figure 10.35 nyquist diagram showing gain and...
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Figure 10.35Nyquist diagram showing gain and phase margins
1. Gain Margin, GM, and Phase Margin, PΜ, indicate the Relative Stability of the closed-loop system.
2. We assume that the system is a Non-minimum Phase system (no GH zeros in the RHP).3. If all the poles of GH are in the LHP, then we can just plot the positive jω axis (Part I) to determine stability using the GM and PM; otherwise, stability needs to be determined first using the Nyquist criterion, Z = P - N.
4.
5. If we multiply GH by GM, the Nyquist plots shifts to where it crosses –1 on the real axis and the system becomes marginally stable. That is, as the GM approaches 1, the system becomes more oscillatory. The GM is less than 1 and positive for stability, i.e., |GH| real-axis crossing is less than 1 for stability.
6. For stability, the PM must be positive. As the PM approaches 0 degrees, the system becomes more oscillatory.
( )
1 oGM where the Angle of GH = ± 180GH
1for Bode plots, GM=20log =-20log GH
GH
oPM +180 + Angle of GH where GH = 1
|GH| = 1/aa = 1/ |GH| =GMArg(GH)= ± 180o
in dBGM
Ogata, Modern Control Engineering, 3rd
Edition
-1
CONDITIONALLY STABLE
MAY BECOME UNSTABLE WITH A SLIGHT GAIN CHANGE
α= PMHighest
Frequency
-
+180 in text
PM =
Figure 10.37Gain and phasemargins on the Bodediagrams
IT IS MUCH EASIER TO FIND THE GM AND PM FROM BODE PLOTS.
THE GM IS FOUND BY FINDING THE MAGNITUDE OF THE COMPOSITE MAGNITUDE WHERE THE COMPOSIT PHASE = -180 DEG.
THE PM IS FOUND BY FINDING THE PHASE OF THE COMPOSITE PHASE WHERE THE COMPOSITE MAGNITUDE = 0dB AND ADDING +180 DEG AS SHOWN ON THE GRAPH AT RIGHT.
GAIN MARGIN & PHASE MARGIN BODE PLOT EXAMPLE % KGH(s)=10/[s(s+1)(0.5s+1)] KGHnum=[10] KGHden=conv([1 1 0],[0.5 1]) % (s^2+s)*(0.5s+1) Disp(‘KGH = ‘) KGH=tf(KGHnum,KGHden) bode(KGH); grid KGH = 10 --------------------- 0.5 s^3 + 1.5 s^2 + s
THE CLOSED-LOOP SYSTEM IS UNSTABLE GAIN & PHASE MARGINS ARE NEGATIVE (Note: Only one of them needs to be negative for the closed-loop system to be unstable.) CLOSED-LOOP POLES ARE: -3.8371 0.4186 + 2.2443i 0.4186 - 2.2443i
Figure 10.36Bodelog-magnitudeand phase diagramsfor the system of Example 10.9
Bode phase plot for G(s) = 40/[(s +2)(s +4)(s +5)]:a. components;b. composite
GM = 20 dB
PM = 180 deg
Figure 10.55Effect of 1 sec delay
1. DELAY ONLY EFFECTS THE PHASE PLOT
2. A T SECOND DELAY IS REPRESENTED BY e-TS
Delay180 θ = −ω π
GH Phase without delay
Composite PhaseGH Phase + θDelay
∠
180-Tjω-Tse = e = 1 - Tω,θ = -Tω degDelay πs=jω
1-Ts -1sGH(s)e = e , T=1 second delays(s+1)(s+2)
-Tjω -1jω1GH(jω)e = ejω(jω+1)(jω+2)
-TjωGH(jω)e =Mag(GH(jω)) with an
-TjωAngle(GH(jω)e ) =
Angle of GH(jω) [ ]
180deg -ωπ
Figure 10.56 Step response forclosed-loop system with
G(s) = 5/[s(s +1)(s + 10)]:
a. with a 1 second delay;
b. without delay
Figure 10.39Representative log-magnitudeplot of Eq. (10.51)
-
≈
2 4 2BW n
Given a closed - loop system :2ωC(s) nG (s) =CL 2 2R(s) s +2ζω +ωn n
Bandwidth is defined as the frequency at which the magnitude of a closed loop system is - 3 dB.
ω =ω (1 - 2ζ ) + 4ζ - 4ζ + 2
dMPeak M, M , when = 0 yields :p dωn
1M =p 22ζ 1-ζ
2ω =ω 1-2ζp n
20log G (jω)CL
Closed-loop System is assumed to approximate a 2nd order system.
Figure 10.41Normalized bandwidthvs. damping ratio for:
a. settling time;
b. peak time;
c. rise time
Figure 10.48Phase margin vs.damping ratio
Closed-loop System is assumed to approximate a 2nd order system.
-1M
2 4
Phase Margin of GH(s)
2ζPM =Φ = tan-2ζ + 1 + 4ζ