Control System Sensitivity Analysis

In control systems, sensitivity is a measure of how changes in a system parameter affect the overall system performance, typically represented by the closed-loop transfer function. For a standard negative feedback control system with forward path gain G and feedback path gain H, the closed-loop transfer function T is given by:

\[ T = \frac{G}{1 + GH} \]

The sensitivity of the closed-loop transfer function T with respect to the forward path gain G is defined as:

\[ S_G^T = \frac{\frac{\partial T}{T}}{\frac{\partial G}{G}} \]

Calculating this derivative for the given closed-loop transfer function, we get the sensitivity formula:

\[ S_G^T = \frac{1}{1 + GH} \]

The question provides the condition that the value of \((1 + GH)\) is less than 1. Let's analyze what this condition implies for the sensitivity \(S_G^T\).

We have \(1 + GH < 1\).

The sensitivity is given by \(S_G^T = \frac{1}{1 + GH}\).

If \(1 + GH\) is a positive value less than 1 (e.g., 0.5, 0.1), then the reciprocal \(\frac{1}{1 + GH}\) will be greater than 1. For example:

• If \(1 + GH = 0.5\), then \(S_G^T = \frac{1}{0.5} = 2\).
• If \(1 + GH = 0.1\), then \(S_G^T = \frac{1}{0.1} = 10\).

In these cases, the magnitude of sensitivity is greater than 1. A sensitivity \(|S_G^T| > 1\) means that a small percentage change in the parameter G will cause a larger percentage change in the closed-loop transfer function T. This indicates that the system is more sensitive to variations in G compared to the open-loop system (where sensitivity of T=G to G is 1).

While \(1 + GH\) could also be negative and less than 1 (e.g., -2, -5), in the context of standard sensitivity analysis for stable systems, \(1 + GH\) is typically considered positive. However, even if \(1+GH\) is negative and less than 1, its magnitude \(|1+GH|\) could still be less than 1 (e.g., \(1+GH = -0.5\), \(|1+GH| = 0.5\)), or greater than 1 (e.g., \(1+GH = -2\), \(|1+GH|=2\)). The condition "less than 1" doesn't strictly mean positive and less than 1. But if we consider the typical scenario where feedback is used to reduce sensitivity, the standard case is \(|1+GH| > 1\). The question explores the scenario where the value of \(1+GH\) is simply "less than 1". Interpreting "less than 1" often implies positive context unless specified otherwise, but if we strictly follow \(1+GH < 1\), it includes negative values. If \(1+GH\) is negative, the system might be unstable. Assuming a context where sensitivity is being evaluated for a potentially functional system, let's consider both positive and negative \(1+GH\) values that are less than 1.

• Case 1: \(0 < 1 + GH < 1\). As shown above, \(S_G^T = \frac{1}{1 + GH}\) will be greater than 1. Sensitivity Increases.
• Case 2: \(1 + GH \le 0\). The system might be unstable if \(1+GH=0\). If \(1+GH\) is negative, \(S_G^T\) is negative. The magnitude \(|S_G^T| = |\frac{1}{1+GH}|\). If, for instance, \(1+GH = -0.5\), then \(S_G^T = -2\), and \(|S_G^T| = 2 > 1\). Sensitivity magnitude Increases. If \(1+GH = -2\), then \(S_G^T = -0.5\), and \(|S_G^T| = 0.5 < 1\). Sensitivity magnitude Decreases.

The question asks about "sensitivity", which often refers to the magnitude of \(S_G^T\) or its effect relative to the open-loop case. Given the standard benefit of feedback is sensitivity reduction (\(|1+GH| > 1\) leading to \(|S_G^T| < 1\)), the condition \(1+GH < 1\) (especially if interpreted as \(0 < 1+GH < 1\)) represents a scenario where this benefit is not realized, or even reversed.

Considering the options and the common understanding of this formula in basic control theory problems, the condition \(1 + GH < 1\) is most likely intended to imply a situation where the feedback mechanism does not provide the usual reduction in sensitivity, leading to increased sensitivity compared to the open-loop case or the desired feedback scenario.

Therefore, if the value of \((1 + GH)\) is less than 1, the sensitivity \(|S_G^T|\) is likely to be greater than 1 (especially if \(0 < 1+GH < 1\)), meaning sensitivity Increases.