# 可控制性格拉姆矩陣

${\displaystyle {\dot {x}}(t)=A(t)x(t)+B(t)u(t)}$

${\displaystyle y(t)=C(t)x(t)+D(t)u(t)\,}$

${\displaystyle W_{c}(t_{0},t_{1})=\int _{t_{0}}^{t_{1}}\Phi (\tau ,t_{0})B(\tau )B^{T}(\tau )\Phi ^{T}(\tau ,t_{0})d\tau }$ ,

## 連續時間，線性非時變系統

${\displaystyle {\dot {x}}(t)=Ax(t)+Bu(t)}$

${\displaystyle y(t)=Cx(t)+Du(t)\,}$

${\displaystyle {\boldsymbol {W_{c}}}(t)=\int _{0}^{t}e^{{\boldsymbol {A}}\tau }{\boldsymbol {B}}{\boldsymbol {B^{T}}}e^{{\boldsymbol {A}}^{T}\tau }d\tau }$

${\displaystyle {\boldsymbol {A}}}$ 若穩定（所有的特徵值實部均為負），可控制性格拉姆矩阵也是以下李亞普諾夫方程的唯一解

${\displaystyle {\boldsymbol {A}}{\boldsymbol {W}}_{c}+{\boldsymbol {W}}_{c}{\boldsymbol {A^{T}}}=-{\boldsymbol {BB^{T}}}}$

${\displaystyle {\boldsymbol {A}}}$ 若穩定（所有的特徵值實部均為負），而且${\displaystyle {\boldsymbol {W}}_{c}}$ 也是正定矩陣，則此系統具有可控制性，也就是${\displaystyle ({\boldsymbol {A}},{\boldsymbol {B}})}$ 矩陣對具有可控制性。

1. ${\displaystyle n\times np}$ 的可控制性矩陣

${\displaystyle {\mathcal {C}}=[{\begin{array}{ccccc}{\boldsymbol {B}}&{\boldsymbol {AB}}&{\boldsymbol {A^{2}B}}&...&{\boldsymbol {A^{n-1}B}}\end{array}}]}$

2. ${\displaystyle n\times (n+p)}$ 矩陣

${\displaystyle [{\begin{array}{cc}{\boldsymbol {A}}{\boldsymbol {-\lambda }}{\boldsymbol {I}}&{\boldsymbol {B}}\end{array}}]}$

## 和李亞普諾夫方程的關係

${\displaystyle {\boldsymbol {A}}{\boldsymbol {W}}_{c}+{\boldsymbol {W}}_{c}{\boldsymbol {A^{T}}}=-{\boldsymbol {BB^{T}}}}$

${\displaystyle {\boldsymbol {W_{c}}}=\int _{0}^{\infty }e^{{\boldsymbol {A}}\tau }{\boldsymbol {BB^{T}}}e^{{\boldsymbol {A}}^{T}\tau }d\tau }$

${\displaystyle {\begin{array}{ccccc}{\boldsymbol {A}}{\boldsymbol {W}}_{c}+{\boldsymbol {W}}_{c}{\boldsymbol {A^{T}}}&=&\int _{0}^{\infty }{\boldsymbol {A}}e^{{\boldsymbol {A}}\tau }{\boldsymbol {BB^{T}}}e^{{\boldsymbol {A}}^{T}\tau }d\tau &+&\int _{0}^{\infty }e^{{\boldsymbol {A}}\tau }{\boldsymbol {BB^{T}}}e^{{\boldsymbol {A}}^{T}\tau }{\boldsymbol {A^{T}}}d\tau \\&=&\int _{0}^{\infty }{\frac {d}{d\tau }}(e^{{\boldsymbol {A}}\tau }{\boldsymbol {B}}{\boldsymbol {B}}^{T}e^{{\boldsymbol {A}}^{T}\tau })d\tau &=&e^{{\boldsymbol {A}}t}{\boldsymbol {B}}{\boldsymbol {B}}^{T}e^{{\boldsymbol {A}}^{T}t}|_{t=0}^{\infty }\\&=&{\boldsymbol {0}}-{\boldsymbol {BB^{T}}}\\&=&{\boldsymbol {-BB^{T}}}\end{array}}}$

### 格拉姆矩陣的性質

${\displaystyle {\boldsymbol {A}}}$ 是穩定矩陣（所有的特徵值實部均為負），可以證明${\displaystyle {\boldsymbol {W}}_{c}}$ 是唯一的。利甪反證法，先假設以下方程有二個不同解

${\displaystyle {\boldsymbol {A}}{\boldsymbol {W}}_{c}+{\boldsymbol {W}}_{c}{\boldsymbol {A^{T}}}=-{\boldsymbol {BB^{T}}}}$

${\displaystyle {\boldsymbol {A}}{\boldsymbol {(W}}_{c1}-{\boldsymbol {W}}_{c2})+{\boldsymbol {(W}}_{c1}-{\boldsymbol {W}}_{c2}){\boldsymbol {A^{T}}}={\boldsymbol {0}}}$

