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1、Automatic Control Theory,College of Information Science and Engineering,CSU 2015.4,2019/5/13,hanhua,2,根据前几章学过的知识，影响系统的运动即性能的主要因素是什么？,闭环极点,2019/5/13,hanhua,3,Chapter 4 Root Locus Method of Linear System (线性系统的根轨迹法),4.1 The Root Locus Concept 4.2 The Root Locus Procedure 4.3 generalized root Locus 4.4
2、 the example of drawing root locus 4.5 the performance analysis of system 4.6 sensitivity and the root 4.7 Compensation by Using Root Locus Method Summary,2019/5/13,hanhua,4,4.1 The Root Locus Concept (根轨迹法的基本概念),The root locus is the path of the roots of the characteristic equation traced out in th
3、e s-plane as a parameter of open-loop system is changed from 0 to infinity. 根轨迹简称根迹，它是开环系统某一参数从零变到无穷时，闭环系统特征方程式的根在 s 平面上变化的轨迹。,1.the Root Locus Concept(根轨迹概念),指闭环极点,2019/5/13,hanhua,5,The root locus method was introduced by Evans in 1948 and has been developed and utilized extensively in control eng
4、ineering practice. The root technique is a graphical method for sketching the locus of roots in the s-plane as a parameter is varied. It is very useful to determine how the roots of the characteristic equation move around the s-plane as we change one parameter.,2. 根轨迹法的特点,2019/5/13,hanhua,6,The root
5、 locus method provides graphical information(根轨迹法提供了图形化信息) 1)stability 2)steay-state accuracy 3)dynamic performance The root locus method can be used to analyze and design a system.,2019/5/13,hanhua,7,Example,2019/5/13,hanhua,8,A second-order system,2019/5/13,hanhua,9,Location of roots for the chara
6、cteristic equation:,K S1 S2 0 0 -2 0.5 -0.3 -1.7 0.75 -0.5 -1.5 1.0 -1.0 -1.0 2.0 -1.0+j1.0 -1.0-j1.0 3.0 -1.0+j1.4 -1.0-j1.4 50.0 -1.0+j7.0 -1.0-j7.0,2019/5/13,hanhua,10,Analysis:,The root locus is a vertical line for K Ka =1.,Ka,return,根轨迹图怎样体现系统的性能？,稳态误差？,稳定性？,动态性能？,2019/5/13,hanhua,11,2.闭环零、极点与开
7、环零、极点之间的关系,前向通道增益,前向通道根轨迹增益,前向通道根轨迹增益,2019/5/13,hanhua,12,The open-loop TF:,f +l=m , q+h=n,The closed-loop TF:,return,闭环零点,反馈通道传递函数极点,开环前向通道传递函数的零点,闭环极点,2019/5/13,hanhua,13,4.2 The Root Locus Procedure,4.2.1 Magnitude and Angle Requirement 4.2.2 Rules for drawing root locus 4.2.3利用MATLAB绘制根轨迹图 4.2.4
8、 闭环极点和传递函数的确定,2019/5/13,hanhua,14,Root locus equation (negative feedback) (根轨迹方程),开环零点,开环极点,4.2.1 Magnitude and Angle Requirement (模值条件和相角条件) (重点),2019/5/13,hanhua,15,magnitude requirement(模值条件),2019/5/13,hanhua,16,angle requirement (相角条件),2019/5/13,hanhua,17,根轨迹上的闭环极点,系统的开环零点,系统的开环极点,2019/5/13,hanh
