2.11.  TRANSFER-FUNCTION CONCEPT

For analysis and design, control systems are usually described by a set of differential equations. A block diagram is a device for displaying the interrelationships of the equations pictorially. Each component is described by its transfer function. This is defined as the ratio of the transform of the output of the component to the transform of the input. The component is assumed to be at rest prior to excitation, and all initial values are assumed to be zero when determining the transfer function.

Consider the block diagram of the simple system shown in Figure 2.7. The only assumption made concerning this system is that the input and output are related by a linear differential equation whose coefficients are constant and can be written in the form

Figure 2.7   Block diagram of a simple linear system.

The Laplace transform of Eq. (2.101), assuming zero initial conditions, can be written as

The ratio C(s)/R(s) is called the transfer function of the system and completely characterizes its performance. Designating the transfer function of the element as G(s), we obtain

Therefore, assuming that the initial conditions are zero, the Laplace transform of the output is

In general, the function G(s) is the ratio of two polynomials in s:

The transfer function G(s) is a property of the system elements only, and is not dependent on the excitation and initial conditions. In addition, transfer functions can be used to represent closed-loop as well as open-loop systems