How To Draw Nyquist Plot

Nyquist Diagram

The term Nyquist diagram is used for a diagram of the line joining the series of points plotted on a polar graph when each point represents the magnitude and phase of the open-loop frequency response corresponding to a particular frequency.

From: Instrumentation and Control Systems (Third Edition) , 2021

Nyquist diagrams

W. Bolton , in Instrumentation and Control Systems, 2004

12.1 Introduction

This chapter follows on from Chapter 11 and presents another method of describing the frequency response of systems and their stability. The method uses Nyquist diagrams ; in these diagrams the gain and the phase of the open-loop transfer function, i.e. the product of the forward path and the feedback path transfer functions, are plotted as polar graphs for various values of frequency. With Cartesian graphs the points are plotted according to their x and y coordinates from the origin; with polar graph the points are plotted from the origin according to their radial distance from it and their angle to the reference axis (Figure 12.1).

Figure 12.1. (a) Cartesian graph with points specified by x and y values, (b) polar graph with points specified by r and θ values

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Nyquist Diagrams

William Bolton , in Instrumentation and Control Systems (Third Edition), 2021

12.2.1 Nyquist Diagrams

The term Nyquist diagram is used for a diagram of the line joining the series of points plotted on a polar graph when each point represents the magnitude and phase of the open-loop frequency response corresponding to a particular frequency. To plot the Nyquist diagram from the open-loop transfer function of a system we need to determine the magnitude and the phase as functions of frequency.

Example

Determine the Nyquist diagram for a first-order system with an open-loop transfer function of 1/(1+ τs).

The frequency response is:

$\frac{1}{1+\mathrm{j}\omega \tau }=\frac{1}{1+\mathrm{j}\omega \tau }\times \frac{1-\mathrm{j}\omega \tau }{1-\mathrm{j}\omega \tau }=\frac{1}{1+{\omega }^2{\tau }^2}-\mathrm{j}\frac{\omega \tau }{1+{\omega }^2{\tau }^2}$

The magnitude is thus:

$\text{Magnitude}=\frac{1}{\sqrt{1+{\omega }^2{\tau }^2}}$

and the phase is:

$\text{Phase}=-{\tan }^{-1}\omega \tau$

At zero frequency the magnitude is 1 and the phase 0°. At infinite frequency the magnitude is zero and the phase is −90°. When ωτ=1 the magnitude is 1/√2 and the phase is −45°. Substitution of other values leads to the result shown in Figure 12.4 of a semicircular plot.

Figure 12.4. Nyquist diagram for a first-order system.

Example

Determine the Nyquist plot for the system having the open-loop transfer function of 1/s(s+1).

The frequency response is:

$G(\mathrm{j}\omega )=\frac{1}{\mathrm{j}\omega (\mathrm{j}\omega +1)}=\frac{1}{\mathrm{j}\omega -{\omega }^2}=\frac{1}{\mathrm{j}\omega -{\omega }^2}\times \frac{-\mathrm{j}\omega -{\omega }^2}{-\mathrm{j}\omega -{\omega }^2}$

$=\frac{-{\omega }^2}{{\omega }^2+{\omega }^4}-\mathrm{j}\frac{\omega }{{\omega }^2+{\omega }^4}$

the magnitude and phase are thus:

$\text{Magnitude}=\frac{1}{\omega \sqrt{{\omega }^2+1}}$

$\text{Phase}={\tan }^{-1}\frac{-1}{-\omega }=180°+{\tan }^{-1}\frac{1}{\omega }$

When ω=∞ then the magnitude is 0 and the phase is 0°. As ω tends to 0 then the magnitude tends to infinity and the phase to 270° or −90°. Figure 12.5 shows the polar plot.

Figure 12.5. Example.

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Feedback Control Systems

Revised by William R. Perkins , in Reference Data for Engineers (Ninth Edition), 2002

Polar Plot (Nyquist Diagram)

The conventional Nyquist diagram must be modified to apply the Nyquist stability criteria to the frequency-response plot. In a linear system, the crucial point on the Nyquist diagram is −1. For nonlinear systems the −[1/ N(ω, X)] locus corresponds to the critical point −1. To evaluate the stability of the system, both −[1/N(ω, X)] and the G(jω) function are plotted on the polar plane. The describing function N(ω, X) generally is a function of both ω and X. If N is only a function of X, there will be one locus −[1/N(x)] plotted as a function of X. If N is also a function of ω, a family of constant-frequency loci are plotted for different values of ω (see Fig. 49).

