Characteristic equation in feedback control: For a closed-loop transfer function written as G1 / (1 + G2), the characteristic equation is 1 + G2 = 0. What does this imply about dependence, stability assessment, and applicability to set-point vs load changes?

Introduction / Context:In classical control theory, the characteristic equation captures the closed-loop poles, which entirely determine stability and transient behaviour. This question probes your understanding of how the characteristic equation relates to the open-loop dynamics and whether it changes with the type of excitation (set-point versus load disturbance).

Given Data / Assumptions:

• Closed-loop form is written so that the denominator is 1 + G2.
• Linear time-invariant system with negative feedback.
• Set-point and load disturbances enter through standard summing junctions.

Concept / Approach:The characteristic equation is obtained by setting the closed-loop denominator to zero. It depends only on the open-loop transfer function combination that appears in the denominator (typically the product of controller and plant). Because closed-loop poles are independent of the input path (set-point or load), the same characteristic equation governs stability for any excitation. Hence, it both depends only on open-loop dynamics and determines stability, and it is common to all input cases.

Step-by-Step Solution:

Verification / Alternative check:Block-diagram manipulations or Mason’s gain formula show that closed-loop poles do not depend on input placement, confirming invariance across set-point and disturbance cases.

Why Other Options Are Wrong:

• Choosing only one of (a), (b), or (c) ignores the full role of the characteristic equation.
• “None of these” contradicts standard control results.

Common Pitfalls:Confusing the characteristic equation (poles) with the full closed-loop transfer function (which includes zeros and may differ for set-point vs load paths).

Final Answer:All (a), (b) & (c)