\[\frac{C(s)}{R(s)}=\frac{1}{\Delta}p_1\Delta_1=\frac{1}{\Delta}G_1G_2G_3\]
当 \(R(s)=0\) 时,前向通路总增益及余因子式为
\[p_1=G_2G_3,\quad \Delta_1=1-L_4=1+G_1G_5\]
由梅森公式系统传递函数为
\[\frac{C(s)}{N(s)}=\frac{1}{\Delta}G_2G_3(1+G_1G_5)\]
因此系统总输出为
\[C(s)=\frac{G_1G_2G_3R(s)+G_2G_3(1+G_1G_5)N(s)}{1+G_1G_2G_3+G_1G_2G_6+G_2G_3G_4+G_1G_5+G_1G_2G_3G_4G_5}\]
2-61 系统结构图如图 2-124 所示,试求传递函数 \(C(s)/R(s)\)。
图 2-124 系统结构图
解 本系统有六个单独回路,各回路增益为
\[L_1=-G_3G_4G_5G_6,\quad L_2=-G_3,\quad L_3=G_5\]
\[L_4=-G_1,\quad L_5=G_4,\quad L_6=-G_1G_4G_5G_6\]
有三个互不接触回路,各回路增益为
\[L_2L_3=-G_3G_5,\quad L_3L_4=-G_1G_5,\quad L_3L_5=G_4G_5\]
有四条前向通路,各总增益及余因子式为
\[p_1=G_3G_4G_5,\quad \Delta_1=1\]
\[p_2=G_1G_4G_5,\quad \Delta_2=1\]
\[p_3=G_3G_4G_2,\quad \Delta_3=1-G_5\]
\[p_4=G_1G_4G_2,\quad \Delta_4=1-G_5\]
流图特征式为
\[\Delta=1-\sum_{i=1}^{6}L_i+L_2L_3+L_3L_4+L_3L_5\]
\[=1+G_3G_4G_5G_6+G_3-G_5+G_1-G_4+G_1G_4G_5G_6\]
\[-G_3G_5-G_1G_5+G_4G_5\]
由梅森增益公式,系统传递函数为
\[\frac{C(s)}{R(s)}=\frac{1}{\Delta}\sum_{i=1}^{4}p_i\Delta_i\]
\[=\frac{(G_1+G_3)G_4(G_2+G_5-G_2G_5)}{1+G_1+G_3-G_4-G_5-G_1G_5-G_3G_5+G_4G_5+(G_1+G_3)G_4G_5G_6}\]