INVERSE FEEDFORWARD CONTROLLER FOR COMPLEX HYSTERETIC NONLINEARITIES IN SMART-MATERIAL SYSTEMS

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1 INVERSE FEEDFORWARD CONTROLLER FOR COMPLEX HYSTERETIC NONLINEARITIES IN SMART-MATERIAL SYSTEMS K. Kuhnen, H. Janocha Laboratory for Process Automaton (LPA), Saarland Unversty, Im Stadtwald, Buldng 13, D Saarbrücken, Germany e-mal: Abstract Undesred complex hysteretc nonlneartes are present to a varyng degree n vrtually all smart materal-based sensors and actuators provded that they are drven wth suffcently hgh ampltudes. In moton and actve vbraton control applcatons, for example, these nonlneartes can excte unwanted dynamcs whch leads n the best case to reduced system performance and n the worst case to unstable system operaton. Ths necesstates the development of purely phenomenologcal models whch characterze these nonlneartes n a way whch s suffcently accurate, amenable to a compensator desgn for actuator lnearzaton and effcent enough for use n real-tme applcatons. To fulfl these demandng requrements the present paper descrbes a new compensator desgn method for nvertble complex hysteretc nonlneartes whch s based on the so-called Prandtl-Ishlnsk hysteress operator. The parameter dentfcaton of ths model 1

2 can be formulated as a quadratc optmzaton problem whch produces the best L 2 2 -norm approxmaton for the measured output-nput data of the real hysteretc nonlnearty. Specal lnear nequalty constrants for the parameters guarantee the unque solvablty of the dentfcaton problem and the nvertablty of the dentfed model. Ths leads to a robustness of the dentfcaton procedure aganst unknown measurement errors, unknown model errors and unknown model orders. The correspondng compensator can be drectly calculated and thus effcently mplemented from the model by analytcal transformaton laws. Fnally the compensator desgn method s used to generate an nverse feedforward controller for the lnearzaton of a magnetostrctve actuator. In comparson to the conventonally controlled magnetostrctve actuator the nonlnearty error of the nverse controlled magnetostrctve actuator s lowered from about 30 % to about 3 %. Key Words Hysteress, Nonlnear Systems, Modelng, Identfcaton, Compensaton 2

3 1. Introducton Complex memory-free nonlneartes or n generalzaton complex hysteretc nonlneartes are present to a varyng degree n vrtually all smart materal-based sensors and actuators provded that they are drven wth suffcently hgh ampltudes. Well-known complex hysteretc nonlneartes are the magnetc nducton - magnetc feld relaton of ferromagnetc materals, the electrcal polarzaton - electrcal feld relaton of ferroelectrc materals and the stress - stran relaton of elasto-plastc materals. The most famlar examples for complex hysteretc nonlneartes n smart materal systems are pezoelectrc, magnetostrctve and shape memoryalloy based actuators and sensors [1]. In many applcatons, these nonlneartes can be lmted through the choce of proper materals and operatng regmes so that lnear sensor and actuator characterstcs can be assumed. In the consequence of more strngent performance requrements a large number of actuators are currently operated n regmes n whch hysteretc nonlneartes are unavodable. In moton and actve vbraton control applcatons, for example, these nonlneartes can excte unwanted dynamcs whch leads n the best case to reduced closed-loop system performance and n the worst case to unstable closed-loop system operaton, see fg _ Lnear feedback controller Hysteretc actuator Lnear plant Lnear sensor fgure 1. Closed-loop control applcaton wth a hysteretc actuator nonlnearty 3

4 Ths necesstates the development of purely phenomenologcal models whch characterze these nonlneartes n a way whch s suffcently accurate, amenable to a compensator desgn for actuator lnearzaton and effcent enough for use n real-tme applcatons. Models of hysteretc nonlneartes have evolved from two dfferent branches of physcs: ferromagnetsm and plastcty theory. The roots of both branches go back to the end of the 19th century. But only at the begnnng of the 1970s was a mathematcal formalsm for a systematc consderaton of hysteretc nonlneartes developed [2]. The core of ths theory s formed by socalled hysteress operators whch descrbe hysteretc transducers as a mappng between functon spaces. But t s only snce the begnnng of the 1990s that engneers employ ths theory on a larger scale to develop modern strateges for the lnearzaton of hysteretc nonlneartes wth an nverse feedforward controller shown n fg 2. + Lnear Inverse feedback feedforward _ controller controller Hysteretc actuator Lnear plant Actuator lnearzaton Lnear sensor fgure 2. Lnearzaton of a hysteretc actuator by an nverse feedforward control strategy Ths type of controller s based on the compensator W -1 of the hysteretc nonlnearty W whch s defned as the operator fulfllng 1 WW [ [ x]]( t) = Ix [ ]( t). (1) In (1) I s the dentty operator whch descrbes the dealzed transfer characterstc 4

