【正文】 omenon,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。 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。vcDRicCLdicdt?Mdiadt?MdibdtCd pmcdt。 (2)whereN is number of rotor teeth. The nonlinearity emphasized in this paper is represented bythe above equations, that is, the fluxlinkages are nonlinear functions of the rotor position.By using the q。d frame is given by [8]TrD23 / ?2 3/ C2 3/ / ?2 3/ C2 3/: (3)For example, voltages in the q。 therefore, theabove transformation from three variables to two variables is allowable. Applying the abovetransformation to the voltage equations (1), the transferred voltage equation in the q。 (7)whereBfis the coefficient of viscous friction, andTlrepresents load torque, which is assumedto be a constant in this paper.In order to constitute the plete state equation of the motor, we need another statevariable that represents the position of the rotor. For this purpose the so called load angle [8] is usually used, which satisfies the following equationd dtD!?!0。u/DAXCFn.X/CBu。 (13)IdD .Vm 0/CNL1!0Iq/=R。 (17)where 1XDT1iq1id1! 1 UT,andAlis defined byAlDACFnXD26666664?RL1?!1?N. 1L1CId/ ?NVmL1 0/!1?RL1NIqNVmL1 0/32N 1J0 ?BfJ000 1 037777775: (18)Assume that all of the eigenvalues of Alhave no zero parts, then the system defined inEquation (10) is stable if and only if all these eigenvalues have negative real parts [9, 10].For a threephase stepper motor whose parameters are shown in Table 1, the calculatedeigenvalues of Alfor some !1based on the first group of equilibra are given in Table 2. Theeigenvalues can also be divided into two groups: one group given by p1and p2, correspondsto the electrical subsystems of the motor, and the other group given by p3and p4correspondsto the mechanical subsystems. When all the real parts of these eigenvalues are negative, theequilibrium is stable and is called an attractor. This indicates that motor is at steadystate operation with the speed!D!1=N. As shown in Table 2,p1andp2are close to?R=L1 j!1tosome extent, and are always stable for all !1. However, the real parts of p3and p4are positivefor some range of!1. The phenomenon that the real parts ofp3andp4bee positive relatesthe observable midfrequency oscillation, which has been analyzed by many authors. At thisOscillation, Instability and Control of Stepper Motors 389Table 2. Eigenvalues of Alevaluated at some !1.!1p1。 0/ at which the following conditions are satisfied:1. The Jacobian matrix of f evaluated at y0and 0has a simple pair of imaginary eigenvalues . 0/D j and no other eigenvalues with zero real part。!11CU and !12T!12?。1!plane when the supplyfrequency is !1D 2826 rad/s, where the initial condition of the trajectory is nonzero. Theboundary between the stable region (basin of attraction) and unstable region of an attractor isalso determined by the separatrices around it. In the following, it is shown that as the supplyfrequency !1increases, the distance between a stable equilibrium and its stability boundarywill tend to zero. This highlights the fact that when supply frequency is high a stepper motorgets many more chances to lose synchronism.394 L. Cao and H. M. SchwartzTaft and Gauthier [3] and Taft and Harned [4] have given a phase portrait including separatrices for stepper motors in the planar case, and have shown that if a trajectory crosses aseparatrix the motor will lose synchronism. Their analysis is very insightful in the sense ofconnecting separatrices with the loss of synchronism phenomenon. However, because in theiranalysis the dynamic behaviours of the motor’s currents were not considered, it cannot beapplied directly to our situation. Here, we have a fourdimensional phase space and the phaseportrait cannot be visualized as in the case of twodimensional space. In our case, it is verydifficult to inspect the separatrices relying on a geometrical grasp of the phase portrait. For thisreason, we develop a very simple analytic method which is relevant and useful in evaluatingthe failure behaviour.Consider the distance from a equilibrium point to the nearest separatrix, which can bewritten asDesm。Bm/D2s 2C VmZ2 sin2.N /。Bm/ as well as Des.Thatis,as!1increases one gets m。B