Why Are Some Hysteresis Loops Shaped Like a Butterfly?

Transcription

1 Wh Are Some Hsteresis Loops Shaped Like a Butterfl? Bojana Drinčić a, Xiaobo Tan b, Dennis S. Bernstein a a Department of Aerospace Engineering, The Universit of Michigan, Ann Arbor, MI b Department of Electrical and Computer Engineering, Michigan State Universit, East Lansing, MI 4884 Abstract The contribution of this paper is a framework for relating butterfl-shaped hsteresis maps to simple (single-looped) hsteresis maps, which are tpicall easier to model and more amenable to control design. In particular, a unimodal mapping is used to transform simple loops to butterfl loops. For the practicall important class of piecewise monotone hsteresis maps, we provide conditions for producing butterfl-shaped maps and eamine the properties of the resulting butterflies. Conversel, we present conditions under which butterfl-shaped maps can be converted to simple piecewise monotone hsteresis maps to facilitate hsteresis compensation and control design. Eamples of a preloaded two-bar linkage mechanism and a magnetostrictive actuator illustrate the theor and its utilit for understanding, modeling, and controlling sstems with butterfl-shaped hsteresis. Ke words: Hsteresis; butterfl hsteresis; unimodal transformation; Prandtl-Ishlinskii model. Introduction Hsteresis is a propert of man nonlinear sstems where the output-versus-input graph forms a nontrivial loop in the stead state, when the input varies periodicall at an asmptoticall low freuenc (i.e., uasi-staticall). Here a nontrivial loop means a loop with nonvanishing interior. In this paper, we refer to the aforementioned input-output loop as the hsteresis map of the sstem. The underling mechanism that gives rise to hsteresis is multistabilit, which refers to the eistence of multiple attracting euilibria. Under uasi-static ecitation, the state of the sstem is attracted to different euilibria depending on the direction of the input (Bernstein, 7). Hsteretic sstems arise in a vast range of applications, such as ferromagnetics, smart materials, biological sstems, and aerodnamics (Tan & Ier, 9; Cross et al., 9; Leang et al., 9; Ier & Tan, 9; Oh et al., 9). Modeling and control of hsteresis is an area of significant interest to the controls communit (Tan & Baras, 4; Wen & Zhou, 7; Chen et al., 9; Wang This paper was not presented at an IFAC meeting. Corresponding author Bojana Drinčić. Tel addresses: bojanad@umich.edu (Bojana Drinčić), btan@egr.msu.edu (Xiaobo Tan), dsbaero@umich.edu (Dennis S. Bernstein). & Su, 6). In some applications (e.g., a thermostat), the input-output map is independent of the freuenc of ecitation, and thus identical to the hsteresis map. In most applications, however, the dnamical sstem response depends on the freuenc of ecitation, and thus the input-output map is freuenc-dependent and approaches the hsteresis map onl as the freuenc of ecitation approaches zero. For details, see Oh & Bernstein (5a). In the present paper, we consider the sstem operating at asmptoticall low freuenc, ignore the transient response, and focus onl on the hsteresis map, that is, on the periodic stead-state response under a uasi-static input. The present paper focuses on butterfl-shaped hsteresis maps, which arise in man applications, such as mechanics, optics, and smart materials (Sahota, 4; Davi, ; Li & Weng, ; Ebine & Ara, 999). A hsteresis map is a butterfl when it consists of two loops of opposite orientation. In some applications, the shape of the hsteresis map is reminiscent of butterfl wings, which eplains the terminolog. The contribution of this paper is a framework that relates butterfl-shaped hsteresis maps to simple (singleloop) hsteresis maps, which are easier to model and more amenable to control design. In particular, unimodal mappings are used to transform simple loops to butterfl loops. For piecewise monotone hsteresis maps, Preprint submitted to Automatica 5 April

