# 狀態轉移矩陣

## 線性系統的解

${\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} (t)\mathbf {x} (t)+\mathbf {B} (t)\mathbf {u} (t),\mathbf {x} (t_{0})=\mathbf {x} _{0}}$,

${\displaystyle \mathbf {x} (t)=\mathbf {\Phi } (t,t_{0})\mathbf {x} (t_{0})+\int _{t_{0}}^{t}\mathbf {\Phi } (t,\tau )\mathbf {B} (\tau )\mathbf {u} (\tau )d\tau }$

## Peano-Baker級數解

${\displaystyle \mathbf {\Phi } (t,\tau )=\mathbf {I} +\int _{\tau }^{t}\mathbf {A} (\sigma _{1})\,d\sigma _{1}+\int _{\tau }^{t}\mathbf {A} (\sigma _{1})\int _{\tau }^{\sigma _{1}}\mathbf {A} (\sigma _{2})\,d\sigma _{2}\,d\sigma _{1}+\int _{\tau }^{t}\mathbf {A} (\sigma _{1})\int _{\tau }^{\sigma _{1}}\mathbf {A} (\sigma _{2})\int _{\tau }^{\sigma _{2}}\mathbf {A} (\sigma _{3})\,d\sigma _{3}\,d\sigma _{2}\,d\sigma _{1}+...}$

## 其他性質

${\displaystyle \mathbf {\Phi } (t,\tau )\equiv \mathbf {U} (t)\mathbf {U} ^{-1}(\tau )}$

${\displaystyle {\dot {\mathbf {U} }}(t)=\mathbf {A} (t)\mathbf {U} (t)}$

${\displaystyle \mathbf {x} (t)=\mathbf {\Phi } (t,\tau )\mathbf {x} (\tau )}$

${\displaystyle {\frac {\partial \mathbf {\Phi } (t,t_{0})}{\partial t}}=\mathbf {A} (t)\mathbf {\Phi } (t,t_{0})}$ and
${\displaystyle \mathbf {\Phi } (\tau ,\tau )=I}$對於所有的${\displaystyle \tau }$，其中${\displaystyle I}$為單位矩陣[3]

${\displaystyle \mathbf {\Phi } }$也有以下的性質：

 1 ${\displaystyle \mathbf {\Phi } (t_{2},t_{1})\mathbf {\Phi } (t_{1},t_{0})=\mathbf {\Phi } (t_{2},t_{0})}$ 2 ${\displaystyle \mathbf {\Phi } ^{-1}(t,\tau )=\mathbf {\Phi } (\tau ,t)}$ 3 ${\displaystyle \mathbf {\Phi } ^{-1}(t,\tau )\mathbf {\Phi } (t,\tau )=I}$ 4 ${\displaystyle {\frac {d\mathbf {\Phi } (t,t_{0})}{dt}}=\mathbf {A} (t)\mathbf {\Phi } (t,t_{0})}$

${\displaystyle \mathbf {\Phi } (t,t_{0})=e^{\mathbf {A} (t-t_{0})}}$

## 參考資料

1. ^ Baake, Michael; Schlaegel, Ulrike. The Peano Baker Series. Proceedings of the Steklov Institute of Mathematics. 2011, 275: 155–159.
2. Rugh, Wilson. Linear System Theory. Upper Saddle River, NJ: Prentice Hall. 1996. ISBN 0-13-441205-2.
3. ^ Brockett, Roger W. Finite Dimensional Linear Systems. John Wiley & Sons. 1970. ISBN 978-0-471-10585-5.