5 Ix [ ]( t) = xt ( ). (2) There are many nonlnear control systems wth hysteress operatng n practce successfully, many of whch have been desgned usng technques as the descrbng functons. But the man drawback of these solutons s the lmtaton to harmonc nput sgnals [3]. In contrast to ths, the lnearzaton of hysteretc nonlneartes W wth an nverse feedforward controller based on ts compensator W -1 s not restrcted to a specal nput functon and thus much more general n ts applcablty. The nverse feedforward control approach for the lnearzaton of hysteretc nonlneartes can be dvded nto two classes. In the frst class the underlyng hysteretc nonlnearty has a local respectvely non-complex memory structure whch means that the present value of the output s only dependent on the present value and the last extremum value of the nput. But nearly all hysteretc nonlneartes whch occur n smart materal based sensor and actuator characterstcs have a non-local respectvely complex memory structure and thus belong to the second class. In ths case the present value of the output s not only dependent on the present value of the nput, but also on more than one extremum value of the nput n the past. In the begnnng manly the well-known Presach operator was used for the modelng and lnearzaton of complex hysteretc nonlneartes occurng n sold-state actuators wth the nverse feedforward control approach [4,5]. But the man drawbacks of the Presach operator are the strong senstvty of the dentfcaton procedure aganst output-nput data and unknown model errors and the fact that n general the compensator of the Presach operator has to be calculated numercally. Recent papers also reference the so-called Prandtl-Ishlnsk operator [6,7,8] whch belongs to an mportant subclass of the Presach operator [9]. The man advantages of ths approach are the reduced model complexty of the Prandtl-Ishlnsk operator n comparson wth the Presach operator and the fact that the compensator of an nvertble Prandtl-Ishlnsk operator 5

6 can be calculated analytcally. Ths allows an effcent mplementaton of the compensator for real-tme applcatons. Developng a consstent phenomenologcal desgn concept for a compensator W -1 of an nvertble complex hysteretc nonlnearty W whch s suffcently flexble n ts modelng capabltes and moreover well suted for real-tme applcatons s n general not a smple task because t covers the followng coupled desgn steps: modelng the real hysteretc nonlnearty, dentfcaton of the model parameters to adapt the model to the real hysteretc nonlnearty and nverson of the model to obtan the desred compensator. Especally the mathematcal complexty of the dentfcaton and nverson problem depends on the phenomenologcal modelng method (for example Presach or Prandtl-Ishlnsk modelng) and nfluences strongly the practcal use of the desgn concept. Another dffculty of the dentfcaton problem follows from the strong senstvty of the model parameters to unknown measurement errors of the output-nput data, unknown model errors and unknown model orders. Due to these effects a parameter dentfcaton can result n the best case to a poor model accuracy or n the worst case to a locally non-nvertble model, and as a consequence the whole compensator desgn fals. Therefore the robustness aganst these effects s an nherent requrement for a consstent phenomenologcal compensator desgn method. To overcome these dffcultes the present paper descrbes a new compensator desgn concept for complex hysteretc nonlneartes based on the Prandtl-Ishlnsk modelng approach whch s robust n the sense mentoned above. The robustness of the new compensator desgn method s reached by the consderaton of lnear nequalty constrants for the free model parameters whch guarantee a search for the best L 2 2 -norm approxmaton of the measured output-nput data only n those parameter ranges where the dentfed model s nvertble. 6

