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Nyquist Stability CriterionEdit

The Nyquist stability criterion is a cornerstone of modern control theory, offering a practical way to judge the stability of a feedback system by looking at how the open-loop response travels through the complex plane. Rather than trying to track all poles of the closed-loop system directly, engineers can inspect the open-loop transfer function and its behavior around the critical point in the complex plane. This approach is especially valuable in the real world where models are approximate and robustness matters.

Named after the late engineer and scientist Harry Nyquist, the criterion links the geometry of the open-loop response to the location of the closed-loop poles. When the system is described by a linear time-invariant model, the open-loop transfer function L(s) = G(s)H(s) captures how disturbances and references are shaped before they are fed back. The heart of the method is the Nyquist diagram, a plot of L(jω) as ω runs from 0 to ∞ (with a consideration of the symmetric part for negative frequencies), and the count of how many times this plot encircles the point −1 in the complex plane. The relation between encirclements and pole locations provides a decisive verdict on stability.

Nyquist stability criterion

Overview and statement

  • The core idea is that the stability of the closed-loop system, represented by T(s) = L(s)/(1+L(s)), can be read off from how the open-loop plot wraps around the critical point −1. If the open-loop transfer function has P poles in the right-half of the complex plane (the RHP), and the Nyquist plot encircles −1 a total of N times in the clockwise sense, then the closed-loop system has Z = P + N right-half-plane poles. A stable closed-loop system requires Z = 0.
  • In practice, engineers typically try to ensure P = 0 (no open-loop RHP poles) and then design so that N = 0. If P > 0, the stability task becomes more delicate, since encirclements must cancel the existing unstable open-loop dynamics to yield a stable closed loop.

Constructing the Nyquist plot

  • The plot is drawn by evaluating L(jω) for ω from 0 to ∞, then including the mirrored path for negative frequencies to form a closed curve. If the open-loop transfer function has poles on the imaginary axis, the contour must detour around those poles with small semicircles, and the resulting encirclement count must account for these detours.
  • The point −1 is the critical point for stability in the sense of BIBO stability for a unity-feedback arrangement. The distance and angle from L(jω) to −1 convey how much gain and phase margin the system possesses.
  • Gains and phases are often discussed in terms of margins: gain margin measures how much gain can be increased before encircling becomes problematic, and phase margin measures how much phase lag can be tolerated before stability is lost. These margins are inherently connected to the Nyquist picture and are closely related to the idea of encirclements around −1.

How to apply in practice

  • Step 1: Write the open-loop transfer function L(s) = G(s)H(s) and identify its pole structure, especially any poles in the right-half plane (P) or on the imaginary axis.
  • Step 2: Construct the Nyquist contour: map the imaginary axis s = jω (ω ∈ [0, ∞)) and close the contour with a large semicircle in the right-half plane. For practical charts, the contribution from the large semicircle often vanishes if the system is proper.
  • Step 3: Draw or compute the Nyquist plot of L(jω) and count how many times the plot encircles −1 in a clockwise sense (N).
  • Step 4: Use the relation Z = P + N to determine how many closed-loop poles lie in the right-half plane. If Z = 0, the system is stable; otherwise, it is unstable.

Special cases and extensions

  • Non-minimum phase and delays: If the open-loop has right-half-plane zeros (non-minimum phase behavior) or significant time delays, the Nyquist plot tends to wrap the −1 point more easily, reducing margins.
  • Imaginary-axis poles: When L(s) has poles on the imaginary axis, the Nyquist contour must avoid them, and the resulting counts must be adjusted accordingly. This often requires careful limiting procedures and sometimes temporary model refinements.
  • Discrete-time systems: The Nyquist criterion has a discrete-time counterpart (often formulated using the unit circle in the z-plane) that serves a similar purpose for digital control loops.

Relationship to other methods

  • Routh–Hurwitz criterion: For many standard problems, the Nyquist criterion is equivalent in scope to the Routh–Hurwitz test for counting unstable poles, but Nyquist provides a more intuitive, graphical view, especially when dealing with parameter variation and gain margins.
  • Root locus and Bode analysis: The Nyquist approach complements root locus plots (which track closed-loop pole movement as a parameter varies) and Bode plots (which show gain and phase response). Together, these methods give a fuller picture of stability and robustness.
  • Robust control considerations: In practice, engineers study how uncertainty in L(s) affects encirclements and margins. The Nyquist perspective is particularly helpful for assessing robustness to model error and unmodeled dynamics.

Practical considerations and examples

  • For a simple, stable open-loop plant with no RHP poles, a positive gain can still destabilize the closed loop if the Nyquist plot encircles −1. The gain margin tells you how far you can push the gain before that encirclement appears. The phase margin tells you how much additional phase lag you can tolerate before the encirclement occurs.
  • Engineers often use software tools to compute Nyquist plots automatically, especially when L(s) is complex or has multiple poles and zeros. The resulting diagrams are then interpreted to verify that N + P equals zero, or to determine the margins needed to stay stable under uncertainty.
  • The approach is widely used in aerospace controls, automotive systems, power electronics, and industrial process control, wherever feedback loops must remain robust in the face of disturbances and model imperfections. See control theory for the broad context and robust control for how practitioners manage uncertainty.

Historical notes

  • The Nyquist criterion has its roots in the work of Harry Nyquist in the early days of control engineering. It built on and complemented earlier stability tests and became a standard tool in the designer’s toolkit for both analog and, with appropriate adaptations, digital control systems. The method is often taught alongside other foundational concepts in systems engineering and signal processing.

See also