a. Calculation of the closed loop transfer function in the form Gcl, (s) = N(s)/D(s):A closed-loop transfer function can be written as follows: Gcl(s)=Gp(s)Gc(s)Gp(s)Gc(s)+Gh(s)Gp(s)Where Gp(s) is the plant transfer function, Gc(s) is the controller transfer function, and Gh(s) is the sensor transfer function. Substituting the provided values, we get the following result.Gc(s) = Kp, Gp(s) = (s+2)/(2s²+2s+1), and Gh(s) = (s+1)/(2s+1)By substituting the provided values, we get the following result.Gcl(s)=Gp(s)Gc(s)/[1+Gh(s)Gp(s)Gc(s)]Gcl(s) = Kp(s+2)/(2s^3+5s^2+5s+2Kp)Therefore, the closed-loop transfer function of the system is Gcl(s) = Kp(s + 2) / (2s^3 + 5s^2 + 5s + 2Kp).b. Calculation of the condition on K that makes the system stable:We will determine the condition for the system to be stable by analyzing the roots of the denominator's characteristic equation, which is 2s^3 + 5s^2 + 5s + 2Kp = 0.By applying Routh-Hurwitz stability criteria to the characteristic equation, we obtain the following conditions.2Kp>0,5>0,1Kp-10>0,2Kp + 5>0By combining all these conditions, we can say that the system will be stable if Kp > 0.5.c. Calculation of the condition on K that sets the stability margin to 1/2:Now, we have to find the condition on K that sets the stability margin to 1/2 if it exists.We will calculate the phase margin using the closed-loop transfer function's magnitude and phase expressions. The phase margin is calculated using the following formula:Phase margin (PM) = ∠Gcl(jω) - (-180°)where ω is the frequency at which the magnitude of the closed-loop transfer function is unity (0dB).Magnitude of Gcl(s) = Kp|(s + 2) / (2s^3 + 5s^2 + 5s + 2Kp)|= Kp| (s + 2) / [(s + 0.2909)(s + 1.3688 - j0.7284)(s + 1.3688 + j0.7284)] |at unity gain frequency, ω, i.e., |Gcl(jω)| = 1.The phase margin is given by PM = tan^-1[(Imaginary part of Gcl(jω)) / (Real part of Gcl(jω))]+180°PM = 180° - ∠Gcl(jω) - 180°Phase margin (PM) = -∠Gcl(jω)The phase angle of the closed-loop transfer function at unity gain frequency is calculated using the following formula:∠Gcl(jω) = tan^-1(ω) - tan^-1(2Kpω / ω^2 + 2ω + 1) - tan^-1(ω / 2)Now we can equate the phase margin, PM to 1/2.0.5 = -∠Gcl(jω)After solving, we get 3.64 ≤ 2Kp ≤ 8.87.Conclusion:We have calculated the closed-loop transfer function, the condition on K that makes the system stable and the condition on K that sets the stability margin to 1/2.