7 2 Hysteress defnton and observable characterstcs What s a scalar hysteretc nonlnearty? To answer ths queston, let us consder a system W,.e. a mappng yt () = Wx []() t (3) for all t [t 0,t E ] wth a scalar nput sgnal x and a scalar output sgnal y, each of whch s contnuously dependent upon the tme t. The transformaton s causal meanng that for a gven set of startng condtons x ( τ) = x ( τ) für τ [ t, t] W[ x ]( t) = W[ x ]( t). (4) The tme-dependent y-x trajectory of the system can be smply graphcally represented by assgnng a pont n the y-x plane to each (x(t),y(t)) par, see fg.3. Let us assume that the nput sgnal x ncreases startng from small values and gong through the values x 1 and x 2 and beyond. Then the par (x(t),y(t)), and consequently the tme-dependent y-x trajectory, follows a path P 1 n the y-x plane. If the nput sgnal changes drecton fallng from large values through x 2 on to x 1 and below, then the par (x(t),y(t)), and consequently the tme-dependent y-x trajectory, follows path P 2. If the nput parameter changes drecton wthn the nterval x 1 < x(t) < x 2, then the par (x(t),y(t)), and consequently the tme-dependent y-x trajectory follows path P 3 nto the so-called hysteretc regon Ω, whch s bounded by the major loop descrbed by paths P 1 and P 2. The major loop need not be closed. The formaton of closed hysteress loops s often observed n practce but s not demanded by the strctly mathematcal defnton of hysteretc nonlnearty [9,10,11]. An observable, nherent feature of hysteretc nonlnearty s, however, the branchng of the tme-dependent y-x trajectory upon reversng the drecton of the nput sgnal. Ths branchng results from the memory effect nherent of hysteretc nonlnearty. As a result of ths effect, the momentary value of the output sgnal depends not only on the momentary value of the nput 7

8 sgnal but also on prevous nput sgnal values as well as the ntal values stored n "memory". Ths means that dfferent nput sgnal hstores and dfferent ntal states wll lead to dfferent paths n the y-x plane. y P 2 P 3 x 1 x 3 x 2 x P 1 Ω fgure 3. Features of hysteretc behavour The branchng characterstc n the y-x trajectory s n tself nsuffcent to ndcate hysteretc nonlnearty snce essentally all causal transfer functons wth memory exhbt branchng behavour. The decsve characterstc dfferentatng hysteretc nonlneartes from all other systems wth memory s ts ndependence of the speed of change of the nput sgnal (ratendependence property). Ths means that the branchng behavour n the y-x plane does not depend on the rate of change of the nput sgnal but only on the order of the nput sgnal ampltudes. Mathematcally, the rate ndependence of the transfer functon can be expressed by the fundamental expresson of the causal operator W Wx [ η]( t) = Wx [ ]( η( t)) (5) 8

9 for all t [t 0,t E ], whereby the propertes η(t 0 ) = t 0 and η(t E ) = t E must apply n order to fulfl the contnuous and monotonous tme transformaton η : [t 0,t E ] [t 0,t E ] [9]. Ths results n Defnton 1: (scalar hysteretc nonlnearty) A scalar hysteretc nonlnearty W, or a scalar hysteress operator W, s a mappng that unquely assgns a scalar tme-dependent nput sgnal x to a scalar tme-dependent output sgnal y and that exhbts the propertes of causalty (4) and rate ndependence (5). The set of all scalar hysteretc nonlneartes can be dvded nto two dfferent classes, namely the class of hysteretc nonlneartes wth local memory and the class of hysteretc nonlneartes wth non-local memory. Hysteretc nonlneartes wth local memory are characterzed by the fact that the value pars (x 0 = x(t 0 ), y 0 = y(t 0 )) Ω and the values of the nput sgnal x(t) for t > t 0 unquely determne the tme dependence of the output sgnal y(t) for t > t 0. Ths means that n hysteretc nonlneartes wth local memory the nfluence of past nput sgnal ampltudes on the future tme dependence of the output sgnal s taken nto consderaton n the momentary value of the par (x(t),y(t)) Ω. The state of hysteretc nonlnear systems wth local memory s therefore gven n the par (x(t),y(t)) Ω and can be represented graphcally n the hysteretc regon Ω by a pont. Therefore, hysteretc nonlneartes wth local memory do not exhbt a real nternal state and can therefore be represented mathematcally n the form yt () = Wxx [, 0, y0 ]() t (6) wth the ntal condtons (x 0,y 0 ) Ω for all t [t 0,t E ]. The ntal system state (x 0,y 0 ) must le n the hysteretc regon Ω. Snce the nput sgnal s ndependent, the assumpton x 0 = x(t 0 ) s always 9

