1. Gm = Ga Consider a closed-loop block diagram with the following transfer functions: Gc= Kc, Gf = 1 Gs Ko S " = 1. (a) Find the closed-loop transfer functions Gsp and G. Express the results in terms of the gain and the appropriate time constant. (b) Consider the servo problem (D = 0) and suppose that ysp (t) = Au(t). Find the closed-loop response and the offset. (c) Consider the regulator problem (Ysp loop response and the offset. = 0) and suppose that d (t) = A u(t). Find the closed- 2. The characteristic equation of a closed loop is q(s) = s + 4s + K. Find the range of values of K for which the closed-loop is stable. Also, find the range of values of K for which the stable response is oscillatory. 3. The characteristic equation of a closed loop is q(s) = S + 2s + 4s + K. Find the range of values of K for which the closed-loop is stable. Also, find the two imaginary poles for the case for which the closed-loop is marginally stable.