In control theory and stability theory, the Nyquist stability criterion or Strecker–Nyquist stability criterion, independently discovered by the German electrical engineer Felix Strecker at Siemens in 1930 and the Swedish-American electrical engineer Harry Nyquist at Bell Telephone Laboratories in 1932, is a graphical technique for determining the stability of a linear dynamical system. Because it only looks at the Nyquist plot of the open loop systems, it can be applied without explicitly computing the poles and zeros of either the closed-loop or open-loop system (although the number of each type of right-half-plane singularities must be known). As a result, it can be applied to systems defined by non-rational functions, such as systems with delays. In contrast to Bode plots, it can handle transfer functions with right half-plane singularities. In addition, there is a natural generalization to more complex systems with multiple inputs and multiple outputs, such as control systems for airplanes. The Nyquist stability criterion is widely used in electronics and control system engineering, as well as other fields, for designing and analyzing systems with feedback. While Nyquist is one of the most general stability tests, it is still restricted to linear time-invariant (LTI) systems. Nevertheless, there are generalizations of the Nyquist criterion (and plot) for non-linear systems, such as the circle criterion and the scaled relative graph of a nonlinear operator. 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
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 by the angle at which the curve approaches the origin. When drawn by hand, a cartoon version of the Nyquist plot is sometimes used, which shows the linearity of the curve, but where coordinates are distorted to show more detail in regions of interest. When plotted computationally, one needs to be careful to cover all frequencies of interest. This typically means that the parameter is swept logarithmically, in order to cover a wide range of values.
Background The mathematics uses the Laplace transform, which transforms integrals and derivatives in the time domain to simple multiplication and division in the s domain. We consider a system whose transfer function is G ( s ) {\displaystyle G(s)} ; when placed in a closed loop with negative feedback H ( s ) {\displaystyle H(s)} , the closed loop transfer function (CLTF) then becomes:
G ( s ) 1 + G ( s ) H ( s ) {\displaystyle {\frac {G(s)}{1+G(s)H(s)}}}
Stability can be determined by examining the roots of the desensitivity factor polynomial 1 + G ( s ) H ( s ) {\displaystyle 1+G(s)H(s)} , e.g. using the Routh array, but this method is somewhat tedious. Conclusions can also be reached by examining the open loop transfer function (OLTF) G ( s ) H ( s ) {\displaystyle G(s)H(s)} , using its Bode plots or, as here, its polar plot using the Nyquist criterion, as follows. Any Laplace domain transfer function T ( s ) {\displaystyle {\mathcal {T}}(s)} can be expressed as the ratio of two polynomials:
T ( s ) = N ( s ) D ( s ) . {\displaystyle {\mathcal {T}}(s)={\frac {N(s)}{D(s)}}.}
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