Nyquist criteria

Vice versa, the Nyquist limit frequency, Vnyq, up to which a process can just be correctly detected is given as 

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Nyquist criterion Nyquist theorem Nystan Nystarescent Nystatin... 

For the condition / = 0, the Nyquist criterion is A closed-loop control system is stable if, and only if, a contour in the G s)H s) plane does not encircle the (—1, jO) point when the number of poles of G s)H s) in the right-hand. v-plane is zero . 

The concept of gain and phase margins derived from the Nyquist criterion provides a general relative stability criterion. Frequency response graphical tools such as Bode, Nyquist and Nichols plots can all be used in ensuring that a control system is stable. As in root locus plots, we can only vary one parameter at a time, and the common practice is to vary the proportional gain. 

For a first order function with deadtime, the proportional gain, integral and derivative time constants of an ideal PID controller. Can handle dead-time easily and rigorously. The Nyquist criterion allows the use of open-loop functions in Nyquist or Bode plots to analyze the closed-loop problem. The stability criteria have no use for simple first and second order systems with no positive open-loop zeros. 

The Fourier frequency bandpass of the spectrometer is determined by the diffraction limit. In view of this fact and the Nyquist criterion, the data in the aforementioned application were oversampled. Although the Nyquist sampling rate is sufficient to represent all information in the data, it is not sufficient to represent the estimates o(k) because of the bandwidth extension that results from information implicit in the physical-realizability constraints. Although it was not shown in the original publication, it is clear from the quality of the restoration, and by analogy with other similarly bounded methods, that Fourier bandwidth extrapolation does indeed occur. This is sometimes called superresolution. The source of the extrapolation should be apparent from the Fourier transform of Eq. (13) with r(x) specified by Eq. (14). 

For a finite sampling interval A<5, more than one superposition of cosine/sine waves can give rise to the recorded interferogram. For the transformed spectrum to be unique, the sampling interval A<5 must be sufficiently small to detect modulations in the interferogram due to the shortest wavelength present in the spectrum, the so-called Nyquist criterion [66] ... 

Examine the stability of the control system described in Example 7.6 using the Nyquist criterion. 

From example 7.6 we know that critical stability occurs for Ac = 1.8, r, = 3.5. Hence, by the Nyquist criterion, when these conditions are applied, the polar plot will pass through the point (-1,0) on the complex plane, i.e. for these values of the controller parameters, 9m (G(i[Pg.631]

It can be shown that there is a Nyquist criterion for sampled data systems which is equivalent to that for continuous systems (see Section 7.10.5) and equation 7.131 can be applied in its comparable r-transformed form(42). In practice it is generally sufficient to ascertain whether the polar plot of G(z) in the complex z-plane encircles the (-1,0) point (as with continuous systems in the j-plane) where 1 + G(r) = 0 is the system z-transformed characteristic equation. The polar plot is constructed from... 

Figure 3.32. Enhanced sensitivity can be realised in favourable cases by oversampling the data and hence reducing digitisation noise. Spectrum (a) shows part of a conventional proton spectrum sampled according to the Nyquist criterion. Oversampling the data by a factor of 24 as in (b) provides a sensitivity gain (all other conditions as for (a)).

Figure 7.4-4 illustrates the Nyquist criterion the signal is sampled at exactly two points per cycle, which is just not good enough. In this borderline case the frequency is recovered, but the amplitudes of the possible sine and cosine components of the signal are not. 

The above illustrates what happens when the signal frequency lies outside the range of frequencies used in the Fourier analysis, in which case the digital Fourier transform will misread that frequency as one within its range. As already indicated, another problem occurs when the frequency lies within the analysis range, and also satisfies the Nyquist criterion (i.e., is sampled more than twice during the repeat cycle of that signal), but has a frequency that does not quite fit those of the analysis, as illustrated below. 

Keep in mind that, because of the Nyquist criterion, the number of available frequencies is only half the number of data points. Or, to put it differently, all frequencies / in the Fourier transform have their negative counterparts at - /The largest number you can therefore select for filtering is half the number of data points in the set, and at that point you would filter out everything ... 

It is possible, though, that the AR or of an open-loop transfer function may not be decreasing continuously with co. In Figure 18.4 we see the Bode plots of an open-loop transfer function where AR and Bode criterion may lead to erroneous conclusions and we need the more general Nyquist criterion which will be discussed in Section 18.4. Fortunately, systems with AR or like those of Figure 18.4 are very few, and consequently the Bode criterion will be applicable in most cases. 

Figure 18.7 shows that curve A does not encircle the point (-1,0), whereas curve B does. Thus, according to the Nyquist criterion, the feedback system with open-loop Nyquist plot the curve A is stable, while curve B indicates an unstable closed-loop system. This in turn implies that for Kc = 1 the system is stable, whereas for Kc = 50 it is unstable. 

As we pointed out in Section 18.1, the Bode stability criterion is valid for systems with AR and monotonically decreasing with a). For feedback systems with open-loop Bode plots like those of Figure 18.4 the more general Nyquist criterion is employed. In this section we present a simple outline of this criterion and its usage. For more details on the theoretical background of the methodology, the reader can consult Refs. 13 and 14. 

To understand the concept of encirclement and therefore correct use of the Nyquist criterion, let us study the following examples. 

State the Nyquist stability criterion and give some examples of stable and unstable feedback control systems different from those presented in this chapter. Explain the concept of encirclement of the point (-1,0) by the Nyquist plot, which is so central for the Nyquist criterion. 

IV.74 Examine the stability of the closed-loop systems whose open-loop transfer functions are given in Problem IV.58. Employ the Nyquist criterion for your analysis. 

IV.75 Use the Nyquist criterion and find the range of Kc values that yield stable closed-loop response for... 

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