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The Nyquist criterion uses the frequency response H(jω) plotted in the complex plane to determine closed-loop stability without explicitly computing poles. Encirclements of the (-1, 0) point indicate instability; gain and phase margins measure robustness to perturbations.
From Bode plot analysis, you know how to read gain and phase margins from frequency response graphs — those margins tell you how far the open-loop response is from the instability boundary at −1. From pole-zero analysis, you know that a closed-loop system is stable if and only if all its poles lie in the left half of the s-plane. The Nyquist criterion unifies these ideas, giving a rigorous test for closed-loop stability based only on the open-loop frequency response, without ever computing the closed-loop poles explicitly.
The mathematical foundation is the argument principle from complex analysis: if a function F(s) is analytic inside a closed contour in the s-plane, the number of times F(s) encircles the origin as s traverses the contour equals Z − P, where Z and P are the numbers of zeros and poles of F(s) inside the contour. For stability analysis, define F(s) = 1 + G(s)H(s) — the characteristic polynomial of the closed loop. The Nyquist contour encloses the entire right half plane (the region of instability). A closed-loop pole in the RHP is a zero of F(s) = 1 + G(s)H(s), which is equivalent to a zero of G(s)H(s) = −1. So counting encirclements of the point (−1, 0) in the G(s)H(s)-plane as s traverses the Nyquist contour gives N = Z − P, where Z is the number of unstable closed-loop poles and P is the number of unstable open-loop poles. For a stable closed loop, Z must equal zero: N = −P (counterclockwise encirclements equal the number of open-loop RHP poles, if any).
The practical recipe: plot G(jω)H(jω) as ω sweeps from 0 to +∞, then mirror (conjugate) to get −∞ to 0, and close the contour at infinity. Count net clockwise encirclements of (−1, 0). For a system with no open-loop RHP poles (P = 0), any clockwise encirclement means instability. For a system with P open-loop RHP poles (an unstable plant, for example), you need exactly P counterclockwise encirclements for stability. This is more powerful than Bode analysis alone: Bode margins implicitly assume a minimum-phase, stable open-loop system, while Nyquist handles non-minimum-phase plants and conditionally stable systems (where increasing gain actually stabilizes the loop) correctly.
Gain margin and phase margin have precise geometric meaning on the Nyquist plot. The gain margin is the factor by which gain can increase before the Nyquist plot crosses through (−1, 0) — equivalently, 1/|G(jω_pc)| where ω_pc is the phase crossover frequency (where phase = −180°). The phase margin is how many additional degrees of phase lag would move the unit-circle crossing of the Nyquist plot to exactly (−1, 0). Both margins measure the distance from the plot to the critical point, giving robustness to gain uncertainty and phase delay respectively. A well-designed feedback system typically targets gain margin > 6 dB and phase margin > 45°, ensuring the system can tolerate significant plant uncertainty or additional actuator lag before going unstable.