In control theory, functions called "transfer functions" are very often used to characterize the input-output relationships of linear time-invariant systems. The concept of transfer functions applies only to linear time-invariant systems, although it can be extended to certain nonlinear control systems.

Transfer Functions. The transfer function of a linear time-invariant system is defined to be the ratio of the Laplace transform of the output (response function) to the Laplace transform of the input (driving function), under the assumption that all initial conditions are zero.

The transfer function is an expression relating the output and input of a linear time-invariant system in terms of the system parameters and is a property of the system itself, independent of the driving function. The transfer function includes the units necessary to relate the input to the output; however, it does not provide any information concerning the physical structure of the system. (The transfer functions of many physically different systems can be identical.)

By using this concept, one can represent the system dynamics by algebraic equations in S.
          Modern Control Engineering by Katsuhiko Ogata, pg. 72. (University of Minnesota.) (©1970)

Derivation of Transfer Function for the Inverted Pendulum

Starting with our characteristic equation from the previous page:


we take the Laplace Transform of both sides:


and do some rearranging:



This is our transfer function for the inverted pendulum.