3. Consider a unity-feedback control system with plant G(s) following figure. 316 and controller C(s), as shown in the Reference Error Controller Plant r(t) e(t) u(t) y(1) C(s) G(s) (a) De oles rela iegr C gain and (b) Саp necessar and sucient. of K nat re (c) Show using the Final Value Theorem that the system with the P controller from (b) can track a init-step reference r(t) = 1 with zero steady-state error lim+o e(t) = 0. (d) Show that it however cannot track a unit-ramp reference r(t) =t with zero steady-state error. Can the error be made arbitrarily small with Kp without losing stability? (e) Can a PI controller C(s) = Kp + AL stabilize the plant G(8) and, at the same time, yield zero steady-state error to both unit-step and unit-ramp references? If so, find the values of Kp and Ki that are necessary and sufficient. Reconsider the unity-feedback control system shown in Problem 3 and let the controller C(s) and plant G(s) be defined as follows. (a) Let C(s) = Kp and G(s) = : Find the value of Kp that yields 10% of overshoot in y(t) when r(t) is a unit-step input. Hint: 2nd-order system. (s+3) (b) Let C(s) = Kp and G(s) = G+US+2+46+5): Sketch the root locus of the open-loop transfer function C(s)G(s) by hand and using MATLAB's rlocus. Derive a condition that Kp must satisfy in order for the closed-loop system to be asymptotically stable. %3D

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HVDC

HVDC stands for high voltage direct current in the electric power transmission system. The high voltage direct current system uses direct current (DC) for the transmission of electrical power through transmission lines in contrast with the alternating current that is the AC system. The HVDC is used to transmit bulk power point-to-point through long distances transmission lines. Power is rapidly transferred because the efficiency of transmission lines is greatly increased over long distances due to HVDC lines.