An electromechanical closed –loop control system has the transfer function C(s)/R(s) = k/s (s² + s +1) (s + 4) +k. Which one of the following is correct? 

Explanation:

Electromechanical Closed-Loop Control System

Problem Statement:

The transfer function of the electromechanical closed-loop control system is given as:

C(s)/R(s) = k / [s × (s² + s + 1) × (s + 4) + k]

We are tasked with determining the stability of the system for various values of the gain parameter \( k \). The correct answer is given as:

Option 1: The system is stable for all positive values of \( k \).

To verify this claim and analyze the other options, we will use the principles of control system stability analysis, specifically the location of the poles in the s-plane and the Routh-Hurwitz criterion.

Solution:

Step 1: Closed-Loop Characteristic Equation

The denominator of the transfer function represents the characteristic equation of the closed-loop system:

Characteristic Equation: \( s × (s² + s + 1) × (s + 4) + k = 0 \)

Expanding this equation:

• \( s × (s² + s + 1) = s³ + s² + s \)
• \( (s³ + s² + s) × (s + 4) = s⁴ + 4s³ + s³ + 4s² + s² + 4s = s⁴ + 5s³ + 5s² + 4s \)

Thus, the characteristic equation becomes:

Denominator: \( s⁴ + 5s³ + 5s² + 4s + k = 0 \)

Step 2: Stability Analysis using Routh-Hurwitz Criterion

The Routh-Hurwitz criterion provides a systematic method to determine the stability of a system by examining the signs of the coefficients in the Routh array. A system is stable if all the roots of the characteristic equation have negative real parts, which corresponds to all the elements in the first column of the Routh array being positive.

Let us construct the Routh array for the characteristic equation:

\( s⁴ + 5s³ + 5s² + 4s + k = 0 \)

Step 3: Construct the Routh Array

The key term in the second row of the Routh array is:

\( \dfrac{5 × 4 - k}{5} \)

For stability, this term must be positive. Thus:

\( 20 - k > 0 \)

\( k < 20 \)

Analysis of Options:

Option 1: The system is stable for all positive values of \( k \).

This is incorrect because the system is stable only for \( k < 20 \). Beyond \( k = 20 \), the system becomes unstable.

Option 2: The system is unstable for all values of \( k \).

Incorrect. The system is stable for \( 0 < k < 20 \).

Option 3: The system is stable for values of \( k \) between zero and 3.36.

Incorrect. The system is stable for a broader range of \( k \), specifically \( 0 < k < 20 \).

Option 4: The system is stable for values of \( k \) between 1.6 and 2.45.

Incorrect. This range is overly restrictive; the system is stable for \( 0 < k < 20 \).

Conclusion:

Based on the Routh-Hurwitz analysis, the correct statement is that the system is stable for \( 0 < k < 20 \), making none of the provided options entirely accurate. The correct answer should specify this range explicitly.