10 fulflled. Ths means that the hysteretc nonlnearty wth local memory can alternatvely be characterzed by the fact that the ntal value of the output parameter (y 0 = y(t 0 )) Σ(x 0 ) = ({x 0 } R) Ω and the values of the nput sgnal x(t) for t t 0 unquely determne the tme dependence of the output sgnal y(t) for t > t 0. In ths way, the nfluence of past nput sgnal ampltudes on the future tme dependence of the output sgnal s ncluded n the momentary value of the output value y(t) Σ(x(t)) = ({x(t)} R) Ω. In ths case, (6) can also be represented by yt () = Wx [, y0 ]() t (7) wth the ntal condtons y 0 Σ(x 0 ) for all t [t 0,t E ]. Σ(x 0 ) = ({x 0 } R) Ω descrbes the nterval llustrated n fg. 4, whch can be ether open, half open or closed dependng on the formaton of the hysteretc regon. Snce the value of the output sgnal y(t 0 ) = W[x, y 0 ](t 0 ) at tme t 0 accordng to (7) s determned by the ntal value of the nput sgnal x(t 0 ) = x 0 and the ndependent ntal value of the output sgnal y 0, then y(t 0 ) = y 0 Σ(x 0 ) and therefore (x 0 = x(t 0 ), y 0 = y(t 0 )) Ω cannot be requred up front. Should y 0 Σ(x 0 ) and (x 0, y 0 ) Ω then the equaton upon whch operators (6) and (7) are based must be augmented by a consstency condton, whch mplements a projecton of y(t 0 ) nto the nterval Σ(x 0 ), see fg. 4. Hysteretc nonlneartes wth non-local memory are characterzed by the fact that not only the ntal value of the output sgnal y 0 = y(t 0 ) Σ(x 0 ) and the values of the nput sgnal x(t) for t t 0 are requred to unquely determne the tme dependence of the output sgnal y(t) for t > t 0, but that also the values of the nput sgnal for tmes t < t 0 nfluence the output sgnal y(t) for t > t 0. Ths nfluence s taken nto consderaton n hysteretc nonlneartes wth non-local memory by the ntal value z 0 of a real nternal state, whch, dependng on the power of the memory, can be of a hgher dmenson or even of nfnte dmenson. 10

11 y y(t 0 ) Σ(x 0 ) y(t 0 ) = W[x,y 0 ](t 0 ) (x(t 0 ),y(t 0 )) Ω x(t 0 ) = x 0 x y 0 (x 0,y 0 ) Ω Ω fgure 4. Consstency condton for ntal values In the case of a fnte dmenson the memory s descrbed by a state vector, n the nfnte dmenson case by a state functon. In the latter case, one speaks of hysteretc nonlneartes wth global memory. Hysteretc nonlneartes wth non-local memory can be descrbed mathematcally by the output-nput mappng yt () = Wxz [, 0 ]() t (8) wth the ntal condton z 0 Σ(x 0 ) for all t [t 0,t E ]. Here, the applcable regon of the hysteretc state s denoted by a hgher dmensonal or nfntely dmensonal hysteretc regon Σ. Addtonally to the rate-ndependent branchng of the output-nput trajectory nearly all smart materal based hysteretc nonlneartes show further mportant characterstcs. At the begnnng of the 20th century Madelung nvestgated expermentally the branchngs and loopngs of ferromagnetc hysteress and stated the followng three rules from hs observatons [8], see fg

12 y C Mnor hysteretc loop A C 2 D B x C 1 Ω Major hysteretc loop fgure 5. Complex hysteretc nonlnearty 1. Any curve C 1 emanatng from a turnng pont A of the output-nput trajectory s unquely determned by the coordnates of A. 2. If any pont B on the curve C 1 becomes a new turnng pont, then the curve C 2 orgnatng at B leads back to the pont A. 3. If the curve C 2 s contnued beyond the pont A, then t concdes wth the contnuaton of the curve C whch led to the pont A before the C 1 -C 2 cycle was traversed. In addton to these three Madelung rules a fourth mportant observaton can be made for ferromagnetc, ferroelectrc, elasto-plastc materals and actuator and sensor characterstcs of smart materals. 4. More than one curve can be pass though a non-turnng pont D wthn the hysteretc regon Ω, see fg. 5. The branch whch was traversed s unquely determned by the relevant past hstory of the nput sgnal. 12

13 It s ths so-called crossng-loop property of real hysteretc nonlneartes n whch the complex or non-local ones dffer from the non-complex or local ones. Ths property can be understood as a generalzaton of the frst Madelung rule to complex hysteretc nonlneartes. 3 Hysteress modelng, compensaton and dentfcaton Because of ts phenomenologcal character the concept of hysteress operators allows a powerful modelng of complex hysteretc nonlneartes wthout takng nto account the underlyng physcs [2]. The basc dea conssts of the modelng of the real complex hysteretc nonlneartes by the weghted superposton of many so-called elementary hysteress operators. Elementary hysteress operators are non-complex hysteretc nonlneartes wth a smple mathematcal structure whch are characterzed by one or more parameters. One of the most famlar and most mportant elementary hysteretc mappng between the nput sgnal x and the output sgnal y s the so called play or backlash operator + yt ( ) = H[ x, y]( t) r R ; y R ; t [ t, t ] (9) r whch s often used to model mechancal play n gears wth one degree of freedom. It s normally defned by the recursve equaton yt ()=max{() xt r,mn{() xt + ryt, ( )}} (10) wth the ntal consstency condton yt ( 0)= max{ xt ( 0) r,mn{ xt ( 0) + r, y0}} (11) for the output sgnal at ntal tme t 0 for pecewse monotonous nput sgnals wth a monotoncty partton t 0 t 1.. t t t +1.. t N = t E [9]. The play operator depends on the ndependent ntal value y 0 R of the output and s characterzed by ts threshold parameter r R + 0. Fg. 6 E 13

