Second-order LTI system — choose the standard canonical transfer function form In classical process control, the standard form of a linear time-invariant second-order system is expressed using the undamped natural frequency (ω_n) and the damping ratio (ζ). Which of the following expressions represents this canonical transfer function (unity static form)?

Difficulty: Easy

ω_n^2 / (s^2 + 2 ζ ω_n s + ω_n^2)
K / (τ s + 1)
K e^(-L s)
none of these
K (s + 1) / (s^2 + s + 1)

Correct Answer: ω_n^2 / (s^2 + 2 ζ ω_n s + ω_n^2)

Explanation:

Introduction / Context:
Recognising the standard (canonical) transfer function of a second-order linear time-invariant (LTI) system is foundational for predicting overshoot, oscillations, settling time, and stability margins. The canonical form uses two interpretable parameters: the undamped natural frequency (ω_n) and the damping ratio (ζ). This question checks your ability to identify that exact standard expression among look-alike distractors that represent other dynamic elements or non-canonical forms.

Given Data / Assumptions:

We are looking for the standard second-order LTI form.
Symbols: ω_n for natural frequency; ζ for damping ratio; s is the Laplace variable.
Unity static form is implied; arbitrary gain factors are not necessary to recognise the canonical structure.

Concept / Approach:
The canonical second-order transfer function is written as G(s) = ω_n^2 / (s^2 + 2 ζ ω_n s + ω_n^2). From this, step and frequency response features are read directly: underdamped (0 < ζ < 1), critically damped (ζ = 1), overdamped (ζ > 1). Any form that does not match this denominator pattern is not the standard second-order canonical expression, even if it is second-order or dynamic in some other way.

Step-by-Step Solution:

Recall canonical form: G(s) = ω_n^2 / (s^2 + 2 ζ ω_n s + ω_n^2).Scan the options for an exact match.Identify that option (a) matches the required numerator and denominator pattern.Reject alternatives that represent first-order lags, pure time delay, or non-canonical numerators.

Verification / Alternative check:
Compare poles: the denominator s^2 + 2 ζ ω_n s + ω_n^2 produces complex poles at −ζ ω_n ± j ω_n sqrt(1 − ζ^2) for ζ < 1, consistent with classical second-order behaviour. Other presented forms do not enforce this pole pattern.

Why Other Options Are Wrong:

K / (τ s + 1) — a first-order lag, not second-order.K e^(−L s) — pure time delay; magnitude 1 and phase lag only.none of these — invalid because the correct canonical form appears in option (a).K (s + 1) / (s^2 + s + 1) — a second-order with a zero; not the standard canonical form using ω_n and ζ in the numerator/denominator as required.

Common Pitfalls:
Confusing “any second-order” with the “canonical second-order form.” The presence of a numerator zero changes transient characteristics and is not the template used to define ζ and ω_n.

Final Answer:
ω_n^2 / (s^2 + 2 ζ ω_n s + ω_n^2)

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