The mathematical relationships of control systems are usually represented by block diagrams, which show the role of various components of the system and the interaction of variables in it.

It is common to use a block diagram in which each component in the system (or sometimes a group of components) is represented by a block. An entire systemmay, then, be represented by the interconnection of the blocks of the individual elements, so that their contributions to the overall performance of the system may be evaluated. The simple configuration shown in Figure is actually the basic building block of a complex block diagram. In the case of linear systems, the input-output relationship is expressed as a transfer function, wahich is the ratio of the Laplace transform of the output to the Laplace transform of the input with initial conditions of the system set to zero. The arrows on the diagram imply that the block diagram has a unilateral property. In other words, signal can only pass in the direction of the arrows.

A box is the symbol for multiplication; the input quantity is multiplied by the function in the box to obtain the output.With circles indicating summing points (in an algebraic sense) and with boxes or blocks denoting multiplication, any linear mathematical expression may be represented by block-diagram notation, as in Figure for the case of an elementary feedback control system.

The block diagrams of complex feedback control systems usually contain several feedback loops, and they may have to be simplified in order to evaluate an overall transfer function for the system. A few of the block diagram reduction manipulations are given in Table 3.4.1; no attempt is made here to cover all the possibilities.