Part B: Frequency Response

In this lab, you will also be guided to design a lead and then a lag compensator for improving the transient response for the output of the closed loop system shown in Figure $1.$

The transfer function of the plant is given in $(2).$

$G(s)=\frac{4}{0\mathrm{.}9s^2+s+1}$

We first analyse the system with only scalar controller:

Type the following to form the transfer function.

$s=tf(s^{\text{'}})$;

$G=\frac{4}{{0\mathrm{.}9}^{**}{s}^2+s+1}$;

$Ks=1$;

$Q=Ks**G$;

figure$(1)$;

bode$(Q)$; $\frac{?}{?}\%to$ produce Bode diagrams for $G(s)**K(s)$

Task $5$ for the report:

$5\mathrm{.}1)$ Copy the bode diagrams plotted above, and present in the report. Then based on the bode diagrams, estimate the phase margin PM and crossover frequency ${}_o$ from the bode diagrams.

$5\mathrm{.}2)$ Derive the closed$-$loop transfer functions $($considering the closed$-$loop system employs unity$-$gain negative feedback control$).[1$ mark$]$

$[$Hint: Note that the closed loop transfer function is $T(s)=\frac{G(s)K(s)}{1+G(s)K(s)}$ and ${}_n$ must be obtained from the denominator of $T(s)$ as it may have non$-$unity DC gain.$]$

$5\mathrm{.}3)$ Check if the closed$-$loop bandwidth $($ ~~${}_n)$ is approximately equal to the crossover frequency $({}_0),$ and the phase margin $($in degrees$)$ approximately equal to the closed$-$loop damping ratio, $100.$