Introduction to the Nyquist Plot

The Nyquist plot remains one of the most powerful graphical tools for assessing feedback system stability. Named after Harry Nyquist, who introduced the stability criterion in 1932, the plot transforms the open‑loop transfer function G(s)H(s) into a polar representation of complex gain as a function of frequency. For graduate engineering students, constructing and interpreting Nyquist plots is not merely an academic exercise; it is a fundamental skill for analyzing robust control systems, dealing with delays, and understanding the trade‑offs between performance and stability. Unlike Bode plots, which separate magnitude and phase, the Nyquist plot preserves the full complex relationship, making it indispensable for systems with non‑minimum phase behavior or right‑half‑plane poles.

The Nyquist stability criterion connects the number of open‑loop poles in the right half‑plane (RHP) to the encirclements of the critical point (−1,0) on the Nyquist plot. Mastering this concept enables engineers to design controllers that ensure closed‑loop stability even when the open‑loop system is unstable or contains delays. This article expands on the theoretical foundations, step‑by‑step construction techniques, and practical interpretation of Nyquist plots, providing a comprehensive resource for advanced study.

Theoretical Foundations of the Nyquist Plot

The construction of a Nyquist plot is rooted in complex analysis, specifically the mapping of contours under analytic functions. The open‑loop transfer function G(s)H(s) is a rational function of the complex variable s = σ + jω. To assess closed‑loop stability, one evaluates G(s)H(s) along a specially chosen closed contour in the s‑plane called the Nyquist contour. This contour encloses the entire right half‑plane, excluding any poles that lie exactly on the imaginary axis (which are handled by small indentations).

The Nyquist Contour and the Argument Principle

The standard Nyquist contour consists of the imaginary axis from −j∞ to +j∞, connected by a semicircle of infinite radius that encircles the right half‑plane. By Cauchy’s argument principle, the number of counter‑clockwise encirclements of the origin by the mapping G(s)H(s) along this closed contour equals Z − P, where Z is the number of zeros and P the number of poles of the function inside the contour. For the closed‑loop characteristic equation 1 + G(s)H(s) = 0, the zeros of 1 + G(s)H(s) correspond to the closed‑loop poles. The Nyquist plot maps the contour through G(s)H(s); the critical point (−1,0) corresponds to the origin of the function 1 + G(s)H(s). Thus, encirclements of (−1,0) reveal information about the closed‑loop pole locations.

Mathematical Expression of the Nyquist Criterion

The Nyquist stability criterion can be stated as: Let N be the number of counter‑clockwise encirclements of the point (−1,0) by the Nyquist plot of G(s)H(s) as s traverses the Nyquist contour in the clockwise direction. Then the number of unstable closed‑loop poles (Z) is given by Z = N + P, where P is the number of open‑loop poles in the right half‑plane. For closed‑loop stability, Z must be zero. Therefore, when the open‑loop system is stable (P = 0), the Nyquist plot must not encircle (−1,0) at all.

This theorem is a direct application of the argument principle and contour integration. The rigorous derivation involves evaluating the change in argument of 1 + G(s)H(s) as s moves along the Nyquist contour. For most engineering purposes, the practical form of the criterion is sufficient: count the encirclements and compare to P.

Step‑by‑Step Construction of the Nyquist Plot

Constructing a Nyquist plot by hand requires evaluating G(jω)H(jω) over a range of frequencies and plotting the real and imaginary parts in the complex plane. The standard process is:

  • Define the open‑loop transfer function G(s)H(s) in factored form (poles and zeros). Identify any poles on the imaginary axis; these require special indentation around them.
  • Substitute s = jω and simplify to obtain G(jω)H(jω) as a complex function of the real frequency ω. Write it as X(ω) + jY(ω) or in polar form M(ω)∠φ(ω).
  • Select a sufficient number of frequency points from ω = 0 to ω → ∞. For most systems, the key regions are near the break frequencies (pole/zero locations) and at very low and very high frequencies. Use a logarithmic spacing for efficiency, but ensure enough points near resonant peaks.
  • Compute gain and phase at each frequency. For multiplication/division of factors, the overall magnitude is the product of magnitudes, and the overall phase is the sum of phases.
  • Plot the points in the complex plane with the real axis horizontal and the imaginary axis vertical. Connect the points smoothly, typically starting at ω = 0+ (just above zero, avoiding any poles at the origin) and ending at ω → ∞. Mirror the plot for negative frequencies using symmetry: the plot for ω → −∞ is the complex conjugate.
  • Handle the infinite semicircle by using the asymptotic behavior of the transfer function as ω → ∞. For proper rational functions, the plot tends to the origin along a specific angle determined by the relative degree.

Dealing with Poles on the Imaginary Axis

If G(s)H(s) has poles on the imaginary axis (e.g., an integrator 1/s or a pole at s = jω₀), the Nyquist contour must detour around them using small semicircles of radius ε → 0. The mapping of these indentations produces large arcs in the Nyquist plot. For a pole at the origin, the plot exhibits an infinite arc that starts at +j∞ and ends at −j∞, contributing 180° of phase shift. For purely imaginary poles at ±jω₀, the detour results in a loop near the critical point, which complicates visual interpretation. In such cases, analytical evaluation is often simpler than graphical construction.

Asymptotic Behavior at High Frequencies

For a strictly proper transfer function (relative degree ≥ 1), as ω → ∞, the magnitude |G(jω)H(jω)| → 0 and the phase approaches −90° × (relative degree). The plot therefore approaches the origin along a radial line. For example, a system with relative degree 2 will approach the origin from the negative real axis (phase −180°). This asymptotic direction helps complete the contour for the infinite semicircle.

