Assessing stability from transfer-function denominators:\nWhich of the following single-input single-output systems is (asymptotically) stable based on its denominator polynomial?

Introduction / Context:
Continuous-time linear systems are asymptotically stable if and only if all poles lie strictly in the left-half complex plane (negative real parts). Recognising stability by inspection of the characteristic polynomial is a core skill for controls and process dynamics.

Given Data / Assumptions:

• Stability means responses decay to zero for bounded inputs after transients.
• We assess the sign of real parts of the roots of each denominator.
• Time delays (e.g., exp(−20s)) do not move poles by themselves but can worsen margins; denominator roots still dictate asymptotic stability.

Concept / Approach:
For a quadratic s^2 + 2ζω_ns + ω_n^2, stability requires ζ > 0 and ω_n > 0. Alternatively, compute roots directly and check real parts. Imaginary-axis poles indicate marginal stability (undamped oscillation) rather than asymptotic stability. Right-half-plane poles imply instability.

Step-by-Step Solution:

Verification / Alternative check:
Routh–Hurwitz for (c): coefficients [1, 2, 2] are positive with no sign changes, confirming stability.

Why Other Options Are Wrong:

• (a): Sustained oscillations violate asymptotic stability.
• (b), (d), (e): Each has at least one pole with positive real part.

Common Pitfalls:
Calling imaginary-axis poles “stable”; they are only marginally stable and sensitive to disturbances.

Final Answer:
1 / (s^2 + 2s + 2)