Q3Control System

The Nyquist stability criterion evaluates closed-loop stability directly from the open-loop frequency response, without needing to factor the closed-loop characteristic equation. The Nyquist contour in the s-plane encircles the entire right-half plane (traversing the imaginary axis from -j∞ to +j∞ and closing with an infinite semicircle on the right), with a small indentation around any open-loop poles that lie exactly on the imaginary axis (such as poles at the origin). Mapping this contour through G(s)H(s) produces the Nyquist plot; the relationship Z = N + P then connects the number of closed-loop right-half-plane poles Z to the number of open-loop right-half-plane poles P and the number of clockwise encirclements N of the critical point (-1, j0) by the plot. Stability requires Z = 0.

Substituting s = jω:

Rationalizing:

More directly, magnitude and phase are:

Illustrative numerical points (taking T = 1 for concreteness): at ω = 0.5, magnitude = k/(0.5×√(1+0.25)) = k/0.559 = 1.789k, phase = -90°-tan⁻¹(0.5) = -90°-26.6°=-116.6°; at ω = 1, magnitude = k/(1×√2)=0.707k, phase=-90°-45°=-135°; at ω=2, magnitude=k/(2×√5)=0.224k, phase=-90°-63.4°=-153.4°; at ω=5, magnitude=k/(5×√26)=0.0392k, phase=-90°-78.7°=-168.7°. These sample points confirm the trend: as ω increases from small to large values, the phase monotonically increases in magnitude from -90° toward -180° while the magnitude of G(jω)H(jω) monotonically decreases toward zero, tracing a smooth spiral that hugs closer and closer to the negative real axis as it shrinks toward the origin, but by the time the phase is close to -180° the magnitude is already very small (well inside the unit circle), which is the graphical reason the plot cannot reach out to encircle the point at (-1, j0).

As ω → 0⁺: magnitude → ∞, phase → -90°. As ω → ∞: magnitude → 0, phase → -180°. This means the Nyquist locus for ω > 0 begins at infinity along the -90° direction (i.e., far down the negative imaginary axis) and spirals clockwise, terminating at the origin tangent to the -180° direction (negative real axis). Because the system has a pole at the origin (Type-1), the standard Nyquist contour must include a small semicircular indentation of radius ε around s = 0, which maps to a large arc of radius k/ε sweeping from -90° to +90° in the G-plane (clockwise), connecting the ω = 0⁻ branch to the ω = 0⁺ branch and closing the plot at infinity in the right half of the G-plane. The full Nyquist plot (for ω from -∞ to +∞, plus the indentation arc) is therefore a large near-semicircular contour occupying primarily the third and fourth quadrants (below the real axis) for positive ω, mirrored above for negative ω, joined by the large clockwise arc at infinity around the indentation.

Stability: The open-loop transfer function G(s)H(s) = k/[s(1+sT)] has one pole at the origin (on the imaginary axis, handled by the indentation) and one pole at s = -1/T (in the left-half plane), so P = 0 (no open-loop poles strictly inside the right-half plane). Examining the plot, since the entire locus (for k, T > 0) stays in the lower half of the G-plane for ω > 0 (phase always between -90° and -180°) and never crosses to the right of or encircles the critical point (-1, j0) — the real-axis crossing only occurs at ω → ∞ where the magnitude has already decayed to zero — the number of encirclements N = 0. By the Nyquist criterion Z = N + P = 0 + 0 = 0, meaning there are no closed-loop poles in the right-half plane. Therefore, the system is stable for all positive values of k and T; this is expected since the open-loop system has only one real pole besides the origin, giving insufficient phase lag (maximum phase approaches but never reaches -180° exactly at finite ω) to destabilize the unity-feedback loop, consistent with Bode analysis for a Type-1, two-pole system.

Design implication. This unconditional stability result is a well-known and important property of second-order Type-1 systems with a single finite pole besides the integrator: since the maximum possible open-loop phase lag is -90° (from the pole at origin) + (up to but never reaching) -90° (from the single real pole as ω→∞), the total phase can only approach -180° asymptotically as ω→∞, precisely where the magnitude has already fallen to zero — so the locus can graze but never actually cross the negative real axis at a magnitude exceeding unity, and the (-1, j0) point can never be encircled for any positive k. This is why simple velocity- or position-control servo loops modeled as k/[s(1+sT)] are inherently robust to gain variation. The situation changes qualitatively the moment a third pole (or additional time constant) is introduced into the loop — as seen in other Bode/root-locus problems in this same paper — because a third pole allows the total phase lag to exceed -180° at a finite frequency where the gain is still greater than unity, at which point the Nyquist plot does encircle (-1, j0) for sufficiently large k, and the system becomes conditionally stable (stable only below a critical gain). This progression from unconditional to conditional stability as poles are added is a central theme connecting the Nyquist, Bode and root-locus techniques covered throughout this paper.

Relation to the root-locus picture. The same conclusion can be reached from a root-locus perspective: for G(s)H(s) = k/[s(1+sT)], there are only two open-loop poles (at s=0 and s=-1/T) and no zeros, so by the standard root-locus rules the two branches simply travel from these two starting poles straight along the real axis toward each other, meet at a single breakaway point exactly midway (at s = -1/(2T)), and then depart vertically as a complex-conjugate pair, moving straight up and down parallel to the imaginary axis forever (since with only two poles and no zeros, the two asymptotes at 90° and 270° are themselves vertical lines through the breakaway point) — critically, these vertical asymptotic branches never bend back to cross into the right-half plane for any finite k, which is the root-locus-domain confirmation of the same unconditional stability property established here via the Nyquist plot.