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