class=skin-invert-image|thumb|300px|right|The Nyquist plot for <math>G(s)=\frac{1}{s^2+s+1} </math> with .

In control theory and stability theory, the Nyquist stability criterion or Strecker–Nyquist stability criterion, independently discovered by the German electrical engineer at Siemens in 1930 Additionally, other stability criteria like Lyapunov methods can also be applied for non-linear systems.

Although Nyquist is a graphical technique, it only provides a limited amount of intuition for why a system is stable or unstable, or how to modify an unstable system to be stable. Techniques like Bode plots, while less general, are sometimes a more useful design tool.

Nyquist plot

class=skin-invert-image|thumb|right|A Nyquist plot. Although the frequencies are not indicated on the curve, it can be inferred that the zero-frequency point is on the right, and the curve spirals toward the origin at high frequency. This is because gain at zero frequency must be purely real (on the X-axis) and is commonly non-zero, while most physical processes have some amount of low-pass filtering, so the high-frequency response is zero.

A Nyquist plot is a parametric plot of a frequency response used in automatic control and signal processing. The most common use of Nyquist plots is for assessing the stability of a system with feedback. In Cartesian coordinates, the real part of the transfer function is plotted on the X-axis while the imaginary part is plotted on the Y-axis. The frequency is swept as a parameter, resulting in one point per frequency. The same plot can be described using polar coordinates, where gain of the transfer function is the radial coordinate, and the phase of the transfer function is the corresponding angular coordinate. The Nyquist plot is named after Harry Nyquist, a former engineer at Bell Laboratories.

Assessment of the stability of a closed-loop negative feedback system is done by applying the Nyquist stability criterion to the Nyquist plot of the open-loop system (i.e. the same system without its feedback loop). This method is easily applicable even for systems with delays and other non-rational transfer functions, which may appear difficult to analyze with other methods. Stability is determined by looking at the number of encirclements of the point (−1, 0). The range of gains over which the system will be stable can be determined by looking at crossings of the real axis.

The Nyquist plot can provide some information about the shape of the transfer function. For instance, the plot provides information on the difference between the number of zeros and poles of the transfer function

Summary

  • If the open-loop transfer function <math>G(s)</math> has a zero pole of multiplicity <math>l</math>, then the Nyquist plot has a discontinuity at <math>\omega = 0</math>. During further analysis it should be assumed that the phasor travels <math>l</math> times clockwise along a semicircle of infinite radius. After applying this rule, the zero poles should be neglected, i.e. if there are no other unstable poles, then the open-loop transfer function <math>G(s)</math> should be considered stable.
  • If the open-loop transfer function <math>G(s)</math> is stable, then the closed-loop system is unstable, if and only if, the Nyquist plot encircle the point −1 at least once.
  • If the open-loop transfer function <math>G(s)</math> is unstable, then for the closed-loop system to be stable, there must be one counter-clockwise encirclement of −1 for each pole of <math>G(s)</math> in the right-half of the complex plane.
  • The number of surplus encirclements (N&nbsp;+&nbsp;P greater than 0) is exactly the number of unstable poles of the closed-loop system.
  • However, if the graph happens to pass through the point <math>-1+j0</math>, then deciding upon even the marginal stability of the system becomes difficult and the only conclusion that can be drawn from the graph is that there exist zeros on the <math>j\omega</math> axis.

See also

  • BIBO stability
  • Bode plot
  • Routh–Hurwitz stability criterion
  • Root locus analysis
  • Gain margin
  • Nichols plot
  • Hall circles
  • Phase margin
  • Barkhausen stability criterion
  • Circle criterion
  • Control engineering
  • Hankel singular value

References

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Further reading

  • Faulkner, E. A. (1969): Introduction to the Theory of Linear Systems; Chapman & Hall;
  • Pippard, A. B. (1985): Response & Stability; Cambridge University Press;
  • Gessing, R. (2004): Control fundamentals; Silesian University of Technology;
  • Franklin, G. (2002): Feedback Control of Dynamic Systems; Prentice Hall,
  • Applets with modifiable parameters
  • EIS Spectrum Analyser - a freeware program for analysis and simulation of impedance spectra
  • MATLAB function for creating a Nyquist plot of a frequency response of a dynamic system model.
  • PID Nyquist plot shaping - free interactive virtual tool, control loop simulator
  • Mathematica function for creating the Nyquist plot
  • The Nyquist Diagram for Electrical Circuits