Routh–Hurwitz stability test in control engineering Which of the following statements correctly describes a limitation or capability of the Routh–Hurwitz criterion applied to a linear time-invariant system?

Introduction / Context:
The Routh–Hurwitz criterion is a classical analytical tool used to assess closed-loop stability of linear time-invariant (LTI) systems. Instead of computing all roots of the characteristic polynomial, it uses a tabular test (the Routh array) to determine whether any roots lie in the right-half complex plane. This question checks your understanding of what Routh–Hurwitz can and cannot do, especially when time delays appear in the loop.

Given Data / Assumptions:

• System is LTI and described by a characteristic equation in s.
• “Transportation lag” refers to a pure time delay with factor e^(−L s).
• Stability is defined by the location of characteristic roots (poles) in the s-plane.

Concept / Approach:
Routh–Hurwitz applies to characteristic equations that are polynomials in s. It reveals how many roots are in the right half-plane (RHP), on the imaginary axis, or in the left half-plane (LHP) by testing the signs/patterns in the first column of the Routh array. However, it does not return the actual root coordinates; it is a counting test, not a root-solver. When a pure time delay e^(−L s) is present, the characteristic equation is not a polynomial. To use Routh, one must first approximate the delay (for example, with a Pade approximation), thereby recovering a rational approximation suitable for the Routh test.

Step-by-Step Solution:

Verification / Alternative check:
Standard control texts advise approximating delays before applying Routh or using frequency-domain methods (Nyquist) that handle delays more naturally.

Why Other Options Are Wrong:

• Provides exact locations: False; Routh only counts, it does not locate.
• Not applicable to polynomials: False; it is specifically for polynomials.
• Counts RHP roots: True statement in (a), but the question asks for the best single descriptor including the delay limitation; option (b) most directly answers the prompt about transportation lag.

Common Pitfalls:
Confusing Routh’s counting ability with root-solving; overlooking the special handling required for time delays.

Final Answer:
It cannot be applied directly when a pure transportation lag is present (e.g., e^(−Ls)) unless the lag is approximated.