【正文】 !0VmZ 2m 。 (10)where XDTiqid! UT, uDT!1TlUTis defined as the input, and !1DN!0is the supplyfrequency. The input matrix B is defined byBD2666400000 ?1=J?1=N 037775:The matrix A is the linear part of F. /, and is given byAD26664?R=L10 ?N 1=L100 ?R=L10032N 1=J 0 ?Bf=J 000 1037775:Fn.X/ represents the nonlinear part of F. /, and is given byFn.X/D26664?Nid!CVm=L1 /Niq!CVm=L1 /0037775:The input term u is independent of time, and therefore Equation (10) is autonomous.There are three parameters 。 (9)where Vmis the maximum of the sine wave. With the above equation, we have changed theinput voltages from a function of time to a function of state, and in this way we can representthe dynamics of the motor by a autonomous system, as shown below. This will simplify themathematical analysis.Oscillation, Instability and Control of Stepper Motors 387From Equations (5), (7), and (8), the statespace model of the motor can be written in amatrix form as followsP。 (8)where !0is steadystate speed of the motor. Equations (5), (7), and (8) constitute the statespace model of the motor, for which the input variables are the voltages vqand vd.Asmentioned before, stepper motors are fed by an inverter, whose output voltages are not sinusoidal but instead are square waves. However, because the nonsinusoidal voltages do notchange the oscillation feature and instability very much if pared to the sinusoidal case(as will be shown in Section 3, the oscillation is due to the nonlinearity of the motor), forthe purposes of this paper we can assume the supply voltages are sinusoidal. Under thisassumption, we can get vqand vdas followsvqDVm /。 (5)386 L. Cao and H. M. SchwartzθωqdabcFigure 2. a, b, c and d, q reference frame.where L1DLCM,and! is the speed of the rotor.It can be shown that the motor’s torque has the following form [2]T D32N 1iq: (6)The equation of motion of the rotor is written asJd!dtD32N 1iq?Bf!?Tl。d framecan be obtained asvqD RiqCL1diqdtCNL1id!CN 1!。c reference, only two variables are independent (iaCibCicD 0)。d reference are given by vqvd DTr24vavbvc35: (4)In the a。c frame to the q。d transformation, the frame of reference is changed from the fixed phaseaxes to the axes moving with the rotor (refer to Figure 2). Transformation matrix from thea。 pmcD 1 C2 =3/。 (1)where R and L are the resistance and inductance of the phase windings, and M is the mutualinductance between the phase windings. pma, pmband pmcare the fluxlinkages of thephases due to the permanent mag, and can be assumed to be sinusoid functions of rotorposition as follow pmaD 1 /。vbDRibCLdibdt?Mdiadt?MdicdtCd pmbdt。 accepted: 1 December 1998)Abstract. A novel approach to analyzing instability in permanentmag stepper motors is presented. It is shownthat there are two kinds of unstable phenomena in this kind of motor: midfrequency oscillation and highfrequencyinstability. Nonlinear bifurcation theory is used to illustrate the relationship between local instability and midfrequency oscillatory motion. A novel analysis is presented to analyze the loss of synchronism phenomenon,which is identified as highfrequency instability. The concepts of separatrices and attractors in phasespace areused to derive a quantity to evaluate the highfrequency instability. By using this quantity one can easily estimatethe stability for high supply frequencies. Furthermore, a stabilization method is presented. A generalized approachto analyze the stabilization problem based on feedback theory is given. It is shown that the midfrequency stabilityand the highfrequency stability can be improved by state feedback.Keywords: Stepper motors, instability, nonlinearity, state feedback.1. IntroductionStepper motors are electromagic incrementalmotion devices which convert digital pulseinputs to analog angle outputs. Their inherent stepping ability allows for accurate positioncontrol without feedback. That is, they can track any step position in openloop mode, consequently no feedback is needed to implement position control. Stepper motors deliver higherpeak torque per unit weight than DC motors。Nonlinear Dynamics 18: 383–404, 1999.169。 1999 Kluwer Academic Publishers. Printed in the Netherlands.Oscillation, Instability and Control of Stepper MotorsLIYU CAO and HOWARD M. SCHWARTZDepartment of Systems and Computer Engineering, Carleton University, 1125 Colonel By Drive,Ottawa, ON K1S 5B6, Canada(Received: 18 February 1998。 in addition, they are brushless machines andtherefore require less maintenance. All of these properties have made stepper motors a veryattractive selection in many position and speed control systems, such as in puter hard diskdrivers and printers, XYtables, robot manipulators, etc.Although stepper motors have many salient properties, they suffer from an oscillation orunstable phenomenon. This phenomenon severely restricts their openloop dynamic performance and applicable area where high speed operation is needed. The oscillation usuallyoccurs at stepping rates lower than 1000 pulse/s, and has been recognized as a midfrequencyinstability or local instability [1], or a dynamic instability [2]. In addition, there is anotherkind of unstable phenomenon in stepper motors, that is, the motors usually lose synchronismat higher stepping rates, even though load torque is less than their pullout torque. This phenomenon is identified as highfrequency instability in t