(Solved):   You may use Matlab code. Consider the unity feedback system shown in Figure Q1 where the t ...

 

You may use Matlab code.

Consider the unity feedback system shown in Figure Q1 where the transfer function of the plant \( G(s) \) and the transfer function of the the controller \( G_{c}(s) \), are, respectively, given by \[ G(s)=\frac{1}{(s+1)(s+2.5)(s+10)} \] \[ G(s)=\frac{K(s+2)}{s+4} \] 1.1 Sketch the root locus of the closed-loop system using MATLAB. 1.2 Compute the asymptotes for \( K \rightarrow \infty \). \( 1.3 \) Extract the approximate value of gain \( K \) so that the closed-loop system has a damping ratio \( \zeta=\sqrt{2} / 2 \). For \( K \) found, derive the dominant poles. Justify the second-order approximation \( 1.4 \) Compute the percent overshoot, settling time (with a \( 2 \% \) criterion), peak time, and natural frequency for the dominant poles. 1.5 Extract the gain \( K \) for a step response with \( 100 \% \) overshoot and the gain \( K \) at a location with multiple equal poles \( 1.6 \) Calculate the range of \( K \) for which the system is stable. \( 1.7 \) Determine the steady state error for a step input for the gain \( K \) found in Question \( 1.3 \), i.e., if \( \zeta=\sqrt{2} / 2 \)