Nyquist Stability Criterion

The Nyquist Stability Criterion is a powerful graphical technique that determines the stability of a closed-loop control system by examining the frequency response of its open-loop counterpart. Unlike root locus or Routh-Hurwitz methods, it can directly analyze systems with time delays and open-loop unstable systems, providing crucial insights from experimentally obtainable frequency data. By transforming a complex algebraic stability problem into a visual plot inspection, it offers both a definitive stability verdict and a quantitative measure of relative stability—how close the system is to instability.

From Open-Loop Response to Closed-Loop Stability

To understand the Nyquist criterion, we must first grasp what it means to map a contour. Consider a complex function $F(s)$. If we let the complex variable $s$ travel along a specific closed path in the s-plane (called the Nyquist path), the values of $F(s)$ will trace out a corresponding closed contour in the $F(s)$-plane. The behavior of this mapped contour, specifically how many times it encircles the origin, is governed by the Principle of the Argument.

This principle states: *For a function $F(s)$ that is analytic (has no poles) on and inside a closed contour in the s-plane, the number of times the mapped contour encircles the origin in the $F(s)$-plane in the clockwise direction is equal to $Z-P$.* Here, $Z$ is the number of zeros of $F(s)$ inside the s-plane contour, and $P$ is the number of poles of $F(s)$ inside the s-plane contour. Encirclements are counted as positive for clockwise (CW) and negative for counter-clockwise (CCW) rotations.

The genius of the Nyquist method is its choice of $F(s)$ and the s-plane contour. For a standard unity-feedback system with open-loop transfer function $G(s)H(s)$, the closed-loop characteristic equation is $1+G(s)H(s)=0$. We therefore define: $$F(s)=1+G(s)H(s)$$ The zeros of $F(s)$ are the closed-loop poles (the roots of the characteristic equation we care about). The poles of $F(s)$ are simply the poles of $G(s)H(s)$, which are the open-loop poles.

The chosen s-plane contour, the Nyquist path, is a massive semicircle that encloses the entire right-half of the s-plane (RHP). It travels up the imaginary axis from $-j\infty$ to $+j\infty$ and then arcs back to the left with an infinite radius. By applying the Principle of the Argument to this path, we can relate the encirclements of the origin by the $F(s)$ contour to the number of closed-loop poles ($Z$) and open-loop poles ($P$) in the RHP.

The Nyquist Contour and the Critical Point (-1,0)

Plotting $F(s)=1+G(s)H(s)$ can be cumbersome. A critical simplification is made by noting that the contour of $F(s)$ is just the contour of $G(s)H(s)$ shifted to the right by 1. Therefore, encirclements of the origin by the $F(s)$ plot are equivalent to encirclements of the point $(-1,0)$ by the $G(s)H(s)$ plot. This point is famously known as the critical point.

Furthermore, because the infinite-radius arc of the Nyquist path typically maps to a single point (often the origin) for physically realizable systems, the only important part of the $G(s)H(s)$ contour is the one generated as $s=j\omega$ travels from $\omega =-\infty$ to $\omega =+\infty$. This is simply the polar plot of the open-loop frequency response $G(j\omega )H(j\omega )$, with the negative-frequency portion being the mirror image of the positive-frequency portion about the real axis.

We can now state the Nyquist Stability Criterion in its standard form: *Let $P$ be the number of open-loop poles of $G(s)H(s)$ located in the Right-Half Plane (RHP). The closed-loop system is stable if and only if the Nyquist plot of $G(j\omega )H(j\omega )$ encircles the critical point $(-1,0)$ exactly $N=-P$ times in the counter-clockwise direction, where a negative $N$ indicates clockwise encirclements.*

In simpler terms: Number of clockwise encirclements of (-1,0) = Number of open-loop RHP poles ($P$) - Number of closed-loop RHP poles ($Z$).

For the common case of a stable open-loop system ($P=0$), the rule simplifies dramatically: The closed-loop system is stable if and only if the Nyquist plot does not encircle the critical point (-1,0). The proximity of the Nyquist plot to this point also gives a measure of stability margins—the gain margin and phase margin.

