Microcontroller based applied digital control (D. Ibrahim, 2006)
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ROOT LOCUS |
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Figure 8.4 Root locus with unit circle
At this point one part of the locus moves towards the zero at z = −0.717 and the other moves towards the zero at −∞.
Figure 8.4 shows the root locus with the unit circle drawn on the same axis. The system will become marginally stable when the locus is on the unit circle. The value of k at these points can be found either from Jury’s test or by using the Routh–Hurwitz criterion.
Using Jury’s test, the characteristic equation is
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0.368(z + 0.717) |
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or |
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z2 − z(1.368 − 0.368K ) + 0.368 + 0.263K = 0. |
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Applying Jury’s test |
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F (1) = 0.631 |
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Also, |
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|0.263K + 0.368| < 1
which gives K = 2.39 for marginal stability of the system.
NYQUIST CRITERION |
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Figure 8.8 Root locus with lines of constant damping factor and natural frequency
and shown in Figure 8.9. The roots at this point are given as s1,2 = 0.55 ± j 0.32. The value of k can now be calculated as
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which gives k = 0.377.
8.5 NYQUIST CRITERION
The Nyquist criterion is one of the widely used stability analysis techniques in the s-plane, based on the frequency response of the system. To determine the frequency response of a continuous system transfer function G(s), we replace s by j ω and use the transfer function G( j ω). In the s-plane, the Nyquist criterion is based on the plot of the magnitude |GH ( j ω)| against the angle GH ( j ω) as ω is varied.
In a similar manner, the frequency response of a transfer function G(z) in the z-plane can be obtained by making the substitution z = e j ωT . The Nyquist plot in the z-plane can then be obtained by plotting the magnitude of |GH (z)|z=e j ωT against the angle GH (z)|z=e j ωT as ω is varied. The criterion is then
Z = N + P,
202 System Stability
Figure 8.9 Point for ζ > 0.6 and ωn > 0.6
where N is the number of clockwise circles around the point −1, P the number of poles of GH(z) that are outside the unit circle, and Z the number of zeros of GH(z) that are outside the unit circle.
For a stable system, Z must be equal to zero, and hence the number of anticlockwise circles around the point –1 must be equal to the number of poles of GH(z).
If GH(z)has no poles outside the unit circle then the criterion becomes simple and for stability the Nyquist plot must not encircle the point −1.
An example is given below.
Example 8.11
The transfer function of a closed-loop sampled data system is given by
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G(z) |
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1 + GH(z) |
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where |
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GH(z) = |
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0.4 |
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Determine the stability of this system using the Nyquist criterion. Assume that T = 1 s.
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NYQUIST CRITERION |
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Solution |
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Setting z = e j ωT = cos ωT + j sin ωT = cos ω + j sin ω, |
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G(z)|z=e j ωT = |
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0.4 |
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0.2) |
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or
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G(z)|z=e j ωT = (cos2 ω − sin2 ω − 0.7 cos ω + 0.1) + j (2 sin ω cos ω − 0.7 sin ω) .
This has magnitude
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|G(z)| =
(cos2 ω − sin2 ω − 0.7 cos ω + 0.1)2 + (2 sin ω cos ω − 0.7 sin ω)2
and phase
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2 sin ω cos ω − 0.7 sin ω |
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cos2 ω − sin2 ω − 0.7 cos ω + 0.1 |
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Table 8.2 lists the variation of the magnitude of G(z) with the phase angle. The Nyquist plot for this example is shown in Figure 8.10. Since N = 0 and P = 0, the closed-loop system has no poles outside the unit circle in the x -plane and the system is stable.
The Nyquist diagram can also be plotted by transforming the system into the w-plane and then using the standard s-plane Nyquist criterion. An example is given below.
Example 8.12
The open-loop transfer function of a unity feedback sampled data system is given by
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Derive expressions for the magnitude and the phase of |G(z)| by transforming the system into the w-plane.
Solution
The w transformation is defined as
z = 1 + w
1 − w
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G(w) |
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0.4) |
2w(0.6 |
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or
1 + w
G(w) = 1.2w + 2.8w2 .
