(Solved): Problem 3: Analysis of Stability Consider the control system shown in Figure 4 Figure 4: Control sy ...

Problem 3: Analysis of Stability Consider the control system shown in Figure 4 Figure 4: Control system. (p3.1) Determine the transfer function of the closed-loop control system \( G(s) \mathrm{Cs}=C(s) / R(s) \) as a function of \( K \) and \( s \). (p3.2) Determine the characteristic equation of the transfer function of the control system. (p3.3) For each value of \( K=1,2,3,4,5,5,7,8,9,10 \) determine the poles, make the rlocus plot, and the plot of the unit-step input, \( r(t)=1 \), and response, \( c(t) \), signals as a function of time, \( t \). Consider a time interval from 0 to 100 seconds. Use the subplot function in Matlab. Then, draw a table where the first column is the value of \( K \), the second column is the values of the poles for each \( K \), and the third column is the type of stability signal: stable, marginally stable, and unstable. (p3.4) For which value of \( K \) does the system change from stable to unstable? Write a proper explanation.