Chapter 8
Frequency-Domain
Analysis
Automatic Control Systems, 9th Edition
F. Golnaraghi & B. C. Kuo
Section 8- 1, p. 409
8-1 Introduction
• For a LTI system: input - steady-state output -• Sinusoidal steady-state analysis:
s = j
•
M(s)
8-1
Section 8- 1, p. 410
Frequency Response
• Closed-loop transfer function:
• Sinusoidal steady-state transfer function:
s = j
Magnitude of M(j):
Phase of M(j):
8-2
Section 8- 1, p. 411
Gain-Phase Characteristics
cut-off frequency
8-3
Section 8- 1, p. 412
Frequency-Domain Specifications
• Resonant Peak Mr:
the maximum value of M ( j)
 the relative stability of a
stable closed-loop system
• Resonant Frequency r:
the frequency at which the
peak resonant Mr occurs
• Bandwidth BW:
the frequency at which M ( j)
drops to 70.7% of, or 3 db down from, its zero-freq. value
 the transient response properties of a control system
• Cutoff Rate:
the slope of M ( j) at high frequency
8-4
Section 8- 2, p. 413
8-2 Mr, r, and Bandwidth of the
Prototype Second-Order System
• The prototype second-order system:
• u = /n:
magnitude:
phase:
8-5
Section 8- 2, p. 414
Resonant Frequency and Resonant Peak
• Resonant frequency r:
ur  0  M ( jur ) is not a truemaximun.
M ( jur )
 Resonant frequency:
• Resonant peak M ( jur ) :
r = 0 if   0.707
Mr = 1 if   0.707
8-6
Section 8- 2, p. 415
Magnification vs Normalized Frequency
u = /n
 = 0  r = n
8-7
Section 8- 2, p. 416
Mr, ur vs Damping Ratio
8-8
Section 8- 2, p. 417
Bandwidth
• Definition:
70.7% or 3 dB
8-9
Section 8- 2, p. 417
Time-Domain Response vs
Frequency-Domain Characteristics
1.
2.
3.
4.
Mr depends on  only.
The maximum overshoot also depends only on .
For   0.707, Mr = 1 and r = 0.
The maximum overshoot is 0 when   1.0.
BW is directly proportional to n and
inversely proportional to .
The rise time tr increases as n decreases.
BW and tr are inversely proportional to each other.
BW and Mr are proportional to each other for
0    0.707.
8-10
Section 8- 2, p. 419
Correlation between Pole Locations
8-11
Section 8- 2, p. 419
Unit-Step and Frequency Responses
8-12
Section 8- 3, p. 418
8-3 Effects of Adding a Zero to the
Forward-Path Transfer Function
•
zero: s = 1/T
closed-loop transfer function:
• Bandwidth:
increase BW
8-13
Section 8- 3, p. 421
Magnification Curves
8-14
Section 8- 3, p. 423
Unit-Step Responses: adding a zero
8-15
Section 8- 4, p. 424
8-4 Effects of Adding a Pole to the
Forward-Path Transfer Function
pole: s = 1/T
Effects:
Decrease BW,
Increase Mr,
Make the closed-loop
system less stable
8-16
Section 8- 4, p. 425
Unit-Step Responses: adding a poles
T   tr   BW 
Mr   yrmax 
8-17
Section 8- 5, p. 426
8-5 Nyquist Stability Criterion: Fundaments
• The Nyquist plot of the loop transfer function G(s)H(s), or
L(s), is done in polar coordinates as  varies from 0 to .
• The Nyquist criterion has the following features:
– In addition to providing the absolute stability, the Nyquist
criterion also gives information on the relative stability of a
stable system and the degree of instability of an unstable
system.
– The Nyquist plot is very easy to obtain.
– The Nyquist plot gives information on the frequencydomain characteristics such as Mr, r, and BW.
– The Nyquist plot is useful for systems with pure time delay.
8-18
Section 8- 5, p. 427
Stability Problem
• Closed-loop transfer function:
• Characteristic equation:
• Stability conditions:
– Open-loop stability: if the poles of the loop transfer
function L(s) are all in the left-half s-plane.
– Closed-loop stability: if the poles of the closed-loop
transfer function or the zeros of 1+L(s) are all in the
left-half s-plane.
