(Solved): 1. Draw the root locus plot for a control system with unity feedback having the forward-path tr ...

1. Draw the root locus plot for a control system with unity feedback having the forward-path transfer function $G(s)=\frac{K}{s(s+1)(s+5)};K?0$ Give all relevant characteristics of the curves that are useful in establishing the locus. Find the least value of $\mathrm{K}$ that can be used before continuous oscillations occur. Find the frequency of the continuous oscillations when $\mathrm{K}$ is just large enough to give this condition. 2. Prove that a combination of two poles $s=?a_1$ and $s=?a_2$ and zero $s=?b$ to the left of both of them on the real axis, results in a root locus whose complex root branches from a circle centered at the zero with radius given by $\sqrt{\left(\left(b?a_1\right)\left(b?a_2\right)\right)}$; the root locus gain varying from zero to $?$. 3. Consider a unity-feedback system with a forward path transfer function, $G(s)=\frac{K(s+3)}{s(s+2)};K?0$ Show that part of the root locus is a circle. Construct the root locus and determine the damping ratio for a maximum oscillatory response. What is the value of $\mathrm{K}$ at this point of the locus?