\[\phi_e(S) = \dfrac{1}{1+\dfrac{1}{S}\cdot\dfrac{1}{S+1}}\]
\[= \dfrac{S(S+1)}{S(S+1)+1}\]
三. 设控制系统如下图所示.
若误差定义为 \(E(S) = R(S) - C(S)\) ,且 \(r(t) = t\) , \(n(t) = 1(t)\) ,试求:
(1) . \(\dfrac{E(S)}{R(S)}\) , \(\dfrac{E(S)}{N(S)}\) ; (2) 系统总的稳态误差 \(e_{ss}(\infty)\) .
解:
\(\because r(t) = t\) , \(n(t) = 1(t)\)
其拉氏变换得:
\(R(S) = \dfrac{1}{S^2}\) , \(N(S) = \dfrac{1}{S}\)
(1). 求 \(\dfrac{C(S)}{R(S)}\) 时 \(N(S)\) 为 0.
\(\therefore\) 得 \(C(S) = \dfrac{S+1}{S^2+S+1} R(S)\)
\(E(S) = R(S) - C(S) = R(S) - \dfrac{S+1}{S^2+S+1} R(S) = \dfrac{S^2}{S^2+S+1} R(S)\)
\(\therefore \dfrac{E(S)}{R(S)} = \dfrac{S^2}{S^2+S+1}\)
求 \(\dfrac{E(S)}{N(S)}\) 时 \(R(S)\) 为 0
\(C(S) = \dfrac{-S^2-S}{S^2+S+1} N(S)\)
\(E(S) = R(S) - C(S) = 0 - C(S) = \dfrac{S^2+S}{S^2+S+1} N(S)\)
\(\therefore \dfrac{E(S)}{N(S)} = \dfrac{S^2+S}{S^2+S+1}\)
(2) 应用劳斯判据.
| \(S^2\) | 1 | 1 |
|---|---|---|
| \(S^1\) | 1 | 0 |
| \(S^0\) | 1 | 0 |
\(\because\) 系统稳定,存在稳态误差.
\[e_{ss}(\infty) = \lim_{S \to 0} S\,E(S) = \lim_{S \to 0} S \cdot \left( \dfrac{S^2}{S^2+S+1} R(S) + \dfrac{S^2+S}{S^2+S+1} N(S) \right)\]
\[= \lim_{S \to 0} \dfrac{S^3}{S^2+S+1}\cdot\dfrac{1}{S^2} + \dfrac{S^3+S^2}{S^2+S+1}\cdot\dfrac{1}{S}\]
\[= 0\]