8-19
Section 8- 5, p. 428
Encircled
• A point or region in a complex function plane is said to be
encircled by a closed path if it is found inside the path
Point A is encircled by 
in the counterclockwise
(CCW) direction.
Point B is not encircled
by the closed path .
8-20
Section 8- 5, p. 428
Enclosed
• A point or region is said to be enclosed by a closed path if
it is encircled in the CCW direction or the point or region
lies to the left of the path when the path is traversed in the
prescribed direction.
8-21
Section 8- 5, p. 429
Number of Encirclements and Enclosures
• When a point is encircled by a closed path ,
N = the number of times it is encircled.
• N is positive for CCW encirclement and negative for CW
B is encircled twice in
encirclement.
A is encircled once
or 2 radians by 
in the CW direction
B is encircled twice
or 4 radians by 
in the CW direction
the CCW direction
A is encircled once in
the CCW direction
8-22
Section 8- 5, p. 429
Locus in the (s)-plane
• Suppose that a continuous closed path s is arbitrarily
chosen in the s-plane, as shown in Fig. 8-17(a).
• If s does not go through any poles of (s), then the
trajectory  mapped by (s) into (s)-plane is also a
closed one, as shown in Fig. 8-17(b).
single-valued mapping
(s): single-valued function
8-23
Section 8- 5, p. 430
Principles of the Argument
• Let (s) be a single-valued function that has a finite
number of poles in the s-plane.
• Suppose that an arbitrarily closed path s is chosen in the
s-plane so that the path does NOT go through any one of
the poles or zeros of (s).
• The corresponding  locus mapped in the (s)-plane will
encircle the origin as many times as the difference
between the number of zeros and poles of (s) that are
encircled by the s-plane locus s.
N=ZP
N = number of encirclements of the origin made by the locus .
Z = number of zeros of (s) encircled by the s-plane locus s.
P = number of poles of (s) encircled by the s-plane locus s.
8-24
Section 8- 5, p. 431
Examples of Determination of N
1. N > 0 (Z>P): the (s)plane locus  will
encircle the origin N
times in the same
direction as that of s.
2. N = 0 (Z=P): the (s)plane locus  will
not encircle the origin
of the (s)-plane.
3. N < 0 (Z<P): the (s)plane locus  will
encircle the origin N
times in the opposite
direction as that of s.
8-25
Section 8- 5, p. 432
Illustrative Example
The net angle traversed
by the (s)-plane:
Z  1, P  0
8-26
Section 8- 5, p. 433
Table 8-1
8-27
Section 8- 5, p. 434
Nyquist Path
• The Nyquist path is defined
to encircled the entire righthalf s-plane.
• The Nyquist path must not
pass through any poles and
zeros of (s).
8-28
Section 8- 5, p. 434
Nyquist Criterion (1/2)
• (s) = 1 + L(s), L(s): loop transfer function
 the origin of the (s)-plane corresponds to
the (1, j0) point in the L(s)-plane.
• Steps of the application of Nyquist criterion to the
stability problem:
1. The Nyquist path s is defined in the s-plane, as shown in Fig. 8-20.
2. The L(s) plot corresponding to the Nyquist path is constructed in
the L(s)-plane.
3. The value of N, the number of encirclement of the (1, j0) point
made by L(s) plot, is obsered.
4. The Nyquist criterion follows from Eq. (8-42),
8-29
Section 8- 5, p. 435
Nyquist Criterion (2/2)
• Stability requirements:
For closed-loop stability, Z must equal zero.
For open-loop stability, P must equal zero.
• The condition of stability according to Nyquist Criterion:
for a closed-loop system to be stable, the L(s) plot must
encircle the (1, j0) point as many times as the number of
poles of L(s) that are in the right-half s-plane, and the
encirclement, if any, must be made in the clockwise
dircetion (if s is defined in the CCW sense).
8-30
Section 8- 6, p. 435
8-6 Nyquist Criterion for Systems with
Minimum-Phase Transfer Functions
Minimum-phase transfer function:
• A minimum-phase transfer function does not have poles or
zeros in the right-half s-plane or on the j-axis, excluding
the origin.
