校正环节的设计

已知单位反馈系统的开环传递函数 $G_{0} (s)$
设计串联校正环节，串联一个零极点校正装置：

\begin{array}{r} G_{c} = \frac{K_{c} (s + z_{c})}{(s + p_{c})} \end{array}

基本步骤：

确定校正装置的零极点位置
设校正后的闭环主导极点 $s_{d}$
$s_{d}$ 在校正后的根轨迹上所以满足相角条件

\begin{array}{r} ∠ (s_{d} - z_{c}) + \sum_{i = 1}^{m} ∠ (s_{d} - z_{i}) = \pm (2 k + 1) π + ∠ (s_{d} - p_{c}) + \sum_{i = 1}^{n} ∠ (s_{d} - p_{i}) \end{array}

$φ = | ∠ (s_{d} - z_{c}) - ∠ (s_{d} - p_{c}) |$
$β = \arctan \frac{\sqrt{1 - ζ^{2}}}{ζ}$

\begin{aligned} \frac{| z_{c} |}{\sin γ} = \frac{ω_{n}}{\sin (π - β - γ)} \\ \frac{| p_{c} |}{\sin (γ + φ)} = \frac{ω_{n}}{\sin (π - β - γ - φ)} \\ α = \frac{| z_{c} |}{| p_{c} |} = \frac{\sin γ \cdot \sin (π - β - γ - φ)}{\sin (π - β - γ) \cdot \sin (γ + φ)} \end{aligned}

\begin{aligned} α = \frac{z_{c}}{p_{c}} \\ \frac{d α}{d γ} = 0 \Rightarrow γ = \frac{1}{2} (π - β - φ) \end{aligned}

校正之后的开环传递函数

\begin{array}{r} G^{'} (s) = G_{0} (s) G_{c} (s) \end{array}

$s_{d}$ 满足模值条件
确定 $K_{c}$