2 we provide conditions on the unimodal functions for producing butterfl-shaped maps and eamine the properties of the resulting butterflies. Conversel, we present conditions under which butterfl-shaped maps can be converted to simple piecewise monotone hsteresis maps to facilitate hsteresis compensation and control design. Although butterfl hsteresis maps are widel observed in the literature, we are not aware of an prior eplanations of the significance or origin of the characteristic shape of these maps. The proposed framework can be used to better understand and model dnamical sstems involving butterfl-shaped hsteresis. To illustrate this point, we consider the preloaded two-bar linkage, which is a classical eample of elastic instabilit (Simitses, 967). The hsteretic nature of this mechanism is studied in Padthe et al. (8), where the hsteresis map is shown to be a simple closed curve in terms of the force input and linkage joint displacement output. In the present paper, we show that if we take the displacement of the spring-loaded mass as output, then the resulting hsteresis map is a butterfl. The mapping from the joint displacement to the displacement of the spring-loaded mass can be given eplicitl and is clearl unimodal. The proposed framework can also facilitate control design for sstems demonstrating butterfl-shaped hsteresis maps. B transforming the butterfl map into a simple hsteresis map, we can eploit various wellstudied hsteresis operators, such as the Preisach operator and the Prandtl-Ishlinskii (PI) operator (Brokate & Sprekels, 996; Kuhnen, 3; Janaideh et al., 8; Visintin, 994; Maergoz, 3; Tan & Baras, 4), and their inverses (Tan & Ier, 9) to develop effective hsteresis compensation and control schemes. We demonstrate this point b considering the butterfl-shaped hsteresis map for a magnetostrictive actuator (Tan, ). In particular, we show that a uadratic law rooted in the phsics of magnetostrictive materials serves as a unimodal function for transforming the butterfl into a simple, piecewise monotone hsteresis map. The latter is then modeled with a generalized PI operator for inverse hsteresis compensation. The contents of this paper are as follows. In Section we describe a framework for mapping simple hsteresis maps into butterflies with unimodal functions, and present conditions that allow such transformations in cases involving piecewise monotone hsteresis maps. In Section 3 we consider a preloaded two-bar linkage mechanism, where the simple hsteresis loop between the linkage joint displacement and the force input is converted to a butterfl loop when a kinematics-based unimodal mapping is applied. In Section 4, the eample of modeling and compensating magnetostrictive hsteresis is presented. Finall, concluding remarks are provided in Section 5. A preliminar version of some of the results of this paper is given in Drinčić & Bernstein (9). Transformation Between Simple and Butterfl- Shaped Hsteresis Maps A hsteresis map is called simple, if it is a simple (oriented) closed curve, which divides the plane into three sets, namel, the interior region, the eterior region, and the curve itself (Guillemin & Pollack, 974). Throughout this paper, let C be a simple hsteresis map and let [, ] [, ] be the smallest rectangle with sides parallel to the - and -aes containing C. We assume that, for each (, ), there eists a uniue pair of points (, min ()), (, ma ()) C such that min () < ma (). The following definitions are needed. Definition A continuous map f : [, ] R is - unimodal if there eists c (, ) such that f is increasing on [, c ) and decreasing on ( c, ]. f is - unimodal if there eists c (, ) such that f is decreasing on [, c ) and increasing on ( c, ]. f is unimodal if it is either -unimodal or -unimodal. Definition A butterfl hsteresis map, or a butterfl, is the union of two oriented simple closed curves with disjoint interiors, a single point of intersection, and opposite orientation, such that the curves are contained in the rectangle [, ] [, ] and for each (, ), the intersection of the curves and the vertical line through consists of at most two points. For f : [, ] R, define f(c) {(, f()) : (, ) C}. The following result is immediate. Lemma 3 Let f : [, ] R be unimodal. Then C = f(c) is a butterfl if and onl if there eist disjoint open intervals I and I such that [, ] = cl(i ) cl(i ), and such that, for all I and all I, [f( min ()) f( ma ())][f( min ( )) f( ma ( ))] <. () We use the notation cl to denote the closure of a set. Note that cl(i ) cl(i ) is a single point. From Lemma 3, it is straightforward to establish the following. Corollar 4 If a simple hsteresis map C is left-right smmetric or up-down smmetric, then there eists no unimodal function f, such that C = f(c) is a butterfl. Net, we focus on a special class of simple hsteresis maps that are piecewise monotonic. This tpe of hsteresis maps is common in smart materials. Definition 5 Let C be a simple hsteresis map such that, for =, there eists a uniue point (, ) C and, for =, there eists a uniue point (, ) C. C is called piecewise monotonicall decreasing if min ()