Fig. 49. Typical polar plots of various N(ω, X).

The stability of the system is determined by the following relationship between the −[1/N((ω, X)] locus and the G(jω) plot (Fig. 50). If the −[1/N(ω, X)] locus lies to the left of the G(jω) plot or is not enclosed, the system is stable. Conversely, if the −[1/N(ω, X)] locus lies to the right of the G(jω) plot or is enclosed, the system is unstable. If the −[1/N(ω, X)] locus intersects with the G(jω) plot, the system may have a sustained oscillation. In the case where N is a function of ω, the condition for sustained oscillation is satisfied if the ω of the G(jω) plot at the intersecting point is the same ω of the −[1/Nω, X)] locus (see Fig. 51).

Fig. 50. Polar-plot stability criteria.

Fig. 51. Typical polar plot of G(jω) and −[1/N (ω, X)] as a function of ω.

The oscillation may be either stable or unstable. If the G(jω) intersects with the −[1/N(ω, X)] locus at one point only, the oscillation is stable (stable limit cycle). If more points of intersection exist, the limit cycle may be either stable or unstable. The stability of the limit cycle is determined by the direction of the two loci at the crossover point.

By establishing the G(jω) locus pointing in the direction of increasing frequency as a reference, if the −[1/N(X)] locus pointing in the direction of increasing amplitude X crosses the G(jω) locus from right to left, the limit cycle is stable. If the crossover occurs from left to right, the limit cycle is unstable. A polar plot with both stable and unstable limit cycles is shown in Fig. 52.

Fig. 52. Polar plot of stable and unstable limit cycles.

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Frequency Domain Analysis

Tony Roskilly , Rikard Mikalsen , in Marine Systems Identification, Modeling and Control, 2015

6.4 Nyquist Diagrams

A Nyquist diagram is a version of the polar plot format for frequency response. It is useful in that it provides a simple graphical procedure for determining the closed-loop stability from the frequency response curves of the open-loop transfer function KG(s).

The closed-loop stability of standard systems can be evaluated in a simplified form (for a full derivation of the Nyquist stability criterion, see, e.g., Burns [1]), known as the "left-hand criterion": If KG(s) has no poles or zeros having positive real parts, then 1 + KG(s) = 0 has no unstable roots. Therefore, if the KG(jω) plot is traced out as ω goes from 0⁺ to $+\infty$ it will always leave the (−1,0) point on its left.

Example

The use of Nyquist diagrams with the left-hand criterion is most easily demonstrated by an example. Plot the Nyquist diagram of the following plant to determine its closed-loop stability:

$G(s)=\frac{1}{(1+0.2s)(1+s)(1+10s)},$

for (a) K = 10, (b) K = 136.8, and (c) K = 500.

First we substitute for jω and work out the open-loop transfer function:

$\begin{array}{ll}\hfill KG(s)=& \frac{K}{(1+0.2s)(1+s)(1+10s)}\hfill \\ \hfill KG(j\omega )=& \frac{K}{(1+0.2j\omega )(1+j\omega )(1+10j\omega )}\hfill \\ \hfill KG(j\omega )=& \frac{K}{(1-12.2{\omega }^2)+(11.2\omega -2{\omega }^3)j}\hfill \end{array}$

This gives the following expressions for magnitude and argument:

$\begin{array}{ll}\hfill M=& \frac{K}{\sqrt{{(1-12.2{\omega }^2)}^2+{(11.2\omega -2{\omega }^3)}^2}}\hfill \\ \hfill \varphi =& {tan}^{-1}\left(\frac{0}{K}\right)-{tan}^{-1}\left(\frac{11.2\omega -2{\omega }^3}{1-12.2{\omega }^2}\right)\hfill \end{array}$

Substituting and working out the magnitude and phase for a set of selected frequencies, we get values as in Table 6.1.

Table 6.1. Magnitude and Phase Values for Nyquist Plot

	K = 10		K = 136.8		K = 500
ω	M	ϕ	M	ϕ	M	ϕ
0	10	0	136.8	0	500	0
1	0.7	-140.6°	19.8	-140.6°	72.5	-140.6°
2	0.2	−172.4°	2.8	-172.4°	10.4	-172.4°
3	0.1	-190.6°	1.2	-190.6°	4.5	-190.6°
$\infty$	0	-270°	0	-270°	0	-270°

The resulting Nyquist plot is sketched in Figure 6.13. The three gain values chosen were not arbitrary but are in fact one value that gives a stable system in the closed loop (K = 10), one that gives a marginally stable system (K = 136.8), and one which would bring the system to instability if closing the loop (K = 500). (You can prove this with the techniques shown in the previous chapters, e.g., the Routh Hurwitz stability criterion.)

