(Solved): Problem 3: Given the following expression for the denominator of the closed-loop transfer function ...

Problem 3: Given the following expression for the denominator of the closed-loop transfer function for a system that uses a proportional controller \( G_{c}(s)=K \) \[ 1+K \frac{1}{(s+2)\left(s^{2}+10 s+34\right)}=0 \] 1. Determine the roots of this characteristic equation when \( K=0 \). 2. To get an idea of where the roots go when \( K \) gets large, determine the imaginary axis crossings. Set \( s=j \omega \) and solve for \( \omega \) and \( K \). 3. Use the MATLAB rlocus command to draw the root locus. 4. Determine the range of \( K \) for which this system is stable.