14 shows the hysteretc regon Ω whch s a strp lne n R 2 and the rate-ndependent output-nput trajectory of ths elementary hysteress operator. Although the three Madelung rules hold for the play operator t can be easly realzed that the ferromagnetc, ferroelectrc or elastc-plastc behavour of real materals and the hysteretc actuator and sensor characterstcs of real smart materals are of much hgher complexty, note also rule 4. y Ω x 0 R -r r 0 y 0 Σ(x 0 ) x y(t 0 ) Σ(x 0 ) Σ(x 0 ) fgure 6. Rate-ndependent characterstc of the play operator H r To obtan a more powerful model for complex hysteretc nonlneartes we ntroduce the socalled Prandtl-Ishlnsk hysteress operator H by the lnear weghted superposton of many play operators wth dfferent threshold values. From ths follows T Hx [ ]( t) : = w Hr [ x, z0 ]( t) (12) wth the vector of weghts w T = (w 0 w 1.. w n ), the vector of thresholds r T = (r 0 r 1.. r n ) wth 0 = r 0 < r 1 <.. < r n < +, the vector of the ntal states z 0 T = (z 00 z 01.. z 0n ) of the play operators and the vector of the play operators 14

15 T Hr[, x z0]() t = ( H [, x z00]() t.. H [, x z0 ]() t ). r0 rn n The hysteretc characterstc of the Prandtl-Ishlnsk hysteress operator s completely defned by the characterstc of the so-called ntal loadng curve. Ths specal branch wll be traversed f the ntal state of the Prandtl-Ishlnsk hysteress operator s zero and t s drven wth a monotonous ncreasng nput sgnal. The ntal loadng curve can be fully characterzed by and therefore equated wth a threshold-dependent pecewse lnear functon ϕ( r) = w ( r r ) ; r r < r 1 ; = 0.. n, (13) j = 0 wth r n+1 = and j j + d ϕ( r) = wj ; r r < r+ 1 ; = 0.. n. (14) dr j = 0 It s called the generator functon of the Prandtl-Ishlnsk hysteress operator [12], see fg. 7 for a Prandtl-Ishlnsk hysteress operator wth a model order of n = 4. y,ϕ H Intal loadng curve and generator functon ϕ(r) x,r n = 4 fgure 7. Intal loadng curve and generator functon ϕ (r) 15

16 Under the consderaton of the lnear nequalty constrants U w u 0 (15) for the weghts wth the matrx ε ε U =...., the vector u = ε and a possbly nfnte small number ε > 0 the generator functon s strongly monotonous for r 0 and therefore the nverse of the generator functon ϕ -1 (r) exsts unquely for r 0. ϕ -1 (r) s pecewse lnear and strongly monotonous and can therefore also be regarded as a generator functon ϕ ( r ) = w ( r r ) ; r r < r 1 ; = 0.. n, (16) j = 0 j j of a Prandtl-Ishlnsk hysteress operator wth r n+1 = and + d ϕ( r ) = w j ; r r < r + 1 ; = 0.. n, (17) dr j = 0 namely the nverse Prandtl-Ishlnsk hysteress operator 1 T H [ y]( t) : = w Hr [ y, z0 ]( t) (18) wth transformed ntal states z 0, threshold values r and weghts w. In ths case the weghts fulfl the same lnear nequaltty constrants U w u 0. (19) The transformaton law r = Ω (r,w) for the thresholds results from the relaton r = ϕ ( r ). From ths follows 16

