Nyquist

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Nyquist stability criterion
Valeri Ougrinovski August 9, 2006

Abstract This note gives a brief introduction to the closed loop system stability analysis based on the frequency response of the system open loop gain.

1

Nyquist polar plots

A polar plot of the frequency response produces a curve with frequency as a parameter relating the points on the polar plot to the appropriate pointson the Bode plots, see Fig. 1. The Nyquist plot contains the same information as the Bode plot. Thus, one can use approximate Bode plots to reconstruct the shape of the Nyquist plot. For instance, the Bode and Nyquist plots of 5(s + 1) G(s) = 2 s + 2s + 10) are shown below in Figure 1. Arrows on the Nyquist plot indicate the direction in which the plot changes as the frequency increases from −∞to +∞ (or equivalently, jω sweeps from the −jω to +jω along the imaginary axis). Note the two symmetric branches of the Nyquist plot corresponding to positive and negative frequencies. Unlike Bode plots, both positive and negative frequencies are used for plotting Nyquist curves; the reasons for this will become clear later. The Matlab command to produce accurate Nyquist plots is as follows: >>nyquist(G,w); The second argument is optional; it allows you to specify an array of frequencies of interest to capture some details of the plot. For example, to produce the Nyquist plot in Fig 1, you need to type >> G=tf(5*[1 1],[1 2 10]); >> nyquist(G); The command nyquist can also be used for computing the frequency response of a system loop gain for positive frequencies only. No plot is produced,hence the command plot must be used: >> G=tf(5*[1 1],[1 2 10]); >> [reG,imG]=nyquist(G,{0.01,1000}); >> plot(squeeze(reG),squeeze(imG))

1

Bode Diagram 10 System: G Frequency (rad/sec): 2.83 Magnitude (dB): 7.96

Magnitude (dB) Phase (deg)

0

−10

−20

−30 45

0

System: G Frequency (rad/sec): 2.83 Phase (deg): −1.66e−14

−45

−90

10

0

10 Frequency (rad/sec)Nyquist Diagram

1

10

2

1.5

1

0.5 Imaginary Axis

0

System: G Real: 5e−06 Imag: 0.005 Freq (rad/sec): −1e+03 System: G Real: 5e−06 Imag: −0.005 Freq (rad/sec): 1e+03

System: G Real: 0.5 Imag: −0 Freq (rad/sec): −0

System: G Real: 2.5 Imag: 0.00186 Freq (rad/sec): −2.83

−0.5

−1

−1.5 −1 −0.5 0 0.5 1 Real Axis 1.5 2 2.5

Figure 1: Bode and Nyquist plots of the loopgain transfer function G(s) =

5(s+1) s2 +2s+10

2

The graphical method can be used to obtain an approximate Nyquist plot. Consider an example. Let 100(−s + 10) G(s) = . (s + 0.1)2 (s2 + 0.1s + 100) This transfer function has the single nonminumum phase zero at s = 10, the double pole at s = −0.1 (real), and the pair of complex conjugate poles at s ≈ −0.05 ± 10j. First let us sketch themagnitude Bode plot. dB 60

40

20 PSfrag replacements 0 0.1 -20 1 10 100 ω

-40

-60 -80

Note that zero at −10 and the pair of complex conjugate poles at −0.05 ± 10j yield the same corner frequency ω = 10 rad/sec. Hence the net change of slope at this corner frequency is +20 − 40 = −20 dB/decade. Also, note that the poles −0.05 ± 10j are very underdamped, hence it is appropriate to drawa peak at the corresponding corner frequency. We now proceed to phase calculations. Write the transfer function in a “standard form”: G(s) = −100(s − 10) . (s + 0.1)2 (s2 + 0.1s + 100)

Note the gain factor of −100. The “−” sign means that the +180 ◦ or −180◦ term should be taken into account, let’s use −180◦ . The phase of G(jω) can be expressed as ∠G(jω) = −180◦ + θ1 − 2θ2 − θ3 − θ4 ; theangles are shown in the Figure 2 below. Note the factor of 2 in front of θ 2 , this factor is to reflect the contribution of the double pole at s = −0.1.

3

θ3 PSfrag replacements jω θ1 θ2

θ4

Figure 2: ω, rad/sec 0 0.1 1 9.5 9.95 10 10.05 10.5 100 ∞ θ1 180◦ 180◦ 180◦ 135◦ 135◦ 135◦ 135◦ 135◦ 90◦ 90◦ θ2 0◦ 45◦ 90◦ 90◦ 90◦ 90◦ 90◦ 90◦ 90◦ 90◦ θ3 -90◦ -90◦ -90◦ -90◦ -45◦ 0◦ 45◦ 90◦ 90◦ 90◦...
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