Stability via the Nyquist diagram - University of Readingsis01xh/teaching/CY2A9/Lecture8.pdf ·...
Transcript of Stability via the Nyquist diagram - University of Readingsis01xh/teaching/CY2A9/Lecture8.pdf ·...
Stability via the Nyquist diagram
Range of gain for stability
Problem: For the unity feedback system be-
low, where
G(s) =K
s(s + 3)(s + 5),
find the range of gain, K, for stability, insta-
bility and the value of K for marginal stability.
For marginal stability, also find the frequency
of oscillation. Use the Nyquist criterion.
Figure above; Closed-loop unity feedback sys-
tem.
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Solution:
G(jω) =K
s(s + 3)(s + 5)|s→jω
=−8Kω − j · K(15 − ω2)
64ω3 + ω(15 − ω2)2
When K = 1,
G(jω) =−8ω − j · (15 − ω2)
64ω3 + ω(15 − ω2)2
Important points:
Starting point: ω = 0, G(jω) = −0.0356 − j∞
Ending point: ω = ∞, G(jω) = 06 − 270◦
Real axis crossing: found by setting the imag-
inary part of G(jω) as zero,
ω =√
15, {− K
120, j0}
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When K = 1, P = 0, from the Nyquist plot,
N is zero, so the system is stable. The real
axis crossing − K120 does not encircle [−1, j0)]
until K = 120. At that point, the system is
marginally stable, and the frequency of oscilla-
tion is ω =√
15 rad/s.
Nyquist Diagram
Real Axis
Imag
inar
y A
xis
−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1−0.05
−0.04
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
0.05
System: GReal: −0.00824Imag: 1e−005Frequency (rad/sec): −3.91
−3 −2.5 −2 −1.5 −1 −0.5 0−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Nyquist Diagram
Real Axis
Imag
inar
y A
xis
(a) (b)
Figure above; Nyquist plots of G(s) = Ks(s+3)(s+5)
;
(a) K = 1; (b)K = 120.
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Gain/phase margin via the Nyquist diagram
We use the Nyquist diagram to define two
quantitative measures of how stable a system
is. These are called gain margin and phase
margin. Systems with greater gain margin and
phase margins can withstand greater changes
in system parameters before becoming unsta-
ble.
Gain margin, GM , The gain margin is the change
in open-loop gain, expressed in decibels (dB),
required at 180◦ of phase shift to make the
closed-loop system unstable.
Phase margin, ΦM , The phase margin is the
change in open-loop phase shift, required at
unity gain to make the closed-loop system un-
stable.
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Figure above; Nyquist diagram showing gain
and phase margins
Problem: Find the gain and phase margin for
the unity feedback system with
G(s) =6
(s2 + 2s + 2)(s + 2).
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Solution: From G(s), we see P = 0. TheNyquist diagram shows that N is zero, so theclosed-loop system is stable.
G(jω) =6
(s2 + 2s + 2)(s + 2)|s→jω
=6[4(1 − ω2) − jω(6 − ω2)]
16(1 − ω2)2 + ω2(6 − ω2)2
The Nyquist diagram crosses the real axis at afrequency of ω =
√6. The real part is found
to be −0.3.
−1 −0.5 0 0.5 1 1.5−1.5
−1
−0.5
0
0.5
1
1.5
Nyquist Diagram
Real Axis
Imag
inar
y A
xis
Figure above; Nyquist diagram for
G(s) =6
(s2 + 2s + 2)(s + 2).
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The gain margin is
GM = 20 log(1/0.3) = 10.45dB.
To find the phase margin find the frequency
for which G(jω) has a unit gain. Using a com-
puting tool, we can find G(jω) has a unit gain
at a frequency of 1.251, at this frequency the
phase angle is −112.3◦. The difference of this
angle with −180◦ is 67.7◦, which is the phase
margin ΦM .
Gain/phase margin via the Bode plots
Figure above; Gain and phase margins on the
Bode plots.
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Problem: Let a unit feedback system have
G(s) =K
(s + 2)(s + 4)(s + 5).
Use Bode plots to determine the range of gain
within which the system is stable. If K = 200
find the gain margin and the phase margin.
The low frequency gain is found by setting s to
zero. Thus the Bode magnitude plots starts
at K/40. For convenience set K = 40, so
that the log-magnitude plots starts at 0dB. At
each break frequency, 2, 4 and 5, a slope of
-20dB/decade is added.
The phase diagram starts at 0◦ until 0.2rad/s
(a decade below the break frequency of 2),
the curve decreases at a slope of 45◦/decadeat each subsequent frequency at 0.4rad/s and
0.5rad/s (a decade below the break frequency
of 4 and 5 respectively). Finally at 20rad/s,
40rad/s and 50rad/s (a decade above the break
frequencies of 2,4,5), a slope of +20dB/decade
is added, until the curve levels out.
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Figure above; Bode log-magnitude and phase
diagram for G(s) = 40(s+2)(s+4)(s+5)
.
The Nyquist criterion tells us that we want zero
encirclement of {−1, j0}. Thus the Bode log-
magnitude plot must be less than unity when
the Bode phase plot is −180◦. Accordingly we
see that at frequency 7 rad/s, when the phase
plot is −180◦. The magnitude is -20dB.
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Thus an increase of 20dB is possible before the
system becomes unstable, which is a gain of
10, so the gain for instability is K > 10× 40 =
400.
If K = 200 (five times greater than K = 40),
the magnitude plot would be 20 log5 = 13.98dB
higher, as the Bodes plots was scaled to a gain
of 40.
At −180◦, the gain is −20+13.98 = −6.02dB,
so GM = 6.02dB.
To find phase margin, we look on the mag-
nitude plot for the frequency where the gain
is 0dB. As the plot should be 13.98dB higher,
so we look at −13.98dB crossing to find the
frequency is 5.5rad/s. At this frequency, the
phase angle is −165◦. Thus
ΦM = −165◦ − (−180)◦ = 15◦.
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