• For a minimum-phase transfer function L(s) with m zeros
and n poles, excluding the poles at s = 0, when s = j and
as  varies from  to 0, the total phase variation of L(j)
is (nm)/2 radians.
• The value of a minimum-phase transfer cannot become
zero or infinity at any nonzero finite frequency.
• A nonminimum-phase transfer function will always have a
more positive phase shift as  varies from  to 0. Or
equally true, it will always have a more negative phase as
 varies from 0 to .
8-31
Section 8- 6, p. 436
Nyquist Criterion for Systems with
Minimum-Phase Transfer Functions
• L(s): minimum-phase type  P = 0
Nyquist criterion  N = 0
• For a closed-loop system with loop transfer function L(s)
that is of minimum-phase type,
– the system is closed-loop stable if the plot of L(s) that
corresponds to the Nyquist path does NOT encircle (or
enclose) the critical point (1, j0) in the L(s) -plane.
– If the (1, j0) point is enclosed by the Nyquist plot, the
system is unstable.
8-32
Section 8- 6, p. 437
Not Strictly Proper Transfer Function
• The characteristic equation of a system:
K: a variable parameter
Leq: the equivalent transfer function
• If Leq does not have more poles than zeros,
– Plot the Nyquist-plot of 1/Leq(s)
the critical point is still (1, j0) for K > 0
the variable parameter on the Nyquist plot is now 1/K.
 the Nyquist criterion can still be applied
8-33
Section 8- 7, p. 437
8-7 Relation between the Root Loci
and the Nyquist Plot
• Both the root locus and the Nyquist criterion deal with the
location of the roots of the characteristic equation of a
linear SISO system.
• Characteristic equation:
– The Nyquist plot of L(s) in the L(s)-plane is the
mapping of the Nyquist path in the s-plane.
– The root locus must satisfy
j  0,  1,  2, ...
The root loci simply represent a mapping of the real
axis of L(s)-plane or the G(s)H(s)-plane onto the splane.
8-34
Section 8- 7, p. 438
Mapping s-plane onto G(s)H(s)-plane
8-35
Section 8- 7, p. 438
Mapping G(s)H(s)-plane onto s-plane
8-36
Section 8- 7, p. 439
Relation between G(s)H(s)- and s-planes
8-37
Section 8- 7, p. 439
Relation between G(s)H(s)- and s-planes
8-38
Section 8- 8, p. 440
8-8 Illustrative Examples: Nyquist
Criterion for Min.-Phase Transfer Func.
• Example 8-8-1:
 minimum-phase type
Sketch of the Nyquist plot of L(j)/K
1. Substitute s = j in L(s):
 270
2. Get the zero-frequency
( = 0) property:
3. Get the infinite-frequency
( = ) property:
 90
8-39
Section 8- 8, p. 441
Example 8-8-1 (cont.)
4. Find the possible intersects on the real axis:
Set the imaginary part of L(j)/K to zero:
( must be positive)
The intersect on the real axis of the L(j)-plane at
8-40
Section 8- 8, p. 441
Example 8-8-1 (cont.)
The intersect on the real axis of the L(j)-plane at

1
240
• K<240: the intersect would
be to the right of (1, j0).
The critical point is not
enclosed  stable.
• K=240: the intersect is at
the 1 point.
 marginally stable.
• K>240: the intersect would be to the left of (1, j0).
 unstable.
• K<0: the critical point (+1, j0) is enclosed  unstable.
8-41
Section 8- 8, p. 442
Example 8-8-1 (cont.)
Stable: 0 < K < 240
8-42
Section 8- 8, p. 442
Example 8-8-2
Characteristic equation:
Sketch the Nyquist plot of Leq(s):
1.
2. Two end points:
3. The possible intersects on the real axis:
 = 0 and
Four imaginary roots
8-43
Section 8- 8, p. 443
Example 8-8-2 (cont.)
Stable: K > 0
8-44
Section 8- 8, p. 444
Example 8-8-2 (cont.)
Stable: K > 0
8-45
Section 8- 9, p. 444
8-9 Effects of Adding Poles and Zeros to
L(s) on the Shape of the Nyquist Plot
• Addition of Poles at s = 0:
• Properties of Nyquist plot:
multiplicity = p:
8-46
Section 8- 9, p. 445
Nyquist Plots
Add a pole
at s = 0 to L(s)
8-47
Section 8- 9, p. 446
Example 8-9-1
8-48
Section 8- 9, p. 447
Example 8-9-1 (cont.)
