In control theory and stability theory, the Nyquist stability criterion or StreckerâNyquist stability criterion, independently discovered by the German electrical engineer Felix Strecker [de] at Siemens in 1930[1][2][3] and the Swedish-American electrical engineer Harry Nyquist at Bell Telephone Laboratories in 1932,[4] is a graphical technique for determining the stability of a 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 plot for G(s)=1s2+s+1{displaystyle G(s)={frac {1}{s^{2}+s+1}}}.
The Nyquist 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. Non-linear systems must use more complex stability criteria, such as Lyapunov or the circle criterion. While 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[edit]
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. The imaginary part is plotted on the Y axis. The frequency is swept as a parameter, resulting in a plot 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 by means of other methods. Stability is determined by looking at the number of encirclements of the point at (â1,0). 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[5] 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[edit]
We consider a system whose open loop transfer function (OLTF) 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 G1+GH{displaystyle {frac {G}{1+GH}}}. Stability can be determined by examining the roots of the desensitivity factor polynomial 1+GH{displaystyle 1+GH}, e.g. 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 GH(s){displaystyle GH(s)} 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)}}.}
The roots of N(s){displaystyle N(s)} are called the zeros of T(s){displaystyle {mathcal {T}}(s)}, and the roots of D(s){displaystyle D(s)} are the poles of T(s){displaystyle {mathcal {T}}(s)}. The poles of T(s){displaystyle {mathcal {T}}(s)} are also said to be the roots of the 'characteristic equation' D(s)=0{displaystyle D(s)=0}.
The stability of T(s){displaystyle {mathcal {T}}(s)} is determined by the values of its poles: for stability, the real part of every pole must be negative. If T(s){displaystyle {mathcal {T}}(s)} is formed by closing a negative unity feedback loop around the open-loop transfer function GH(s)=A(s)B(s){displaystyle GH(s)={frac {A(s)}{B(s)}}}, then the roots of the characteristic equation are also the zeros of 1+GH(s){displaystyle 1+GH(s)}, or simply the roots of A(s)+B(s)=0{displaystyle A(s)+B(s)=0}.
Cauchy's argument principle[edit]
From complex analysis, a contour Îs{displaystyle Gamma _{s}} drawn in the complex s{displaystyle s} plane, encompassing but not passing through any number of zeros and poles of a function F(s){displaystyle F(s)}, can be mapped to another plane (named F(s){displaystyle F(s)}plane) by the function F{displaystyle F}. Precisely, each complex point s{displaystyle s} in the contour Îs{displaystyle Gamma _{s}} is mapped to the point F(s){displaystyle F(s)} in the new F(s){displaystyle F(s)} plane yielding a new contour.
The Nyquist plot of F(s){displaystyle F(s)}, which is the contour ÎF(s)=F(Îs){displaystyle Gamma _{F(s)}=F(Gamma _{s})} will encircle the point s=â1+k{displaystyle s={-1+k}} of the F(s){displaystyle F(s)} plane N{displaystyle N} times, where N=PâZ{displaystyle N=P-Z} by Cauchy's argument principle. Here are Z{displaystyle Z} and P{displaystyle P} respectively the number of zeros of 1+kF(s){displaystyle 1+kF(s)} and poles of F(s){displaystyle F(s)} inside the contour Îs{displaystyle Gamma _{s}}. Note that we count encirclements in the F(s){displaystyle F(s)} plane in the same sense as the contour Îs{displaystyle Gamma _{s}} 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.
The Nyquist criterion[edit]
We first construct the Nyquist contour, a contour that encompasses the right-half of the complex plane:
The Nyquist contour mapped through the function 1+G(s){displaystyle 1+G(s)} yields a plot of 1+G(s){displaystyle 1+G(s)} in the complex plane. By the Argument Principle, the number of clock-wise encirclements of the origin must be the number of zeros of 1+G(s){displaystyle 1+G(s)} in the right-half complex plane minus the number of poles of 1+G(s){displaystyle 1+G(s)} in the right-half complex plane. If instead, the contour is mapped through the open-loop transfer function G(s){displaystyle G(s)}, the result is the Nyquist Plot of G(s){displaystyle G(s)}. By counting the resulting contour's encirclements of -1, we find the difference between the number of poles and zeros in the right-half complex plane of 1+G(s){displaystyle 1+G(s)}. Recalling that the zeros of 1+G(s){displaystyle 1+G(s)} are the poles of the closed-loop system, and noting that the poles of 1+G(s){displaystyle 1+G(s)} are same as the poles of G(s){displaystyle G(s)}, we now state The Nyquist Criterion:
Given a Nyquist contour Îs{displaystyle Gamma _{s}}, let P{displaystyle P} be the number of poles of G(s){displaystyle G(s)} encircled by Îs{displaystyle Gamma _{s}}, and Z{displaystyle Z} be the number of zeros of 1+G(s){displaystyle 1+G(s)} encircled by Îs{displaystyle Gamma _{s}}. Alternatively, and more importantly, if Z{displaystyle Z} is the number of poles of the closed loop system in the right half plane, and P{displaystyle P} is the number of poles of the open-loop transfer function G(s){displaystyle G(s)} in the right half plane, the resultant contour in the G(s){displaystyle G(s)}-plane, ÎG(s){displaystyle Gamma _{G(s)}} shall encircle (clock-wise) the point (â1+j0){displaystyle (-1+j0)}N{displaystyle N} times such that N=ZâP{displaystyle N=Z-P}.
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-plane must be zero. Hence, the number of counter-clockwise encirclements about â1+j0{displaystyle -1+j0} 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[edit]
The above consideration was conducted with an assumption that the open-loop transfer function G(s){displaystyle G(s)} does not have any pole on the imaginary axis (i.e. poles of the form 0+jÏ{displaystyle 0+jomega }). 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).
