\[\Delta = 1 - (L_1 + L_2 + L_3) + L_1 L_2 = 1 + a_{11}G_1(s) + a_{22}G_2(s) + (a_{11}a_{22} - a_{12}a_{21})G_1(s)G_2(s)\]
\[p_1 = G_1(s), \qquad L_2 与 p_1 不接触, \qquad \Delta_1 = 1 + a_{22}G_2(s)\]
由梅森增益公式可得传递函数为
\[\frac{C_1(s)}{R_1(s)} = \frac{\sum p_i \Delta_i}{\Delta} = \frac{G_1(s)[1+a_{22}G_2(s)]}{1+a_{11}G_1(s)+a_{22}G_2(s)+(a_{11}a_{22}-a_{12}a_{21})G_1(s)G_2(s)}\]
(2) 求 \(C_1(s)/R_2(s)\)。
与(1)相比,梅森增益公式中 \(\Delta\) 不变,只是 \(\sum p_i\Delta_i\) 发生变化,此时系统的前向通道
\[p_1 = -a_{21}G_1(s)G_2(s), \qquad \Delta_1 = 1\]
由梅森增益公式可得传递函数为
\[\frac{C_1(s)}{R_2(s)} = \frac{\sum p_i \Delta_i}{\Delta} = \frac{-a_{21}G_1(s)G_2(s)}{1+a_{11}G_1(s)+a_{22}G_2(s)+(a_{11}a_{22}-a_{12}a_{21})G_1(s)G_2(s)}\]
(3) 求 \(C_2(s)/R_1(s)\)。
此时系统的前向通道
\[p_1 = -a_{12}G_1(s)G_2(s), \qquad \Delta_1 = 1\]
由梅森增益公式可得传递函数为
\[\frac{C_2(s)}{R_1(s)} = \frac{\sum p_i \Delta_i}{\Delta} = \frac{-a_{12}G_1(s)G_2(s)}{1+a_{11}G_1(s)+a_{22}G_2(s)+(a_{11}a_{22}-a_{12}a_{21})G_1(s)G_2(s)}\]
(4) 求 \(C_2(s)/R_2(s)\)。
此时系统的前向通道
\[p_1 = G_2(s), \qquad L_1 与 p_1 不接触, \qquad \Delta_1 = 1 + a_{11}G_1(s)\]
由梅森增益公式可得传递函数为
\[\frac{C_2(s)}{R_2(s)} = \frac{\sum p_i \Delta_i}{\Delta} = \frac{G_2(s)[1+a_{11}G_1(s)]}{1+a_{11}G_1(s)+a_{22}G_2(s)+(a_{11}a_{22}-a_{12}a_{21})G_1(s)G_2(s)}\]
又解 可用结构图简化的方法对结果进行验证,以 \(C_1(s)/R_1(s)\) 为例。
在求取 \(C_1(s)/R_1(s)\) 时,图2-107系统的等效结构图如图2-109所示。
经过反馈和串联等效,可得图2-110;再经过反馈等效,可得图2-111。
图 2-109 系统等效结构图(\(C_1(s)/R_1(s)\))
图 2-110 系统简化结构图(\(C_1(s)/R_1(s)\))
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