8-49
Section 8- 9, p. 447
Addition of Finite Nonzero Poles
s = 1/T2 (T2 > 0)
• Properties of Nyquist plot:
Adding nonzero poles to
the loop transfer function
reduces stability of the
closed-loop system.
8-50
Section 8- 9, p. 448
Addition of Zeros (Example 8-9-2)
Stable:
Stable:
Stabilize the closed-loop system
8-51
Section 8- 10, p. 449
8-10 Relative Stability: Gain Margin
and Phase Margin
• Relative stability is used to indicate how stable a system is.
– Time domain: the maximum overshoot or the damping
ratio
– Frequency domain: the resonant peak or how close the
Nyquist plot of L(j) is to the (1, j0) point.
8-52
Section 8- 10, p. 450
Correlation among Nyquist Plots, Step
Responses, and Frequency Responses
The loop gain K is low
K is increased
8-53
Section 8- 10, p. 450
Correlation among Nyquist Plots, Step
Responses, and Frequency Responses
K is increased further
K is relatively very large
8-54
Section 8- 10, p. 451
Phase Crossover and Gain Margin
For minimum-phase loop transfer functions:
• Phase Crossover: a point at which
the Nyquist plot intersects the
negative real axis.
• Phase-Crossover Frequency:
the frequency at the phase
crossover (p) or
• Gain Margin:
8-55
Section 8- 10, p. 452
Physical Significance of Gain Margin
• Gain margin is the amount of gain in decibels (dB) that
can be added to the loop before the closed-loop system
becomes unstable.
• L(j) plot does not intersect the negative real axis:
The loop gain can be increased
to infinity before instability
• L(j) plot intersects the real axis between 0 and 1:
• L(j) plot passes through the (1, j0) point:
The loop gain can no longer be
increased (margin of instability)
• L(j) plot encloses the (1, j0) point:
The loop gain must be reduced
by gain margin to achieve stability
8-56
Section 8- 10, p. 453
Discussion of Gain Margin
• Gain margin indicates system
stability with respect to the
variation in loop gain only.
• A system with a large gain
margin should always be
relatively more stable than one
with smaller gain margin.
• Fig. 8-38: Systems A and B
have the same gain margin,
but A is more stable than B.
 System B has a larger
Mr than system A.
8-57
Section 8- 10, p. 453
Phase Margin
• Gain Crossover: a point
at which L(j) = 1.
• Gain-Crossover Frequency:
the frequency at the gain
crossover (g) or
• Gain Margin:
8-58
Section 8- 10, p. 453
Definition & Significance of Phase Margin
• Definition:
Phase margin is defined as the angle in degree through
which the L(j) plot must be rotated about the origin so
that the gain crossover pass through the (1, j0) point.
• Significance:
Phase margin is the amount of pure phase delay that can
be added before the system becomes unstable.
8-59
Section 8- 10, p. 454
Example 8-10-1
• Loop transfer function:
• Gain crossover g
= 6.22 rad/sec
Phase crossover p
= 15.88 rad/sec
 The system is stable.
8-60
Section 8- 11, p. 455
8-11 Stability Analysis with Bode Plot
Advantages:
• Bode diagram can be sketched by approximating the
magnitude and phase with straightline segment.
• Gain crossover, phase crossover, gain margin, and phase
margin are more easily determined on Bode plot than from
Nyquist plot.
• The effects of adding controllers and their parameters are
more easily visualized on Bode plot than on Nyquist plot.
Disadvantage:
• Absolute and relative stability of only minimum-phase
systems can be determined from Bode plot.
8-61
Section 8- 11, p. 456
8-62
Section 8- 11, p. 457
Example 8-11-1
• Loop transfer
function:
• Gain crossover g
= 6.22 rad/sec
Phase crossover p
= 15.88 rad/sec
• GM = 14.82 dB
PM = 31.72°
 The system is stable.