3 and ma () are decreasing functions of. C is piecewise monotonicall increasing if min () and ma () are increasing functions of. C is piecewise monotonic if it is either piecewise monotonicall decreasing or piecewise monotonicall increasing. The following lemma is used in the proof of Theorem 7. Lemma 6 Let S be a closed polgonal region in a plane with vertices A, B, C, D, labeled consecutivel. Let C be a continuous curve that connects A to C and satisfies C \ {A, C} int(s). Let C be a continuous curve that connects B to D and satisfies C \ {B, D} int(s). Then C C. If, in addition, there eist coordinate aes with respect to which C is monotonicall increasing (resp. decreasing) function and C is monotonicall decreasing (resp. increasing) function, then C C consists of a single point. Proof. Because C is a continuous curve connecting A to C, it divides S into two open disjoint regions R and R, where B R and D R. Since the curve C connects points B and D it must cross from region R to region R. From the Jordan curve lemma C must cross the boundar between these regions, that is, the curve C. It is straightforward that there eists a uniue point of intersection between C and C if, with respect to some aes, these curves are monotonic with opposite monotonicit. Theorem 7 Let C be a piecewise monotonic simple hsteresis map. Furthermore, let f be a -unimodal function with its minimum point c ( c ) such that c (, ) or a -unimodal function with its maimum point c ( c ), such that c (, ). Then C = f(c) is a butterfl. Proof. We assume that the map f is -unimodal and that C is monotonicall increasing; the remaining three cases are analogous. Let J = (, c ), where c (, ) and ma ( c ) = c, and let J = ( c, ), where c (, ) and min ( c ) = c. Note that c < c. Because f is -unimodal, f( a ) < f( b ) for a, b [, c ] such that a > b, and f( a ) > f( b ) for a, b [ c, ] such that a > b. Thus, for all J, f( ma ()) < f( min ()), and, for all J, f( ma ()) > f( min ()). Let J 3 = [ c, c ]. Since C and f are piecewise monotonic, C = {(, f( ma ())) : J 3 } is monotonicall increasing and C = {(, f( min ())) : J 3 } is monotonicall decreasing. Furthermore, it follows from Lemma 6 that C C consists of a uniue point (, ). Now, for all I = (, ), f( ma ()) < f( min ()) while, for all I = (, ), f( ma ()) > f( min ()). Thus, () is satisfied for all I and all I, and thus C = f(c) is a butterfl. We now further investigate the properties of the butterfl map created b appling a unimodal map to a simple hsteresis map C. The following definition is needed. Definition 8 Let C be a simple hsteresis map or a butterfl, and let [, ] [, ] be the smallest rectangle containing C. A point (, ) C is a minimum of C. A point (, ) C is a maimum of C. The following result, which is not restricted to the class of piecewise monotone simple hsteresis maps, is illustrated b Figure. Theorem 9 Let f : [, ] R be unimodal and let C be a simple hsteresis map defined on the rectangle [, ] [, ]. Assume that, for each (, ), there eists a uniue pair of points ( min (), ), ( ma (), ) C such that min () < ma (), and assume that C f(c) is a butterfl. If f is -unimodal, then C has eactl two minima, which are eual. Alternativel, if f is -unimodal, then C has eactl two maima, which are eual. Proof. We assume that the map f is -unimodal; the -unimodal case is analogous. Let [, ] [ c, ] be the smallest rectangle containing C. Let c (, ) satisf c = f( c ). B Definition, c is the global minimizer of f. B assumption, there eist eactl two points ( min ( c ), c ) and ( ma ( c ), c ) C such that min ( c ) < ma ( c ). Appling f to these points ields ( min ( c ), f( c )), ( ma ( c ), f( c )) C. Hence these points are minima of the curve C and, since min ( c ) ma ( c ), it follows that these points are distinct. Thus, the butterfl map C has eactl two minima of eual value c. The following theorem is related to Theorem 9 and represents the dual of Theorem 7, and informs when a butterfl can be transformed into a simple, piecewise monotone hsteresis map. Theorem Let C be a butterfl and let [, ] [, ] be the smallest rectangle containing C, with sides parallel to the - and -aes. Decompose C as the union of two branches, namel, branch B + associated with increasing and branch B associated with decreasing. Furthermore, define g + : [, ] [, ] and g : [, ] [, ], such that B + and B are the graphs of g + and g, respectivel. Assume that C has eactl two minima ( a, c ), ( b, c ), with < a < b < and c =. Furthermore, assume that g + (resp., g ) is decreasing on [, b ] and increasing on [ b, ], and g (resp., g + ) is decreasing on [, a ] and increasing on [ a, ]. Then, for each unimodal function f with minimum value c, there eist a piecewise monotonicall increasing simple hsteresis map C with counterclockwise (resp., clockwise) ori- 3

4 ( min ( c ), c ) ( ( ( ), ) c, c ) ma c c where the continuous functions f and f + are strictl decreasing and increasing, respectivel (Figure ). Let f and f+ denote the inverse functions of f and f +, respectivel, as illustrated in Figure (c). Note that f is continuous and strictl decreasing, and f+ is continuous and strictl increasing, with c = f ( c ) = f+ ( c ). Define a curve C on the plane as the union of g () g + () f + () f (). ( ( ), ) min c c ( ( ), ) ma c c c a b c c..5.5 (c) c f + () c f (g + ()) f (g ()) f (g+ ()) f (g+ + ()) Fig.. Illustration of the proof of Theorem 9. The simple hsteresis map C and points ( min ( c ), c ) and ( ma ( c ), c ) are shown in. The -unimodal map f and the point ( c, c ) are shown in. The butterfl map C and the points ( min( c), c) and ( ma( c), c) are shown in (c). entation and a piecewise monotonicall decreasing simple hsteresis map C with clockwise (resp., counterclockwise) orientation, such that C = f(c ) = f(c ). Assume that C has eactl two maima ( a, c ), ( b, c ), with < a < b < and c =. Furthermore, assume that g + (resp., g ) is increasing on [, b ] and decreasing on [ b, ], and g (resp., g + ) is increasing on [, a ] and decreasing on [ a, ]. Then, for each unimodal function f with maimum value c, there eist a piecewise monotonicall increasing simple hsteresis map C with counterclockwise (resp., clockwise) orientation and a piecewise monotonicall decreasing simple hsteresis map C with clockwise (resp., counterclockwise) orientation, such that C = f(c ) = f(c ). Proof. We prove the case in where g + is decreasing on [, b ] and increasing on [ b, ], and g is decreasing on [, a ] and increasing on [ a, ], as illustrated in Figure. The other case in and the two cases in can be proven analogousl. Let f be a unimodal function, such that f( c ) = c for some c, and f() = { f (), if c, f + (), if c, c (c) f () a b (d) Fig.. Illustration of the proof of Theorem. A butterfl C is shown in, and a -unimodal function f is shown in. The inverse functions of f + and f in are shown in (c). The constructed simple hsteresis map C is shown in (d). four directed segments that are the corresponding graphs of = f (g + ()) as increases from to b, () = f+ (g + ()) as increases from b to, (3) = f+ (g ()) as decreases from to a, (4) = f (g ()) as decreases from a to, (5) as illustrated in Figure (d). Since f (g + ( b )) = f ( c ) = c = f+ (g + ( b )), f+ (g + ( )) = f+ (g ( )) (from continuit of C ), f+ (g ( a )) = c = f (g ( a )), and f (g ( )) = f (g + ( )) (from continuit of C ), C is continuous and closed. Define = h + () on [, ] using () and (3), and define = h () on [, ] using (4) and (5). Namel, h + and h represent the two branches of C associated with increasing and decreasing, respectivel. Using the properties of f± and g ±, it can be shown that both h + and h are strictl increasing functions of. Furthermore, for each (, ), h + () < h (). Therefore, C is a simple hsteresis map with counterclockwise ori- 4