Mathematical Techniques for Plot Construction

Beyond direct evaluation, several mathematical tools simplify constructing and interpreting Nyquist plots:

Cauchy’s Argument Principle

The argument principle is the underlying engine of the Nyquist criterion. It states that the change in argument of a function f(s) as s traverses a closed contour equals 2π(Z − P), where Z and P are the number of zeros and poles inside the contour. When applied to f(s) = 1 + G(s)H(s), the encirclements of the origin by the mapping correspond to the zeros of the characteristic equation. This principle allows engineers to compute stability indirectly without solving for closed‑loop poles.

Using M‑Circles and M‑Contours

For a more quantitative assessment, constant magnitude circles (M‑circles) and constant phase contours can be overlaid on the Nyquist plot. These loci indicate the closed‑loop magnitude and phase for a unity feedback system. The intersection of the Nyquist plot with an M‑circle gives the frequency at which the closed‑loop gain reaches a certain value, aiding in the design of phase‑lead or phase‑lag compensators.

Relationship to Bode Plots

The Nyquist plot is intimately related to the Bode plot. The gain and phase from the Bode plot at each frequency directly correspond to a point in the Nyquist plane. For many systems, it is easier to first sketch the Bode plot and then transfer the data to the polar plot. The Nyquist plot, however, provides a more direct view of stability margins and the proximity to (−1,0).

Interpreting the Nyquist Plot: Stability Margins

The most important feature on a Nyquist plot is the location of the point (−1,0). The distance from this point to the Nyquist curve defines two critical measures:

  • Gain Margin (GM): The factor by which the gain can be increased before the closed‑loop system becomes unstable. It is the reciprocal of the magnitude at the frequency where the phase is −180° (the phase crossover frequency). On the Nyquist plot, the gain margin is the distance along the real axis from (−1,0) to the point where the plot crosses the negative real axis. Mathematically, GM = 1/|G(jωπ)H(jωπ)|, where ωπ is the phase crossover frequency.
  • Phase Margin (PM): The additional phase lag that would cause instability. It is the phase difference from −180° at the frequency where the magnitude is 1 (the gain crossover frequency). On the Nyquist plot, the phase margin is the angle between the negative real axis and the line from the origin to the point where the Nyquist curve intersects the unit circle.

Relative Stability and Robustness

Larger gain and phase margins generally indicate a more robust system, but excessive margins may degrade performance. Typical design specifications for control systems are PM = 30°–60° and GM = 6–12 dB. The Nyquist plot provides a single‑view summary: a plot that passes close to (−1,0) suggests poor relative stability, while a plot that loops well away from it indicates a well‑damped system.

Applications in Control System Design

Graduate engineers use Nyquist plots in several advanced design contexts:

Controller Synthesis Using Lead/Lag Compensators

When designing a phase‑lead compensator, the goal is to increase the phase margin at a desired gain crossover frequency. The Nyquist plot helps visualize how the compensator shifts the open‑loop response. By adding phase lead, the Nyquist curve rotates clockwise, moving the intersection with the unit circle to a higher phase. Similarly, lag compensators increase the low‑frequency gain and can improve steady‑state accuracy without significantly affecting the stability margins.

Robustness to Model Uncertainties

In robust control, the Nyquist plot is used to evaluate the system's sensitivity to parameter variations. The distance between the Nyquist curve and the critical point is a measure of robustness. The smaller the distance, the more likely small modelling errors could destabilize the system. Techniques such as the small‑gain theorem and the stability margin are often derived from Nyquist‑based reasoning.

Systems with Time Delays

Time delays introduce an additional phase lag of −ωτ, which degrades stability margins. On the Nyquist plot, a delay causes the curve to spiral inward as frequency increases, because the phase decreases without bound while the magnitude remains unchanged (for a pure delay). This spiral can lead to multiple encirclements of (−1,0) if the delay is large enough. Nyquist analysis is one of the few methods that handles delays directly without approximation.

Advanced Considerations and Common Pitfalls

Multiple Encirclements and Non‑minimum Phase Systems

When the open‑loop transfer function has RHP poles (P > 0), the Nyquist criterion requires the plot to encircle (−1,0) exactly P times counter‑clockwise for stability. Interpreting such plots can be challenging because the encirclement direction and number must be carefully counted. For non‑minimum phase systems (systems with RHP zeros), the Nyquist plot often shows unusual loops that may cross the negative real axis multiple times. Despite this, the criterion remains valid.

Using Computational Tools

While hand sketching is educational, modern engineering practice relies on software like MATLAB, Python (control library), or Octave to generate accurate Nyquist plots. These tools automatically handle the frequency sweep, the infinite semicircle, and even indentations around imaginary‑axis poles. However, a deep theoretical understanding is necessary to interpret the plots correctly, especially when numerical errors arise from poles on the imaginary axis or from systems with high relative degree. For example, the MATLAB nyquist function uses a numerical contour that may produce artifacts near poles; the user must know to specify the correct frequency range or to use the nyquist1 variant for proper handling.

External resources for further study include:

Conclusion

Mastering the theoretical basis of Nyquist plot construction transforms an engineer's ability to analyze and design feedback control systems. From the argument principle to the practical counting of encirclements, this knowledge bridges abstract complex analysis with concrete stability assessment. For graduate engineering students, the Nyquist plot is not just a tool but a conceptual framework that reveals the fundamental relationship between open‑loop characteristics and closed‑loop behavior. Continuing to explore advanced topics such as multiple encirclements, time‑delay systems, and robustness margins will deepen this understanding and prepare students for research and industrial applications where stability is paramount.