Accurate expression and high-precision estimation of $\left\Vert e^{At}\right\Vert_{2}$

The state transition matrix
$e^{At}$
and its norm
$\Vert e^{At}\Vert_{2}$
play an indispensable role in the analysis and design of control systems. Existing literature has provided two kinds of estimation results of
$\Vert e^{At}\Vert_{2}$
. However, because of rather low precision, they can only be applied for stability, local controllability, observability analysis, and it is incapable for them to estimate the dynamic performance indexes. Main contributions of our work are summarized as two aspects: 1) the accurate expressions of
$\Vert e^{At}\Vert_{2}$
while
$A$
is normal and 2) the high-precision estimation results of
$\Vert e^{At}\Vert_{2}$
while
$A$
is nonnormal. Furthermore, these results above not only effectively reduce a lot of computation load but also significantly improve estimation precision, and they can be used to estimate the dynamic performance index of control systems.