Modeling for control of a three degrees-of-freedom Magnetic. Levitation System

Transcription

1 Modelin for control of a three derees-of-freedom Manetic evitation System Rafael Becerril-Arreola Dept. of Electrical and Computer En. University of Toronto Manfredi Maiore Dept. of Electrical and Computer En. University of Toronto in s Collee Road in s Collee Road Toronto ON M5S 3G4 Toronto ON M5S 3G4 Systems Control Group Report No. 4 March 7, 3 Abstract This paper presents the derivation of the model for a hih-precision positionin system usin manetic levitation, which is achieved by an arranement of Permanent Manet inear Synchronous Motors. QR set-point stabilization is performed as an introduction to the more advanced nonlinear control desin presented in [9].

2 Stators Movers z x Platen Fiure : Confiuration with three SMs to achieve three derees of freedom Introduction We consider the problem of preventin the production of free particles on hih precision positionin systems, like those used in photolithoraphy. To this end, replacement of bearins witanetic levitation is proposed. This yields a similar solution to the one shown in [6]. However, the alternative presented here involves the use of already existin technoloies while spawnin an attractive control desin problem. Almost invariably, linear motors are used under the constraint of a constant airap enforced by means of bearins. Manetic and dynamic models of linear motors have been developed within this context. Nevertheless, in order to achieve levitation, airap control must be performed beside positionin control. In this document, a detailed analytical model considerin a variable airap is introduced. As shown later, this model is hihly nonlinear. For this purpose, the results developed for a fixed airap in [] are revisited in order to take into account a variable airap. Afterwards, the model is enhanced by includin the effect of the stator slots on the field distribution, this is accomplished by buildin on top of the fundamentals iven in [4]. For the sake of illustration, we briefly present the results of a linear control desin. A riorous nonlinear control desin approach based on this model can be found in [9].

3 Problem statement Consider the setup shown in Fiure, where three PMSMs (Permanent Manet inear Synchronous Motors) drive a floatin platen. The system employs three identical motors equally spaced alon a straiht line and perpendicularly oriented with respect to each other. The windins of motors and 3 (those at the ends of the structure) are connected in parallel whereas motor is fed independently. In order to allow lon travel, the movers of the motors are sinificantly smaller than their stators.. In each of the three motors, the permanent manets are placed on the surface of a flat structure of ferromanetic material (the mover), as shown in Fiure. Every stator made of lonitudinal laminations and transversally slotted in order to house three-phase windins, as shown in Fiure. In addition to the previous specifications, the desin takes into account the followin constraints. Assumption. The motors are far enouh from each other so that their fields do not interfere with each other. In addition, the model developed in this chapter is valid provided that the movement of the platen never drives the permanent manets outside of the areas covered by the stators manetic fields. Moreover, if the movers never et too close to the borders of such reions, modellin can assume uniform effects at the ends of the movers. Makin sure that such conditions are always met is a control desin objective. Remark. Since the forces are periodically exerted all over the surfaces of the movers, the resultant torque sums zero. Therefore, rotation is prevented and translational dynamics yield a complete model of the system. 3 Modelin In order to obtain a model describin each of the motors, a detailed electromanetic analysis is required. The components of the electromanetic forces actin on the mover of the motor, i.e. the thrust and the normal force, can be obtained by independently computin the fields produced by the permanent manets and the stator windins. The obtained analytical expressions for the resultant manetic field are then used to compute the forces exerted on the surfaces of the permanent manets. For most of the followin development, we use the results in [] and [4] by, to some extent, merin them. Some Actually, [6] presents the only one experimental test bed for this problem with technoloy similar to the one we use. 3

4 STATOR aminated ferromanetic material Slots pm pm pm pm Back Iron Windins MOVER Fiure : Confiuration of a sinle PMSM stator z x S N N S S N N S p mover x i x i y Fiure 3: Frame-set considered in the analysis details are included primarily for convenience of the reader, but also to emphasize the differences between [] and this development. Hereafter, the manetic field density produced by the permanent manets B pm will be found by first computin the correspondin manetic field intensity H pm. This is achieved by relyin of the imaes method, which allows us to easily find the manetic potential of the permanent manets, Ψ pm. The frame-set under which all quantities are measured is depicted in Fiure 3. By establishin an analoy, some manetic problems can be solved by applyin the methods developed for electric fields. We use this analoy in order to find the manetic field density produced by the permanent manets on the mover. Considerin a permanent manet with parallepipedal shape and a uniform distribution of Ψ pm, by virtue of the mentioned analoy and from standard 4

