47. The purpose of an AutomaticVoltage Regulator is to maintainconstant the voltage generated in anelectrical power system, despite loadand line variations, in an electricalpower distribution system (Gozde,2011). Figure P5.31 shows

the block diagram of such a system. Assuming Ka \begin{array}{l} =10, T_{a}=0.1, K_{e}=1, T_{e}=0.4, K_{g}=1 \\ T_{g}=1, K_{S}=1, T_{s}=0.001, \text { and the } \\ \text { controller, } G_{P I D}(s)=1.6+\frac{0.4}{s}+0.3 s \end{array} find the closed-loop transfer function, T(s)=\frac{\Delta V_{t}(s)}{\Delta V_{r e f}(s)}, \text { of the system } expressing it as a rational function.