3.2 根轨迹超前校正系统设计与实验
3.2 System Design andExperiment of Root Locus Lead Compensation
基于根轨迹的超前校正设计是通过增加一个串联超前校正环节，即增加开环极点和零点，对原来的系统进行调整，得到所需的根轨迹。校正环节的传递函数通常取下面的公式实现：
Lead compensator based on root locus is designed by adding a series component，namely by adding the open-loop poles and zeros to adjust the original system，which will obtain the desired root locus. The transfer function of compensation is usually taken as following formula：
其中|ZD|＜|PD|；
Where|ZD＜PD|；
用根轨迹校正的基本思想就是假设控制系统有一对闭环主导极点，这样系统的动态性能就可以近似地用这对主导极点所描述的二阶系统来表征。
The main idea of compensation by root locus is to assume that the closed-loop control system has a pair of dominant poles，thus the dynamic performance of the system should be approximately equal to second-order system described by the dominant poles.
（1）根据期望的性能指标计算出阻尼系数ξ和自然振荡频率ωn。
（1）Calculate damping coefficient ξand natural oscillation frequency ωn according to expected performance index.
（2）计算一对主导极点：
（2）Calculate the pair of dominant poles：
这里求得近似主导极点为
s1，2=-6±5.3j
here，the approximate dominant pole is
s1，2=-6±5.3j
（3）确定超前校正器的零点和极点（如图3.7所示）：
（3）Determine zeros and poles of the lead compensator（It shown in Figure 3.7）：
1）直接在期望的闭环极点位置下方增加一个相位超前网络的实零点。这里取z=-6。
1）Add directly a real zero of phase advance network underneath the expected close-loop poles. Take z=-6.
2）利用校正网络极点的相角，使得系统在期望主导极点上满足根轨迹的相角条件：
2）Using phase angle of pole in the compensator network，make sure that the expected dominant pole satisfy phase angle conditions in the root locus：
式中，∑θj为所有各其他开环极点到该极点的向量的相位夹角之代数和，∑φi为所有开环零点到该极点的向量的相位夹角之代数和。
Where ∑θjis algebraic sum of phase angle of vector from all other open-loop poles to pole in question，∑φiis algebraic sum of phase angles of vectors from all other open-loop zeros to the pole.
3）根据幅值条件得到K：
3）According to the amplitude condition，we could obtain K：
Figure 3.7 Determine Zeros and Poles of the Lead Compensator
图3.7 超前校正零极点的确定
由MATLAB编程实现（见附录3.2），我们可以得到下面的参数：
tp=0.0730；
yss=1.0000；
ct=0.064；
ts=0.3892；
sp=102.2；
k=26.0853。
According to the MATLAB programming（See attached Appendix 3.2），we get some parameters as following：
tp=0.0730；
yss=1.0000；
ct=0.064；
ts=0.3892；
sp=102.2；
k=26.0853.
综上得到校正网络传递函数为：
Transfer function of the compensator is：
即极点为 sp=102.2，系数k=26.0853，超调量 P. O.%=6.84%，调节时间为 ts=0.3892，满足性能指标要求。
That is：pole sp=102.2，parameter K=26.0853，overshoot P. O.%=6.84%，Setting time ts=0.3892. all meet the performance requirement.
校正后的根轨迹如图3.8所示，阶跃响应曲线如图3.9所示。
Root locus after compensation is shown in Figure 3.8，System step response after compensation is shown in Figure 3.9：
Figure 3.8 Root Locus after Compensation
图3.8 校正后的根轨迹图
Figure 3.9 System Unit Step Response after Compensation
图3.9 校正后的单位阶跃响应曲线