To be able to analyze systems with poles on the imaginary axis, the Nyquist Contour can be modified to avoid passing through the point 0+jÏ{displaystyle 0+jomega }. One way to do it is to construct a semicircular arc with radius râ0{displaystyle rto 0} around 0+jÏ{displaystyle 0+jomega }, that starts at 0+j(Ïâr){displaystyle 0+j(omega -r)} and travels anticlockwise to 0+j(Ï+r){displaystyle 0+j(omega +r)}. Such a modification implies that the phasor G(s){displaystyle G(s)} travels along an arc of infinite radius by âlÏ{displaystyle -lpi }, where l{displaystyle l} is the multiplicity of the pole on the imaginary axis.
Mathematical derivation[edit]
A unity negative feedback system G with scalar gain denoted by K
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 the OLHP).
We suppose that we have a clockwise (i.e. negatively oriented) contour Îs{displaystyle Gamma _{s}} enclosing the right half plane, with indentations as needed to avoid passing through zeros or poles of the function G(s){displaystyle G(s)}. Cauchy's argument principle states that
Die Stabilitat Der Kreiszylinderschale Meaning
Where Z{displaystyle Z} denotes the number of zeros of D(s){displaystyle D(s)} enclosed by the contour and P{displaystyle P} denotes the number of poles of D(s){displaystyle D(s)} by the same contour. Rearranging, we haveZ=N+P{displaystyle Z=N+P}, which is to say
We then note that D(s)=1+kG(s){displaystyle D(s)=1+kG(s)} has exactly the same poles as G(s){displaystyle G(s)}. Thus, we may find P{displaystyle P} by counting the poles of G(s){displaystyle G(s)} that appear within the contour, that is, within the open right half plane (ORHP).
We will now rearrange the above integral via substitution. That is, setting u(s)=D(s){displaystyle u(s)=D(s)}, we have
We then make a further substitution, setting v(u)=uâ1k{displaystyle v(u)={frac {u-1}{k}}}. This gives us
We now note that v(u(Îs))=D(Îs)â1k=G(Îs){displaystyle v(u(Gamma _{s}))={{D(Gamma _{s})-1} over {k}}=G(Gamma _{s})} gives us the image of our contour under G(s){displaystyle G(s)}, which is to say our Nyquist plot. We may further reduce the integral
by applying Cauchy's integral formula. In fact, we find that the above integral corresponds precisely to the number of times the Nyquist plot encircles the point â1/k{displaystyle -1/k} clockwise. Thus, we may finally state that
We thus find that T(s){displaystyle T(s)} as defined above corresponds to a stable unity-feedback system when Z{displaystyle Z}, as evaluated above, is equal to 0.
Summary[edit]
See also[edit]References[edit]
Further reading[edit]
External links[edit]
Retrieved from 'https://en.wikipedia.org/w/index.php?title=Nyquist_stability_criterion&oldid=902409039'
In this article, the author looks back on the shell buckling research activities during his 20âyears as Full Professor for Steel Structures at the University of Essen, Germany. The main research projects of that time are described very shortly. They included unstiffened circular cylindrical shells under external pressure or meridional compression or combined loading, made of structural or stainless steel at ambient or high temperatures, optionally with wall openings. Stiffened circular cylindrical shells under external pressure or meridional compression were also dealt with, as well as unstiffened conical shells, either as part of cone/cylinder assemblies or as individual truncated cones. Furthermore, the special case of very thin-walled open cylindrical tank shells was investigated under axisymmetric external pressure or wind-like pressure distribution or transverse shear. All projects combined realistic experiments on steel specimens with comparative or enhancing numerical investigations, and all of them were aimed at contributing directly to the improvement of practical shell stability design rules. Last but not least, it is reported on early efforts to develop guidelines for shell buckling design by global numerical analysis, and the authorâs credo argued that physical experiments are still indispensable in practice-oriented structural research, particularly in a complex field like shell stability. The article has been written in honor of John Michael Rotter, Professor Emeritus of the University of Edinburgh, who will celebrate his 70th birthday in October 2018.
Keywords experiments, metal shells, practice-oriented research, shell buckling, shell stability, steel shells, structural design rules
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Sorgfaltig bearbeiteter Nachdruck der Originalausgabe aus dem Jahr 1906.
Published June 20th 2013 by Dogma
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Max Born (was a German-British physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a number of notable physicists in the 1920s and 30s. Born won the 1954 Nobel Prize in Physics (shared with Walther Bothe).
Thin cylindrical shells subjected to high bending loads normally buckle by forming a number of diamond- shaped panels on part of their compressed surfaces. The values of the buckling loads show considerable scatter but are in general substantially less than those predicted by simple linear theory.
This paper describes an attempt made a few years ago to improve the buckling strength and to reduce the scatter in the buckling loads of such shells by means of internal pressure. It was expected and confirmed that internal pressure would increase the buckling loads, but little improvement was obtained in the amount of scatter. Considerable modifications of the buckling deformations were observed; at sufficiently high pressures the diamond pattern disappeared entirely and was replaced by a single ridge of deformation extending circumferentially around the compressed side of the shell.
An account is given of the experimental investigation together with some excerpts from the theoretical work. Camtasia studio 9 crackeado 64 bits. Most of the shells were formed from thin steel sheets bent into the shape of a cylinder then soldered down a generator. They were filled with water, plugged at their ends and put under pressure by means of a hand-pump. Observations made during bending included load, pressure, strain, deflections and mode of buckling. These are discussed and compared with the results of other investigations.
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