3.3 根轨迹滞后校正设计与实验
3.3 System Design and Experiment of Root Locus Lag Compensation

若系统稳态性能不能满足要求，也可以采用串联滞后校正以提高系统的稳态精度，同时保持其动态性能满足性能指标要求。

If the system cannot fit the steady-state performance requirements，we can design a lag compensator to increase the steady-state accuracy，while keep its required dynamic performance.

滞后校正采用增加开环偶极子来增大系统增益。滞后校正网络的传递函数为

Lag compensator increases the system gain by adding open- loop dipole. The transfer function of lag compensator is：

其中|z_c|＞|p_c|。

Where|z_c|＞|p_c|.

已知系统的开环传递函数为

After knowing open loop transfer function

如果要求其超调量P. O.%＜10%，调节时间t_s＜0.5秒，单位斜坡响应的稳态误差是e_ss≤0.1。

the required overshoot P. O.%＜10%，setting time t_s＜0.5 s，the steady-state error of unit ramp response e_ss≤0.1.

校正后的系统开环传递函数为：

The open loop transfer function after lag compensation is：

由单位斜坡响应稳态误差为：

As the steady-state error of unit ramp response is：

可得k₀≥2.176。按照要求，偶极子的零点和极点比值应为10，因此我们可以选择z=0.15，p=0.015。在确定零极点值时，可采用实验的方法，获得校正环节的传递函数如式（3.7）所示。滞后实验方法确定参数的程序如附录3.3所示。

so k₀≥2.176. As requested，the ratio between zero and pole of dipole should be 10，so we could choose z=0.15，p=0.015. In determining the pole and zero value，the experimental method can be used to obtain the compensator transfer function as shown in formula（3.7）. And the programming of experimental of determine parameters in lag compensator is shown in Appendix 3.3.

即超调量为P. O.%=8.16%，调节时间t_s=0.4141 s，稳态误差e_ss=0.0077＜0.1，满足性能指标要求。

That is：overshoot P. O.%=8.16%，setting time t_s=0.4141 s，steady-state error e_ss=0.0077＜0.1. All meet the required performance index.

校正后的单自由度机械臂系统的根轨迹如图3.10所示：校正后的系统阶跃响应如图3.11所示。

The root locus after lag compensation for POFR-Arm is shown in figure 3.10，and its unit step response curve is shown in figure 3.11.

本文以上介绍的校正环节可通过软件仿真实现，也可以通过RC电路实现。

Compensators which above mentioned should be done both by simulation and by RC circuit.