17 r = w ( r r ) ; = 0.. n (20) j = 0 j for the threshold-dscrete case, see fg. 8. j ϕ,ϕ ϕ(r) ϕ(r ) ϕ (r ) ϕ (r ) r r r,r fgure 8. Generator functons ϕ (r) and ϕ (r ) The transformaton law w = Ξ(w) for the weghts results from the relaton d ϕ( r ) dr = 1( d ϕ( r) dr), see fg. 8. From ths follows w = 0 1 w 0 w and w = 1 ; = 1.. n. (21) ( w + w )( w + w ) 0 j 0 j= 1 j= 1 j The transformaton law z 0 = Ψ(z 0,w) for the ntal states results from the relaton ( z z ) ( r r ) = ( z z ) ( r r) whch s the threshold-dscrete counterpart to the relaton d( zr )dr = d()d zr rfor the thresholdcontnuous case dscussed n [12]. From ths follows the transformaton law z = w z + w z ; = 0.. n (22) 0 j 0 j 0 j j = 0 j= + 1 n 17

18 for the ntal states. The Prandtl-Ishlnsk hysteress operator has the followng more or less obvous propertes: 1. Because the Madelung rules persst under lnear superposton, they hold also for the Prandtl- Ishlnsk hysteress operator. Moreover due to the n > 1 nner hysteretc state varables dfferent branches can be traversed from a non-turnng pont D whch s n agreement wth rule The closed loops whch wll be traversed for nput sgnals oscllatng between maxmum and mnmum values have an even symmetry to the center pont of the correspondng loop. Ths even symmetry property s a property of the play operator and perssts also under lnear superposton. 3. The nverson operaton whch s gven by the transformaton laws does not change the structure of the Prandtl-Ishlnsk hysteress operator and ts nequalty constrants for the weghts. Property 1 agrees at least qualtatvely wth expermental observatons for complex hysteretc nonlneartes. Property 3 leads to a drect formulaton and thus to a very effcent mplementaton of the correspondng compensator whch s feasble for real-tme control applcatons. The amount of calculaton for the Prandtl-Ishlnsk operator H and ts compensator H -1 n dependence of the model order n s shown n table 1. Table 1 Amount of calculaton for the Prandtl-Ishlnsk operator H and ts compensator H -1 operator addton multplcaton comparson H 2(n+1)+n n + 1 2(n+1) H -1 2(n+1)+n n + 1 2(n+1) 18

19 The even symmetry property 2 whch s an nherent model characterstc s the man drawback of ths Prandtl-Ishlnsk modelng approach n comparson wth the Presach hysteress modelng approach. But n many practcal cases ths property s often fulflled. Well-known examples are pezoelectrc and magnetostrctve actuators drven n operatng regmes wth moderate nput ampltudes. The dentfcaton procedure whch s used to obtan the compensator of the real hysteretc nonlnearty s dvded nto three parts. In the frst part the thresholds r and the ntal states z 0 of the Prandtl-Ishlnsk hysteress operator are determned by the formulas r = max{ xt ( ) } ; = 0.. n (23) n + 1 t0 t t e and z0 = 0 ; = 0.. n. (24) (24) assumes the start of the hysteretc state evoluton from the so-called demagnetzed state. The dentfcaton of the weghts w of the Prandtl-Ishlnsk hysteress operator whch s the object of the second part can be formulated as an L 2 2 -norm mnmzaton of the so-called output error model T Ex [, y ]( w,t) : = w H [ x, z0 ]( t) yt ( ) (25) r whch s lnearly dependent on the weghts. Ths leads to the quadratc optmzaton problem T T T 2 mn { w Hr[ x, z ]( t) Hr[ x, z ]]( t) d t w y( t) Hr[ x, z ]]( t) d t w+ y( t) d t} n R (26) w te te te t0 t0 t0 wth the lnear nequalty constrants (15) U w u 0 19

20 whch has one global soluton and whch ensures the nvertablty of the dentfed Prandtl- Ishlnsk hysteress operator. Ths guarantees a unque best L 2 2 -norm approxmaton of the measured hysteretc characterstc n that space of the weghts whch leads to an nvertble Prandtl-Ishlnsk hysteress operator. Step 1 Threshold and ntal state dstrbuton r = x ; z0 = ; =.. n n Error model Exy [, ]( w, t) = w T H [ x, z ]( t) yt ( ) r 0 Step 2 Quadratc program mn Ex [, y]( w) n w R wth U w u 0 2 (r, w, z 0 ) Hysteress model T Hx [ ]( t) = w H [ x, z ]( t) r 0 Step 3 Transformaton laws w = Φ (w) r = Ψ(w,r) z 0 = Θ(w,z 0 ) (r, w, z 0 ) Compensator 1 T H [ y]( t) = w H [ y, z ]( t) r 0 fgure 9. Compensator desgn procedure 20