5 entation. Finall, f(c ) can be defined as f () = f (f (g + ())) = g + () (6) as increases from to b, f + () = f + (f + (g + ())) = g + () (7) as increases from b to, f + () = f + (f + (g ())) = g () (8) as decreases from to a, f () = f (f (g ())) = g () (9) as decreases from a to, c a b c f () f () c c and, thus, C = f(c ). Following the same line of reasoning as above, it can be proven that C is a piecewise monotonicall decreasing, simple hsteresis map with clockwise orientation, and C = f(c ). In Section 3 and Section 4, we show phsical eamples where the butterfl hsteresis loops satisf the assumptions of Theorem, and the unimodal functions linking the butterfl and simple loops are given b the phsical models. If a butterfl hsteresis loop satisfies the assumptions of Theorem, and et a proper phsical model is not available, we can choose an arbitrar - unimodal ( -unimodal, resp.) function f with minimum (maimum, resp.) value of c as shown. Such an f guarantees successful transformation of the butterfl loop into a simple, piecewise monotone loop. The proof of Theorem is constructive, and thus it not onl establishes the claims, but also illustrates how the simple loop is constructed with the chosen f. To illustrate, we transform the butterfl shown in Figure 3 into a simple loop using the two -unimodal functions shown in Figure 3. The resulting simple loops are shown in Figure 3(c). Note that functions f and f have the same minimum value c, but at two different values of, namel c and c. 3 Hsteresis in a Preloaded Two-bar Linkage Mechanism In this section we analze the dnamics of a two-bar linkage with joints P, Q, and R and preloaded b a spring with stiffness constant k as shown in Figure 4. The purpose of this discussion is to show that we can transform a simple hsteresis map into a butterfl through a unimodal map. Additional details of derivations related to the simple hsteresis map are given b Padthe et al. (8). A constant vertical force F is applied at Q, where the two bars are joined b a frictionless pin. Let θ denote the counterclockwise angle that the left bar makes with the horizontal, and let denote the distance between c c a b (c) Fig. 3. Transformation of a butterfl loop into a simple loop with using two different -unimodal functions. shows the butterfl loop, shows the -unimodal functions f () = 3 (+.5).5 (solid) and f () =.5(+.5).7 (dashed). Appling the two different -unimodal functions f (solid) and f (dashed) to the butterfl loop ields the two distinct simple butterfl loops shown in (c). P l θ F Q l Fig. 4. A preloaded two-bar linkage with a vertical force F acting at the joint Q. The word preloaded refers to the presence of the spring with stiffness constant k, which is compressed when the two-bar linkage is in the horizontal euilibrium. the joints P and R. When F =, the linkage has three euilibrium configurations. In the first two θ and are ±θ and = l cos θ, respectivel, and the spring k is relaed. For the third euilibrium, both bars are horizontal with θ =. Note that is the vertical distance from the joint Q to the horizontal euilibrium, and is the horizontal distance from joint P to joint R as shown in Figure 4. If θ is known, then and can be determined from respectivel. R m = l sin θ, = l cos θ, () The euations of motion for the preloaded two-bar link- k c 5

6 age are given b ( (ml m barl ) sin θ m barl ) θ + (ml m barl )(sin θ)(cos θ) θ () + kl (cos θ cos θ)(sin θ) + cl (sin θ) l cos θ θ = F, where m bar is the inertia of each bar. Using () the nonlinear dnamics () can be epressed in terms of the displacement or. We use () and () to simulate the linkage dnamics under the periodic eternal force F = sin(.t) N with parameter values k = N/m, m = kg, c = N-s/m, m bar =.5 kg, and l = m. As shown in Figure 5 there eists a nontrivial clockwise hsteresis map from the vertical force F to the vertical displacement at low freuencies. The presence of a nontrivial loop at asmptoticall low freuencies constitutes hsteresis. Figure 5 shows the input-output map between the vertical force F and horizontal displacement. At asmptoticall low freuencies this input-output map is a smmetric butterfl with two loops of opposite orientation. Displacement [m] Force