8-63
Section 8- 11, p. 458
Bode Plots of Systems with
Pure Time Delays
Example 8-11-2:
Fig. 8-43: K = 1, Td = 0,1
• Gain crossover g
= 0.446 rad/sec
Phase crossover p
= 1.416 rad/sec
• GM = 15.57 dB, PM = 53.4°
 The system is stable.
• Negative phase shift caused
by time delay = Td
 critical value Td=2.09 sec.
0.66
1.416
without time delay
p = 0.66 rad/sec
GM = 4.5 dB
with time delay
phase shift
= Td
8-64
Section 8- 12, p. 459
8-12 Relative Stability Related to the
Slope of Magnitude Curve of Bode Plot
• For a minimum-phase transfer function, the relation
between its magnitude and phase is unique.
• If the loop gain is increased, the gain-crossover frequency
g is increased, and the slope of the magnitude curve is
more negative.  a smaller phase margin
 the system is less stable
• The negative slope of the magnitude curve is a result of
having more poles than zeros in the transfer function.
 the corresponding phase is also negative.
• The steeper the slope of the magnitude curve,
the more negative the phase.
8-65
Section 8- 12, p. 459
Conditionally Stable System (Ex. 8-12-1)
corner freq.
K = 1:
• g = 1 rad/sec
 PM =  78°
• p = 25.8 & 77.7 rad/sec
 69 & 85.5 dB
• g falls in Slope = 60 dB
 PM < 0  unstable
• g falls in Slope = 20 dB
 stable
• g falls in Slope = 40 dB
 stable if PM > 0
8-66
Section 8- 12, p. 461
Example 8-12-1 (cont.)
8-67
Section 8- 13, p. 462
8-13 Stability Analysis with the
Magnitude-Phase Plot
• The critical point is the intersect
of the 0-dB-axis and 180°-axis.
• The phase crossover is where the
locus intersects the 180°-axis.
• The gain crossover is where the
locus intersects the 0-dB-axis.
• The gain margin is the vertical
distance in dB measured from the
phase crossover to the critical
point.
• The phase margin is the
horizontal distance in degree
measured from the gain
crossover to the critical point.
8-68
Section 8- 14, p. 463
8-14 Constant-M Loci in
Magnitude-Phase Plane: Nichols Chart
The closed-loop transfer function:
• sinusoidal steady state:
• magnitude:
M ( j )  M
a circle
center: x  Re G( j)  M 2 (1  M 2 ) ,
radius: r  M (1  M 2 )
y0
8-69
Section 8- 14, p. 464
Constant-M Clrcles
M>1
0<M<1
8-70
Section 8- 14, p. 465
Nquist Plot + Constant-M Circles
• Resonant peak (Mr): the smallest
circle that is tangent to G(j)
curve
• Resonant frequence (r): the
point of tangency
• When the system is unstable, the
constant-M circle and Mr no
longer have any meaning.
marginally
stable
stable
8-71
Section 8- 14, p. 466
Nichols Chart
• Nichols Chart: the constant-M loci are plotted in
magnitude-phase coordinates.
8-72
Section 8- 14, p. 467
Example 8-14-1
K = 7.248, 14.5, 181.2,
and 273.57.
• Gain crossover,
Phase crossover,
GM, and PM
8-73
Section 8- 14, p. 468
Example 8-14-1
(cont.)
• Resonant Peak,
Resonant frequency,
Bandwidth,
GM, and PM
8-74
Section 8- 14, p. 469
Example 8-14-1 (cont.)
8-75
Section 8- 15, p. 469
8-15 Nichols Chart Applied to
Nonunity-Feedback Systems
• Closed-loop transfer function of the system with nonunity
feedback (H(s)  1):
 the numerator of M(s) does not contain H(j)
• Modification:
8-76
Section 8- 16, p. 470
8-16 Sensitivity Studies in Freq. Domain
• A linear control system with unity feedback:
• The sensitivity of M(s) with respect to the loop gain K:
dM ( s) dG( s) dM ( s) G ( s)
SGM ( s) 

M ( s)
G( s)
dG( s) M ( s)
k: a positive real number
8-77
Section 8- 16, p. 470
Magnitude Plot of Sensitivity
8-78
Section 8- 16, p. 471
G(j) vs 1/G(j)
8-79