Figure 6.13. Sketch of Nyquist plot for $G(s)=\frac{1}{(1+0.2s)(1+s)(1+10s)}$ .

It can be seen that for a stable system, the Nyquist plot passes to the left of the − 1 point on the real axis. At marginal stability, the plot passes through the − 1 point, whereas for an unstable system, the Nyquist contour will encircle the − 1 point.

6.4.1 Stability margins in the Nyquist diagram

Similarly, as in the Bode plot, stability margins can be read off the Nyquist diagram directly, as shown in Figure 6.14. The point at which the Nyquist contour crosses the real axis is equal to the inverse of the gain margin. (Note that this may need to be converted into decibels if comparing it to a value read off a Bode plot.) The phase margin is the angle of the unity magnitude response to the real axis, i.e., the angle at the point where the magnitude crosses a unit circle.

Figure 6.14. Determining gain and phase margin from Nyquist plots.

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Introduction

László Keviczky , Csilla Bányász , in Two-Degree-of-Freedom Control Systems, 2015

Phase Margin

Let us draw the Nyquist diagram of the open loop for positive frequencies. Let us then determine the intersection point of the Nyquist diagram with the circle of unity radius. The frequency belonging to this point is called cut-off frequency and is denoted by ω_c. Let us connect the origin and the intersection point with a straight line. The angle of this straight line formed with the negative real axis is called the phase margin (Figure 1.3.2).

Figure 1.3.2. Interpretation of the phase margin.

(1.3.10) ${\mathrm{\phi }}_{\text{t}}=\mathrm{\phi }\left({\mathrm{\omega }}_{\mathrm{c}}\right)+{180}^{\circ }=\arg L\left(j{\mathrm{\omega }}_{\mathrm{c}}\right)+{180}^{\circ }$

If the phase margin is positive, the system is stable. If the phase margin is zero, the system is on the stability limit. If the phase margin is negative, the system is unstable.

Thus for the stability of the control system the following statements can be made:

(1.3.11) $\begin{array}{l}{\mathrm{\phi }}_{\mathrm{t}}>0\text{Stable}\text{system}\\ {\mathrm{\phi }}_{\mathrm{t}}=0\text{Boundary}\text{of}\text{stability}\\ {\mathrm{\phi }}_{\mathrm{t}}<0\text{Unstable}\text{system}\end{array}$

The stability of the system can be evaluated based on the phase margin as a single measure only if the Nyquist diagram of the open loop crosses the unit circle only once.

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Linear Reactor Process Dynamics with Feedback

JEFFERY LEWINS PhD (Cantab), PhD (MIT) , in Nuclear Reactor Kinetics and Control, 1978

STABILITY OF CONTROLLED SYSTEM AT LOW POWER

Figure 4.2 is the Nyquist diagram representation of the same information on the transfer function G_n (jω) in another form. To illustrate questions of stability treated via the Nyquist diagram let us suppose that (at low power) the neutron level was to be controlled by a feedback system with an idealised characteristic of having an arbitrary gain K and zero phase lag/lead at all frequencies as in the block diagram of Fig. 4.3. Would this system be stable?

FIG. 4.2. Nyquist diagram for neutronics transfer function.

FIG. 4.3. Block diagram for low power stability with idealised feedback.

The open loop transfer function is KG_n and the closed loop transfer function is KG_n /[1 – KG_n ], where we have absorbed the power level into the definition of the feedback gain K. The control system envisages a measurement of the actual neutron level n, a comparison with the demand level n ₀ and the provision of a reactivity control signal ρ_i proportional to the difference δ_n via say the movement of a control rod. If we consider a small error and the known direction of the effect of reactivity on n, it is elementary to see that stability will require the correction reactivity to be of opposite sign to the error signal δn, i.e. that K will have to be negative for static stability. There remains conceptually the possibility of dynamic instability, that at too high a gain |K| the feedback system becomes unstable.

