Questions: Deriving Transfer Functions from Differential Equations

Question 1

A control engineer derives a transfer function G(s) for a motor system, but the motor starts from rest with a non-zero initial velocity. Why can't G(s) alone be used to predict the full response?

• A. The transfer function only applies to nonlinear systems
• B. G(s) assumes zero initial conditions, so it captures only the forced response; the free response from stored initial energy must be added separately
• C. The Laplace transform cannot be applied to systems with initial velocity
• D. G(s) requires the input to be a unit step function

Answer: (B)

The transfer function is derived by applying the Laplace transform and invoking zero initial conditions, which allows d^n y/dt^n to simplify to s^n Y(s) with no extra boundary terms. If initial conditions are non-zero, extra terms appear that break the clean G(s) = Y(s)/U(s) form. The transfer function describes forced response only; free response from initial energy must be computed separately and added.

Question 2

What does the pole-zero structure of a transfer function G(s) tell you about a system's stability?

• A. Stability is determined by the zeros of the numerator polynomial
• B. A system is stable if and only if all zeros of G(s) lie in the left half of the s-plane
• C. A system is stable if and only if all poles of G(s) lie in the left half of the s-plane
• D. Stability requires that the number of poles equals the number of zeros

Answer: (C)

Poles — the roots of the denominator polynomial — determine the natural (unforced) behavior of a system. A pole at s = −a produces a decaying exponential e^{−at}; poles with positive real parts produce growing exponentials, i.e., instability. Stability requires all poles to be in the left half of the s-plane (negative real parts). Zeros affect how the system responds to particular inputs but do not determine stability.

Question 3

The transfer function G(s) = Y(s)/U(s) fully describes the complete response of a system, including how it responds to stored initial energy.

• T. True
• F. False

Answer: False

G(s) is derived under the assumption of zero initial conditions. It captures only the input-output (forced) response to an applied input U(s). If the system has stored energy at t=0 — such as initial velocity or initial capacitor charge — the resulting free response is not captured by G(s)·U(s). Complete response requires the forced and free contributions to be calculated and summed separately.

Question 4

Cascading two LTI systems in series is equivalent to multiplying their individual transfer functions.

• T. True
• F. False

Answer: True

One of the key advantages of transfer functions is that convolution in the time domain becomes multiplication in the s-domain. If Y(s) = G₁(s)·U(s) and Z(s) = G₂(s)·Y(s), then Z(s) = G₂(s)·G₁(s)·U(s) — the cascaded transfer function is simply the product. This algebraic property is why transfer functions make system design tractable: cascading, feedback, and parallel combinations all reduce to rational arithmetic.

Question 5

Why must zero initial conditions be assumed when deriving a transfer function, and what physical interpretation does this assumption carry?

Model answer: Zero initial conditions ensure that the Laplace transform of d^n y/dt^n reduces cleanly to s^n Y(s), with no boundary terms. Physically, this means the system starts with no stored energy — no initial voltage on a capacitor, no initial velocity of a mass. The transfer function then captures only the system's response to the applied input, not to pre-existing internal state. It characterizes the system itself (its poles, zeros, and gain) rather than a particular run of the system from a specific starting state.

If initial conditions were non-zero, the transformed equations would include extra terms that depend on the specific starting conditions, preventing formation of the clean Y(s)/U(s) ratio. The zero-initial-conditions assumption is therefore not a limitation of the technique but a deliberate choice to isolate the system's inherent input-output character — the same character regardless of how the system was set up before the input arrived.