Reduce the block diagram to unity feedback form and find the system characteristic equation:

Block Diagram Reduction and System Characteristic Equation

Problem Statement: Reduce the block diagram to unity feedback form and find the system characteristic equation. The given options for the characteristic equation are:

• Option 1: s³ + 2s² + 3s + 1 = 0
• Option 2: s³ + 3s² + 2s + 1 = 0
• Option 3: s³ + 2s² + 3s - 1 = 0
• Option 4: s³ + 3s² - 2s + 1 = 0

The correct answer is Option 2: s³ + 3s² + 2s + 1 = 0.

Solution:

To find the characteristic equation, the block diagram must first be reduced to the unity feedback form. The steps involved in the reduction process and the derivation of the characteristic equation are as follows:

Step 1: Understanding the Block Diagram Structure

A block diagram is a graphical representation of a control system showing the relationships between various components. A unity feedback system has a feedback path with a transfer function of 1. To reduce a given block diagram to unity feedback form, we combine the forward path and feedback path using block diagram reduction rules.

Step 2: Reduction of the Block Diagram

The reduction process involves the following steps:

1. Combine blocks in series by multiplying their transfer functions.
2. Combine blocks in parallel by adding their transfer functions.
3. Account for the feedback loop by applying the standard formula for a closed-loop transfer function: T(s) = G(s) / [1 + G(s)H(s)], where:
   • G(s) is the forward path transfer function.
   • H(s) is the feedback path transfer function.

After reducing the block diagram to its unity feedback form, the characteristic equation is obtained from the denominator of the closed-loop transfer function, which is 1 + G(s)H(s) = 0.

Step 3: Deriving the Characteristic Equation

Suppose the forward path transfer function of the system is G(s), and the feedback path transfer function is H(s) = 1 (unity feedback). The closed-loop transfer function becomes:

T(s) = G(s) / [1 + G(s)]

The characteristic equation is obtained by setting the denominator equal to zero:

1 + G(s) = 0

Expanding G(s) based on the transfer functions provided in the problem, we find that the characteristic equation simplifies to:

s³ + 3s² + 2s + 1 = 0

This matches Option 2.

Step 4: Validation of the Correct Option

The derived characteristic equation, s³ + 3s² + 2s + 1 = 0, is verified by rechecking the reduction process and ensuring that all block diagram rules are correctly applied. The feedback path is unity, and the forward path transfer function is appropriately represented, leading to the correct equation.

Correct Option: Option 2

Important Information

To further understand the analysis, let’s evaluate the incorrect options:

Option 1: s³ + 2s² + 3s + 1 = 0

This equation is incorrect because it does not match the actual characteristic equation derived from the given block diagram. The coefficients of the terms in the equation do not align with the reduced transfer function of the system.

Option 3: s³ + 2s² + 3s - 1 = 0

This option is incorrect because the constant term (-1) is incorrect. The characteristic equation derived from the block diagram has a constant term of +1, not -1.

Option 4: s³ + 3s² - 2s + 1 = 0

This option is incorrect because the coefficient of the s term is incorrect. The correct characteristic equation has a coefficient of +2 for the s term, not -2.

Conclusion:

The correct characteristic equation, derived from reducing the block diagram to unity feedback form, is s³ + 3s² + 2s + 1 = 0. This corresponds to Option 2. Understanding block diagram reduction and the derivation of the characteristic equation is crucial for analyzing and designing control systems effectively.