Handling Special Cases: Open-Loop Poles on the Imaginary Axis

A practical complication arises when $G(s)H(s)$ has poles directly on the imaginary axis (e.g., an integrator with a pole at $s=0$). These poles lie exactly on the Nyquist path, violating the "analytic on the contour" requirement of the Principle of the Argument. We resolve this by modifying the s-plane contour to indent around these singularities with an infinitesimally small semicircle into the left-half plane.

On the Nyquist plot, this indentation maps to an arc of infinite radius. The procedure is: 1) Identify the imaginary axis poles. 2) Imagine a small detour around them to the left. 3) When plotting $G(j\omega )H(j\omega )$, approach the pole frequency ${\omega }_0$ from below and from above; the plot will show a discontinuous jump, connecting with a large-magnitude arc whose direction (CW or CCW) you must determine. For a pole at the origin, the infinite-radius arc typically connects the plot from $\omega =0^{-}$ to $\omega =0^{+}$. Correctly accounting for this mapping is essential for an accurate encirclement count.

The Power of Nyquist: Unstable Open-Loop Systems and Time Delays

The Nyquist criterion shows its true strength where other methods falter. First, it can directly analyze systems that are open-loop unstable ($P>0$). The Routh-Hurwitz method can struggle to provide design guidance here, but Nyquist gives a clear visual target: the plot must achieve a specific number of counter-clockwise encirclements to "cancel out" the unstable open-loop poles and render the closed-loop system stable. This is a critical tool for designing controllers for inherently unstable plants, like inverted pendulums or certain aircraft.

Second, it handles time delays with elegance. A pure time delay $e^{-sT}$ contributes phase lag ($-\omega T$ radians) without affecting magnitude. On a Bode plot, this wreaks havoc on phase margins. On a Nyquist plot, the increasing phase lag causes the frequency response to spiral inward toward the origin. This visualization makes it immediately apparent how even a small delay can cause the plot to encircle (-1,0), destabilizing an otherwise stable system. This graphical intuition is far more direct than trying to solve transcendental characteristic equations.

Common Pitfalls

1. Miscounting Encirclements: The most frequent error is incorrectly counting $N$. Remember to consider the net encirclements of the point (-1,0). Draw a vector from (-1,0) to a point on the plot and follow it as $\omega$ goes from $-\infty$ to $+\infty$. The net number of full 360° rotations of this vector is $N$ (positive for CCW). For systems with $P=0$, simply check if (-1,0) is to the left of the plot as you travel in the direction of increasing $\omega$.

1. Misinterpreting "Open-Loop Poles" ($P$): $P$ is the number of poles of the transfer function $G(s)H(s)$ in the Right-Half Plane. Do not include poles on the imaginary axis in this count; they are handled separately via the indentation rule. Confusing $P$ will lead to an incorrect stability prediction.

1. Ignoring the Negative-Frequency Plot: The Nyquist plot is defined for $\omega$ from $-\infty$ to $+\infty$. For real-valued systems, the plot for negative frequencies is the mirror image of the positive-frequency plot across the real axis. Forgetting to draw this mirror image can change the apparent enclosure of the (-1,0) point. Always sketch the full, symmetrical contour.

1. Incorrectly Mapping Imaginary Axis Poles: When $G(s)H(s)$ has a pole at $s=0$ (an integrator), the plot at very low frequencies will have near-infinite magnitude. The connection between $\omega =0^{-}$ and $\omega =0^{+}$ is an infinite-radius arc. Assuming the plot is discontinuous or drawing this arc in the wrong angular direction (e.g., CW instead of CCW) will lead to an error in the encirclement count $N$.

Summary

• The Nyquist Stability Criterion is a graphical method that determines closed-loop stability by analyzing the open-loop frequency response plot $G(j\omega )H(j\omega )$.
• The key rule is $Z=P-N$, where $Z$ is the number of unstable closed-loop poles, $P$ is the number of unstable open-loop poles, and $N$ is the net clockwise encirclements of the critical point (-1,0) by the Nyquist plot.
• For the common case of an open-loop stable system ($P=0$), stability requires no encirclements of the (-1,0) point.
• The criterion uniquely handles open-loop unstable systems ($P>0$) and systems with time delays, providing essential design insight where algebraic methods are cumbersome.
• Special care must be taken to correctly apply the indentation rule and map the contour when open-loop poles lie on the imaginary axis (e.g., integrators).
• The proximity of the Nyquist plot to the (-1,0) point directly indicates the system's gain and phase margins, quantifying its relative stability.