留数定理
$F(s)=\frac{s+5}{s^2+3s+2}=\frac{K_1}{s+1}+\frac{K_2}{s+2}$
$$K_1=\lim_{s\to -1}[(s+1)\frac{s+5}{(s+1)(s+2)}]$$
$$K_2=\lim_{s\to -2}[(s+2)\frac{s+5}{(s+1)(s+2)}]$$
$F(s)=\frac{2(s+3)}{(s+2)^2(s+1)}=\frac{K_1}{s+1}+\frac{K_2}{(s+2)^2}+\frac{K_3}{(s+2)}$
$$K_1=\lim_{s\to -1}[(s+1)\frac{2(s+3)}{(s+2)^2(s+1)}]$$
$$K_2=\lim_{s\to -2}[(s+2)^2\frac{2(s+3)}{(s+2)^2(s+1)}]$$
$$K_3=\frac{d}{ds}\lim_{s\to -2}[(s+2)^2\frac{2(s+3)}{(s+2)^2(s+1)}]=\frac{d}{ds}[\frac{2s+6}{s+1}]_{s\to{-2}}$$
常用Z变换
| 拉氏变换 E(s) | Z变换E(z) |
|---|---|
| 1 | 1 |
| $e^{-nst}$ | $z^{-n}$ |
| $\frac{1}{s}$ | $\frac{z}{z-1}$ |
| $\frac{1}{s^2}$ | $\frac{Tz}{(z-1)^2}$ |
| $\frac{1}{s^3}$ | $\frac{T^2z(z+1)}{2(z-1)^3}$ |
| $\frac{1}{s^4}$ | $\frac{T^3z(z^2+4z+1)}{6(z-1)^4}$ |
| $\frac{1}{s+a}$ | $\frac{z}{z-e^{-aT}}$ |
| $\frac{1}{(s+a)^2}$ | $\frac{Tze^{-aT}}{(z-e^{-aT})^2}$ |