5 electrostatic analysis, the manetic potential at a point P x y z in the vicinity of the manet is iven by Ψ pm x y z 4 S σ m ds r S σ m ds r () Where σ m is the manetic chare, S and S are the two surfaces perpendicular to the manetization vector of the PM, and r and r are the lenths of the vectors from S and S to the point P, respectively. The manetic field intensity is then found by notin that H pm Ψ pm H pm 4 S σ m r ds r 3 S σ m r ds r 3 () et be the heiht of the manets and be the airap between the manets and the stator (see Fiure 3). By knowin the chare in the interior of the manet and the potential on its boundary, it is possible to determine the potential on the surroundin space. This is accomplished by findin the Green function which satisfies the Poisson s equation for the manetic potential. However, due to the hih symmetry of the problem, the simpler method of imaes can be applied (see, e., [7]). The method of imaes is commonly applied to find the electric potential due to a chare in presence of an equipotential surface, such as a conductor. Its main idea is to replace the equipotential reion by a chare which is the mirror of the actual chare with respect to such reion. In our case, and due to the ferromanetic material, the surfaces of the mover and the stator are equipotential surfaces. When the chare is located between two equipotential parallel surfaces, an infinite set of imaes is obtained. For the ith pole, the chare of its kth imae is σ m i k k i σ m. et h k be the position of the kth imae and σ m its chare. et e c be the effective airap due to the effect of the slots in the stator and c the Carter s coefficient, which is a parameter containin information relative to the topoloy of the motor. c is iven by 4 c γ γ arctan ln (3) 5

6 .5..5 B pmy z.5.5 x Fiure 4: Manetic field density on the stator s surface bein the slot pitch and the slot aperture, as shown in Fiure 3. The positions of the imaes, h k are h k k k e k when k is odd k e when k is even (4) et A be the depth of the poles alon the z direction, be the permanent manets pole pitch (i.e, the distance between poles centers), p m be the number of permanent manets and p be the permanent manets pole arc (see Fiure 3). We are ready to calculate the manetic potential due to the permanent manets at any iven point P x y z in the airap. The y component of H pm is iven by where H pmy x y z σ m 4 p m i k k i arctan x y x z h k D k z x x i z A x x i z A (5) D k x x y h k z z x i i α x i i α α p Evaluatin (5) on a rid of values of x z with y and m, and multiplyin by the free space permeability, 6

7 µ, the plot of B pmy x y z in Fiure 4 is obtained. Since a closed form for (5) is impossible to obtain, we fix the value of the airap,, and proceed to numerically averae the field intensity. This is achieved by averain alon the z axis after substitution of, thus obtainin a function H pmyav thus obtainin x y. Next, in order to find the field intensity on the surface of the stator, we let y H pmyav A x A H pmy x y z dz (6) Numerical results revealed that (6) describes an almost sinusoidal function. Hence, H pmyav can be approximated by calculatin its Fourier series first harmonic without considerable loss of accuracy. The coefficient of the first harmonic is iven by H pmy 4 n n H pmyav x sin! x " dx (7) with n p m, where p m is the number of poles. By notin that, in the airap, B µ H, we conclude that the y component of the flux density due to the permanent manets can be approximated by B pmy # B pmy µ H pmy sin! x " (8) Fiure 5 shows the field intensity distribution alon x and its fundamental harmonic. Note that the sinusoidal approximation is rather accurate in the rane of interest. Remark. As mentioned above, (8) holds for a fixed value of the airap. Since we need to consider a variable airap, one has that B pmy is also a function of. We therefore numerically calculate the coefficient µ H pmy for several different values of in the rane of interest and approximate the unknown function B pmy by polynomial interpolation, as shown in Fiure 6. Once the manetic field density pertainin to the permanent manets is known, we need to calculate the one produced by the stator. Here, we aain closely follow the development in [], where the manetomotive force (mmf) of the travelin manetic field is found in order to calculate Ψ m, which will be used to obtain B. et d denote the 7

8 .5 B pmy B pmy B pmy /B pmy B pmy () x [m] Fiure 5: Manetic field density alon x axis Fiure 6: Manetic field density as function of relative position of the mover with respect to the stator. The coefficient of the first harmonic of the mmf enerated by the travelin manetic field of the stator of a three-pahse SM is F s 6$ WI a k w c p sin % x d & (9) where W is the number of turns of wire on each phase, I a is the armature current, p is the number of poles, w c is the coil pitch, and k wn is the windin factor. Once F s has been found, it can be used to compute the first harmonic of the manetic potential, which is iven by Ψ s F s ' cosh! y " cosh ( *) sinh ( ) sinh! y ",+ sin! x " () Then, the components of the first harmonic of the manetic field density produced by the stator are B sx µ H sx µ Ψ s x B sy µ H sy µ Ψ s y µ F s µ F s sinh ( sinh ( y ) h m -) cosh ( y ) sinh ( *) cos! sin! x " x " () The difference between the reluctances of the slots and the teeth of the stator (see Fiure 7) produces spatial modulation on the distribution of the field. This phenomenon, referred to as the slots effect, is accounted for by 8

