An open-loop system of an actuator, as shown in Figure 2(a), is satisfactory as far as the transient response and steady-state gain are concerned. The effects of unwanted disturbances can be reduced by converting the system to that of a closed loop as shown in Figure 2(b). (i) If the transfer function of the controller is K(s)=k (constant, independent on s), find the transfer function of the closed-loop system for the input T(s)=Y(s)/R(s), and the transfer function for the disturbance Ta(s)=Y(s)/D(s). Determine the values of A and k so that the steady-state gain of this system is the same as for the open-loop system, and that the steady-state output due to a step disturbance is 100 times smaller than that in the open-loop system. (ii) Now, consider the "proportional+integral" controller, with K(s)=k[1 + 1/(sT)]. Derive the closed-loop system transfer function for the input T(s)=Y(s)/R(s), and the transfer function for the disturbance Ta(s)=Y(s)/D(s). Determine the values of A and k required to have the steady-state gain of the closed-loop system equal to that of the open-loop system, and the time constant of the closed-loop system 100 times smaller than that in the open-loop system. Also, find the steady-state output of this system if the disturbance is a unit step function, and if it is a unit ramp function.[12 marks]

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