9、ua,18,要能绘制根轨迹，哪些必须是已知的？闭环传递函数一般是已知的吗？可根据什么来绘制根轨迹的草图？,如何由已知的开环零、极点的分布及开环根轨迹增益，通过图解的方法找出闭环极点。,相角条件和幅值条件,2019/5/13,hanhua,19,幅值条件可以用来确定特定的闭环极点处的根轨迹增益 K* (because s is vary with K* ). 检验点S是否位于根轨迹上，只需满足相角条件： 绘制根轨迹时，只有相角条件是充分必要条件。,结论,2019/5/13,hanhua,20,当以 K* 为参数时，其根轨迹应满足相角条件：,对于二阶系统,Im,2019/5/13,hanhua,2
10、1,特征方程,闭环极点,Im,2019/5/13,hanhua,22,根据幅值条件可以求出特定闭环极点出（系统）的根轨迹增益 K*,2019/5/13,hanhua,23,绘制根轨迹时的参数可以是根轨迹增益也可以是其它参数！,常规根轨迹,参数根轨迹,2019/5/13,hanhua,24,绘制根轨迹时的参数可以是大于零也可以是小于零！,常规根轨迹,零度根轨迹,2019/5/13,hanhua,25,由于相角条件是绘制根轨迹的充要条件，因此下面从相角条件出发，将导出一系列绘制根轨迹的规则！,2019/5/13,hanhua,26,4.2.2 Rules for Drawing Root Locu
11、s(绘制根轨迹的法则)（重点）,Rule 1： The Starting Points and the End Points of the Root Locus（根轨迹的起始点和终止点）,2019/5/13,hanhua,27,由开环传递函数,特征方程,2019/5/13,hanhua,28,1.根轨迹的起始点,根轨迹 起始于 开环极点,2019/5/13,hanhua,29,2.根轨迹的终止点,根轨迹 起始于 开环极点,2019/5/13,hanhua,30,零点与极点个数 不同怎么办?,2019/5/13,hanhua,31,个根终止于何处,个根起始于何处,终止于无穷远处,起始于无穷远处,
12、2019/5/13,hanhua,32,2019/5/13,hanhua,33,The locus begins at the open-loop poles and ends at the open-loop zeros (根轨迹起于开环极点,终于开环零点).,结论,2019/5/13,hanhua,34,Rule 2：The Symmetry and branches of the Root Locus (根轨迹的对称性和分支) 根轨迹对称于实轴.,根轨迹有 n条分支. 其中有 nm条伸向无穷远处. K* 的连续性 决定了根轨迹的连续性.,为什么？,2019/5/13,hanhua,35,
13、Rule 3：The Asymptotes of the Root Locus (根轨迹的渐近线),绘制根轨迹时，什么情况下需要借助渐进线？如何判断有几条渐进线？,2019/5/13,hanhua,36,渐近线与实轴的交点,渐近线与实轴的夹角,2019/5/13,hanhua,37,2019/5/13,hanhua,38,2019/5/13,hanhua,39,2019/5/13,hanhua,40,Rule 4：The Root Locus on the Real Axis (实轴上的根轨迹),根据什么来确定实轴上的根轨迹？,相角条件,2019/5/13,hanhua,41,根轨迹在实轴上的
14、分布。实轴上的某一区域，若其 右边开环实数零、极点个数之和为奇数，则该区域 必是根轨迹。,结论,2019/5/13,hanhua,42,Rule 5：根轨迹的分离点与分离角,两条或两条以上根轨迹分支在s 平面上相遇又立即 分开的点，称为根轨迹的分离点,分离点处的根有何特点？如何求取？知道分离点后要继续画根轨迹还要知道什么？,根据定义，你能判断什么情况下存在分离点吗？,2019/5/13,hanhua,43,分离角,2019/5/13,hanhua,44,Breakaway point,-0.586,-3.414,2019/5/13,hanhua,45,怎样求分离点吗？,所以闭环特征方程,方法一
15、,假设分离点s=d,2019/5/13,hanhua,46,方法二：,2019/5/13,hanhua,47,2019/5/13,hanhua,48,2019/5/13,hanhua,49,分离点一定位于实轴上吗？,分离点,分离点,2019/5/13,hanhua,50,注意,大多数分离点位于实轴上,分离点也可以是共轭复根,所有的分离和汇合点只有经过验证其不在根轨迹上才能舍去,2019/5/13,hanhua,51,分离角,重根的个数（分离点处根轨迹分支数）,2019/5/13,hanhua,52,2019/5/13,hanhua,53,Rule 6： 复数极点的出射角(起始角)和复数零点的入
16、射角(终止角),为什么要求起始角和终止角？,2019/5/13,hanhua,54,相角条件,充分必要条件,2019/5/13,hanhua,55,Re,Im,相角条件,2019/5/13,hanhua,56,2019/5/13,hanhua,57,The Angle of departure at complex Poles:,The Angle of arrival at complex Zeros,结论,2019/5/13,hanhua,58,2019/5/13,hanhua,59,例,2019/5/13,hanhua,60,2019/5/13,hanhua,61,Rule 7：The Root Locus Cros
