Consider the system shown in Fig. Determine the value of 'a' such that the damping ratio is 0.5. Also obtain the values of the rise time 't_r' and maximum overshoot 'M_p' in its step response.

The open loop transfer function of a unity feedback open-loop system is $$\frac{2s^2+6s+5}{\left(s+1\right)^2\left(s+2\right)}$$. The characteristic equation of the closed loop system is

Consider a feedback control system with loop transfer function $$$G\left(s\right)H\left(s\right)=\frac{K\left(1+0.5s\right)}{s\left(1+s\right)\left(1+2s\right)}$$$ The type of the closed loop system is

Draw a signal flow graph for the following set of algebraic equations: $$$\begin{array}{l}y_2=ay_1-\;gy_3\\y_3=ey_2+\;cy_4\\y_4=by_2-dy_4\end{array}$$$ Hence, find the gains $$\frac{y_2}{y_1}$$ and $$\frac{y_3}{y_1}$$.