3.1 被控对象的传递函数与根轨迹图分析
3.1 Analysis for the Transfer Function and Root Locus of The Plant
根轨迹是指系统的增益K由零到正无穷大时的闭环特征方程的根在s平面上的变化轨迹。在系统的增益K由零变化到负无穷大时闭环特征方程的根的轨迹为补根轨迹。系统闭环特征方程的根轨迹与补根轨迹称为全根轨迹。通常情况下根轨迹是指增益K由零到正无穷大下的根的轨迹。
Root locus Means the closed-loop char-acteristic square’s root locus in s plane when gain K change from zero to positive infinity.When gain K from zero to negative infinity called complement root locus. And it is called full root locus when K changes from negative infinity to positive infinity. Normally，root locus is considered as the gain K varies from zero to positive infinity.
根据第2章的分析，被控对象的传递函数为：
According to the conclusion in Chapter 2，the transfer function of DC motor is as following：
假定式（3.1）的增益K是可变化（可调节的）的，则式（3.1）可以写为式（3.2）：
Assume that gain K in（3.1）is adjustable，and the formula can be written as（3.2）：
校正反馈控制系统的实物构成如图3.2所示。运用MATLAB工具可以很容易地画出式（3.2）中当K从零变化到无穷大时的根轨迹图，如图3.3所示：
The structure diagram of compensation feedback control system is shown in Figure 3.2. Using MATLAB tools，the root locus diagram in Figure 3.3 should be drawn when K selected different data（K is changed from zero to positive infinity）.
Figure 3.2 The Structure Diagram of Compensation Feedback Control System
图3.2 校正反馈控制系统的实物构成示意图
Figure 3.3 The Root Locus of Plant（POFR-Arm）
图3.3 被控对象（便携式单自由度机械臂）的根轨迹图
由这个系统可以看出，被控对象单自由度机械臂是一个自身稳定的系统，也就是说，无论K怎么变化，都不会导致闭环系统的根落入右半s平面而引起不稳定。我们可以通过调整K的变化观察其对系统性能指标的影响。即，调节 K值的大小，来改善系统的超调量、过渡过程时间、稳态误差等值。
It is obviously that the POFR-Arm is a stabilizing system，that is to say，no matter how K changes，the root of the closed-loop system should not fall in right s plane and cause instability. By adjusting the parameter of K，we can observe how it affects system performance，that is，adjusting the parameter of K will improve the values of system overshoot，setting time，steady state error etc.
进行以下3个实验：
The following three experiments should be done：
1）取K=1；
1）Take K=1；
2）取K=84.42；
2）Take K=84.42；
3）取K=200。
3）Take K=200。
分别画出对应的阶跃响应曲线如图3.4、图3.5、图3.6所示。
Draw the step response curve as Figure 3.4，Figure 3.5，Figure 3.6.
由图3.4可以看出，当K=1，单自由度机械臂的系统没有超调，稳态误差趋向于零。
It can be seen in Figure 3.4，when K=1，the POFR-Arm is a system without overshoot，steady-state error tends to zero.
Figure 3.4 Step Response Curve of POFR-Arm when K=1
图3.4 K=1时单自由度机械臂的单位阶跃响应曲线
Figure 3.5 Step Response Curve of POFR-Arm when K=84.42
图3.5 K=84.42时单自由度机械臂的单位阶跃响应曲线
取K=84.42，系统的响应曲线如图3.5所示。
Let K=84.42，the system response curve can be obtained as shown in Figure 3.5.
当K=84.42，可以看出一自由度机械臂的单位阶跃响应曲线依然是没有超调，稳态误差趋向于零。但响应速度比图3.4快（图3.5的坐标尺度为0.2 s，而3.4的坐标尺度为20 s）。
It can be seen when K=84.42，the POFR-Arm system is still a system without overshoot；steady-state error tends to zero，but response faster than Figure 3.4（Scale interval in Figure 3.5 is 0.2 seconds，Scale interval in Figure 3.4 is 20 seconds）.
Figure 3.6 Step Response Curve of POFR-Arm when K=200
图3.6 K=200单自由度机械臂的单位阶跃响应曲线
从图3.6可以看出，当K=200，一自由度机械臂系统的超调量近似为0.0731，稳态误差最终趋向于零，但响应速度比图3.4和图3.5更快（注意横坐标的尺度，图3.6为0.2 s；图3.5为0.2 s，图3.4为20 s）。
It can be seen in Figure 3.6，when K=200，the POFR-Arm system is a system with 0.0731 percent overshoot，steady-state error tends to zero，but response faster than Figure 3.4 and Figure 3.5（The hori-zontal scale interval in Figure 3.6 is 0.2 seconds，scale interval in Figure 3.5 is 0.2 seconds，scale interval in Figure 3.4 is 20 seconds）.
由上面的图形可以看出，当K取不同数值时，将会导致不同的控制性能指标。附录3.1是关于MATLAB的程序实现。
It is obviously in the curves，when K takes different values，it will cause the different performance in control system. There is MATLAB programming shown in Appendix 3.1.