The Nyquist plot for .

The Nyquist stability criterion, named for Harry Nyquist, provides a simple test for stability of a closed-loop control system by examining the open-loop system's Nyquist plot. Stability of the closed-loop control system may be determined directly by computing the poles of the closed-loop transfer function. In contrast, the Nyquist stability criterion allows stability to be determined without computing the closed-loop poles. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Harry Nyquist (February 7, 1889 - April 4, 1976) was an important contributor to information theory. ... Look up stability in Wiktionary, the free dictionary. ... A Proportional-Integral-Derivative controller is a standard feedback loop component in industrial control applications. ... It has been suggested that this article or section be merged with Control theory. ... A Nyquist plot is a graph used in signal processing in which the magnitude and phase of a frequency response are plotted on orthogonal axes. ... A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. ...

Background

We consider a system whose open loop transfer function (OLTF) is G(s); when placed in a closed loop with feedback H(s), the closed loop transfer function (CLTF) then becomes G/GH+1. The case where H=1 is usually taken, when investigating stability, and then the "Characteristic Equation', used to predict stability, becomes G+1=0. Stability can be determined by examining the roots of this equation eg using the Routh array, but this method is somewhat tedious. Conclusions can also be reached by examining the OLTF, using its Bode plots or, as here, polar plot of the OLTF using the Nyquist criterion , as follows. The Routh-Hurwitz stability criterion is a necessary (and frequently sufficient) method to establish the stability of a single-input, single-output (SISO), linear time invariant (LTI) control system. ... The Bode plot for a first-order Butterworth filter A Bode plot, named after Hendrik Wade Bode, is usually a combination of a Bode magnitude plot and Bode phase plot: A Bode magnitude plot is a graph of log magnitude against log frequency often used in signal processing to show...

Any Laplace domain transfer function can be expressed as the ratio of two polynomials In mathematics, the Laplace transform is a powerful technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems, to name just a few. ...

.

We define:

• Zero: the zeros of are the roots of N(s) = 0, and

• Pole: the poles of are the roots of D(s) = 0.

Stability of is determined by its poles or simply the roots of the characteristic equation: D(s) = 0. For stability, the real part of every pole must be negative. If is formed by closing a negative feedback loop around the open-loop transfer function , then the roots of the characteristic equation are also the zeros of , or simply the roots of A(s) + B(s).

Cauchy's argument principle

From complex analysis, specifically the argument principle, we know that a contour Γ_s drawn in the complex s plane, encompassing but not passing through any number of non-analytic points, can be mapped to another plane (the F(s) plane) by a function F(s). The resulting contour Γ_F(s) will encircle the origin of the F(S) plane N times, where N = Z − P. Z and P are the number of zeros and poles of F(s), respectively. Note that we count encirclements in the F(s) plane in the same sense as the contour Γ_s and that encirclements in the opposite direction are negative encirclements. The contour C (black), the zeros of f (blue) and the poles of f (red). ...

Instead of Cauchy's argument principle, the original paper by Harry Nyquist in 1932 used 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. Harry Nyquist (February 7, 1889 - April 4, 1976) was an important contributor to information theory. ... Dr. Hendrik Wade Bode Hendrik Wade Bode (pronounced Boh-dee in English, Boh-dah in Dutch),[1] (24 December 1905 Madison, Wisconsin – 21 June 1982 Cambridge, Massachusetts) was a gifted researcher, prolific inventor and eloquent and nuanced engineer, author and scientist, an American of Dutch ancestry. ... Bell Telephone Laboratories or Bell Labs was originally the research and development arm of the United States Bell System, and was the premier corporate facility of its type, developing a range of revolutionary technologies from telephone switches to specialized coverings for telephone cables, to the transistor. ...

The Nyquist criterion

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

• a path traveling up the jω axis, from to .
• a semicircular arc, with radius , that starts at and travels clock-wise to .

The Nyquist Contour mapped through the open-loop transfer function F(s) yields a Nyquist plot for F(s). By the Argument Principle, the number of clock-wise encirclements of the origin must be the number of zeros of F(s) in the right-half complex plane minus the poles of F(s) in the right-half complex plane. If we look at the contour's encirclements of -1 instead of the origin, we find the difference between the number of poles and zeros in the right-half complex plane of 1 + F(s). Recalling that the zeros of 1 + F(s) are the poles of the close-loop system, and noting that the poles of 1 + F(s) are same as the poles of F(s), we now state The Nyquist Criterion: A Nyquist plot is a graph used in signal processing in which the magnitude and phase of a frequency response are plotted on orthogonal axes. ...

Given a Nyquist contour Γ_s, let P be the number of poles of F(s) encircled by Γ_s, and Z be the number of zeros of F(s) encircled by Γ_s -- therefore the number of poles of enclosed by Γ_s. The resultant contour in the F(s)-plane, Γ_F(s) shall encircle (clock-wise) the point (-1 + j0) N times such that N = Z − P. For Stability of a System, we must have Z = 0 , ie. the number of closed loop poles in the right half of s-plane must be zero. Hence, the number of anticlockwise encirclements about − 1 + j0 must be equal to P, the number of open loop poles in the right half plane.

Summary:

• If the open-loop transfer function F(s) is stable, then the closed-loop system is unstable for any encirclement of the point -1.
• If the open-loop transfer function F(s) is unstable, then there must be one counter clock-wise encirclement of -1 for each pole of F(s) in the right-half of the complex plane.
• The number of surplus encirclements (greater than N+P) is exactly the number of unstable poles of the closed-loop system
• However, if the graph happens to pass through the point − 1 + j0, 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 jω axis.

In the theory of dynamical systems, a linear time-invariant system is marginally stable if every eigenvalue in the systems transfer-function is non-positive, and all eigenvalues with zero real value are simple roots. ...

See also

• Nyquist plot
• Routh Hurwitz Stability Criterion
• Transfer function
• Control engineering

A Nyquist plot is a graph used in signal processing in which the magnitude and phase of a frequency response are plotted on orthogonal axes. ... The Routh Hurwitz Stability Criterion is a necessary, and frequently sufficient method to establish the stability of a Single-Input, Single-Output, Linear Time Invarient (LTI) control system. ... A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. ... Control engineering is the engineering discipline that focuses on the mathematical modelling systems of a diverse nature, analysing their dynamic behaviour, and using control theory to make a controller that will cause the systems to behave in a desired manner. ...

References

• Faulkner, E.A. (1969): Introduction to the Theory of Linear Systems; Chapman & Hall; ISBN 0-412-09400-2
• Pippard, A.B. (1985): Response & Stability; Cambridge University Press; ISBN 0-521-31994-3