9 (x) /./././././././././././././. /./././ /././. ////////////// b t //// /// ////////////// ///////// // ////////////// ///////// // x b t + Fiure 7: Relationship between the actual slots and the function x multiplyin the field density, B, by the relative permeance λ on the forces expressions. Remark 3. In order to obtain the expression for the relative permeance, we adapt the concept presented in [4] for permanent manet DC motors and develop the linear variable airap version. As stated in [4], the relative permeance λ is iven by h m µ λ rec h x 3 m () bein the oriinal ap (considered as a constant for this part of the analysis) and x the shortest distance that the flux lines have to travel throuh the slots before reachin the teeth when leavin the manets from a point x (see Fiure 7). In order to find an analytical expression for this function, we need the followin assumption. Assumption. The manetic flux lines travel vertically throuh the airap and horizontally in the slots. In other words, we assume that the pitch of the slot is reater than its depth. Elementary calculations lead to the followin expression for (x) x k 54 x 4 k k 6 x 6 6 x b 6 k k (3) 9

10 Permeance Relative permeance st harmonic x Fiure 8: Relative permeance and its first harmonic where k is the index of each slot, the width of the slot and w c the coil pitch. Since (3) is a piecewise continuous function, we obtain its Fourier series representation. Fiure 8 shows a plot of the relative permeance, and its first harmonic approximation (dashed line). To simplify calculations, we define the function h x as follows h x x 7 (4) where is the relative recoil permeability of the permanent manets. Moreover, since we are dealin with a periodic function that will be approximated by its Fourier series, without loss of enerality, calculations can be performed for k. Hence, h x is reduced to h x hm µrec hm µrec 8 x 8 hm b hm µrec : µrec : 6 x 6 6 x 6 (5) In terms of h x, the relative permeance is iven by λ h x (6)

11 # After few calculations, the Fourier series of h x is obtained as h x a a n cos n 4 n x b h m cos n n ; cos n x < (7) Thus, the relative permeance is iven by b λ 4 cos n n 5 cos n x (8) Remark 4. Now, we use (8) to find expressions for the thrust and the normal force, accordin to the method described in [] but, unlike [], allowin for a variable airap. Reardin the stator surface as a surface S of chared particles, we can find the motors thrust by considerin the x component of the travelin manetic field, iven in (), which is affected by the relative permeance iven in (8). The thrust manitude is iven by p F x # p m A σ m λb p sx dx (9) ρ c sinh ( ) ' η ζ b h m + I a sin! d " () where ρ $ k w p m A σ m µ sinh = > p () Λ ζ b 4 η Λ () η sin! n cos p " (3) n 5 Λ n (4)

12 & & & & Λ n n sin %! " p & n sin %! n " n p &? (5) The normal force can be found from the definition of the Maxwell stress tensor (see [8]) and from the resultant manetic field on the airap, which is iven by B y B sy B pmy B sy B pmy cos! d " (6) Recall that the density fields on the riht hand side of (6) is a function of. In order to ease the calculations, only the fundamental component λ and the first harmonic of the permeance λ are considered. Thus the normal force is F y # # ϒ B p m A % λ B sin! µ ϕ I a c coth % x " & & dx B pmy 7 ϕi a B pmy c coth % & cos! d "DC Γ (7) where ϒ p m A µ (8) ϖ ς b 4 6t Γ b 4 6 sin ϑ b 4 ξ β ϕ 6$ k w µ p ϑ cos ξ β cos ξ sin sin % 9 : β ς 5 sin % 9 : ϖt 4 sin % 9 : 4 (9) (3) (3) cos sin % 9 : (3)? (33) (34) (35) Γ is the term that accounts for the effect of the slots. If that effect is to be nelected, Γ Γ.

13 FGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFG FEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFE FGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFG FEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFE FGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFGFG z F Fy y y F x Motor Motor Motor 3 a b Fiure 9: Forces in the three SMs confiuration In order to derive the dynamic model of the manetic levitation system shown in Fiure, let M p be the mass of the platen, G the ravity constant, and i d, i q be the direct and quadrature currents which are rearded as the control inputs. For x IH ġ x ẋ z żj T and the forces defined as in Fiure 9, Newton laws yield the followin equations M p M p G F y z M p ẍ F x M p z F z x F y x z (36) where x and z are the displacements in the x and z directions respectively, F y F y are the forces enerated by motor and 3, and F y F y x x 5 i d i q and F z F z x x 5 i d i q x x 3 i d i q and F x F x x x 3 i d i q are the forces produced by motor. F x, F y, F y and F z are iven by expressions () and (7). The dynamics become x M G F y M p F y M p ẋ x 4 F x M p (37) x 6 F z M p 3

