2次系の伝達関数の離散化
ここでは，ラプラス変換された2次系の伝達関数を，近似的にz変換により離散化する。前進オイラー，後退オイラー，双一次変換（パデ）近似を適用する。
※以下は正しいかどうか未検証（間違いご指摘歓迎）
$ \frac{Y(s)}{U(s)} = \frac{b_{0}}{a_{0}+a_{1}s+a_{2}s^{2}} $ \longrightarrow (a_{0}+a_{1}s+a_{2}s^{2})Y(s) = b_{0} U(s)
前進オイラー近似
$ z \simeq 1+sT $ \longrightarrow s=\frac{1-z^{-1}}{Tz^{-1}}
$ \left\{a_{0}+a_{1}\frac{1-z^{-1}}{Tz^{-1}}+a_{2}\left(\frac{1-z^{-1}}{Tz^{-1}}\right)^{2}\right\}Y(s) = b_{0} U(s)
$ \left\{a_{0}T^{2}z^{-2}+a_{1}T(1-z^{-1})z^{-1}+a_{2}\left(1-z^{-1}\right)^{2}\right\}Y(s) = b_{0}T^{2}z^{-2} U(s)
$ \left\{ a_{2} + (a_{1}T-2a_{2})z^{-1} + (a_{0}T^{2}-a_{1}T+a_{2})z^{-2} \right\}Y(s) = b_{0}T^{2}z^{-2} U(s)
漸化式に変換
$ a_{2}y_{n} + (a_{1}T-2a_{2})y_{n-1} + (a_{0}T^{2}-a_{1}T+a_{2})y_{n-2} = b_{0}T^{2}u_{n-2}
$ y_{n} = \frac{- (a_{1}T-2a_{2})y_{n-1} - (a_{0}T^{2}-a_{1}T+a_{2})y_{n-2} + b_{0}T^{2}u_{n-2}}{a_{2}}
後退オイラー近似
$ z \simeq \frac{1}{1-sT} $ \longrightarrow s=\frac{1-z^{-1}}{T}
$ \left\{ a_{0}+a_{1}\frac{1-z^{-1}}{T}+a_{2} \left( \frac{1-z^{-1}}{T} \right) ^{2} \right\}Y(s) = b_{0} U(s)
$ \left\{a_{0}T^{2}+a_{1}T(1-z^{-1})+a_{2}\left(1-z^{-1}\right)^{2}\right\}Y(s) = b_{0}T^{2} U(s)
$ \left\{ a_{0}T^{2}+a_{1}T+a_{2} - (a_{1}T+2a_{2})z^{-1} + a_{2}z^{-2} \right\}Y(s) = b_{0}T^{2} U(s)
漸化式に変換
$ (a_{0}T^{2}+a_{1}T+a_{2})y_{n} - (a_{1}T+2a_{2})y_{n-1} + a_{2}y_{n-2} = b_{0}T^{2} u_{n}
$ y_{n} = \frac{(a_{1}T+2a_{2})y_{n-1} - a_{2}y_{n-2} + b_{0}T^{2} u_{n}}{a_{0}T^{2}+a_{1}T+a_{2}}
双一次変換（パデ近似）
$ z \simeq \frac{1+\frac{sT}{2}}{1-\frac{sT}{2}} $ \longrightarrow s=\frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}}
$ \left\{a_{0}+a_{1}\frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}}+a_{2}\left(\frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}}\right)^{2}\right\}Y(s) = b_{0} U(s)
$ \left\{a_{0}T^{2}(1+z^{-1})^{2}+2a_{1}T(1-z^{-2})+4a_{2}(1-z^{-1})^{2}\right\}Y(s) = b_{0}T^{2}(1+z^{-1})^{2} U(s)
$ \left\{ (a_{0}T^{2}+2a_{1}T+4a_{2}) + (2a_{0}T^{2}-8a_{2})z^{-1} + (a_{0}T^{2}-2a_{1}T+4a_{2}) z^{-2} \right\}Y(s) = b_{0}T^{2}(1+2z^{-1}+z^{-2}) U(s)
漸化式に変換
$ (a_{0}T^{2}+2a_{1}T+4a_{2})y_{n} + (2a_{0}T^{2}-8a_{2})y_{n-1} + (a_{0}T^{2}-2a_{1}T+4a_{2}) y_{n-2} = b_{0}T^{2} u_{n} + 2b_{0}T^{2} u_{n-1} + b_{0}T^{2} u_{n-2}
$ y_{n} = \frac{- (2a_{0}T^{2}-8a_{2})y_{n-1} - (a_{0}T^{2}-2a_{1}T+4a_{2}) y_{n-2} + b_{0}T^{2} u_{n} + 2b_{0}T^{2} u_{n-1} + b_{0}T^{2} u_{n-2}}{a_{0}T^{2}+2a_{1}T+4a_{2}}
＜参考＞連続系を離散化する際の誤差の話