Questions: Nyquist Criterion and Stability from Frequency Response

Question 1

A feedback system has an open-loop transfer function with 2 unstable poles (P = 2). The engineer plots the Nyquist diagram and counts 1 clockwise encirclement of the (-1, 0) point. What does this imply about the closed-loop system?

• A. The closed-loop system is stable because only 1 encirclement is present
• B. The closed-loop system is unstable: N = Z - P gives Z = N + P = 1 + 2 = 3 unstable closed-loop poles
• C. The result is inconclusive — the Bode plot must also be checked before drawing conclusions
• D. The closed-loop system is marginally stable since the Nyquist plot did not pass through (-1, 0)

Answer: (B)

The Nyquist stability criterion states N = Z - P, where N is the number of net clockwise encirclements of (-1, 0), Z is the number of unstable closed-loop poles, and P is the number of unstable open-loop poles. With N = 1 and P = 2: 1 = Z - 2, so Z = 3. Three unstable closed-loop poles means the system is unstable. For stability, Z = 0 is required, which needs N = -P = -2 — two counterclockwise encirclements.

Question 2

Why is the point (-1, 0) specifically the critical point in the Nyquist criterion, rather than the origin or any other reference?

• A. It is chosen by convention to make the gain margin equal to 1 at marginal stability
• B. A closed-loop instability occurs when 1 + G(jω)H(jω) = 0, i.e., G(jω)H(jω) = -1, which is exactly the point (-1, 0) — the gain and phase condition for marginal closed-loop instability
• C. The (-1, 0) point is where the phase of the open-loop transfer function crosses 0°, marking the natural frequency
• D. (-1, 0) is the point where the open-loop gain equals the phase margin by definition

Answer: (B)

The closed-loop characteristic equation is 1 + G(s)H(s) = 0. A root on the imaginary axis at s = jω means G(jω)H(jω) = -1, which corresponds to (-1, 0) in the complex plane. The argument principle applied to 1 + GH maps encirclements of its origin to encirclements of (-1, 0) in the GH-plane (they differ by a shift of 1 on the real axis). The (-1, 0) point is not a convention — it is the exact condition under which the closed-loop system is on the verge of instability.

Question 3

The Nyquist criterion can correctly determine closed-loop stability even when the open-loop transfer function has poles in the right half plane.

• T. True
• F. False

Answer: True

This is one of Nyquist's key advantages over Bode analysis. The formula N = Z - P explicitly accounts for open-loop RHP poles (P ≠ 0). For stability, Z = 0 is required, so exactly P counterclockwise encirclements are needed to compensate. Bode gain and phase margins implicitly assume P = 0 (minimum-phase, stable open loop) and therefore cannot reliably analyze systems with open-loop instability.

Question 4

Gain margin and phase margin derived from a Bode diagram provide an equivalent and equally reliable stability assessment to the Nyquist criterion, including for systems with open-loop unstable poles.

• T. True
• F. False

Answer: False

Bode margins are valid only when the open-loop system is stable and minimum-phase (no RHP poles or zeros). For systems with open-loop RHP poles, the Bode margin interpretation fails: the crossover frequencies may suggest adequate margins while the closed loop is actually unstable, or a conditionally stable system may require a specific range of gains that Bode margins do not reveal. The Nyquist criterion handles non-minimum-phase plants and open-loop unstable systems correctly through explicit N = Z - P accounting.

Question 5

Why does the Nyquist criterion analyze encirclements of (-1, 0) rather than simply checking whether the open-loop transfer function has any poles in the right half plane?

Model answer: The stability question is about the closed-loop poles, not the open-loop poles. A system can be open-loop unstable but closed-loop stable — this is precisely how feedback is used to stabilize an inherently unstable plant. The Nyquist criterion applies the argument principle to count RHP zeros of 1 + G(s)H(s), which are the closed-loop poles, by counting encirclements of (-1, 0) in the GH-plane. This converts a closed-loop pole-counting problem (which would require computing the closed-loop transfer function explicitly) into an encirclement-counting problem on the open-loop frequency response (which is directly measurable). The formula N = Z - P relates encirclements to closed-loop RHP poles after accounting for known open-loop RHP poles.

This is also why Nyquist works on experimentally identified frequency responses: you do not need an analytic transfer function — you can measure G(jω) directly by sweeping a sinusoidal input and plot the Nyquist diagram from the measured data.