When control blocks are in series, how do their transfer functions combine?

When control blocks are connected in series, the overall transfer function of the system is found by multiplying the individual transfer functions of each block. This reflects how the output of one block becomes the input for the next block in the series.

Each block applies its own transformation to the input signal, and the effect of these transformations combines multiplicatively. For example, if you have two transfer functions, represented as H1(s) and H2(s), the total transfer function H(s) for the series connection would be expressed as H(s) = H1(s) * H2(s).

This multiplication preserves the system dynamics and appropriately represents the cumulative effect of the control blocks in influencing the final output based on the inputs. Therefore, the multiplication of transfer functions in series is fundamental to control theory and helps in analyzing complex systems systematically.