F [N] Displacement [m] Force F [N] Fig. 5. Input-output maps of the two-bar linkage model () for F = sin(.t). shows the hsteresis map with the output variable, while shows the butterfl hsteresis map with the output variable. The parameter values are k = N/m, m = kg, c = N-s/m, m bar =.5 kg, and l = m. The euilibrium set E for the preloaded two-bar linkage is the set of points (F, ) that satisf ( ) l cos θ = F l 4k. () Alternativel, the set E can be epressed as the set of points (F, ) that satisf ± ( (4l ) l cos θ ) = F k. (3) The euilibrium sets E defined b ()-(3) are shown in Figure 6. The corresponding hsteresis maps (as the Displacement [m] force varies uasi-staticall) are also shown in Figure 6. The set E is useful for analzing the hsteresis of the preloaded two-bar linkage. It is shown in Oh & Bernstein (5b) that a sstem that ehibits hsteresis has a multi-valued euilibrium map and that the hsteresis map is a subset of the euilibrium map. The onl parts of the hsteresis map that do not belong to the euilibria set are the vertical sections Hst. map Euil. set Force F [N] Displacement [m] Hst. map Euil. set Force F [N] Fig. 6. Comparison of the euilibrium sets E and the hsteresis maps for the preloaded two-bar linkage. The output variable is in and in. The hsteresis map is a subset of E ecept for the vertical segments at the bifurcation points. The parameter values are given in Figure 5 with F (t) = sin(.t) N. Thus the hsteresis map is a simple closed curve when the output variable is and a butterfl when the output variable is. Furthermore, we note that the butterfl in Figure 5 satisfies the conditions of Theorem, which implies that f should be -unimodal. From the kinematics euation () we find the unimodal map f = 4(l ) shown in Figure 7, which is indeed - unimodal and transforms the simple hsteresis map into a butterfl. Displacement [m] Displacement [m] Fig. 7. The -unimodal mapping function f() = 4(l ), which transforms the simple hsteresis map of the buckling mechanism into a butterfl. 4 Hsteresis in a Magnetostrictive Actuator In this section we present butterfl hsteresis data obtained from a magnetostrictive actuator. Based on material phsics, we find the unimodal function that relates the butterfl hsteresis map to a simple hsteresis map that is piecewise monotone. The latter can be modeled 6

7 with a hsteresis operator such as generalized Prandtl- Ishlinskii model, which can then be inverted for compensation of the hsteresis effect in the magnetostrictive actuator. A Terfenol-D magnetostrictive actuator manufactured b Etrema Products, Inc. ehibits displacement-tocurrent butterfl hsteresis shown in Figure 8. We note that the observed butterfl has two minima that are approimatel eual, and can be taken as satisfing assumptions of Theorem. In order to transform the butterfl into a simple hsteresis map, we adopt the following from Tan (). L (µ m) I (A) Fig. 8. Eperimental displacement-to-current butterfl hsteresis in a Terfenol-D magnetostrictive actuator (Tan, ). Definition Let L be the change in the length of the magnetostrictive rod, and let L rod be the length of the demagnetized rod. Then the magnetostriction in the rod is λ L L rod. (4) Based on the phsics of magnetostrictive materials (Brown, 966) the magnetostriction λ and the magnetization M along the rod direction can be approimatel related through a uadratic law stated in Tan () where a = λ s M s λ = a M + b, (5), b is a constant, λ s is the saturation magnetostriction, and M s is the saturation magnetization. The input current I and the magnetic field H are related through H = c I + H bias, (6) λ where c is the coil factor and H bias is the bias field produced b a dc current. Actuator specifications state that M s = A/m, L rod = 5.3 m, c =.54 4 m ; the remaining parameters are eperimentall identified to be λ s =.33 3 and H bias =.3 4 A/m. Combining Definition 4 with (5) and (6) we transform the butterfl curve in Figure 8 into a simple hsteresis curve between the magnetic field H and the magnetization along the rod M. Using the unimodal mapping (5), we obtain λ b M = ±. (7) a The sign of M is chosen such that the H M hsteresis curve is piecewise monotonicall increasing, as dictated b the