Fundamentals of the Nyquist Plot

The Nyquist plot is a cornerstone of frequency‑domain analysis in control engineering. It is a polar representation of a system’s open‑loop transfer function G(s)H(s) evaluated along the Nyquist contour in the complex s‑plane. As the frequency ω varies from -∞ to +∞, the plot traces the magnitude and phase of the transfer function. Unlike the Bode plot, which separates magnitude and phase into two graphs, the Nyquist plot presents both in a single complex plane, making it especially useful for stability analysis using the Nyquist stability criterion.

The traditional Nyquist plot is drawn for ω from 0 to +∞; the complete closed contour (including negative frequencies) is obtained by symmetry about the real axis. The critical point in this analysis is -1 + 0j, often called the critical point or the -1 point. The number of times the plot encircles this point determines the closed‑loop stability of the feedback system.

For a deeper theoretical foundation, MIT OpenCourseWare provides lecture notes on the Nyquist stability criterion as part of its feedback control systems curriculum.

The Nyquist Stability Criterion

The Nyquist stability criterion is a graphical method that relates the number of encirclements of the -1 point to the number of unstable poles in the closed‑loop system. The criterion is mathematically expressed as:

N = Z - P

where:

  • N is the net number of counterclockwise encirclements of the -1 point by the Nyquist plot.
  • Z is the number of closed‑loop poles in the right‑half plane (RHP).
  • P is the number of open‑loop poles in the RHP.

For a stable closed‑loop system, Z must equal 0. Therefore, the net number of counterclockwise encirclements must satisfy:

N = -P  (if P > 0, then N must be negative, meaning clockwise encirclements)

If the open‑loop system is stable (P = 0), then stability requires N = 0, i.e., the Nyquist plot must not encircle the -1 point at all.

Direction and Counting of Encirclements

Encirclements are counted by drawing a ray from the -1 point to infinity in any direction and observing the number of times the Nyquist plot crosses that ray. Each crossing in the clockwise direction contributes -1 to N; each crossing in the counterclockwise direction contributes +1. The net sum gives N.

A common technique is to observe the phase change as the plot passes near the -1 point. If the plot goes around the -1 point, an encirclement occurs. Care must be taken to include only full encirclements, not mere approaches.

Relationship with Open‑Loop Poles

The number of open‑loop poles in the RHP (P) is usually known from the plant model. For a minimum‑phase system with no RHP poles (P = 0), the Nyquist plot must avoid the -1 point entirely to guarantee stability. For systems with open‑loop unstable poles (P > 0), the plot must encircle the -1 point exactly P times in the clockwise direction to cancel the unstable poles in the closed loop.

Example: If a system has two open‑loop RHP poles (P = 2), then a stable closed loop requires exactly two clockwise encirclements (N = -2). A different number of encirclements would indicate instability.

Interpreting Encirclements in Practice

Engineers use the Nyquist plot and its encirclements to quickly assess stability margins and the effect of controller gains.

Stable System (P = 0, No Encirclement)

Consider a simple first‑order system G(s) = 1/(s+1). Its Nyquist plot starts at (1, 0) at ω = 0 and ends at the origin at ω → ∞. The plot stays entirely in the right half of the complex plane and never encloses the -1 point. Consequently, the closed‑loop system is stable for any positive gain. This is the simplest case.

Unstable System Due to Gain Increase

For a second‑order system with a resonance peak, increasing the gain can cause the Nyquist plot to cross the -1 point. When the plot passes through -1 exactly, the system is marginally stable (oscillatory). A small further increase makes the plot encircle -1, indicating instability. The critical gain at which encirclements begin is the gain margin.

Conditionally Stable Systems

Some systems are stable only for a certain range of gains. For example, a non‑minimum phase system may have a Nyquist plot that encircles -1 at low gains, then becomes stable as gain increases, and finally becomes unstable again at very high gains. The number and direction of encirclements change with gain, requiring careful analysis. These systems are called conditionally stable. The Nyquist plot reveals these transitions visually.

Extending to Systems with Poles on the Imaginary Axis

When the open‑loop transfer function has poles on the imaginary axis (e.g., integrators 1/s or pure oscillators 1/(s²+ω²)), the Nyquist contour must be modified to avoid those poles. A small semicircular indentation to the right of the pole is added. This indentation maps to a circular arc of infinite radius in the Nyquist plot. The number of encirclements caused by this arc must be accounted for.

