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Closed loop transfer function

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A closed-loop transfer function in control theory is a mathematical expression (algorithm) describing the net result of the effects of a closed (feedback) loop on the input signal to the circuits enclosed by the loop.

Contents

Overview

The closed-loop transfer function is measured at the output. The output signal waveform can be calculated from the closed-loop transfer function and the input signal waveform.

An example of a closed-loop transfer function is shown below:

The summing node and the G(s) and H(s) blocks can all be combined into one block, which would have the following transfer function:

Y ( s ) X ( s ) = G ( s ) 1 + G ( s ) H ( s )

Derivation

We define an intermediate signal Z shown as follows:

Using this figure we write:

Y ( s ) = Z ( s ) G ( s ) Z ( s ) = X ( s ) − Y ( s ) H ( s ) X ( s ) = Z ( s ) + Y ( s ) H ( s ) X ( s ) = Z ( s ) + Z ( s ) G ( s ) H ( s ) ⇒ Y ( s ) X ( s ) = Z ( s ) G ( s ) Z ( s ) + Z ( s ) G ( s ) H ( s ) Y ( s ) X ( s ) = G ( s ) 1 + G ( s ) H ( s )

References

Closed-loop transfer function Wikipedia
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