可观测性格拉姆矩阵

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

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

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

連續時間，線性非時變系統

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

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

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

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

${\displaystyle {\boldsymbol {A^{T}}}{\boldsymbol {W}}_{o}+{\boldsymbol {W}}_{o}{\boldsymbol {A}}=-{\boldsymbol {C^{T}C}}}$

${\displaystyle {\boldsymbol {A}}}$ 若穩定（所有的特徵值實部均為負），而且${\displaystyle {\boldsymbol {W}}_{o}}$ 也是正定矩陣，則此系統有可观测性。

離散時間，線性非時變系統

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

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

${\displaystyle {\boldsymbol {A}}}$ 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

${\displaystyle W_{do}-{\boldsymbol {A^{T}}}{\boldsymbol {W}}_{do}{\boldsymbol {A}}={\boldsymbol {C^{T}C}}}$

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

參考資料

• Chen, Chi-Tsong. Linear System Theory and Design Third Edition. New York, New York: Oxford University Press. 1999. ISBN 0-19-511777-8.