Stability may be determined in terms of the Nyquist criterion, however, making use of the Nyquist diagram (Fig. 4.4). Assuming first K negative we are to investigate 1 + |K|G_n around the origin or equivalently, |K|G_n around the point –1,0. Because of the behaviour of G_n as ω→0, however, it is necessary to "close" the Nyquist diagram and we note that as the variable p tends to zero,

FIG. 4.4. Nyquist stability construction.

$\underset{\rho \to o}{ℒ}{G}_n\left(p\right)=\frac{\lambda }{\beta p}=\frac{\lambda }{\beta }\frac{1}{v}{e}^{-j\theta }$

where we have substituted the polar form for p = re^jθ . The counter-clockwise semicircle in the p-plane around the origin (see Fig. 1.9) leads to 1/p traversing a clockwise semicircle of large radius, 1/v e^−jθ . The completed Nyquist diagram is seen not to enclose the test point for any magnitude of gain. Since the open loop system has no poles in the right half p-plane we have shown that the closed system is stable at all gains for negative feedback control of the idealised sort postulated.

In the sense of the linearisation of the neutronics equations it is indeed true that the open loop system has no poles within the right half-plane and the claim is therefore substantiated. Note, however, that the open loop is unstable (the transfer function tends to ∞ as ω tends to zero) so that in a somewhat special sense the closed system is only conditionally stable.

If the gain is positive, we are investigating 1 – |K|G_n around – 1, 0 and the effect of the change of sign is to require a double reflection of |K|G_n in real and imaginary axis. The "closure" construction is still a clockwise semicircle of large radius and it can be left to the reader to sketch the resulting figure and see that it surrounds the test point; the system with positive feedback is unstable for all values of the gain.

The example is of limited practical interest and serves chiefly to introduce the Nyquist criterion to a nuclear reactor problem because we have ignored other feedback mechanisms and we have assumed no lead or lag associated with K. The control mechanism would have to be astonishingly rapid to justify this in relation to the natural frequencies in G_n . However, there are occasions in dealing with slower natural phenomena such as xenon poisoning, where it is more reasonable to assume that the feedback has a much shorter time scale than the effect (xenon) being studied and this form of power coefficient can then be employed.

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Nichols-Krohn-Manger-Hall Chart

Yazdan Bavafa-Toosi , in Introduction to Linear Control Systems, 2019

8.5 M- and N-Contours

As in the case of drawing the Nyquist diagram, the problem associated with the polar plots and the use of M- and N-circles is that a change in the gain (i.e., multiplication or division of the gain by a constant) results in a certain amount of deformation of the polar plot. There we saw that this problem was rectified by the use of log magnitude in place of magnitude. This idea was also used by Nichols and Krohn who suggested the Lm versus phase diagram. They originally used the name ψ-contours ³ instead of N-contours. However, then the community well-nigh unanimously adopted the name N-contours, to go with M-contours, and we do so as well. Moreover, they directly started from ψ-contours, i.e., they did not introduce ψ-circles, although they did introduce M-circles. Anyway, by this transformation of the vertical axis, the M- and N-circles are also transformed. The results are called the M- and N-contours, depicted in the sequel Figs. 8.6 and 8.7. The figure repeats every $360\mathrm{deg}$ and is symmetric with respect to either of the multiples of $\pm 180\mathrm{deg}$ line. It looks like Fig. 8.6. In this figure $M_1<M_2<0<M_3$ and $-{\alpha }_i$ is used instead of $360-{\alpha }_i$ for simplicity of notation—they refer to the same point; the same for $-{\beta }_i,-{\gamma }_i$ . If these contours are superimposed on the open-loop Lm versus phase grid of the system then the resulting plane is the Nichols chart which may also be called the NKMH due to the contributions of all. An expansion of the $[-320°0°]$ interval of the NKMH chart is delineated in Fig. 8.7. Note that once upon a time before the MATLAB® era such sheets were commercially available.

Figure 8.6. M- and N-contours.

Figure 8.7. Expansion of part of the NKMH chart plane.

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Electrochemical Measurement Methods and Characterization on the Cell Level

Têko W. Napporn , ... Viktor Hacker , in Fuel Cells and Hydrogen, 2018

9.8.1 Graphical Representation of Impedance Spectra

Common graphical methods for visualizing the impedance spectrum are the Bode diagram and the Nyquist diagram. They provide an informative overview of the impedance and the state of the corresponding regions inside the electrochemical element. The Cole-Cole diagram is, in a narrow sense, the preferred choice for visualizing the permittivity. The Nyquist diagram and Cole-Cole diagram are common terms for graphic presentations in the complex ( z) plane.

