北航自控原理课件7(英文版).ppt

7.2 The root locus procedure Sum of all roots of closed-loop characteristic equation * Chapter 7 The root locus method 重点掌握 根轨迹的绘制 通过根轨迹分析闭环系统的性能 Main contents 1、The root locus concept and root locus equation 2、The root locus procedure 3、Zero degree root loci 4、Estimating the specifications by the dominant poles 7.1 The root locus concept and root locus equation The roots of the closed-loop system Definition The roots locus is the path of the roots of the characteristic equation traced out in the s-plane as a system parameter is changed. 根轨迹是指系统中某个参数由0→∞变动时，闭环特征根在s平面上移动的轨迹。 G(s) H(s) R(s) C(s) D(s)=1+G(s)H(s)=0 The characteristic equation is Where K : the root locus gain Zeros of open loop transfer function. Poles of open loop transfer function. The equation may be rewritten in polar form as Then Magnitude equation Phase equation Step 1:Write the characteristic equation as Parameter that is changed G(s) H(s) R(s) C(s) example when Characteristic equation is So When Characteristic equation is Step 2:Factor P(s),write the polynomial in the form of poles and zeros Step 3:Locate the poles and zeros of P(s) on s-plane with selected symbols z2 The locus of the roots of the characteristic equation 1+KP(s)=0 begins at the poles of P(s) and end at the zeros of P(s) as K increases form 0 to infinity. Step 4:Locate the segments of the real axis that are root loci. The root locus on the real axis always lies in a section of the real axis to the left of an odd number of poles and zeros. Step 5 :The number of separate loci. The number of separate loci is equal to the number of poles. Step 6: The root loci must be symmetrical with respect to the horizontal real axis. n阶系统的特征方程有n个特征根，当K（由0→∞）变动，则n个特征根跟随变化，在s平面上必然出现n条根轨迹。 Step 7 :Asymptotes of the root loci Asymptote centroid Angle of the asymptotes 渐近线与实轴正方向的夹角为： 渐近线与实轴相交点的坐标为： Step 8: Determine the point at which the locus crosses the imaginary axis. 根轨迹与虚轴相交，表明系统闭环特征方程有纯虚根，系统处于临界稳定状态。 求解方法1： 将