21 Therefore the nvertablty of the Prandtl-Ishlnsk hysteress operator and ts nverse s always guaranteed durng the optmzaton and thus the desgn process for the model and the correspondng compensator s consstent and robust aganst unknown measurement errors of nput-output data, unknown model errors and unknown model orders. The parameters of the compensator result n the thrd step by applyng the transformaton equatons (20-22) to obtan the threshold, weghts and ntal states of the nverse Prandtl-Ishlnsk operator. Fg. 9 shows the whole compensator desgn procedure based on the Prandtl-Ishlnsk approach. 4 Results In ths secton the performance of the presented compensator desgn method for complex hysteretc nonlneartes wll now be demonstrated by means of the dsplacement-current relaton of a magnetostrctve transducer, see fg ,5 µm 4,5 D C s 1,5-1,5 A C 2 C 1 B -4,5 Ω -7,5-0,5-0,3-0,1 0 0,1 0,3 A 0,5 I fgure 10. Measured complex hysteretc dsplacement-current relaton n the operatng range To get a strongly monotonous relaton between the dsplacement and current whch s fundamental for actuator operaton, the magnetostrctve transducer s used wth an addtonal 21

22 bas current. The bas current whch amounts to 1 A n ths example determnes the operatng pont of the magnetostrctve actuator. Ths operatng pont concdes wth the orgn of the s-iplane n fg. 10 whch shows the strongly monotonous hysteretc dsplacement-current relaton n the moderate sgnal operatng range of the magnetostrctve actuator. It s manly characterzed by strongly monotonous branches and nearly symmetrcal hysteretc loops wth a counterclockwse orentaton. Addtonally the real hysteretc characterstc fulfls the Madelung rules and exhbts the crossng loop property. Therefore the modelng, dentfcaton and compensaton of ths real complex hysteretc nonlnearty can be realzed wth the Prandtl-Ishlnsk approach. 7,5 µm 4,5 Measured characterstc 7,5 µm 4,5 Model order n = n = 0 1,5 1,5 s -1,5 H[I] -1,5-4,5-4,5-7,5-0,5-0,3-0,1 0 0,1 0,3 A 0,5 I -7,5-0,5-0,3-0,1 0 0,1 0,3 A 0,5 I fgure 11. Measured and modeled hysteretc dsplacement-current relaton The model order n = 0 leads to a lnear rate-ndependent operator model and thus the dentfcaton procedure determnes the best lnear L 2 2 -norm approxmaton of the real hysteretc nonlnearty, see fg. 11. The nonlnearty error defned by max{ HI []() t st ()} t0 t te max{ HI [ ]( t)} t0 t te (27) 22

23 amounts n ths case up to 29.8 %. Fg. 11 shows the loopng and branchng behavour of the real complex hysteretc characterstc of the magnetostrctve actuator and the Prandtl-Ishlnsk hysteress operator wth a model order of n = 14 as a result of the dentfcaton procedure. The nonlnearty error amounts n ths case to 3.0 % whch s nearly ten tmes smaller as for the best lnear L 2 2 -norm approxmaton. Due to unknown model errors a further ncreasng of the model order doesn t mprove the nonlnearty error n ths case. For the compensaton of the real hysteretc nonlnearty a nverse feedforward controller s used whch s based on the nverse Prandtl-Ishlnsk hysteress operator, see fg. 12. s c (t) s the gven dsplacement sgnal value. 0,5 A 0,3 I 0,3-0,3-0,3-0,5-7,5-5,0-2,5 0 2,5 5,0 µm7,5 s c fgure 12. Inverted hysteretc dsplacement-current relaton The nverse Prandtl-Ishlnsk hysteress operator s obtaned from the Prandtl-Ishlnsk hysteress operator usng the transformaton laws for the thresholds (20), the weghts (21) and the ntal states (22). It s realzed by a dgtal sgnal processor wth a samplng rate of up to 10 khz and a dsplacement controlled current source. The loopng and branchng characterstc of the nverse Prandtl-Ishlnsk hysteress operator s shown n fg

24 As a fnal result fg. 13 shows the compensated characterstc of the overall system gven by the seral combnaton of the nverse feedforward controller and the magnetostrctve actuator. 7,5 µm 4,5 s 1,5-1,5-4,5-7,5-7,5-5,0-2,5 0 2,5 5,0 µm7,5 s c fgure 13. Compensated hysteretc dsplacement-gven dsplacement relaton In ths example the control error defned by max{ s ( t) s( t)} t0 t te c max{ s ( t)} t0 t te c (28) wll be strongly reduced to about 3 % due to the nverse feedforward control strategy. 5 Conclusons The man contrbuton of ths paper s to extend the Prandtl-Ishlnsk modelng approach for complex hysteretc nonlneartes to a robust compensator desgn method for nvertble complex hysteretc nonlneartes of the Prandtl-Ishlnsk type. For ths purpose the threshold-dscrete verson of the Prandtl-Ishlnsk hysteress operator was formulated wth lnear nequalty constrants for the model parameters whch guarantee the nvertablty of the model. Based on these lnear nequalty constrants and an error model whch s lnearly dependent on the model 24

