Problem 2. A cascaded control system as seen on the right R(s) C(s) has a plant...

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Problem 2. A cascaded control system as seen on the right R(s) C(s) has a plant transfer function, G.(s) G,(s) 1 Gp(s) = (s - 1)(s – 3) (a) When the controller is G.(s) = K, which is a simple P, i.e. proportional controller, sketch the root locus to show that the closed loop system is always unstable. (10 pts) (b) When the controller has a zero and a pole as given below sketch the new root locus (10 pts) K(s + 2) s+ 20 G(s) = and determine the range of K for which the closed loop system is stable. (5 pts) (c) Determine the value of K and the imaginary poles at jw crossings. (5 pts) Hint: When sketching the root locus, if necessary, make use of the asymptotes finding oa and Oa that are the intersecting point and angles with the real axis, respectively, with the following formula, Σintepoles-Σ inite zeros da = (2k+1)n and ea ,where k = 0,+1, ±2,.. %3D #finite poles-#finite zeros #finite poles-#finite zeros Problem 2. A cascaded control system as seen on the right R(s) C(s) has a plant transfer function, G.(s) G,(s) 1 Gp(s) = (s - 1)(s – 3) (a) When the controller is G.(s) = K, which is a simple P, i.e. proportional controller, sketch the root locus to show that the closed loop system is always unstable. (10 pts) (b) When the controller has a zero and a pole as given below sketch the new root locus (10 pts) K(s + 2) s+ 20 G(s) = and determine the range of K for which the closed loop system is stable. (5 pts) (c) Determine the value of K and the imaginary poles at jw crossings. (5 pts) Hint: When sketching the root locus, if necessary, make use of the asymptotes finding oa and Oa that are the intersecting point and angles with the real axis, respectively, with the following formula, Σintepoles-Σ inite zeros da = (2k+1)n and ea ,where k = 0,+1, ±2,.. %3D #finite poles-#finite zeros #finite poles-#finite zeros