1- [a] For positive values of K, plot the root locus for a unity negative feedback control system having the following open-loop transfer function: K G(s) =- (s+1)(s+4)(s+7) For what values of gain K does the system become unstable? Find also the value of k at which the damping ratio is 0.5 and the closed loop poles. (25%) [b] The characteristic equations of linear control systems are given below. Apply Routh-Hurwitz criterion to determine the root distribution and the system stability. (15%) 1) S' + 3S + 3S+ s*+ S'+ 3S? + 3S +1 =0 2) s +8s* +2s +4s +2s+4 =0 3) S+s* + 2S* + 2s + 3S + 2S +4 = 0 2- [a] For the system that have the following transfer function 4(s+2)(s+7) U(s) (s+3)(s+ 4)(s+5) Y(s) Give the state space in pole-zero form, then find the matrix T to converted into diagonal form then find the new system. (30%) [b] For the system has the transfer function 5 , assuming unity negative s+s+6 feedback 1) Compute the rise time and the percentage overshoot for the step input 2) The error constants (15%)