The process of deriving the transfer function of a control system often involves reducing its block diagram to a single block. This simplification can be achieved through a series of strategic operations, including relocating branch points and comparators. These operations preserve the overall function of the system while allowing for easier manipulation and combination of blocks.

The first step in this process is the identification and relocation of a branch point. A branch point, where a signal splits into multiple paths, must be shifted to a new location that does not alter the system's behavior. This involves selecting an appropriate new position and ensuring the mathematical relationships between the signals before and after the move remain consistent. If the output signals remain unchanged, the relocation is deemed successful. This operation facilitates the subsequent combination of multiple blocks by aligning them in a manner that simplifies the overall structure.

Next, a comparator, which subtracts one signal from another, is moved to a new position within the block diagram. The relocation of the comparator also requires a careful examination of the mathematical relationships before and after the move. By ensuring that the system's output remains unchanged, the relocation confirms the integrity of the system's functionality. The precise positioning of the comparator is critical to maintaining the correct feedback and feedforward paths within the diagram.

This shift aligns the paths so that multiple blocks can be combined more easily. Following this, blocks representing various system components are merged, reducing the overall complexity of the diagram. The final stage involves removing several feedback loops, further simplifying the diagram. Each feedback loop typically introduces a recursive relationship that complicates the overall transfer function.