A series connection of three RC-circuits, shown in Fig. 9.20, with time constants of 1   ms, 10   μs, and 100   ns, is used here as an example. Simulations were done using Matlab and Simulink. The circuit and its component values were selected on purpose to clearly show the relationship between circuit diagram, Nyquist Plot, and Bode diagram.

Fig. 9.20. Impedance Z comprising three different RC time constants.

A Nyquist plot is a graphical presentation of the real part and the imaginary part of an impedance Z over a specified frequency range. In this example (see Fig. 9.21), the real and imaginary part of Z were calculated for frequencies ranging from 0   Hz (right side, Re   =   30   Ω, Im   =   0   Ω) to 10   KHz (left side, Re   =   0   Ω, Im   =   0   Ω).

By looking at Fig. 9.20, it becomes clear why this pattern occurs in Fig. 9.21. If the frequency is very low, the impedance of all capacitors is very high. Thus, all current flows only through the resistors, which results in 30   Ω. On the other hand, the impedance is zero for very high frequencies due to the very low impedance of the capacitors. In the middle frequency range, there is always the real part of the impedance of the respective resistances, and the imaginary part of the impedance of their respective capacitors, which leads to the three arcs pictured in Fig. 9.21.

Fig. 9.21. Nyquist Plot of the circuit in Fig. 9.20.

Nyquist plots have one shortcoming—the exact frequency information is missing. Low frequency data points are on the right side, and higher frequencies are on the left of the plot. (Not all are true of all circuits).

The Bode plot (see Fig. 9.22) is another widely used method of visualizing the transfer function of two terminal-pair networks. Amplitude (dB) and phase (degree) response is delivered over a logarithmic frequency scale. It is especially useful for stability analysis in feedback control systems, but also well suited for impedance characterization of two-terminal circuits.

Fig. 9.22. Bode plot of an RC circuit. Left: impedance of a parallel circuit of a resistor and a capacitor; middle: sketch of the parallel circuit; right: bode plot of the circuit.

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Introductory Review

JEFFERY LEWINS PhD (Cantab), PhD (MIT) , in Nuclear Reactor Kinetics and Control, 1978

Linear System Stability

Consider a linear system operating in steady state at an operating point (of which there is only the one in a linear system) determined of course by equating the rates of change ${\dot{x}}_i$ to zero. Inevitably in a real system there will be small disturbances and we study the resulting transients. If these disturbances, whatever their initial form, die out, the system is stable. If they grow or even if only some of them grow, the linearity implies that they will go on growing without bound; the system is unstable. There is the possibility of a behaviour on the limit of stability, where an oscillatory behaviour is maintained whose amplitude neither increases nor decreases from the initial disturbance.

The behaviour of systems with simple poles, of any finite order, in the Laplace transform representation illustrates these different possibilities, as in Fig. 1.8.

FIG. 1.8. Time and Laplace transform representation for stability.

Any system whose Laplace transform G(p) leads to a pole in the right half-plane is unstable because the corresponding time dependent behaviour corresponds to increasing exponentials whether or not these exponentials are coupled with oscillatory terms. Single poles on the imaginary axis correspond to the limit of stability, while poles in the left hand plane indicate a stable behaviour with decreasing (though possibly oscillatory) behaviour. A pole at the origin is a little more complicated since a single pole corresponds to the step function h(t) = 1 for t > 0, while repeated poles on the imaginary axis indicate further terms in t, t ², etc., that would denote instability.

Of course the Laplace transform G(p) may have terms in it other than simple poles of finite order, but it may be stated as a general result of complex analysis that if there are poles in the right half p-plane of any form, the time dependent behaviour shows an increase without bound and the system is unstable.

If we attack the question of the stability of a system head on from this point of view, we would try to find all the poles in G(p) and see whether any lie in the right half-plane. But this is as difficult as finding the complete time dependent behaviour. We might approach the question obliquely with more efficiency by considering merely the question of whether any poles exist in the right half p-plane without actually finding their number or location. If we limit ourselves to transforms G(p) that are rational polynomials of the form g(p)/f(p), then there are several techniques for determining stability from this oblique viewpoint.

