页面顶部残留笔迹(与本页主要内容无关,书本页边缘)
\[\frac{\pi}{2}\]
\[y\]
\[e^{-(4+\eta)}(\eta-9)\]
红笔批注区(页面上方,疑为对下方"五"题D(s)的重新推算/纠错)
\[G_1G_2(s) = \frac{2}{s(s+1)(s+2)} + 1 = \frac{D(s)}{s(s+1)(s+2)}\]
① \(D(s) = s(s^2+3s+2)+2\)
\(= s^3+3s^2+2s+2\)
劳斯表:
| \(s^3\) | 1 | 2 |
|---|---|---|
| \(s^2\) | 3 | 2 |
| \(s^1\) | \(\frac{4}{3}\) | \(\frac{6-2}{3}\) |
(此处红笔字迹不清,似写"俶穷0")
$E(s) = $ (未写完)
五. (15分)已知单位反馈控制系统的 \(G_1(s)=\dfrac{2}{s(s+2)}\);\(G_2(s)=\dfrac{2}{(s+1)}\);
\(n(t)=1,r(t)=2+0.5t\),试求系统的稳态误差。
\[R(s)=\frac{2}{s}+\frac{0.5}{s^2}\qquad N(s)=\frac{1}{s}\]
(1) \(\Phi(s) = \dfrac{G_1(s)G_2(s)}{1+G_1(s)G_2(s)} = \dfrac{2}{s(s+2)(s+1)+2} = \dfrac{2}{s^3+3s^2+2s+2}\)
\(D(s)=s^3+3s^2+2s+2\)
用劳斯方程:
| \(s^3\) | 1 | 2 | 0 |
|---|---|---|---|
| \(s^2\) | 3 | 2 | |
| \(s^1\) | \(\frac{4}{3}\) | 0 | |
| \(s^0\) | 2 |
由上图可知,稳定。
(2) 首先求N(s)?
\[\Phi_{ER} = \frac{E(s)}{R(s)} = \frac{R(s)-\dfrac{G_1(s)G_2(s)}{1+G_1(s)G_2(s)}R(s)}{R(s)} = \frac{1}{1+G_1(s)G_2(s)}\]
\[\Phi_{EN} = \frac{E(s)}{N(s)} = -\frac{C(s)}{N(s)} = -\frac{G_2(s)}{1+G_1(s)G_2(s)}\]
\[= -\frac{2s(s+2)}{s^3+3s^2+2s+2}\cdot\frac{1}{s}\]
\[E(s)=\Phi_{ER}R(s)+\Phi_{EN}N(s)= \text{(圈起部分,字迹被涂改圈盖,无法辨认)}\]
(下方公式跨页/被截断,末行公式含分母 \(s^3+3s^2+2s+2\),分子及等号右侧内容因原稿在页面底部被截断,无法完整转写)