In the application of the Routh-Hurwitz criterion, two specific scenarios can arise that complicate stability analysis.

The first scenario occurs when a singular zero appears in the first column of the Routh table. This situation creates a division by zero issues. To resolve this, a small positive or negative number, denoted as epsilon (∈), is substituted for the zero. The stability analysis proceeds by assuming a sign for ∈. If ∈ is positive, any sign change in the first column of the Routh table indicates that the system is unstable, with two poles located in the right-half s-plane. Conversely, if ∈ is negative, the same conclusion of instability is reached.

The second scenario arises when an entire row in the Routh table consists solely of zeros. This occurrence suggests that the original polynomial has an even polynomial as a factor. To address this, an auxiliary polynomial is constructed using the coefficients from the row above the zero row. This auxiliary polynomial is then differentiated, and the coefficients from the derivative replace the zeros in the Routh table. The standard procedure for constructing the remaining Routh table continues from this point.

When dealing with even and odd polynomials separately, the total number of poles in the right-half s-plane is determined by summing the poles found in the Routh tables of the even and odd polynomials. This method ensures a comprehensive analysis of the system's stability.

These special cases within the Routh-Hurwitz criterion are crucial for accurately determining the stability of a system. By carefully managing zeros in the first column and rows of zeros, engineers can avoid misinterpretation of the system’s stability. This approach allows for precise identification of right-half s-plane poles, ensuring robust and reliable system design and analysis.