If the numerator g(p) is bounded, poles in G(p) arise only by virtue of the zeros of f(p), i.e. the roots of the equation f(p) = 0. If the polynomial form f(p) is known explicitly, then Routh's criteria test for the existence of poles in the right half p-plane is via the roots of f(p). Suppose f(p) to be in the form

$f\left(p\right)=p^n+a_1{p}^{n-1}+\dots a_n=0$

We shall not give the full criteria for arbitrary n but state it explicitly for n = 1, 2 and 3 in Table 1.3. (See problem 1.12 and ref. 1 for extensions.) Clearly for f(p) = p + a ₁ we have the trivial result that p ₁ = -a ₁, so that if a ₁ is negative the system is unstable with the pole lying in the right half plane. The criterion requires the row of numbers each to be greater than zero for stability, terminated as shown for the various orders 1, 2, 3.

TABLE 1.3. Routh's array for low orders

a 1	First order
a 1, a 2	Second order
a 1, a 1 a 2-a 3, a 3	Third order

However, Routh's approach in any order requires knowledge of the polynomial form. Whilst we may suppose that the transform could be expressed as a rational polynomial, it may not be available in that form. For example, we may have measured the transfer function experimentally or we may have added several transfer functions graphically from their Bode diagrams. It would be advantageous to have a direct test via the Fourier transform or transfer function G(jω) for the open loop case or, more important, via 1 + G(jω)H(jω) in the case of feedback where stability is more suspect. The latter case is available via the Nyquist criterion for stability.

NYQUIST STABILITY CRITERION (negative feedback convention)

We are examining the behaviour of the feedback expression $\overline{x}=G/\left[1+GH\right]\overline{s}$ which may again be written as $\overline{x}=\left[\overline{g}/\overline{f}\right]\overline{s}=F\left(p\right)\overline{s}$ , say. We assume that the over-all transfer function F(p) is again a rational polynomial and, furthermore, we assume that the order of the denominator is greater than that of the numerator, or in more physical terms that the system is such as to attenuate high frequencies. There will be poles in the right half p-plane if either (a) g(p) has poles in the right half plane or (b) f(p), i.e. 1 + GH, has zeros in the right half plane. To start with we assume that the open loop system was stable so that there were no poles from g(p) = GH and we are concerned only with the zeros of 1 + GH. If GH has no poles then 1 + GH has no poles and we are dealing with a polynomial function that can be entirely represented as the product of its factors p – p_i leading to the poles in F(p) at the roots of f(p). Consider individual roots leading to terms such as 1/[p – p_i ] in F(p). Suppose (Fig. 1.9) we construct a loop around such a pole in the right half-plane.

FIG. 1.9. Construction of Nyquist stability criterion.

The vector p from the origin traces such a loop enclosing a zero of $\overline{f}\left(p\right)$ and a pole of F(p). The complex number p – p_i is shown as a vector and as p traces the loop, p – p_i is seen to rotate. That is, the polar form p – p_i = Re^jθ rotates such that the number returns to the same R value but with an angle θ increased by 2π. The corresponding pole 1/[p – p_i ] is 1/Re ^−jθ and this also rotates, in the opposite direction, by −2π. If, on the other hand, there is no pole p_i within the loop (compare the pole at p_k ), then there is no net rotation. Therefore we plot the complex number F(p) = 1/[p – p_i ] as p passes round the loop; F(p) will show a rotation if p_i lies inside the loop.

If we extend the loop to cover the whole of the right half p-plane save possibly for any poles on the imaginary axis whose effect must be separately determined, we have tested the whole of the right half p-plane against the presence of a pole that might lie in it. If F(p) rotates, there is a pole. If it does not rotate, but θ returns to its original value and not 2π different, then no pole was contained. This construction of the Nyquist criterion to test 1 + GH is shown in Fig. 1.10.

FIG. 1.10. Nyquist criterion using the transfer function.

The loop so indicated passes up the imaginary axis of the p-plane, and so in the 1 + GH plane we are actually studying the behaviour of the transfer function 1 + G(jω) H(jω) for ω from -∞ to ∞. The completion of the loop in the p-plane calls for a semicircle at large radius, but by our assumption of low pass, attenuated high frequencies, the transfer function goes to the origin for large magnitudes of ω. We have the distinct advantage of determining stability in terms of the rotation of the (closed) transfer function.

