Nyquist Plot Construction

A Nyquist plot is more than just a polar graph—it is a powerful visual tool for assessing the stability of closed-loop control systems directly from their open-loop behavior. Unlike other frequency response methods, it provides a definitive stability criterion by mapping the entire frequency response, from $-\infty$ to $+\infty$, onto the complex plane. Mastering its construction is essential for control engineers to predict system robustness and avoid instability without solving for closed-loop poles directly.

The Fundamental Concept: Mapping Frequency to the Complex Plane

At its core, a Nyquist plot is a parametric plot of the open-loop transfer function $G(j\omega )H(j\omega )$ as the frequency $\omega$ varies. You evaluate the transfer function by substituting $s=j\omega$, which yields a complex number for each frequency: a magnitude and a phase. This complex number is then plotted as a point in the complex plane, with the real part on the x-axis and the imaginary part on the y-axis. As $\omega$ is swept from $0$ to $+\infty$, a curve is traced. The complete Nyquist plot also includes the mirror image of this curve for frequencies from $-\infty$ to $0$, which is the complex conjugate of the positive-frequency plot. This full contour is called the Nyquist path or contour.

The genius of the Nyquist Stability Criterion lies in how this plot relates to stability. It states that the number of unstable closed-loop poles, $Z$, is given by $Z=N+P$. Here, $P$ is the number of open-loop poles in the right-half of the s-plane (unstable open-loop poles), which you know from the system model. $N$ is the net number of times the Nyquist plot of $G(j\omega )H(j\omega )$ encircles the critical point $(-1,0j)$ in the clockwise direction. Your goal in construction is to create an accurate plot to correctly determine $N$.

Step-by-Step Construction Procedure

Constructing a clear and accurate plot follows a logical sequence. Begin by determining the low-frequency asymptote. As $\omega \to 0$, evaluate $G(j\omega )H(j\omega )$. For most practical systems with integrators ($1/s$ terms), this asymptote will have a phase of $-90^{\circ }$ per integrator and a magnitude that approaches infinity. Conversely, find the high-frequency asymptote as $\omega \to \infty$. The phase will typically approach $-90^{\circ }\times (n-m)$, where $n$ and $m$ are the denominator and numerator orders, and the magnitude approaches zero.

Next, calculate the real and imaginary axis crossings, as these define the plot's shape. The real axis crossing is found by setting the imaginary part of $G(j\omega )H(j\omega )$ to zero and solving for the frequency $\omega$. Substitute this frequency back into the real part to find the coordinate. Similarly, find the imaginary axis crossing by setting the real part to zero. These crossings are crucial for determining how close the plot comes to the critical point $(-1,0)$.

A critical step is handling systems with poles on the imaginary axis, such as integrators ($s=0$) or oscillatory modes ($s=\pm j{\omega }_n$). The standard Nyquist path cannot go through these poles, so you must detour around them with an infinitesimally small semicircle into the right-half plane. You must evaluate the transfer function along this detour. For an integrator ($1/s$), as you detour around $s=0$, the plot will show a large semicircular arc of infinite radius. The direction of this arc (clockwise or counterclockwise) is determined by the phase change along the detour and is vital for getting the encirclement count $N$ correct.

Interpreting the Plot for Stability Analysis

Once the plot is constructed, stability analysis is methodical. First, identify $P$, the number of open-loop poles in the right-half plane. For a stable open-loop system, $P=0$. Then, plot the complete contour for $G(j\omega )H(j\omega )$, including the detours and the negative frequency mirror image. Finally, count the net clockwise encirclements, $N$, of the point $(-1,0j)$.

Apply the formula $Z=N+P$. Here, $Z$ represents the number of closed-loop poles in the right-half plane. For the closed-loop system to be stable, $Z$ must equal zero. This means the number of encirclements must exactly counteract any inherent open-loop instability. A common result for stable open-loop systems ($P=0$) is that you require $N=0$ for closed-loop stability—meaning the Nyquist plot must not encircle $(-1,0)$ at all. The proximity of the plot to this critical point also gives qualitative measures of stability margins: the gain margin is determined by the distance to the real axis crossing left of -1, and the phase margin is related to the angle at the unit circle crossing.

Worked Example: A Standard Second-Order System

Let's construct the Nyquist plot for $G(s)H(s)=\frac{5}{s(s+2)}$. This is a type-1 system with a pole at the origin.

