# 赫維茲矩陣

## 赫維茲矩陣和赫維茲穩定性準則

${\displaystyle p(z)=a_{0}z^{n}+a_{1}z^{n-1}+\cdots +a_{n-1}z+a_{n}}$

${\displaystyle n\times n}$方块矩阵

${\displaystyle H={\begin{pmatrix}a_{1}&a_{3}&a_{5}&\dots &\dots &\dots &0&0&0\\a_{0}&a_{2}&a_{4}&&&&\vdots &\vdots &\vdots \\0&a_{1}&a_{3}&&&&\vdots &\vdots &\vdots \\\vdots &a_{0}&a_{2}&\ddots &&&0&\vdots &\vdots \\\vdots &0&a_{1}&&\ddots &&a_{n}&\vdots &\vdots \\\vdots &\vdots &a_{0}&&&\ddots &a_{n-1}&0&\vdots \\\vdots &\vdots &0&&&&a_{n-2}&a_{n}&\vdots \\\vdots &\vdots &\vdots &&&&a_{n-3}&a_{n-1}&0\\0&0&0&\dots &\dots &\dots &a_{n-4}&a_{n-2}&a_{n}\end{pmatrix}}.}$

{\displaystyle {\begin{aligned}\Delta _{1}(p)&={\begin{vmatrix}a_{1}\end{vmatrix}}&&=a_{1}>0\\[2mm]\Delta _{2}(p)&={\begin{vmatrix}a_{1}&a_{3}\\a_{0}&a_{2}\\\end{vmatrix}}&&=a_{2}a_{1}-a_{0}a_{3}>0\\[2mm]\Delta _{3}(p)&={\begin{vmatrix}a_{1}&a_{3}&a_{5}\\a_{0}&a_{2}&a_{4}\\0&a_{1}&a_{3}\\\end{vmatrix}}&&=a_{3}\Delta _{2}-a_{1}(a_{1}a_{4}-a_{0}a_{5})>0\end{aligned}}}

## 赫維茲穩定矩陣

${\displaystyle \mathop {\mathrm {Re} } [\lambda _{i}]<0\,}$

${\displaystyle {\dot {x}}=Ax}$

${\displaystyle G(s)}$是（矩陣型的）传递函数，此传递函数稱為赫維茲传递函数的條件是若${\displaystyle G}$中所有元素的极点都有負的實部。此條件不需要${\displaystyle G(s)}$在特定的${\displaystyle s}$下為赫維茲矩陣，${\displaystyle G(s)}$也不需要是方陣。

${\displaystyle {\dot {x}}(t)=Ax(t)+Bu(t)}$
${\displaystyle y(t)=Cx(t)+Du(t)\,}$