Stability & Frequency Response

Routh, root locus, Bode, Nyquist and compensators. Will the loop be stable, and by how much? Routh–Hurwitz, root locus, Bode and Nyquist analysis, gain and phase margins, and classical lead/lag compensator design by loop shaping.

• Stability — A stable system returns to equilibrium after a disturbance. Learn BIBO stability, pole locations, the Routh–Hurwitz criterion, marginal stability and critical gain, and relative stability measures.
• Root Locus — The root locus plots how closed-loop poles move as a gain varies. Learn the sketching rules, asymptotes, breakaway points and jω crossings, and how to pick a gain for a desired damping ratio.
• Bode Plot — Bode plots show gain (dB) and phase versus log frequency. Learn asymptotic sketching, corner frequencies, bandwidth, gain and phase margins from the open-loop plot, and loop-shaping design.
• Nyquist — The Nyquist criterion determines closed-loop stability from the open-loop frequency response — even for open-loop unstable plants and systems with time delay. Learn the polar plot, encirclements of −1, and margins.
• Compensator Design: Lead, Lag & Loop Shaping — Designing controllers in the frequency domain. Learn gain and phase margins as design specs, the lead compensator (adds phase, speeds up), the lag compensator (raises low-frequency gain, reduces error), lead-lag, loop-shaping rules, bandwidth vs crossover, and a complete lead design computed in Python — the classical techniques behind servo and aerospace control.