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Nyquist CriterionEdit

Nyquist Criterion is a fundamental tool in control theory for assessing the stability of feedback systems using frequency-domain information. Named after the engineer Harry Nyquist, it connects the behavior of an open-loop transfer function along the imaginary axis to the location of closed-loop poles in the complex plane. In practice, engineers sketch or compute a Nyquist plot — the image of the open-loop transfer function L(jω) as ω runs from 0 to ∞ (and around the appropriate contour to account for any right-half-plane dynamics). By examining how this plot encircles the critical point −1, one can infer whether the feedback loop will remain stable once it is closed. The method is widely used in aerospace, automotive, electronics, and many other engineering disciplines to judge how changes in gain, phase, or sensor/actuator dynamics might affect stability.

The core idea is straightforward but powerful: stability of the closed-loop system is governed by the topology of the open-loop response in the complex plane, not by time-domain simulations alone. The Nyquist plot serves as a compact, informative map of how the loop gain behaves across frequencies, and it yields immediate intuition about gain and phase margins. For many practical systems, this approach complements time-domain testing and other frequency-domain tools such as Bode plots and Nichols charts. It also underpins design methods that ensure robust performance in the presence of modeling error or parameter variation.

Theory and Formulation

Open-loop and closed-loop descriptions

  • Let L(s) denote the loop transfer function, typically written as L(s) = G(s)H(s), where G(s) is the plant and H(s) is the feedback path (s is a complex frequency variable).
  • The closed-loop transfer function is T(s) = L(s) / [1 + L(s)], and the closed-loop poles are the roots of 1 + L(s) = 0.
  • The right-half-plane (RHP) poles of L(s) and the right-half-plane zeros of 1 + L(s) determine stability. See also right-half-plane and zero (mathematics).

Nyquist plot and encirclements

  • The Nyquist plot traces L(jω) in the complex plane as ω goes from 0 to ∞, with appropriate detours to account for any poles of L(s) on or near the imaginary axis.
  • The critical point −1 in the complex plane plays a key role: encirclements of −1 by the Nyquist plot indicate how the closed-loop poles move when the loop is closed.
  • The standard convention counts encirclements of −1 in the clockwise direction as positive. Let P be the number of poles of L(s) in the right-half plane (RHP). Let N be the number of clockwise encirclements of −1 by the Nyquist plot. Then the Nyquist criterion states that the number Z of right-half-plane poles of 1 + L(s) (i.e., unstable closed-loop poles) satisfies Z = P − N.
  • Stability is achieved when Z = 0, which, under this convention, means N must equal P. In particular, if the open-loop transfer function has no RHP poles (P = 0), stability requires that the Nyquist plot encircle −1 zero times (N = 0).

Practical use and margins

  • From a single Nyquist plot, engineers can read gain margins and phase margins, which quantify how much gain increase or phase shift the loop can tolerate before losing stability. See also gain margin and phase margin.
  • For systems with multiple inputs and outputs (MIMO), the Nyquist criterion extends to more general encirclement arguments in higher-dimensional complex mappings, though the visualization becomes more complex.
  • Real-world factors such as sensor saturation, actuator limits, and unmodeled dynamics are often considered through robustness analyses that complement the basic Nyquist check.

Extensions and variants

  • Discrete-time Nyquist criterion replaces the imaginary axis with the unit circle in the complex z-plane, analyzing stability of digital control loops through the mapping of the frequency response on the unit circle. See Nyquist criterion (discrete-time).
  • The approach can be extended to non-minimum-phase systems (which have right-half-plane zeros), where the encirclement patterns become more intricate but still informative for stability assessment.
  • Related tools include Bode plot and root locus methods, which offer alternative viewpoints on how system poles move with gains and how margins behave.

History and context

Nyquist introduced the criterion in the 1930s, building on foundational work in frequency-domain analysis. The method provided a robust bridge between the time-domain intuition of feedback and the frequency-domain realities of resonances and phase shifts. Over the decades, engineers have refined and extended the criterion to cover a wide range of control problems, from simple single-loop systems to complex multivariable architectures. See also control theory.

Applications and limitations

  • The Nyquist criterion is a mainstay in the design of autopilots, power electronics controllers, and industrial process controls, where fast frequency-domain judgments can prevent unstable behavior and reduce design iteration.
  • Limitations include the need for a valid open-loop model L(s) with well-understood analytic properties, and the fact that the criterion provides conditions for stability rather than guaranteeing optimal performance. In practice, designers often use Nyquist analysis in concert with time-domain simulations and robust-control techniques.

See also