Nyquist Stability Criterion

Why This Matters

The Nyquist Stability Criterion lets you determine closed-loop stability by examining only the open-loop transfer function. Instead of factoring the characteristic equation or tracking pole locations directly (like root locus), Nyquist uses a graphical method rooted in complex analysis: you count encirclements on a polar plot to reveal how many unstable closed-loop poles exist.

This criterion is especially useful when you can't easily factor the characteristic equation, or when you need stability margins that quantify how close your system is to going unstable. The core concepts here, including gain margin, phase margin, right-half plane poles, and encirclement counting, show up constantly in both exams and real-world controller design. Don't just memorize the encirclement rule; make sure you understand why encirclements correspond to unstable poles and how the Nyquist contour captures all relevant frequency information.

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The Mathematical Foundation

The Nyquist criterion builds on the argument principle from complex analysis, which relates contour encirclements to zeros and poles of a function.

Open-Loop Transfer Function

The loop transfer function $G(s)H(s)$ is the product of the plant $G(s)$ and controller $H(s)$, evaluated with the feedback path open. To get the frequency response, you substitute $s = j\omega$, which gives you magnitude and phase information across all frequencies. The poles and zeros of $G(s)H(s)$ directly determine the shape of the Nyquist plot and the outcome of your stability analysis.

Nyquist Contour

The Nyquist contour is a closed path that encloses the entire right-half of the $s$-plane. It consists of:

1. The imaginary axis from $-j\infty$ to $+j\infty$
2. A semicircular arc of infinite radius closing the path through the right-half plane

This contour captures every potentially unstable frequency, ensuring that any right-half plane pole of the closed-loop system falls inside it.

When $G(s)H(s)$ has poles on the imaginary axis (at $s = 0$ or $s = j\omega_0$), you must indent the contour around them using small semicircles to avoid the singularities.

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Constructing and Interpreting the Plot

Nyquist Plot Construction

To build a Nyquist plot, follow these steps:

1. Evaluate $G(j\omega)H(j\omega)$ as $\omega$ varies from $0$ to $+\infty$, plotting each value as a point in the complex plane.
2. Reflect the curve across the real axis to account for negative frequencies. This works because real-coefficient transfer functions have conjugate symmetry: $G(-j\omega)H(-j\omega) = \overline{G(j\omega)H(j\omega)}$.
3. Connect the endpoints using the mapping of the infinite semicircular arc (this portion often collapses to the origin or a single point for strictly proper systems).

At each frequency, the magnitude $|G(j\omega)H(j\omega)|$ sets the distance from the origin, and the phase $\angle G(j\omega)H(j\omega)$ sets the angle. The result is a polar-style trajectory through the complex plane. Low-frequency behavior starts near the DC gain value, and high-frequency behavior typically approaches the origin since most physical systems roll off at high frequencies.

Encirclements of the Critical Point

The key to the entire criterion: you count encirclements of the point $(-1, 0)$, not the origin.

Why $(-1, 0)$? The closed-loop characteristic equation is $1 + G(s)H(s) = 0$, which means $G(s)H(s) = -1$. So the critical point corresponds exactly to the boundary of instability.

The encirclement rule is:

$Z = N + P$

• $N$ = number of clockwise encirclements of $(-1, 0)$ (counterclockwise counts as negative)
• $P$ = number of right-half plane poles of $G(s)H(s)$ (open-loop unstable poles)
• $Z$ = number of right-half plane poles of the closed-loop system

For closed-loop stability, you need $Z = 0$, which means $N = -P$. In other words, counterclockwise encirclements must exactly cancel out the open-loop RHP poles.

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Stability Margins and Robustness

Beyond a binary stable/unstable answer, the Nyquist plot tells you how much margin you have before instability occurs.

Gain Margin

Gain margin is measured where the Nyquist plot crosses the negative real axis. At that crossing, the phase is exactly $-180°$, and the frequency is called the phase crossover frequency $\omega_{pc}$.

If the magnitude at that crossing is $a = |G(j\omega_{pc})H(j\omega_{pc})|$, then:

• Gain margin = $1/a$
• In decibels: $GM = 20\log_{10}(1/a)$

This tells you how much you could multiply the loop gain before the plot passes through $(-1, 0)$ and the system goes unstable. For example, if the plot crosses the negative real axis at $-0.25$, then $a = 0.25$, the gain margin is $1/0.25 = 4$, and $GM = 20\log_{10}(4) \approx 12$ dB. You could quadruple the gain before losing stability.

Phase Margin

Phase margin is measured at the gain crossover frequency $\omega_{gc}$, where $|G(j\omega)H(j\omega)| = 1$ (the plot crosses the unit circle).

$PM = 180° + \angle G(j\omega_{gc})H(j\omega_{gc})$

This represents the additional phase lag needed to rotate the plot to the critical point. If the phase at gain crossover is $-135°$, then $PM = 180° - 135° = 45°$.

Typical design targets are $PM > 45°$ and $GM > 6$ dB for adequate robustness against modeling errors and parameter variations.

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Stability Determination and Special Cases

Closed-Loop Stability Analysis

The stability condition is $Z = N + P = 0$. How this plays out depends on the open-loop system:

• Minimum-phase systems ($P = 0$): All open-loop poles are in the left-half plane. Stability simply requires no encirclements of $(-1, 0)$, since $N$ must equal zero.
• Non-minimum-phase systems ($P > 0$): The open-loop system is already unstable. Feedback must stabilize it, which requires exactly $P$ counterclockwise encirclements ($N = -P$) to bring $Z$ to zero.

Right-Half Plane Poles and Zeros

Open-loop RHP poles ($P > 0$) mean the plant is open-loop unstable. The feedback loop must generate the right number of counterclockwise encirclements to cancel them out.

Open-loop RHP zeros don't appear directly in the $Z = N + P$ formula, but they still matter. RHP zeros cause non-minimum phase behavior, where the step response initially moves in the wrong direction before correcting. More importantly, each RHP zero imposes a fundamental limit on achievable closed-loop bandwidth. You cannot push the bandwidth past the RHP zero frequency without risking instability.

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Limitations and Practical Considerations

Assumptions and Constraints

The Nyquist criterion applies under specific conditions:

• The system must be linear and time-invariant (LTI). Nonlinear systems require other approaches, such as describing function methods.
• The transfer function must be rational with finite gain at all frequencies. Pure integrators and differentiators require careful contour modifications (the small semicircular indentations mentioned earlier).
• Time delays of the form $e^{-sT}$ add phase lag without changing magnitude. On the Nyquist plot, a delay spirals the curve inward at high frequencies, which can introduce encirclements that wouldn't exist without the delay. This is why phase margin is so important for systems with time delay.

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Quick Reference Table

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Self-Check Questions

1. A system has two open-loop RHP poles ($P = 2$) and the Nyquist plot makes one clockwise encirclement of $(-1, 0)$. How many closed-loop poles are in the RHP? Is the system stable?

2. Which stability margin would you rely on more heavily when designing a controller for a system with significant unmodeled time delay: gain margin or phase margin? Why?

3. A Nyquist plot crosses the negative real axis at $-0.5$. What is the gain margin in dB, and by what factor could you increase the loop gain before instability?

4. Why does a non-minimum-phase system ($P > 0$) require counterclockwise encirclements for stability, while a minimum-phase system simply needs to avoid the critical point?

5. If a Nyquist plot passes exactly through $(-1, 0)$, what does this tell you about the closed-loop system's poles and its stability?