensation to suppress hysteresis. As the loop command frequency increased to 10 Hz when the hysteresis effects are relative significant, the controller via the hysteresis cancellation and robust H? control leads to much smaller loop areas
than the other two cases, showing that the proposed control scheme indeed improves the performance substantially by canceling the hysteresis effects in the microscopic level.
8 Conclusions
This study proposes a control scheme consisting of microscopic hysteresis cancellation and an accompanied robust H? controller to overcome resulted cancellation
errors. The proposed modeling is novel in the addition of a polarization term into linear constitutive equations of piezoelectric materials. Preisach model is then used to describe the polarization and employ finite element mod-eling afterward. In this way, one is able to predict macro-scopic hysteresis effects of piezoelectric structures in microscopic level. The proposed modeling and control technique is applicable to piezoelectric structures in com-plicated geometry and varied sizes. Simulation and experiment are conducted to validate controller perfor-mance. The conclusive remarks based on the simulations and experimental results are summarized as follows. 1. The microscopic polarization is successfully modeled
by Preisach model and further identified via the built finite element model. In identification, a quasi-static process of voltage application is employed. The obtained prediction errors are within acceptable 3% in percentage.
2. For step commands, simulated and experimental responses with the hysteresis cancellation assisted by a robust H?compensator are close to each other with
non-notable positioning error within 0.05 s, even for a long range of 100 lm.
3. For sinusoidal commands, the designed hysteresis cancellation assisted by a robust H? compensator
renders much smaller positioning errors for 1 Hz command than 10 Hz. This is due to the fact that the Preisach model is originally identified based on a quasi-static process. The positioning errors would be magnified as the frequencies of sinusoidal commands increase.
4. Compared to the conventional PI and double-lead controller without hysteresis cancellation, the pro-posed microscopic hysteresis cancellation and a H?
Fig. 18 Experimental and simulated steady-state voltage-displace-ment loops with the microscopic hysteresis cancellation assisted by robust H?compensation and subjected to sinusoidal commands in frequencies 1 and 10 Hz
Fig. 19 Experimental results for the cases (1) with microscopic hysteresis cancellation and H? compensation design, (2) with a conventional PI and double-lead compensator designed but without hysteresis cancellation, and (3) no feedback control; furthermore, the cases are subjected to sinusoidal commands in frequencies 1 and 10 Hz
compensator renders much better control performance for 10 Hz, showing that the proposed control scheme improves the performance substantially by canceling the hysteresis effects in the microscopic level.
In the future, the identification on polarization would be performed following non-quasi-static processes, i.e. con-sidering dynamic applied voltage, to reflect effects of dynamic hysteresis effects. Furthermore, the proposed control design would be applied to micro-sized actuators in production hard disk drives.
Acknowledgments The authors are indebted to the National Sci-ence Council of ROC for the financial support through the contacts NSC 95-2221-E-009-367, NSC 95-2745-E-033-004-URD, and NSC 97-2221-E-009-057-MY3. The authors are also grateful to National Chip Implementation Center (CIC) of Taiwan for help implement the controllers. This work was supported in part by the National Science Council, Taiwan, on Establishing ‘‘International Research-Intensive Centers of Excellence in Taiwan’’ (IRiCE Project) under Contract NSC 99-2911-I-010-101, and in part by the Aiming for the Top University Plan of National Chiao Tung University, the Ministry of Education, Taiwan, under Contract 99W962.
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