1. Low Frequency ($\omega \to 0^{+}$): $G(j\omega )\approx \frac{5}{j\omega \cdot 2}=-\frac{2.5j}{\omega }$. This is a large-magnitude point on the negative imaginary axis (phase = -90°).
2. High Frequency ($\omega \to \infty$): Magnitude goes to zero, phase approaches $-180^{\circ }$ (from two poles).
3. Real Axis Crossing: Find where the imaginary part is zero.

$$G(j\omega )=\frac{5}{j\omega (j\omega +2)}=\frac{5}{-{\omega }^2+j2\omega }$$ Multiply numerator and denominator by the conjugate: $$G(j\omega )=\frac{5(-{\omega }^2-j2\omega )}{{\omega }^4+4{\omega }^2}=\frac{-5{\omega }^2}{{\omega }^4+4{\omega }^2}+j\frac{-10\omega }{{\omega }^4+4{\omega }^2}$$ Set the imaginary part to zero: $\frac{-10\omega }{{\omega }^4+4{\omega }^2}=0$. This only occurs as $\omega \to \infty$, telling us the plot only touches the real axis at the origin (high-frequency end).

1. Imaginary Axis Crossing: Set the real part to zero: $\frac{-5{\omega }^2}{{\omega }^4+4{\omega }^2}=0$. This also only occurs as $\omega \to \infty$. At finite frequencies, the real part is always negative.
2. Detour around $s=0$: On the small detour, $s=ϵe^{j\theta }$ with $\theta$ going from $-90^{\circ }$ to $+90^{\circ }$. Then $G(s)\approx \frac{5}{ϵe^{j\theta }\cdot 2}=\frac{2.5}{ϵ}{e}^{-j\theta }$. As $ϵ\to 0$, the magnitude is enormous. The angle $\theta$ sweeps from $-90^{\circ }$ to $+90^{\circ }$, causing $-j\theta$ to sweep from $+90^{\circ }$ to $-90^{\circ }$. This results in an infinitely large clockwise semicircle in the complex plane, connecting the low-frequency asymptote from the negative imaginary axis to the positive imaginary axis.

The resulting plot starts at infinity on the negative imaginary axis (due to the detour connection), comes in from the positive imaginary side as frequency increases from zero, and spirals into the origin at -180° as $\omega \to \infty$. The negative frequency plot is the mirror image. The critical point $(-1,0)$ is on the negative real axis. This plot does not encircle $(-1,0)$. With $P=0$ (no open-loop right-half-plane poles) and $N=0$, we have $Z=0$, indicating a stable closed-loop system.

Common Pitfalls

1. Ignoring the Negative Frequency Contour: A common mistake is only plotting for $\omega$ from $0$ to $\infty$. The stability criterion requires the complete contour for $\omega \in (-\infty ,\infty )$. The negative frequency part is simply the complex conjugate mirror of the positive frequency plot, but omitting it can lead to an incorrect assessment of encirclements, especially for plots that cross the real axis multiple times.

1. Incorrect Detour around Imaginary Axis Poles: Improperly handling poles at the origin or on the $j\omega$-axis is the most frequent source of error. The detour must be into the right-half plane, and you must correctly calculate the phase change along the detour arc. For an integrator, forgetting to include the large clockwise semicircle can completely change the encirclement count $N$. Always sketch the s-plane Nyquist path first to visualize the detour.

1. Miscalculating Axis Crossings: Solving for the wrong part of the complex function can lead to incorrect crossing points. Remember: to find the real axis crossing, set the imaginary part of $G(j\omega )H(j\omega )$ to zero. To find the imaginary axis crossing, set the real part to zero. Double-check your algebra, as the expressions can become cumbersome.

1. Confusing Open-Loop and Closed-Loop Stability: The Nyquist plot is of the open-loop transfer function $G(s)H(s)$, but the encirclement criterion tells you about closed-loop stability. A stable open-loop system ($P=0$) can still produce an unstable closed-loop system if the Nyquist plot encircles $(-1,0)$. Do not assume that because the individual components $G(s)$ and $H(s)$ are stable, the feedback system will be stable.

Summary

• A Nyquist plot is a polar plot in the complex plane of the open-loop frequency response $G(j\omega )H(j\omega )$ for $\omega$ from $-\infty$ to $+\infty$, used to determine closed-loop stability.
• Construction requires analyzing low and high frequency asymptotes, calculating real and imaginary axis crossings, and making proper contour detours around any poles on the imaginary axis.
• The Nyquist Stability Criterion uses the formula $Z=N+P$, 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 point $(-1,0)$.
• The plot's proximity to $(-1,0)$ provides insights into gain and phase margins, which quantify the system's relative stability and robustness.
• Avoiding pitfalls like neglecting the negative-frequency contour or mishandling detours is critical for performing a correct stability analysis.