In a Routh array, what indicates an unstable system?

• A. All elements in the first column are positive
• B. All elements in the first column are negative
• C. Any sign change in the first column
• D. The presence of a zero in the last row

Answer: (C)

In a Routh array, an unstable system is indicated by any sign change in the first column. This is because the Routh-Hurwitz stability criterion is based on the formation of the Routh array, where the signs of the elements in the first column provide information about the location of the poles of the characteristic equation in the complex plane.

If all elements in the first column are positive, it implies that all poles have negative real parts, indicating stability. Conversely, if all elements are negative, that would mean all poles have positive real parts, indicating instability. The presence of a zero in the last row does not directly indicate instability, but rather a potential issue that may need to be addressed, such as further analysis using auxiliary equations.

Thus, any sign change in the first column signifies that there are poles with positive real parts, which correspond to unstable behavior in the system. This makes sign changes a key factor in assessing system stability through the Routh array method.