Rotation is a question of drawing a vector from the origin whose tip traces the completed transfer function loop in the Nyquist diagram. Rather than plot 1 + GH and determine the rotation about the origin, it is common to plot GH alone and determine the rotation about the point −1, 0 on the real axis. Es egal. ^†

If 1 + GH has more than one pole and is of the form 1/[p – p ₁][p – p ₂] ···, etc., then the polar form is Re^jnθ and the rotations caused by the loop will be 2nπ, where n is the number of poles within the loop, i.e. in the right half p-plane. Thus the number of rotations counts the number of zeros in the return difference 1 + GH and hence the number of poles of the transfer function G/1 + GH in the right half p-plane. If, contrary to assumption, the open loop system had been unstable, there would be right half poles of G and hence in the return difference, of the form p – p_i = Ae^jθ , leading to positive rotations in the opposite sense of the zeros investigated. This would have tended to cancel the count and invalidated the result; hence the assumption. In the more general case this effect can be accounted for if we know the number of poles of the open loop transfer function and require a corresponding number of rotations of the closed loop transfer function in the Nyquist diagram accordingly.

Figure 1.11 illustrates two possible situations assuming negative feedback in the form of Nyquist diagrams. Figure 1.11(a) itself indicates stability. If the feedback system depends on a gain or parameter K as 1 + KGH for the return difference, then the same diagram for GH will serve subject to a simple radial scaling according to the factor 1/K. It is likely, therefore, that at low K the system will be stable as shown in Fig. 1.11(a), while at increasing K or with increasing feedback the system will be unstable. It is convenient to distinguish, therefore, between static instability, which is essentially a need for the feedback term to be negative, and dynamic instability, the onset of the instability caused by too high a gain. Static instability, of course, is easily understood as arising when a small departure from the desired state is returned positively to increase the divergence. Dynamic instability given a statically stable system is linked with the existence of a range of frequencies where a phase lag of π essentially turns negative feedback into positive feedback.

FIG. 1.11. Stability in Nyquist diagrams.

In Fig. 1.11(b), however, we have a situation where the system is currently stable but the decrease as well as the increase of a gain K would drive the system unstable. Such a situation is conventionally called conditionally stable. Its disadvantage comes from the possibility that when combined with other stable systems the combination may be (unexpectedly) unstable because of an over-all reduction of gain.

ROOT LOCUS METHOD

The Nyquist criterion has the advantage of working with the graphical transfer function data but it does not give detailed information directly on the transient behaviour (though we will quote design rules for phase and stability margin). Such detail requires knowledge of the poles of G(p)/[1 + G(p)H(p)] throughout the p-plane. Furthermore, the situation often involves an adjustable parameter, e.g. the gain of a proportional feedback controller, and it is desirable to see the effect of different values of this parameter directly.

Evans ⁽⁴⁾ (and independently Westcott) suggested a graphical method for studying the roots of the return difference 1 + GH as a function of a linear parameter. The variation of the roots with the parameter leads to a root locus in the p-plane. Suppose we can write the transfer function in the form of rational polynomials using this time the assumption of positive feedback

(1.20) $\overline{x}=\frac{G}{1-GH}\overline{s}=\frac{h\left(p\right)}{f\left(p\right)-\mu g\left(p\right)}\overline{s}\equiv F\left(p\right)\overline{s}$

Where f, g and h are polynomials of the form f(p) = pⁿ + ···, g(p) = p^m + ··· and μ is a real parameter. ^† Then the poles arise solely from the zeros of f – μg. Note again that the expression is not written for the conventional difference of a feedback control system and the actual sign of μ must be allowed for.

We assume that the polynomials can be written in terms of their roots as f(p) = [p – p ₁][p – p ₂] ··· and g(p) = [p – z ₁][p – z ₂] ··· so that we have at a pole of F(p):

(1.21) $\frac{\left[p-p_1\right]\left[p-p_2\right]\cdots \left[p-p_n\right]}{\left[p-z_1\right]\left[p-z_2\right]\cdots \left[p-z_m\right]}=\mu$

The left hand side is the product/division of complex numbers whose phases must add or subtract to the phase of the right hand side, which is (2k + 1)π for μ < 0 and 2kπ for μ > 0, where k = 0, 1, 2 ···. We also note that when μ is zero, the required poles are the p_i , poles of the open loop transfer function, and as μ becomes large in magnitude the roots are the zeros z_i of the open loop transfer function.

