Q6Control System
Question
Q.6. State the necessary condition for stability.
Answer
The necessary (but not sufficient) condition for stability is that all coefficients of the characteristic equation must be present (non-zero) and must have the same sign; if any coefficient is missing or of different sign, the system is definitely unstable.
For a characteristic equation a0sⁿ + a1sⁿ⁻¹ + ⋯ + an = 0, a necessary condition for all roots to lie in the left-half s-plane (stability) is that all the coefficients a0, a1, …, an must be real, non-zero (no missing/zero terms), and of the same sign. This condition is necessary but not sufficient — satisfying it does not guarantee stability, but violating it (a missing term or a sign change) immediately guarantees instability, without needing to construct the full Routh array.