__Closed - Loop Transfer Function__

The closed - loop transfer function is defined as the overall transfer function of the entire control system. It is a mathematical expression (algorithm) describing the net result of the effects of a closed (feedback) loop. Consider a closed loop transfer function or feedback loop consisting of several elements as shown on Fig.,

Fig Closed-loop transfer function. |

Now, for the forward path element, we know that

where K G (D ) = F1 (D ) X F2 (D ) X F3 (D)

On rearranging, we get

θo = K G ( D )θi − K G ( D)θo

or [1 + K G ( D )] θo = K G ( D)θi

The above expression shows the transfer function for the closed-loop control system. Thus the block diagram may be further simplified as shown in Fig., where the entire system is represented by a single block.

Fig. Simplified closed-loop system. |

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