Block diagram of a closed loop system

The output C(s) is fed back to the summing point, where it is compared with reference input R(s). The closed loop nature is indicated in fig1.3. Any linear system may be represented  by a block diagram consisting of blocks, summing points and branch points. A branch is the point from which the output signal from a block diagram goes concurrently to other blocks or summing points.

 When the output is fed back to the summing point for comparison with the input, it is necessary  to convert the form of output signal to that of he input signal. This conversion is followed by the feed backelement whose transfer function is H(s) . Another important role of the feed back element is to modify the output  before it is compared with the input.

                                       C(s)[1 + G(s)H(s)] = G(s)R(s) 

 The ratio of the feed back signal B(s) to the actuating error signal E(s) is called the open loop transfer function

open loop transfer function = B(s)/E(s) = G(s)H(s)

 The ratio of the output C(s) to the actuating error signal E(s) is called the feed forward transfer function .

Feed forward transfer function =  C(s)/E(s) = G(s)

If the feed back transfer function is unity, then the open loop and feed forward transfer function are the same. For the system shown  the output C(s) and input R(s) are related as follows. 

C(s) = G(s) E(s)

E(s) = R(s) - B(s)

        = R(s) - H(s)C(s)        but B(s) = H(s)C(s) 

                     Eliminating E(s) from these equations

 C(s) = G(s)[R(s) - H(s)C(s)] 

C(s) + G(s)[H(s)C(s)] = G(s)R(s)

C(s)[1 + G(s)H(s)] = G(s)R(s)

C(s)/R(s) is called the closed loop transfer function

The output of the closed loop system clearly depends on both the closed loop transfer function and the nature of the input. If the feed back signal is positive.