${\displaystyle e^{{\boldsymbol {A}}t}[{\boldsymbol {A}}{\boldsymbol {(W}}_{c1}-{\boldsymbol {W}}_{c2})+{\boldsymbol {(W}}_{c1}-{\boldsymbol {W}}_{c2}){\boldsymbol {A^{T}}}]e^{{\boldsymbol {A^{T}}}t}={\frac {d}{dt}}[e^{{\boldsymbol {A}}t}[({\boldsymbol {W}}_{c1}-{\boldsymbol {W}}_{c2})e^{{\boldsymbol {A^{T}}}t}]={\boldsymbol {0}}}$

${\displaystyle 0}$ 積分到${\displaystyle \infty }$

${\displaystyle [e^{{\boldsymbol {A}}t}[({\boldsymbol {W}}_{c1}-{\boldsymbol {W}}_{c2})e^{{\boldsymbol {A^{T}}}t}]|_{t=0}^{\infty }={\boldsymbol {0}}}$

${\displaystyle {\boldsymbol {0}}-({\boldsymbol {W}}_{c1}-{\boldsymbol {W}}_{c2})={\boldsymbol {0}}}$

${\displaystyle {\boldsymbol {x^{T}W_{c}x}}=\int _{0}^{\infty }{\boldsymbol {x}}^{T}e^{{\boldsymbol {A}}t}{\boldsymbol {BB^{T}}}e^{{\boldsymbol {A}}^{T}t}{\boldsymbol {x}}dt=\int _{0}^{\infty }\left\Vert {\boldsymbol {B^{T}e^{{\boldsymbol {A}}^{T}t}{\boldsymbol {x}}}}\right\Vert _{2}^{2}dt}$

## 離散時間，線性非時變系統

${\displaystyle {\begin{array}{c}{\boldsymbol {x}}[k+1]{\boldsymbol {=Ax}}[k]+{\boldsymbol {Bu}}[k]\\{\boldsymbol {y}}[k]={\boldsymbol {Cx}}[k]+{\boldsymbol {Du}}[k]\end{array}}}$

${\displaystyle {\boldsymbol {W}}_{dc}=\sum _{m=0}^{\infty }{\boldsymbol {A}}^{m}{\boldsymbol {BB}}^{T}({\boldsymbol {A}}^{T})^{m}}$

${\displaystyle {\boldsymbol {A}}}$ 若穩定（所有的特徵值絕對值均小於1），也是以下離散李亞普諾夫方程的解

${\displaystyle W_{dc}-{\boldsymbol {A}}{\boldsymbol {W}}_{dc}{\boldsymbol {A^{T}}}={\boldsymbol {BB^{T}}}}$

${\displaystyle {\boldsymbol {A}}}$ 若穩定（所有的特徵值絕對值均小於1），而且${\displaystyle {\boldsymbol {W}}_{dc}}$ 也是正定矩陣，則此系統有可控制性。

## 線性時變系統（LTV）

${\displaystyle {\begin{array}{c}{\dot {\boldsymbol {x}}}(t){\boldsymbol {=A}}(t){\boldsymbol {x}}(t)+{\boldsymbol {B}}(t){\boldsymbol {u}}(t)\\{\boldsymbol {y}}(t)={\boldsymbol {C}}(t){\boldsymbol {x}}(t)\end{array}}}$

${\displaystyle {\boldsymbol {W}}_{c}(t_{0},t_{1})=\int _{_{0}}^{^{\infty }}{\boldsymbol {\Phi }}(t_{1},\tau ){\boldsymbol {B}}(\tau ){\boldsymbol {B}}^{T}(\tau ){\boldsymbol {\Phi }}^{T}(t_{1},\tau )d\tau }$

### 格拉姆矩陣的性質

${\displaystyle {\boldsymbol {W}}_{c}(t_{0},t_{1})={\boldsymbol {W}}_{c}(t_{0},t)+{\boldsymbol {\Phi }}(t,t_{0}){\boldsymbol {W}}_{c}(t,t_{0}){\boldsymbol {\Phi }}^{T}(t,t_{0})}$

${\displaystyle {\boldsymbol {\Phi }}(t_{0},t_{1})={\boldsymbol {\Phi }}(t_{1},\tau ){\boldsymbol {\Phi }}(\tau ,t_{0})}$

## 參考資料

1. ^ Chen, Chi-Tsong. Linear System Theory and Design Third Edition. New York, New York: Oxford University Press. 1999: 145. ISBN 0-19-511777-8.
2. ^ Chen, Chi-Tsong. Linear System Theory and Design Third Edition. New York, New York: Oxford University Press. 1999: 169. ISBN 0-19-511777-8.
3. ^ Chen, Chi-Tsong. Linear System Theory and Design Third Edition. New York, New York: Oxford University Press. 1999: 176. ISBN 0-19-511777-8.