25 parameters the dentfcaton problem can be formulated as a quadratc program whch provdes always the best nvertble L 2 2 -norm approxmaton of the measured output-nput data. The correspondng compensator can be drectly calculated and thus effcently mplemented from the model by analytcal transformaton laws. Fnally the compensator desgn method s used to generate an nverse feedforward controller for a magnetostrctve actuator. In comparson to the conventonally controlled magnetostrctve actuator the nonlnearty error of the nversely controlled magnetostrctve actuator s lowered from about 30 % to about 3 %. In future works the method wll be extended to hysteress operators whch are also able to model complex hysteretc nonlneartes wth asymmetrcal hysteretc loops. These type of nonlneartes occur f magnetostrctve or pezoelectrc actuators are drven wth hgher nput ampltudes. 6 Acknowledgements The results of ths work are obtaned wthn the European project "Magnetostrctve Equpment and Systems for more electrc Arcraft" (MESA) and the German project "Investgatons of Extended Presach Models for sold-state Actuators". The authors thank the European Communty and the Deutsche Forschungsgemenschaft (DFG) for the fnancal support of ths work and Dr. Pavel Krejc from the Weerstraß-Insttute for Appled Analyss and Stochastcs (WIAS) n Berln for hs openness for dscusson to the mathematcal aspects of ths work. References [1] H.T. Banks, R.C. Smth, Hysteress Modelng n Smart Materal Systems, Journal of Appled Mechancs and Engneerng, 5(1), 2000,

26 [2] M.A. Krasnosel'sk, A.V. Pokrovsk, Systems wth hysteress (Berln: Sprnger, 1989). [3] O. Föllnger, Nchtlneare Regelungen I (München: Oldenbourg, 1989). [4] P. Ge, M. Jouaneh, Generalzed Presach Model for Hysteress Nonlnearty of Pezoceramc Actuators, Journal of Precson Engneerng, 20, 1997, [5] J. Schäfer, H. Janocha, Compensaton of Hysteress n sold-state Actuators, Sensors and Actuators, Physcal A, 49, 1995, [6] M. Dmmler, U. Holmberg, R. Longchamp, Hysteress Compensaton of Pezo Actuators, Proceedngs of the European Control Conference, Karlsruhe, 1999, Rubcom GmbH (CD- ROM). [7] M. Goldfarb and N. Celanovc, Modelng Pezoelectrc Stack Actuators for Control of Mcromanpulaton, IEEE Control Systems, 1997, [8] K. Kuhnen and H. Janocha, Adaptve Inverse Control of Pezoelectrc Actuators wth Hysteress Operators, Proceedngs of the European Control Conference, Karlsruhe, 1999, Rubcom GmbH (CD-ROM). [9] M. Brokate and J. Sprekels, Hysteress and phase transtons (Berln, Hedelberg, NewYork: Sprnger, 1996). [10] I.D. Mayergoyz, Mathematcal Models of Hysteress (New York: Sprnger 1991). [11] A. Vsntn, Dfferental Models of Hysteress (Berln, Hedelberg, New York: Sprnger, 1994). [12] P. Krejc, Hysteress, Convexty and Dsspaton n Hyperbolc Equatons (Tokyo: Gakuto Int. Seres Math. Sc. & Appl. vol. 8., Gakkotosho, 1996). 26

27 Klaus Kuhnen, born n 1967, receved the Dpl.-Ing. degree n electrcal engneerng at the Unversty of Saarland n Followng graduaton, he has been workng there as a scentfc collaborator at the Laboratory for Process Automaton (LPA) n the felds of sold-state actuators and control of systems wth hysteress and creep. Hs research nterests are n the control of systems wth hysteress and creep, adaptve control theory and nonlnear sgnal processng. Hartmut Janocha, born n 1944, studed electrcal engneerng at the Unversty of Hanover. Snce 1989, he has been head of the Laboratory for Process Automaton (LPA) at the Unversty of Saarland n Saarbrücken, Germany. Here, the man felds of work are new actuators wth system and sgnal-processng concepts, calbraton methods for mprovng the postonng accuracy of ndustral robots and measurement of 3D- geometry usng CCD vdeo cameras. 27