Nyquist Stability Criterion

Why This Matters

The Nyquist Stability Criterion represents one of the most powerful tools in your control theory toolkit because it lets you determine closed-loop stability by examining only the open-loop transfer function. You're being tested on your ability to connect frequency-domain analysis, complex plane geometry, and feedback system behavior—all in one elegant graphical method. Unlike root locus, which tracks pole locations directly, Nyquist works through the principle of argument, counting encirclements to reveal what's happening inside the closed-loop system.

This criterion becomes essential when you're dealing with systems where you can't easily factor the characteristic equation, or when you need to assess stability margins that tell you how close your system is to going unstable. The concepts here—gain margin, phase margin, right-half plane poles, and encirclement counting—appear repeatedly in exam questions and real-world controller design. Don't just memorize the encirclement rule; know 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 complex analysis, specifically the argument principle, which relates contour encirclements to zeros and poles of a function.

Open-Loop Transfer Function

• $G(s)H(s)$ defines the loop transfer function—the product of plant $G(s)$ and controller $H(s)$ without the feedback path closed
• Frequency response is obtained by substituting $s = j\omega$, yielding magnitude and phase information across all frequencies
• Poles and zeros of $G(s)H(s)$ directly determine the shape of the Nyquist plot and the stability analysis outcome

Nyquist Contour

• A closed path enclosing the entire right-half plane—consisting of the imaginary axis from $-j\infty$ to $+j\infty$ plus a semicircular arc of infinite radius
• Captures all potentially unstable frequencies by ensuring every right-half plane pole of the closed-loop system is inside the contour
• Indentations around imaginary axis poles are required when $G(s)H(s)$ has poles at $s = 0$ or $s = j\omega_0$, using small semicircles to avoid singularities

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Constructing and Interpreting the Plot

Understanding how to build and read a Nyquist plot is fundamental to applying the stability criterion correctly.

Nyquist Plot Construction

• Plot $G(j\omega)H(j\omega)$ as $\omega$ varies from $0$ to $+\infty$—then reflect across the real axis for negative frequencies, since the transfer function of real systems has conjugate symmetry
• Magnitude determines distance from origin while phase determines angle, creating a polar-style trajectory through the complex plane
• Low-frequency behavior starts near the DC gain value; high-frequency behavior typically approaches the origin as most physical systems roll off

Encirclements of the Critical Point

• The point $(-1, 0)$ is critical—encirclements of this point, not the origin, determine closed-loop stability
• Clockwise encirclements are counted as positive ($N$), and the criterion relates these to right-half plane poles of $G(s)H(s)$ ($P$) and closed-loop unstable poles ($Z$)
• The fundamental relationship is $Z = N + P$—for closed-loop stability, you need $Z = 0$, meaning $N = -P$ (counterclockwise encirclements must cancel out open-loop RHP poles)

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Stability Margins and Robustness

Beyond binary stable/unstable determination, the Nyquist plot reveals how much margin you have before instability occurs.

Gain Margin

• Measured where the Nyquist plot crosses the negative real axis—the gain margin is $1/|G(j\omega_{pc})H(j\omega_{pc})|$, where $\omega_{pc}$ is the phase crossover frequency
• Expressed in decibels as $GM = 20\log_{10}(1/a)$ where $a$ is the magnitude at the real-axis crossing
• Indicates how much loop gain can increase before the plot passes through $(-1, 0)$ and the system goes unstable

Phase Margin

• Measured at the gain crossover frequency $\omega_{gc}$—where $|G(j\omega)H(j\omega)| = 1$, meaning the plot crosses the unit circle
• Phase margin equals $180° + \angle G(j\omega_{gc})H(j\omega_{gc})$—the additional phase lag needed to reach the critical point
• 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

Applying the criterion correctly requires understanding both the standard case and important exceptions.

Closed-Loop Stability Analysis

• Stability requires $Z = N + P = 0$—the number of clockwise encirclements plus open-loop RHP poles must equal zero
• For minimum-phase systems ($P = 0$), stability simply requires no encirclements of $(-1, 0)$
• For non-minimum-phase systems ($P > 0$), you need exactly $P$ counterclockwise encirclements to achieve stability

Right-Half Plane Poles and Zeros

• Open-loop RHP poles ($P > 0$) mean the system is open-loop unstable—feedback must stabilize it through counterclockwise encirclements
• Open-loop RHP zeros don't appear directly in the encirclement formula but constrain achievable bandwidth and cause non-minimum phase behavior
• Each RHP zero limits performance—you cannot achieve arbitrarily fast response without risking instability

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Limitations and Practical Considerations

The Nyquist criterion has boundaries—knowing when it applies (and when it doesn't) is exam-critical.

Assumptions and Constraints

• Requires linear time-invariant (LTI) systems—nonlinear systems need describing function methods or other approaches
• Transfer function must be rational with finite gain at all frequencies—pure integrators and differentiators require careful contour modifications
• Time delays add phase lag without changing magnitude—a delay $e^{-sT}$ rotates the Nyquist plot, potentially causing encirclements at high frequencies

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Quick Reference Table

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Self-Check Questions

1. If 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. Compare gain margin and phase margin: which one would you rely on more heavily when designing a controller for a system with significant unmodeled time delay, and 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 (with RHP zeros) require counterclockwise encirclements for stability, while a minimum-phase system simply needs to avoid the critical point?

5. If an FRQ presents a Nyquist plot that passes exactly through $(-1, 0)$, what does this indicate about the closed-loop system's poles, and how would you describe the system's stability status?