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Wikipedia / Nyquist stability criterion

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 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 systemss, 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.

, the closed loop transfer function (CLTF) then becomes:

Stability can be determined by examining the roots, e.g. using the Routh array, using its Bode plots or, as here, its polar plot using the Nyquist criterion, as follows.

Any Laplace domain can be expressed as the ratio of two polynomials:

are called the zeros are the poles are also said to be the roots of the characteristic equation.

is formed by closing a negative unity feedback loop around the open-loop transfer function,

.

Cauchy's argument principle

From complex analysis, can be mapped plane yielding a new contour.

by Cauchy's argument principle and that encirclements in the opposite direction are negative encirclements. That is, we consider clockwise encirclements to be positive and counterclockwise encirclements to be negative.

Instead of Cauchy's argument principle, the original paper by Harry Nyquist in 1932 uses a less elegant approach. The approach explained here is similar to the approach used by Leroy MacColl (Fundamental theory of servomechanisms 1945) or by Hendrik Bode (Network analysis and feedback amplifier design 1945), both of whom also worked for Bell Laboratories. This approach appears in most modern textbooks on control theory.

Definition

We first construct the Nyquist contour, a contour that encompasses the right-half of the complex plane:

  • .

  • .

, the result is the Nyquist Plot, we now state the Nyquist Criterion:.If the system is originally open-loop unstable, feedback is necessary to stabilize the system. Right-half-plane (RHP) poles represent that instability. For closed-loop stability of a system, the number of closed-loop roots in the right half of the s must be equal to the number of open-loop poles in the RHP. Any clockwise encirclements of the critical point by the open-loop frequency response (when judged from low frequency to high frequency) would indicate that the feedback control system would be destabilizing if the loop were closed. (Using RHP zeros to "cancel out" RHP poles does not remove the instability, but rather ensures that the system will remain unstable even in the presence of feedback, since the closed-loop roots travel between open-loop poles and zeros in the presence of feedback. In fact, the RHP zero can make the unstable pole unobservable and therefore not stabilizable through feedback.)

The Nyquist criterion for systems with poles on the imaginary axis

). This results from the requirement of the argument principle that the contour cannot pass through any pole of the mapping function. The most common case are systems with integrators (poles at zero).

is the multiplicity of the pole on the imaginary axis.

Mathematical derivation

Our goal is to, through this process, check for the stability of the transfer function of our unity feedback system with gain k, which is given by

That is, we would like to check whether the characteristic equation of the above transfer function, given by

has zeros outside the open left-half-plane (commonly initialized as OLHP).

. Cauchy's argument principle states that

, which is to say

that appear within the contour, that is, within the open right half plane (ORHP).

, we have

. This gives us

, which is to say our Nyquist plot. We may further reduce the integral

by applying Cauchy's integral formula clockwise. Thus, we may finally state that

, as evaluated above, is equal to0.

Importance

The Nyquist stability criterion is a graphical technique that determines the stability of a dynamical system, such as a feedback control system. It is based on the argument principle and the Nyquist plot of the open-loop transfer function of the system. It can be applied to systems that are not defined by rational functions, such as systems with delays. It can also handle transfer functions with singularities in the right half-plane, unlike Bode plots. The Nyquist stability criterion can also be used to find the phase and gain margins of a system, which are important for frequency domain controller design.

Summary

  • should be considered stable.

  • is stable, then the closed-loop system is unstable, if and only if, the Nyquist plot encircle the point −1 at least once.

  • is unstable, then for the closed-loop system to be stable, there must be one counter in the right-half of the complex plane.

  • The number of surplus encirclements (N+P greater than 0) is exactly the number of unstable poles of the closed-loop system.

  • , then deciding upon even the marginal stability axis.

See also

References

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,