Question

In the Routh-Hurwitz criterion, a system is stable if:

• A. All elements in the first row are positive
• B. All elements in the first column have the same sign
• C. All roots are on the right-half plane
• D. The determinant is zero

Hint:

A necessary (but not sufficient) condition for stability is that all coefficients of the characteristic equation must be present and have the same sign.
The sufficient condition is that all elements in the first column of the Routh array must have the same sign.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the condition of system stability as defined by the Routh-Hurwitz stability criterion.
The Routh-Hurwitz criterion is an algebraic method used to determine the absolute stability of a linear time-invariant (LTI) system by evaluating its characteristic equation.

Step 2: Key Formula or Approach:
Let the characteristic equation of the system be:
\[ a_n s^n + a_{n-1} s^{n-1} + \dots + a_1 s + a_0 = 0 \] We construct the Routh array using these coefficients.
The stability of the system is determined strictly by analyzing the signs of the coefficients in the first column of the completed Routh array.

Step 3: Detailed Explanation:
The rules of the Routh-Hurwitz stability criterion are outlined below:

• First Column Sign Rule:
- A system is stable if and only if all the closed-loop poles lie in the Left Half of the s-plane (LHP).
- According to the Routh-Hurwitz criterion, the number of roots of the characteristic equation with positive real parts (located in the right half of the s-plane) is exactly equal to the number of sign changes in the first column of the Routh array.
- Therefore, for a system to be stable, there must be zero sign changes in the first column.
- This condition requires that all elements in the first column of the Routh array must have the same sign (typically all positive).

• Sign Change Interpretation:
- If the first column contains elements with values like \( [1, 2, -3, 4] \), there are two sign changes (from +2 to -3, and from -3 to +4), indicating two unstable poles in the right-half s-plane.

Step 4: Final Answer:
Under the Routh-Hurwitz criterion, a system is stable if all elements in the first column of the Routh array have the same sign.