For example, a system with a single integrator has a Nyquist plot that goes to infinity at ω = 0⁺. The indentation contributes a half‑circle in the Nyquist plot, which may or may not encircle -1 depending on the system’s phase. This is why the Nyquist criterion for systems with poles on the imaginary axis often requires counting the net phase change along the indented contour.

The Control Tutorials for MATLAB website offers a clear example of Nyquist plotting for systems with integrators.

Practical Applications in Control Design

Understanding encirclements directly informs control design choices, particularly for gain selection and compensation.

Gain and Phase Margins from Encirclements

The distance between the Nyquist plot and the -1 point is quantified by the gain margin (GM) and phase margin (PM). The GM is the factor by which the gain can be increased before the plot passes through -1. The PM is the amount of additional phase lag required to reach -1. Both margins can be read directly from the Nyquist plot:

  • Gain margin: the reciprocal of the magnitude |G(jω)| at the frequency where the phase is -180°.
  • Phase margin: 180° plus the phase of G(jω) at the gain crossover frequency (where |G(jω)| = 1).

While margins are often obtained from Bode plots, the Nyquist plot provides a visual cross‑check: a plot that passes very close to -1 indicates low margins and potential instability even if no encirclement occurs.

Lead‑Lag Compensation and Encirclements

When designing lead or lag compensators, the goal is often to reshape the Nyquist plot to avoid encirclements or to achieve a desired number of encirclements for a given P. A lead compensator adds phase near the crossover frequency, shifting the plot away from -1 and increasing phase margin. A lag compensator reduces high‑frequency gain, lowering the chance of encirclement at high gains. Both techniques rely on understanding how the compensator’s poles and zeros affect the encirclement count.

Robustness and Uncertainty

In real systems, model uncertainty means the exact Nyquist plot is unknown. Robust control techniques ensure that even under worst‑case variations, the Nyquist plot of the perturbed system does not encircle -1 incorrectly. The small‑gain theorem and Nyquist robust stability criterion use a forbidden region around -1 that the plot must avoid. Encirclements in the nominal plant may be allowed if the uncertainty does not change the net count, but safety margins are built in.

Common Mistakes and How to Avoid Them

Engineering students and practitioners often make errors when counting encirclements or applying the Nyquist criterion. The following pitfalls are worth noting:

  • Misidentifying encirclement direction: Counterclockwise is positive; clockwise is negative. Use a reference ray to double‑check each crossing.
  • Ignoring multiple encirclements: The plot may encircle -1 more than once, especially in high‑order systems. Count each full revolution.
  • Forgetting poles on the imaginary axis: Always modify the contour and account for the infinite‑radius arc. The encirclement from that arc can change N.
  • Confusing Nyquist plot with polar plot: The complete Nyquist plot includes the negative frequency portion (mirror image). The mirror image is essential for correct encirclement counting.
  • Using the wrong sign for N: Remember the formula N = Z - P. If Z = 0, then N = -P. Do not accidentally set N = P.

For a comprehensive walkthrough of these pitfalls, the textbook Feedback Control of Dynamic Systems by Franklin, Powell, and Emami‑Naeini contains an extensive appendix on Nyquist plot interpretation. Additionally, MATLAB’s documentation on the nyquist function includes examples of encirclement analysis.

Summary

The concept of encirclements in Nyquist plot stability analysis is central to control engineering. By applying the Nyquist criterion N = Z - P, engineers can determine closed‑loop stability from the open‑loop frequency response. Correctly counting the number and direction of encirclements of the -1 point is essential for reliable system design. Understanding how to handle poles on the imaginary axis and how to interpret gain and phase margins from the plot further strengthens this analysis. Whether designing a simple gain controller or a robust compensator for an uncertain plant, mastery of encirclements is a prerequisite for practical, production‑level control system work.

For further reading, the University of Michigan’s Control Tutorials for MATLAB offer interactive examples, and the classic textbook Modern Control Systems by Dorf and Bishop provides an in‑depth treatment of the Nyquist criterion in a real‑world context.