【文章內(nèi)容簡介】 Tl , (7) 如果 Bf是粘性摩擦系數(shù)，和 Tl 代表負(fù)荷扭矩（在本文中假定為恒定）。 為了構(gòu)成完整的電動機(jī)的狀態(tài)方程，我們需要另一種代表轉(zhuǎn)子位置的狀態(tài)變量。為此，通常使用滿足下列方程的所謂的負(fù)荷角 δ [8] Dδ /dt = ω ?ω 0 , (8) 其中 ω 0是電動機(jī)的穩(wěn)態(tài)轉(zhuǎn)速。方程（ 5），（ 7），和（ 8）構(gòu)成電動機(jī)的狀態(tài)空間模型，其輸入變量是電壓 vq和 ，步進(jìn)電機(jī)由逆變器供給，其輸出電壓不是正弦電波而是方波。然而，由于相比正弦情況下非正弦電壓不能很大程度地改變振蕩特性和不穩(wěn)定性（如將在第 3部分顯示的，振蕩是由于電動機(jī)的非線性），為了本文的目的我們可以假設(shè)供給電壓是正弦波。根據(jù)這一假設(shè)，我們可以得到如下的 vq和 vd vq = Vmcos(Nδ ) , vd = Vmsin(Nδ ) , (9) 其中 Vm是正弦波的最大值。上述方程，我們已經(jīng)將輸入電壓由時間函數(shù)轉(zhuǎn)變?yōu)闋顟B(tài)函數(shù)，并且以這種方式我們可以用自控系統(tǒng)描繪出電動機(jī)的動態(tài)，如下所示。這將有助于簡化數(shù)學(xué)分析。 根據(jù)方程（ 5），（ 7），和（ 8），電動機(jī)的狀態(tài)空間模型可以如下寫成矩陣式 ? = F(X,u) = AX + Fn(X) + Bu , (10) 其中 X = [iq id ω δ] T, u = [ω 1 Tl] T 定義為輸入，且 ω 1 = Nω 0 是供應(yīng)頻率。輸入矩陣 B被定義為 矩陣 A是 F(.)的線性部分，如下 Fn(X)代表了 F(.)的線性部分，如下 輸入端 u獨(dú)立于時間，因此， 方程（ 10）是獨(dú)立的。 在 F(X,u)中有三個參數(shù)，它們是供應(yīng)頻率 ω 1，電源電壓幅度 Vm和負(fù)荷扭矩 Tl。這些參數(shù)影響步進(jìn)電機(jī)的運(yùn)行情況。在實(shí)踐中，通常用這樣一種方式來驅(qū)動步進(jìn)電機(jī)，即用因指令脈沖而變化的供應(yīng)頻率 ω 1來控制電動機(jī)的速度，而電源電 壓保持不變。因此，我們應(yīng)研究參數(shù) ω 1的影響。 ， 設(shè) ω =ω 0，得出方程（ 10）的平衡 且 φ 是它的相角， φ = arctan(ω 1L1/R) . (16) 方程（ 12）和（ 13）顯示存在著多重均衡，這意味著這些平衡永遠(yuǎn)不能全局穩(wěn)定。人們可以看到，如方程（ 12）和（ 13）所示有兩組平衡。第一組由方程（ 12）對應(yīng)電動機(jī)的實(shí)際運(yùn)行情況來代表。第二組由方程（ 13）總是不穩(wěn)定且不涉及到實(shí)際運(yùn) 作情況來代表。在下面，我們將集中精力在由方程（ 12）代表的平衡上。 注意：請將外文文獻(xiàn)原文復(fù)印件附在后面。 Oscillation, Instability and Control of Stepper Motors LIYU CAO and HOWARD M. SCHWARTZ Department of Systems and Computer Engineering, Carleton University, 1125 Colonel By Drive, Ottawa, ON K1S 5B6, Canada (Received: 18 February 1998。 accepted: 1 December 1998) Abstract. A novel approach to analyzing instability in permanentmag stepper motors is presented. It is shown that there are two kinds of unstable phenomena in this kind ofmotor: midfrequency oscillation and highfrequency instability. 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 are used to derive a quantity to evaluate the highfrequency instability. By using this quantity one can easily estimate the stability for high supply frequencies. Furthermore, a stabilization method is presented. A generalized approach to analyze the stabilization problem based on feedback theory is given. It is shown that the midfrequency stability and the highfrequency stability can be improved by state feedback. Keywords: Stepper motors, instability, nonlinearity, state feedback. 1. Introduction Stepper motors are electromagic incrementalmotion devices which convert digital pulse inputs to analog angle outputs. Their inherent stepping ability allows for accurate position control 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 higher peak torque per unit weight than DC motors。 in addition, they are brushless machines and therefore require less maintenance. All of these properties have made stepper motors a very attractive selection in many position and speed control systems, such as in puter hard disk drivers and printers, XYtables, robot manipulators, etc. Although stepper motors have many salient properties, they suffer from an oscillation or unstable phenomenon. This phenomenon severely restricts their openloop dynamic performance and applicable area where high speed operation is needed. The oscillation usually occurs at stepping rates lower than 1000 pulse/s, and has been recognized as a midfrequency instability or local instability [1], or a dynamic instability [2]. In addition, there is another kind of unstable phenomenon in stepper motors, that is, the motors usually lose synchronism at higher stepping rates, even though load torque is less than their pullout torque. This phenomenon is identified as highfrequency instability in this paper, because it appears at much higher frequencies than the frequencies at which the midfrequency oscillation occurs. The highfrequency instability has not been recognized as widely as midfrequency instability, and there is not yet a method to evaluate it. Midfrequency oscillation has been recognized widely for a very long time, however, a plete understanding of it has not been well established. This can