In control theory, a transfer function G(s) is defined as what?

In control theory, a transfer function G(s) is defined as what?

• A. G(s) = Input(s)/Output(s)
• B. G(s) = Output(s) × Input(s)
• C. G(s) = ΔOutput(s)/ΔInput(s)
• D. G(s) = Output(s)/Input(s) for linear time-invariant systems

Answer: (D)

The transfer function describes how the system’s output relates to its input in the Laplace domain for a linear time-invariant system. It is defined as the ratio of the Laplace transform of the output to the Laplace transform of the input (assuming zero initial conditions), so G(s) = Output(s)/Input(s). This compact form captures the system’s dynamics—its memory and how it responds across different frequencies—through its poles and zeros, and it lets us analyze stability and frequency response by working in the s-domain. Choosing Output(s)/Input(s) for linear time-invariant systems is the right way to express this relationship. Inverting it to Input(s)/Output(s) would misrepresent how the system propagates input signals to produce output, and using a product of signals or a simple ratio of small changes, ΔOutput/ΔInput, misses the full dynamic behavior and time-domain memory that the transfer function embodies.

The transfer function describes how the system’s output relates to its input in the Laplace domain for a linear time-invariant system. It is defined as the ratio of the Laplace transform of the output to the Laplace transform of the input (assuming zero initial conditions), so G(s) = Output(s)/Input(s). This compact form captures the system’s dynamics—its memory and how it responds across different frequencies—through its poles and zeros, and it lets us analyze stability and frequency response by working in the s-domain.

Choosing Output(s)/Input(s) for linear time-invariant systems is the right way to express this relationship. Inverting it to Input(s)/Output(s) would misrepresent how the system propagates input signals to produce output, and using a product of signals or a simple ratio of small changes, ΔOutput/ΔInput, misses the full dynamic behavior and time-domain memory that the transfer function embodies.