Suppose the input

$u=-Kx+k_{\text{r}\text{e}\text{f}}r$

is applied to the open$-$loop system

$x^{}=Ax+Bu$

$y=Cx+Du$

Find $K$ and $k_{\text{r}\text{e}\text{f}}$ that achieve a peak time of $p_{\text{t}\text{i}\text{m}\text{e}}$ and a peak overshoot of $p_{\text{o}\text{v}\text{e}\text{r}}$ in the step

response with zero steady$-$state error. All problem data are given as follows:

matlab

import numpy $asnp$

$A=np.$array$([[0\mathrm{.}40,0\mathrm{.}40],[-1\mathrm{.}00,0\mathrm{.}40]])$

$B=np.$array$([[0],[1]])$

$C=np.$array$(\left[\begin{array}{cc}1& 0\end{array}\right])$

$D=np.$array$(\left[\begin{array}{c}0\end{array}\right])$

pTime $=1\mathrm{.}90$

pOver $=0\mathrm{.}33$