phsics. The resulting plot of λ versus M is shown in Figure 9. Euation (6) is used to calculate the values of the magnetic field H corresponding to the input current. The resulting H M hsteresis map is shown in Figure 9. The vertical jump in the hsteresis map at the point where it crosses the -ais is due to the fact that, because of measurement error, the two local minima of the butterfl map in Figure 8 are not eactl eual. However, the eperimental data approimatel meet the assumptions of Theorems 9 and M (A/m) 5 M (A/m) H (A/m) 4 Fig. 9. Transformation from the butterfl to a simple closed curve. The -unimodal relationship between M and λ obtained from (7) is shown in. shows the simple hsteresis curve between the magnetic field H and magnetization M along the rod. A vertical jump due to uneual local minima of the butterfl map is visible at the point where the map crosses the H-ais. The piecewise monotone hsteresis map in Figure 9 can be modeled with various operators, such as the Preisach operator and the Prandtl-Ishlinskii (PI) operator. As an eample, we use the generalized PI operator given in Janaideh et al. (8) and shown to accuratel characterize hsteresis in magnetostrictive actuators. Since the generalized PI model has nonlocal memor, it can be used to model minor and major hsteresis loops and thus can easil be fit and inverted. The generalized PI model consists of generalized pla operators that are defined b the input u, threshold r, and envelope function γ. As in Janaideh et al. (8), we use the envelope function γ = c tanh(c u + c ) + c 3 and the densit function p(r) = ρe τr. We assume r = βj, where j =,...,, and we minimize the error between the 7

8 Parameter Value Parameter Value c 4.94 c.777 m/a c -.66 c ρ 4.64 A/m τ A/m β.39 Table Identified parameters of the generalized PI model from the least suares optimization routine. data and the model with a least suares optimization routine. The output of the optimization is summarized in Table. Comparison of the output of the identified generalized PI model with parameters in Table and eperimental data in Figure 9 is shown in Figure. Using (4)- (6) we convert the output of the generalized PI model from Figure into a butterfl hsteresis curve. The comparison of this butterfl map and the eperimentall measured data from Figure 8 is shown in Figure. M (A/m) Data Fit 6 H (A/m) L (µ m) Data Fit I (A) Fig.. Comparison of the eperimental data and the simulated output of the PI model. compares the eperimental data shown in Figure 9 and the output of the generalized PI model with parameters defined in Table. compares the butterfl map from eperimental data in Figure 8 and the butterfl map obtained b appling (5), (6) to the output of the generalized PI model with parameters defined in Table. In the eample above, we focused on onl the major hsteresis loop. Ferromagnetic materials and man other smart materials demonstrate minor loops inside major loops. The same unimodal function used to reduce a major butterfl loop to a simple loop can be used to reduce minor butterfl loops without needing to choose a different unimodal functions for the transformation of minor loops within a major loop. Conversel, appling a unimodal function to a piecewise monotone hsteresis operator that generates minor loops will lead to a hsteresis operator capable of generating minor butterfl loops. 5 Conclusions We studied the relationship between simple and butterfl hsteresis maps and showed that the can be linked through unimodal functions in almost all cases. Depending on the contet, such unimodal functions can be given b kinematics or underling phsics or can be chosen to facilitate analsis or control design. In particular, the proof of Theorem 9 provides a procedure to construct the simple hsteresis map given a butterfl and a chosen unimodal function. Two eamples involving a nonlinear buckling sstem and a smart material actuator were used to illustrate the utilit of the proposed framework in understanding, modeling, and compensating for butterfl hsteresis. In the eample of magnetostrictive hsteresis, the simple M H hsteresis map obtained from the butterfl carries phsical meaning (ferromagnetic hsteresis), and can in theor be eperimentall