14 O & & Expression (37) can be expanded by substitutin the full expressions of the forces, yieldin ẋ x ẋ G ϒΓ x ẋ 3 x 4 ẋ 4 ϒΓ x M p ϕ M p c coth x % x B 3B ϕb pmy x 7 pmy x coth % c x ρ M p c 9 x : sinhh 9 x : J η ζ x hm µrec ( i d i q 7 i d i q) i q x & H i d i d J C (38) ẋ 5 x 6 ẋ 6 ρ M p c 9 x : sinhh 9 x : J η ζ x hm µrec i q Introducin the chane in notation iven by N ϒ M p ϕ χ 3ϒ M p ν ϕϒ M p O ρ M p κ N Γ x φ x coth ( x ) c x νγ x ψ x B pmy x coth % x c x γ x ζ η c x sinh ( x *) x κ χ x χγ x B pmy x 4

15 and definin u i q, u i q, u 3 i d, u 4 i d, those equations are described in a more compact form by ẋ x ẋ G φ x H u u u 3 u 4J χ x ψ x H u 3 u 4 J ẋ 3 x 4 (39) ẋ 4 γ x u ẋ 5 x 6 ẋ 6 γ x u The values of the constants used in simulation are µ 4 P 7 p m 4.57 A.5 p.86 B r.55 H c 836 µ r.. p Table : Parameters of the motor considered for numerical calculations 4 inear Control Desin Consider the problem of movin the platen to a desired set point in the space, i.e, x d H ḡ x z J. For the problem to be well-posed, one has to determine whether there exists a value of the control input u QH u RS u 4 J T u d that renders x d an equilibrium condition. For the previously specified constants, one finds that the problem is well-posed whenever ḡ is restricted within the rane 35 6 ḡ 6 5. We desined an QR controller based on the linearization of (39) around the equilibrium condition x d u d with 5

16 .6.3 x x x 3 x 4 x 5 x position [m] speeds [m/s] time [s] time [s] Fiure : Position of the platen under QR control for x d 55m and x 47m Fiure : Speed of the platen under QR control for x d 55m and x 47m. x x. x 3 x 5.8 x 4 x position [m].4. speeds [m/s] time [s] time [s] Fiure : Position of the platen under QR control for x d 5m and x 5m Fiure 3: Speed of the platen under QR control for x d 5m and x 5m 6

17 the weihtin matrices Q dia and R I 4. The resultin feedback ain matrix is (4) Simulation results are shown in Fiures and, where the desired set point is x dth 6 J T and the initial condition is x IH 47 5 H 5 J T. The equilibrium inputs are u H 5 83J. Fiures and 3 show the simulation results with the equilibrium point set to x duh 5 J T, u d 5 778J, and the initial conditions set to x H J T. 5 Conclusions The positionin device here proposed demonstrated to be linearly controllable within certain subset of the state space. Hence, smooth nonlinear control stabilization can be performed for this system. A riorous nonlinear control law, developed on [9], has produced improved results as compared to the linear controller. Currently, a robust controller is bein explored. Experimental implementation is under way. References [] S.A Nasar and G. Xion, Analysis of fields and forces in a permanent manet linear synchronous machine based on the concept of manetic chare, in IEEE Transactions on Manetics, vol. 5, no. 3, May 989. [] S.A Nasar and G. Xion, Determination of the field of a permanent-manet disk machine usin the concept of manetic chare, in IEEE Transactions on Manetics, vol. 4, no. 3, May 988. [3] I. Boldea and S.A. Nasar, inear electric actuators and enerators, Cambride University Press, U, 997. [4] Z.Q. Zhu and D. Howe, Instantaneous manetic field distribution in brushless permanent manet DC motors, part III: effect of stator slottin, in IEEE Transactions on Manetics, vol. 9, no., Jan

18 [5] J.F. Gieras and Z.J. Piech, inear synchronous motors, CRC Press C, USA,. [6] W. im and D.. Trumper, Hih-precision levitation stae for photolithoraphy, in Precision Enineerin vol., 998, USA 998 [7] E. Weber, Electromanetic Theory: Static fields and their mappin, Dover Publications, USA, 965 [8] J.F. Gieras and M. Win, Permanent Manet Motor Technoloy, Desin and Applications, nd edition, Marcel Dekker Inc, USA, [9] R. Becerril-Arreola, M. Maiore, Nonlinear Stabilization of a 3 derees-of-freedom Manetic evitation System, accepted for publication at the American Control Conference 3. 8