Question

Which one of the following statements is True?

• A. The number of sign changes present in the first column of Routh Array shows the number of poles of the closed loop system in the right half of S plane.
• B. The number of sign changes present in the first column of Routh Array shows the number of zeros of the closed loop system in the right half of S plane.
• C. The number of sign changes present in the first column of Routh Array shows the number of zeros of the open loop system in the right half of S plane.
• D. The number of sign changes present in the first column of Routh Array shows the number of poles of the open loop system in the right half of S plane.

Hint:

Routh-Hurwitz works entirely with the closed-loop characteristic equation.
Thus, it only provides information about closed-loop poles, never about open-loop poles or zeros.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This is a standard theorem-based question about the Routh-Hurwitz stability criterion.

Step 2: Key Formula or Approach:
The Routh-Hurwitz criterion determines the stability of a system by looking at the characteristic equation of the closed-loop system:
\[ 1 + G(s)H(s) = 0 \]

Step 3: Detailed Explanation:

• The Routh array is constructed using the coefficients of the closed-loop characteristic equation.

• According to Routh's stability theorem, a system is stable if and only if all elements in the first column of the Routh array have the same sign.

• If there are sign changes in the first column, the system is unstable.

• The theorem specifically states that the number of sign changes in the first column is exactly equal to the number of roots of the characteristic equation (which are the poles of the closed-loop system) located in the right-half of the s-plane ($\text{RHP}$).

Step 4: Final Answer:
The number of sign changes in the first column indicates the number of closed-loop poles in the right-half s-plane, which corresponds to Option (A).