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）.

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.

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）.

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.