式展開: 状態空間表現，バネマスダンパー系，状態行列の基底の変換行列

{\begin{array}{rclrc} \frac{d}{d t} x & = & A x & + & B u \dots A : 状 態 行 列, B : 入 力 行 列, x : 状 態 ベ ク ト ル, u : 入 力 ベ ク ト ル \\ y & = & C x & + & D u \dots C : 出 力 行 列, D : 直 達 行 列, y : 出 力 ベ ク ト ル \end{array}

{\begin{array}{rcl} \frac{d}{d t} [\begin{array}{c} x \\ v \end{array}] & = & [\begin{array}{c} 0 & 1 \\ - \frac{k}{m} & - \frac{c}{m} \end{array}] [\begin{array}{c} x \\ v \end{array}] \dots k : バ ネ, m : マ ス, c : ダ ン パ ー, x : 位 置, v = \frac{d x}{d t} : 速 度 \\ = & [\begin{array}{c} 0 & 1 \\ - ω_{0}^{2} & - 2 γ \end{array}] [\begin{array}{c} x \\ v \end{array}] \dots γ = \frac{c}{2 m}, ω_{0} = \sqrt{\frac{k}{m}} \\ [\begin{array}{c} y \end{array}] & = & [\begin{array}{c} 1 & 0 \end{array}] [\begin{array}{c} x \\ v \end{array}] \end{array}

\begin{array}{rcl} T & = & [\begin{array}{c} v_{1} & v_{2} \end{array}] \\ = & [\begin{array}{c} [\begin{matrix} v_{11} \\ v_{12} \end{matrix}] & [\begin{matrix} v_{21} \\ v_{22} \end{matrix}] \end{array}] \\ = & [\begin{array}{c} v_{11} & v_{21} \\ v_{12} & v_{22} \end{array}] \\ = & [\begin{array}{c} 1 & 1 \\ - γ + ξ & - γ - ξ \end{array}] \\ \dots v_{1} = [\begin{array}{c} v_{11} \\ v_{12} \end{array}] = t [\begin{array}{c} 1 \\ - γ + ξ \end{array}] \\ \dots v_{2} = [\begin{array}{c} v_{21} \\ v_{22} \end{array}] = t [\begin{array}{c} 1 \\ - γ - ξ \end{array}] \\ \dots ξ = \sqrt{γ^{2} - ω_{0}^{2}}, \end{array}

\begin{array}{rcl} T^{- 1} & = & \frac{1}{| \begin{array}{c} T \end{array} |} [\begin{array}{c} v_{22} & - v_{21} \\ - v_{12} & v_{11} \end{array}] \\ \dots A = [\begin{array}{c} a & b \\ c & d \end{array}], A^{- 1} = \frac{1}{| A |} [\begin{array}{c} d & - b \\ - c & a \end{array}] \\ = & \frac{1}{1 \cdot (- γ - ξ) - 1 \cdot (- γ + ξ)} [\begin{array}{c} - γ - ξ & - 1 \\ - (- γ + ξ) & 1 \end{array}] \\ = & \frac{1}{- γ - ξ + γ - ξ} [\begin{array}{c} - γ - ξ & - 1 \\ γ - ξ & 1 \end{array}] \\ = & \frac{1}{- 2 ξ} [\begin{array}{c} - γ - ξ & - 1 \\ γ - ξ & 1 \end{array}] \end{array}

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