Responses
4 Frequency Responses and Their Plotting
4.3 Nyquist Plot Since
G(jω) = M(ω)ejφ(ω) = X(ω) + jY(ω) (4.11)
every choice of ω gives a point in a complex plane either plotted in polar coordinates for the M, φ form or in rectangular coordinates in X, Y form. Joining the points together as ω is varied produces a locus with ω as a parameter which is known as a polar or Nyquist plot. To obtain analytical results one needs to be able to work in both polar and rectangular coordinates, since one may be more appropriate than the other for a particular evaluation. From consideration of the individual elements of a transfer function in the Bode approach of the previous section one should be able to estimate the shape of a Nyquist plot. Important points in this respect are the high and low frequency limits, that is the value of G(jω) as ω→ ∞ and 0. For large s the limit of the general transfer function G(s) of equation (2.9) will tend to 1/s(n-m),
that is 1 over s to the relative degree. Thus for a strictly proper transfer function the gain will tend to zero and the phase to -90(n – m)° as ω→∞. For a proper transfer function with n = m the gain will tend to a finite value and the phase to zero. At low frequencies the transfer function, G(0) will tend either to a constant or s to the power of the number of differentiation terms minus integration terms in the transfer function. Typically only integration terms exist in transfer functions for control systems so the behaviour at low frequencies depends on the number of integrators and G(s) tends to 1/si where i is the number
of integrators. Thus at low frequencies for i > 0 the magnitude tends to infinity and the phase to -90i°. This phase result does not mean that the locus starts on an axis as sketches in many books incorrectly show. As a simple example of this point consider the transfer function
2
)
1
(
1
)
(
+
=
s
s
s
G
(4.12)then putting s = jω, and writing G(jω) in the form X(ω) + jY(ω) gives X(ω) = (1 2)2
2 ) ( ω ω + − = X and Y(ω) = 222 ) 1 ( ) 1 ( ) ( ω ω ω + − − = Y .
Clearly as ω → 0, X(ω) → –2, not the imaginary axis, although the phase does tend to -90°. This will always be the case that the locus for a transfer function with one or more integrators will tend to an asymptote which in principle can be calculated. The Nyquist plot of this transfer function is obtained with the instruction nyquist(G). It is shown in Figure 4.4, which is obtained by the following single instruction defining the transfer function of G in the Nyquist statement:-
Control Engineering Frequency Responses and Their Plotting
Information about where a Nyquist plot cuts the axes can be obtained from the facts that the real axis is cut when Y(ω) = 0 or arg G(jω) = 0° or 180°, and the imaginary axis when X(ω) = 0 or arg G(jω) = -90° or +90°. Which are the easiest calculations can depend on the transfer function. For the above example it is easily seen from Y(ω) that the real axis is cut when ω = 1 and the imaginary axis is only reached as ω tends to infinity. However for G(s) = 1/(1 + s)6 then where it cuts the axes is best obtained using arg G(jω), which is simply equal to – 6 tan-1ω.
Figure 4.4 Nyquist plot of 1/s(s + 1)2.
Three further comments must be made here about the plot of Figure 4.4:-
1) For reasons to be explained later the graph is drawn for both positive and negative frequencies. The labelling of these has been added to the plot afterwards.
2) It can be shown for all transfer functions that X(ω) is an even function and Y(ω) an odd function of ω. Thus the negative frequency part of the plot is a reflection of the positive frequency plot in the real axis.
3) MATLAB does not label the frequencies automatically on the plot but they can be selected by use of the cursor as has been done to obtain one frequency point on this plot.
4) The frequency response plot instructions bode(G) and nyquist(G) in MATLAB
automatically select the frequency range. This can be done by the user by selecting a vector
ω, typically on a log scale using the instruction ω = logspace(a,b,n), which generates n
points on a log scale between 10a and 10b. If n is omitted the default is 50 points. The plot instructions are then bode(G, ω)and nyquist(G, ω).
The last instruction in (4) has been used with the ω vector generated by logspace for a = -0.5 and b = 1 to show a more detailed plot near the origin in Figure 4.5. From the two plots it can be clearly seen that at low frequencies, where the gain tends to infinity because of the single integration, the locus starts from the asymptote at X(ω) = -2 with a phase of -90°, crosses the real axis, that is has a phase of -180°, at X = -0.5 and tends to the origin (zero gain) at high frequencies with a phase of -270° (relative degree of 3 times -90°). The real axis crossing occurs at a frequency of unity.
Control Engineering Frequency Responses and Their Plotting
A final comment on Nyquist plots is that sometimes Inverse Nyquist plots are drawn, these are simply the Nyquist plot of the inverse of the transfer function, i.e a Nyquist plot of G(jω)-1.
4.4
Nichols Plot
The Nichols plot is similar to the Nyquist plot in that it is a locus as a function of ω, the difference being the chosen axes. On a Nichols plot these are the magnitude in dB on the ordinate and the phase in degrees on the abscissa. The origin is chosen, for reasons which will be explained, later as 0 dB and -180°. The Nichols plot for the same transfer function as the Nyquist plot of Figure 4.4 is obtained by the instruction nichols(G) and is shown in Figure 4.6. The grid is obtained by typing ngrid. As expected the plot shows the magnitude decreasing monotonically with increase in frequency, the arrow for which was added to the plot, and the phase changing from -90° to -270°.