脉冲传递函数

Z-Transfer Function z 传递函数

线性定常离散系统的脉冲传递函数定义为：系统输出采样信号的 z 变换与输入采样信号的 z 变换之比。用于描述系统的差分方程：

\begin{aligned} G (z) & = \frac{Y (z)}{U (z)} = \frac{Z [y (k T)]}{Z [u (k T)]} = \frac{\sum_{k = 0}^{\infty} y (n T) z^{- k}}{\sum_{k = 0}^{\infty} u (n T) z^{- k}} \end{aligned}

\begin{array}{r} Y (s) = G_{1}^{*} (s) G_{2}^{*} (s) U^{*} (s) \Rightarrow \frac{Y (z)}{U (z)} = G_{1} (z) G_{2} (z) \end{array}

\begin{array}{r} Y (s) = [G_{1} (s) G_{2} (s)]^{*} U^{*} (s) \Rightarrow \frac{Y (z)}{U (z)} = G_{1} G_{2} (z) \end{array}

\begin{array}{r} G_{h} (s) = \frac{1 - e^{- T s}}{s} G (z) = \frac{Y (z)}{U (z)} = Z [(1 - e^{- T s}) \cdot \frac{G_{o} (s)}{s}] = (1 - z^{- 1}) Z [\frac{G_{o} (s)}{s}] \end{array}

\begin{aligned} Y (s) = \frac{G_{1} (s) G_{2} (s)}{1 + G_{1} (s) G_{2} (s) F (s)} R (s) \\ \Rightarrow & Y^{*} (s) = \frac{[G_{1} (s) G_{2} (s)]^{*}}{1 + [G_{1} (s) G_{2} (s) F (s)]^{*}} R (s) \\ \Rightarrow & \frac{Y (z)}{R (z)} = \frac{G_{1} G_{2} (z)}{1 + G_{1} G_{2} F (z)} \end{aligned}

\begin{aligned} Y (s) = \frac{G_{1} (s) G_{2} (s) G_{3} (s)}{1 + G_{1} (s) G_{2} (s) G_{3} (s) H (s)} R (s) \\ \Rightarrow & Y^{*} (s) = \frac{{[G_{1} (s) R (s)]}^{*} G_{2}^{*} (s) G_{3}^{*} (s)}{1 + G_{2}^{*} (s) [G_{1} (s) G_{3} (s) H (s)]^{*}} \\ \Rightarrow & \frac{Y (z)}{R (z)} = \frac{G_{2} (z) G_{3} (z) G_{1} R (z)}{1 + G_{2} (z) G_{3} H G_{1} (z)} \frac{1}{R (z)} \end{aligned}

Important

一定要注意采样开关的位置！！！