please solve parts (c) and (d) using MATLAB and please upload the code

Thank you

The transfer function for a closed-loop system with a single feedback control action is givern by T(s)H(S) 1+G(s)H (s) ' where H(s) is the open-loop transfer function (obtained from the ODE defining the basic dynamical system) and G(s) is the controller function. For this project, suppose H(s) and G(s) are given by the functions H(s) and G(s) = s* +272,572) S + a Required: a) Find the general analytic expression for the transfer function T(s). [10 pts.] Choose the values a=y2 , =1/2 , a =1/10 and construct a root-locus diagram (i.e., construct a plot of the roots of the denominator of T(s) in the complex s-plane) as the gain parameter K varies from K-1 to K-8. [25 pts.] b) Note: You should present your computed values of the roots on a single graph of the complex s-plane as K is varied in increments of 0.5 c) Determine the value of the gain K for which the system is stable. [20 pts.] Note: You should determine the critical value of K, say K-Ko, to at least two significant digits. This may require that you choose smaller increments in K as you vary the gain near the critical value Kc. d) Repeat the calculation for the critical gain Ke with the following sets of parameters i) =34 and ii) =1/4 . Construct a plot showing how the critical gain K, varies with the damping parameter of the system. [20 pts.] The transfer function for a closed-loop system with a single feedback control action is givern by T(s)H(S) 1+G(s)H (s) ' where H(s) is the open-loop transfer function (obtained from the ODE defining the basic dynamical system) and G(s) is the controller function. For this project, suppose H(s) and G(s) are given by the functions H(s) and G(s) = s* +272,572) S + a Required: a) Find the general analytic expression for the transfer function T(s). [10 pts.] Choose the values a=y2 , =1/2 , a =1/10 and construct a root-locus diagram (i.e., construct a plot of the roots of the denominator of T(s) in the complex s-plane) as the gain parameter K varies from K-1 to K-8. [25 pts.] b) Note: You should present your computed values of the roots on a single graph of the complex s-plane as K is varied in increments of 0.5 c) Determine the value of the gain K for which the system is stable. [20 pts.] Note: You should determine the critical value of K, say K-Ko, to at least two significant digits. This may require that you choose smaller increments in K as you vary the gain near the critical value Kc. d) Repeat the calculation for the critical gain Ke with the following sets of parameters i) =34 and ii) =1/4 . Construct a plot showing how the critical gain K, varies with the damping parameter of the system. [20 pts.]