Without proof we state the construction rules that follow from these observations:

(1)

Plot the poles p_i and the zeros z_i of the open loop transfer function.

(2)

n loci, symmetric about the real axis, emerge from the n poles p_i .

(3)

Every zero z_i is a terminator for a locus; the remaining n-m loci terminate at "infinity" either along the real axis or along asymptotes that intersect the real axis at [Σp_i – Σz_i ]/[n-m], at angles (2k + 1)π/[n-m] (if μ < 0) or 2kπ/[n-m] (if μ > 0).

(4)

A locus lies on the real axis if the sum of zeros and poles to its right (including the pole it came from) is odd (μ < 0) or even (μ > 0).

(5)

Loci join or leave the real axis vertically at a point p_c , where Σ1/[p_c – p_i ] = Σ1/[p_c – z_i ].

The following example of a simple second order system can, of course, be solved exactly as a quadratic but illustrates the application of the rules. We take G(p) = 1/p and H(p) = μ/[p + τ]. We seek, therefore, the zeros of p ² + τp – μ = 0. There are no open loop zeros z_i : m = 0. There are two open loop poles: p ₁ = 0 and p ₂ = -τ; n = 2. Rules (3) and (5) give the same point, − $\frac{1}{2}$ τ. It is seen (Fig. 1.12) that the nature of the transient response and stability depend substantially on the sign of the parameter.

FIG. 1.12. Elementary root locus example.

We shall employ the method in Chapter 4 for a system of more complexity. It is to be remarked, however, that even more complicated systems very commonly are well approximated by a second or third order system in the range of interest, which is usually close to the imaginary axis. Thus simple graphical methods are indeed helpful for a preliminary analysis, and with the availability of computer aid design it can be expected that even more complicated systems will yield to a root locus study of the effect of varying one or even more parameters.

STABILITY MARGINS

The Nyquist criterion itself determines a limit of stability, sustained oscillations. A satisfactory system must be offset from this limit with sufficient damping of transients to ensure they die out in acceptable times. Fortunately, the Nyquist diagram serving the Nyquist criterion can itself be utilised to determine approximate transient conditions in the form of stability margins, at least if we omit conditionally stable systems. Figure 1.13 illustrates the conventional designer's guide to ensure adequate transient behaviour of a stable system. Phase and gain margin definitions are shown.

FIG. 1.13. Stability margins.

The setting of margins is open to some choice. Widely used margins in control practice are to take a gain margin g of 8 db and a phase margin θ of 35°. However, in the design of a nuclear power station for a life of, say, 30 years and to allow for the deterioration of components, it might be wiser to adopt values of 12 db and 50° respectively in order to see that temperature transients are well damped and do not lead to overshoots above critical temperatures, etc.

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Control Systems

JO Flower Bsc(Eng), PhD, DSc(Eng), CEng, FIEE, FIMarE, MSNAME , EA Parr MSc, CEng, MIEE, MInstMC , in Electrical Engineer's Reference Book (Sixteenth Edition), 2003

13.30.2 Ultimate cycle methods

The basis of these methods is determining the controller gain which just supports continuous oscillation, i.e. point A and gain K on the Nichols chart and Nyquist diagram of Figure 13.86 . The method is based on work by J. G. Ziegler and N. B. Nichols and is often called the Ziegler Nichols method.

Figure 13.86. Basis of the ultimate cycle test. Point A determines the frequency at which continuous oscillations will occur when gain K is applied: (a) Nichols chart; (b) Nyquist diagram

The integral and derivative actions are disabled to give proportional only control, and the control output manually adjusted to bring P _v near the required value. Auto control is selected with a low gain.

Step disturbances are now introduced and the effect observed. One way of doing this is to go back into manual, shift O _p by, say 5%, then reselect automatic control. At each trial the gain is increased. The increasing gain will give a progressively underdamped response and eventually continuous oscillation will result. Care must be taken in these tests to allow all transients to die away before each new value of gain is tried.

If the value of gain is too high, the oscillations will increase. The value of gain which gives constant oscillations neither increasing or decreasing is called the ultimate gain, or P _u (expressed as proportional band). The period of the oscillations T _u should also be noted from the chart recorder (or with a watch).

The required controller settings are:

Proportional only control

PI Control

PID Control

T _i = 4 T _d is a useful rule of thumb.

Other recommended settings for a PID controller are:

and

All of these values should be considered as starting points for further tests.

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How To Draw Nyquist Plot

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