$2024$ Mechanical Systems Seminar III Control Engineering $($Week $3)$

The positi

The equation of mation of the controllaci object is

$m{y}^{}(t)=-c{y}^{}(t)+u(t)$

Answer the following questions when $m=1$ and $c=10.$ Show not only your answer but also your derivation process. Fill in the Bode plot on the semi$-$logarithmic graph on the answer sheet and briefly explain how you drew it$.$

$(1)$ Find the transfer function $P(s)$ from the control input $U(s)$ to the output $Y(s)$ of the controlled object.

$(2)$ Draw a Bode plot of $P(s)$ using a broken line approximation.

$(3)$ Control is performed using the following Pl controller, where $K>0,1>0.$ Draw a Bode plot of $C(s)$ using a broken line approximation when $K=100$ and $w1=0\mathrm{.}1.$

$C(s)=K(1+\frac{{}_{\text{I}}}{s})$

$(4)$ Find the open$-$loop transfer function $L(s)=P(s)C(s)$ of the feedback system when $K=100$ and ${}_1=0\mathrm{.}1.$

$(5)$ Draw a Bode plot of $L(s)$ when $K=100,$ wl $=0\mathrm{.}1$ using a broken line approximation.

$(6)$ From the broken$-$line approximation of the Bode plot, find the gain crossover frequency and phase margin of $L(s)$ when $K=100,w_1=0\mathrm{.}1.$

$(7)$ Find the value of K that maximizes the phase margin when $w_i=0\mathrm{.}1.$

$(8)$ Find the range of $"w"$ for which the feedback system is stable when $K=100.$