Stability Analysis

A linear system is stable if all closed-loop poles have negative real parts.

• Routh–Hurwitz — determines how many roots of the characteristic equation are in the right half-plane, without solving it: count sign changes in the first column of the Routh array.
• Root locus — plots how closed-loop poles move as gain $K$ varies from 0 to ∞.
• Bode plot — gain margin (how much gain can increase before instability, measured where phase = −180°) and phase margin (extra phase lag tolerable at the gain-crossover frequency). A phase margin of 45–60° is typical.
• Nyquist criterion — $Z = N + P$: closed-loop RHP poles = encirclements of −1 + open-loop RHP poles.