validated using the measurement of M. However, in general, when we reduce a butterfl map to a simple hsteresis map for control purposes, the latter does not have to carr specific phsical interpretation, and thus no validation is necessar. Future work includes the etension of these results to include hsteretic unimodal maps and multibutterflies. The use of hsteretic unimodal maps ma give more fleibilit in the tpes of butterflies that can be obtained from a given simple hsteresis map and enable easier fits to eperimental data in control applications. Acknowledgements This research is supported in part b a National Defense Science and Engineering Graduate Fellowship and NSF Grants and References Bernstein, D. S. (7). Ivor ghost. IEEE Control Sstems Magazine, 7, 6 7. Brokate, M., & Sprekels, J. (996). Hsteresis and Phase Transitions. New York, NY: Springer. Brown, W. F. (966). Magnetoelastic Interactions. Springer-Verlag. Chen, X., Hisaama, T., & Su, C.-Y. (9). Pseudoinverse-based adaptive control for uncertain discrete time sstems preceded b hsteresis. Automatica, 45, Cross, R., Grinfeld, M., & Lamba, H. (9). Hsteresis and economics: Taking the economic past into account. IEEE Control Sstems Magazine, 9, Davi, F. (). On domain switching in deformable ferroelectrics, seen as continua with microstructure. Zeitschrift fr Angewandte Mathematik und Phsik, 5, Drinčić, B., & Bernstein, D. S. (9). Wh are some hsteresis loops shaped like a butterfl? In Proc. Amer. Contr. Conf. (pp ). St. Louis, MO. 8

9 Ebine, N., & Ara, K. (999). Magnetic measurement to evaluate material properties of ferromagnetic structural steels with planar coils. IEEE Transaction on Magnetics, 35, Guillemin, V., & Pollack, A. (974). Differential Topolog. Englewood Cliffs, NY: Prentice-Hall, Inc. Ier, R. V., & Tan, X. (9). Control of hsteretic sstems through inverse compensation. IEEE Control Sstems Magazine, 9, Janaideh, M. A., Mao, J., Rakheja, S., Xie, W., & Su, C. Y. (8). Generalized Prandtl-Ishlinskii hsteresis model: Hsteresis modeling and its inverse for compensation in smart actuators. In Proc. IEEE Conf. Dec. Contr. (pp ). Cancun, Meico. Kuhnen, K. (3). Modeling, identification and compensation of comple hsteretic nonlinearities: A modified Prandtl-Ishlinskii approach. European Jour. of Cont., 9, Leang, K. K., Zou, Q., & Devasia, S. (9). Feedforward control of piezoactuators in atomic force microscope sstems. IEEE Control Sstems Magazine, 9, 7 8. Li, J., & Weng, G. J. (). A micromechanics-based hsteresis model for ferroelectric ceramics. Journal of Intelligent Material Sstems and Structures,, Maergoz, I. D. (3). Mathematical Models of Hsteresis and Their Applications. Amsterdam: Elsevier. Oh, J., & Bernstein, D. S. (5a). Semilinear Duhem model for rate-independent and rate-dependent hsteresis. IEEE Trans. Autom. Contr., 5, Oh, J., & Bernstein, D. S. (5b). Step-convergence analsis of nonlinear feedback hsteresis models. In Proc. Amer. Contr. Conf. (pp ). Portland, OR. Oh, J., Drincic, B., & Bernstein, D. S. (9). Nonlinear feedback models of hsteresis. IEEE Control Sstems Magazine, 9, 9. Padthe, A. K., Chaturvedi, N. A., Bernstein, D. S., Bhat, S. P., & Waas, A. M. (8). Feedback stabilization of snap-through buckling in a preloaded two-bar linkage with hsteresis. Int. J. Non-Linear Mech., 43, Sahota, H. (4). Simulation of butterfl loops in ferroelectric materials. Continuum Mechanics and Thermodnamics, 6, Simitses, G. J. (967). An Introduction to the Elastic Stabilit of Structures. Englewood Cliffs, NJ: Prentice Hall. Tan, X. (). Control of Smart Actuators. Ph.D. thesis Universit of Marland College Park, MD. Tan, X., & Baras, J. S. (4). Modeling and control of hsteresis in magnetostrictive actuators. Automatica, 4, Tan, X., & Ier, R. V. (9). Modeling and control of hsteresis. IEEE Control Sstems Magazine, 9, 6 9. Visintin, A. (994). Differential Models of Hsteresis. New York: Springer-Verlag. Wang, Q., & Su, C.-Y. (6). Robust adaptive control of a class of nonlinear sstems including actuator hsteresis with Prandtl-Ishlinskii presentations. Automatica, 4, Wen, C., & Zhou, J. (7). Decentralized adaptive stabilization in the presence of unknown backlash-like hsteresis. Automatica, 43,