Routh stability criterion: the tabulation and test are performed using the characteristic polynomial of which transfer function—open-loop or closed-loop?

Introduction / Context:
The Routh–Hurwitz stability test determines whether all roots of a polynomial lie in the left half of the s-plane. In control systems, we apply it to the characteristic equation of the closed-loop system to assess stability without computing roots explicitly.

Given Data / Assumptions:

• Unity or non-unity feedback is allowed; the approach is unchanged.
• The characteristic equation arises from the denominator of the closed-loop transfer function: 1 + L(s) = 0.
• No numerical coefficients provided; this is a conceptual identification.

Concept / Approach:
Stability of the actual, operating loop depends on the poles of the closed-loop transfer function. The Routh array is built from the coefficients of that closed-loop characteristic polynomial. While open-loop transfer functions are used to derive the closed-loop characteristic equation (e.g., 1 + L(s) = 0), the Routh test itself is applied to the resulting closed-loop denominator.

Step-by-Step Solution:
Form L(s) from plant and controller.Write characteristic equation 1 + L(s) = 0 and expand to get the polynomial.Construct the Routh array with those coefficients to test sign changes in the first column.

Verification / Alternative check:
Textbooks consistently define the “characteristic equation” as the closed-loop denominator; examples of gain margin design via Routh all start from the closed-loop polynomial.

Why Other Options Are Wrong:
Open-loop alone does not reflect feedback stability.“Either/Neither” misstates the criterion’s basis.Controller-only polynomials ignore plant dynamics.

Common Pitfalls:
Applying Routh to the numerator or to open-loop polynomials; forgetting to include dead time approximations (if any) in the characteristic polynomial used.

Final Answer:
Closed-loop transfer function