【正文】 hat of being able to independently control the two linear motors using a single inverter. The outline of the rest of the paper is as follows: Section 2 briefly describes the modeling of PM synchronous motors, Section 3 develops a linear PM motor model from the rotary model, presents the door model and summarizes the standard PM synchronous motor control algorithm, Section 4 considers the control of two linear PM motors using a single inverter for both the parallel and series connection. Finally, Section 5 offers some conclusions. 外文翻譯（ 原 文） 4 2. Modeling and control of PM synchronous motors A linear permanent mag motor may be modeled by considering an equivalent threephase permanent mag (PM) rotary synchronous motor. To do so, let x, ? denote the position and speed of the linear motor, m denote the mass of the linear motor, req denote the radius of the equivalent rotary motor (., the linear motor travels a distance 2πreq for each plete revolution of the rotary motor), m is the mass of the linear motor, and F, FL denote the force produced by the linear motor and the load force on the motor. Then, for the rotary motor, it follows that the angular position is h = x/req, the angular speed is given by ω= ?/req, the moment of inertia is J = r2eq m, the torque τ= req F, the load torque τL = req FL. A model of a threephase PM synchronous (rotary) motor is [5]. )s i n(n p11321 ??mssssss KiRvdtdiMdtdiMdtdiLs ????? )32s i n(n p22321 ??? ??????? mssssss KiRvdtdiMdtdiLsdtdiM )34s i n(n p33321 ??? ??????? mssssss KiRvdtdiLsdtdiMdtdiM Lsmsm τiKiKdtdJ )34s i n(niK)32s i n(n)s i n(n ps3mp2p1 ?????? ????? － ???dtd Here LS is the selfinductance of a stator winding, M is the coefficient of mutual inductance between the phases, Km is the torque/backemf constant (so that KM = Km/req is the force/backemf constant in the linear motor), RS the resistance of a stator winding, np is the number of pole pairs (or the number of rotor teeth for a stepper motor). If the phases were perfectly coupled, one would have M = 21 LS. The threephase to twophase transformations for currents and voltages are defined by ??????????????????????????????????3210 212121232302121132SSSsbsaiiiiii which transforms the original model into the equivalent model 外文翻譯（ 原 文） 5 busV2?)s i n(23)( ?? pmsaSsasas nKiRvdtdiML ???? )s i n(23)( ?? pmsaSsasbs nKiRvdtdiML ???? For a balanced threephase system assumed here, it follows that v0=(vs1+vs2+vs3)/ 3 =0, i0=(vs1+vs2+vs3)/ 3 =0 so that one obtains the twophase equivalent model given by sapeqsaSsa vnKiRdtdiL ???? )s i n( ?? sbpeqsbS vnKiRdtdi bL ???? )c os ( ?? Here L= Ls + M(≈23 Ls),Keq = 23 Km ia and ib are the equivalent currents in phases a and b, respectively. Letting Vbus denote the bus voltage into a threephase inverter. The maximum voltage out of the inverter is obtained when it is run in six step mode and the peak of the fundamental of the sixstep waveform is vmax = This is taken to be the maximum limit of the phase voltage. Finally, with i max, vmax denoting the limits of the phase currents and voltages of the threephase motor, the corresponding limits Imax,Vmax for the equivalent twophase motor are then The directquadrature or dq transformation is defined by where id, iq and vd, vq are the transformed currents and voltages, respectively in the dq (for direct and quadrature) reference frame. The definition of the dq reference system assumes that the daxis is aligned with the rotor,s magic axis when ? = that ???????????????????? sbsapp ppqd iinn nnii )c os ()s i n( )s i n()c os ( ?? ?? 外文翻譯（ 原 文） 6 ???????????????????? sbsapp ppqd vvnn nnvv )c os ()s i n( )s i n()c os ( ?? ?? when ? =0, the daxis is aligned the iaaxis which in turn is the same as the iS1axis. The statespace model in the dq coordinates is ???dtd (5) This model assumes that the rotor is smooth (nonsalient) and that the magics are linear. 3. Motor specifications The motor parameters are specified for a linear motor and are converted to an equivalent rotary motor. The linear motor parameters are stator inductance LS = , stator resistance RS = ? , coefficient of mutual inductance M = = , motor mass m = kg, force constant KM = , distance between poles dp = , np = 1 (no. of primary polepairs). The maximum dc bus voltage to the inverter is Vmax = 320V resulting in a peak fundamental waveform to the motor of vmax = Vma x= phase currents are limited to Imax = 10A (peak) and the maximum (linear) force put out by the motor is 320N. The radius of an equivalent rotary motor satisfies 2preq = np2dp ? req = . The torque constant of an equivalent threephase rotary motor is found from the linear force constant by setting Km = req KM = ()(32)Nm/A = and the moment of inertia is J =r2eqm =() =? 103kgm2. The parameters LS, M, RS, np are the same as for the linear motor. Here x = 0 for the linear motor corresponds to the magic axis of its rotor phase a being lined up with the magic axis of stator phase a and similarly for the equivalent rotary motor. The corresponding equivalent twophase parameters are then L = LS + M = , RS = ? , Keq = 23 Km=, Imax (continuous) = 23 imax =,Vmax= 23 vmax= linear force put out by this motor is F = Keq iq / req= 23 Km iq / req . 外文翻譯（ 原 文） 7 . Door model The door model is from the technical report of He [4] and is of the form dx/dt = Ax+bu y =Cx where A ? R8? 8 , b?R8 , C?R8? 8 . The values of the triple {A,b,C} are given in [4]. Here x1 is the door position, x2 is the door speed and the input u to the door is the linear force F = Keq iq / req put out by the motor. The state variables x1, x2 are the two measured/p