Closed-loop stability inferred from loop shapes

Task 11.1 · Nyquist plot and the critical point

L(s) = 1(s+1)³  closed with a gain k, negative feedback

Z = N + P. N counts clockwise encirclements of −1/k by the full Nyquist plot, P the open-loop poles in the right half-plane, Z the closed-loop poles there. Stable iff Z = 0.

L(iω) = (1 − 3ω²)/(1+ω²)³ + i (ω³ − 3ω)/(1+ω²)³. It starts at 1 (ω = 0), crosses the imaginary axis at ω = 1/√3, the negative real axis at −1/8 (ω = √3), and ends at the origin with phase −270°.

Task 11.2 · A pole on the D-contour

L₁(s) = 1s (s+1)²

The pole at s = 0 lies on the contour, so the contour must detour around it (segment D4). The detour maps to a huge arc that closes the Nyquist plot. Which side you detour on changes N and P, but never Z.

L₁(iω) = −2/(1+ω²)² + i (ω² − 1)/(ω(1+ω²)²). For ω → 0⁺: Re → −2, Im → −∞. Crosses the real axis at −1/2 (ω = 1). On the detour s = εe^iθ: L₁ ≈ e^−iθ/ε − 2, an arc of radius 1/ε (here ε = 0.05).

Robustness · gain margin and phase margin

GM: factor by which |kL| may grow before |kL| = 1 at the phase crossover ω₁₈₀. PM: how far the phase is above −180° at the gain crossover ω_c. k scales the magnitude only; the phase curve never moves.

Task 11.3 · Spacecraft with a round-trip delay

G(s) = 1s+1  K(s) = 1T_I s  L(s) = K(s)G(s) e^{−2T_t s}

A delay leaves |L(iω)| untouched and subtracts the phase 2T_tω. The Nyquist curve spirals clockwise and eats the phase margin. Stability is lost when the lag at ω_c equals the phase margin.

The closed loop from r to the physical output is e^−T_ts/(e^−2T_ts + T_Is(s+1)). Only the round-trip delay in the denominator matters for stability.