Consider an electrical power grid, where stability is essential to prevent blackouts. The Routh-Hurwitz criterion is a valuable tool for assessing system stability under varying load conditions or faults. By analyzing the closed-loop transfer function, the Routh-Hurwitz criterion helps determine whether the system remains stable.

To apply the Routh-Hurwitz criterion, a Routh table is constructed. The table's rows are labeled with powers of the complex frequency variable s, starting from the highest power. The variable s represents a complex number, typically in the form

where σ is the real part and jω is the imaginary part. The first row is filled horizontally with every other coefficient of the denominator polynomial, beginning with the highest power. The second row is then populated with the skipped coefficients from the first row.

Subsequent entries in the Routh table are calculated using the determinants of the preceding rows. Specifically, each entry is determined by taking the negative determinant of the two entries above it and dividing it by the first entry in the column directly above it. This process continues until the entire table is filled.

To illustrate, consider a system for which the Routh table rows are computed. If needed, each row is scaled independently by a positive constant to simplify calculations. According to the Routh-Hurwitz criterion, the number of roots of the polynomial in the right-half s-plane corresponds to the number of sign changes in the first column of the Routh table. These sign changes indicate instability.

Conversely, a system is stable if all the poles are in the left-half s-plane, meaning there are no sign changes in the first column of the Routh table. Ensuring that no sign changes occur confirms that the system will remain stable under various operating conditions.

Understanding and applying the Routh-Hurwitz criterion is crucial for maintaining the stability of complex systems like electrical power grids. By verifying that all poles are located in the left-half s-plane, engineers can ensure that the system will operate reliably, preventing issues like blackouts and ensuring a continuous power supply. This method provides a systematic approach to stability analysis, identifying and mitigating potential instability in the system.