Module 14
Nyquist Analysis and
Relative Stability
(1 hours)
Gain Margin
Phase Margin
14.1 Conditional Stability (条件稳定性）
)13)(12()( sss
KsF
1)3(1)2( 22

KM
3290 11 tgtgGH
Example 1 （ P272)
1 8 0?w h e n
KM 5661
)83.0(156 KKw h e n
The system is stable.
1,Write the open-loop transfer function in Bode form,and
write down expressions for the magnitude and phase.
2,Sketch the Nyquist diagram for an arbitrary value of gain to
determine whether the critical point passes to the left or right of
an observer moving along the frequency response curve in the
direction of increasing frequency.
3,Using computer method,if necessary,determine the frequency
that makes the corresponding angle -180o.
4,Substitute this value of frequency into the magnitude equation and
determine the corresponding magnitude.
Conclusion ( P274,1,~ 5,)
5,If the magnitude is less than unity and the critical point
passes to the left,the system is stable; otherwise it is not.
14.2 Gain and Phase Margins
Ex,1 Ex,2
Great marginSmallmargin
No Margin
ψ3
Great margin
)(1 8 0 ujGHPM
Gain margin
The increase in the system gain when phase
= — 180othat will result in a marginally
stable system with intersection of the -1+ j0
point on the Nyquist diagram.
Phase margin
The amount of phase shift of the GH(jω) at
unity magnitude that will result in a
marginally stable system with intersections
of the - 1 + j0 point on the Nyquist diagram.
Other Example
)1)(1()( 21 sTsTs
KsGH
)1)(1()( 21 sTsTs
KsGH
)()1( )1()( 21
2
2
1 TT
sTs
sTKsGH?

)1()( 2 Tss
KsGH
3)( s
KsGH? 3 21
)1)(1()(
s
sTsTKsGH
)1)(1)(1)(1(
)1)(1()(
4321
65


sTsTsTsTs
sTsTKsGH
)1(1)(
1
KsT KsGH
)1(1)(
1
KsT KsGH
)1()1()( KTss KsGH )1(1)(
1
KsT KsGH
Instructional objectives:
At the end of this lecture students
should be able to
Determine the stability of a feedback
control system by calculating the gain
margin and phase margin of system