1. Frequency Response Time-domain vs Frequency-domain ?
Adapted from Ch.9 of Romagnoli & Palazoglu’s book
See also Ch.17 and 18 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice”
Revision 12.5 of June 4, 2019 by Michele Miccio

2. Frequency Response: definition
A curve representing the output-to-input relationship of a dynamic system as a function of frequency (e.g., with the input being the sine function)
In simplest terms, if a sine wave is inputed to a system at a given frequency, a linear system will respond with another oscillating wave at that same frequency with a certain magnitude and a certain phase angle with respect to the input.

3. Summary of Linear Dynamic Systems
Transfer Function
Response to the input Asin(ωt)
1st order
purely capacitive 1st order
2nd order
dead time
= = =
PID controller B
y(t) = Bsin(ωt + Φ)
amplitude ratio:
AR=B/A

4. Frequency Response: First-Order Process
s=jω
Let’s calculate Re[G(j)] and Im[G(j)], then mod G(j))and arg(G(j))
arg(G(j)) =
= tan-1 [Im[G(j)]/Re[G(j)]=
= tan-1 (-)
By comparison to amplitude ratio AR=B/A and from the response to A sin(ωt):
Introduction to Process Control Romagnoli & Palazoglu

5. Frequency Response: a lucky circumstance !
for a 1st order system:
AR=B/A=mod[G1st(jω)]
Φ=arg G1st(jω)
This result is GENERAL !
take G(s)
place s = jω
determine mod G(jω) and arg G(jω)
AR=B/A=mod[G(jω)]
Φ=arg G(jω)
Definizione di AR
see ch.17

6. Bode-Nyquist Diagrams
The response of a linear constant coefficient system to a sinusoidal input signal is an output sinusoidal signal at the same frequency as the input.
The frequency response of a system is characterized by its
Amplitude Ratio AR
phase angle Φ
However, the magnitude and phase of the output signal differ from those of the input sinusoidal signal, and the amount of difference is a function of the input frequency.
 frequency ω is a parameter
Bode Diagrams
(Logarithmic Plot)
Nyquist Diagram
(Polar Plot)
Introduction to Process Control Romagnoli & Palazoglu

7. Bode Diagrams

8. Bode Diagrams  originally conceived by Hendrik Wade Bode in the 1930s10 -2 -1 1 2 AR w
Bode Diagrams consist of a pair of plots showing:
How AR varies with frequency
Howvaries with frequency 10 -2 -1 1 2
-100
-50 f w 
sometimes a normalized AR/K is plotted and a normalized ω is used
Introduction to Process Control Romagnoli & Palazoglu

9. Bode Diagrams First-Order Process
1st order system
Low frequency asymptote
High frequency asymptote
Example 1
Corner frequency:
inflection point
at ϕ = -45°
ωc = 1/P
ωc = 1/3 rad/s
Introduction to Process Control Romagnoli & Palazoglu

10. Bode Diagrams Example 1 - First-Order Process
Low frequency asymptote
High frequency asymptote
Corner Fequency: ωc = 1/P
Bode Diagrams
slope -1
Introduction to Process Control Romagnoli & Palazoglu

11. Bode Diagrams First-Order Purely Capacitive Process
Low frequency asymptote
High frequency asymptote
slope -1
Corner Fequency:
none

12. Bode Diagrams Second-Order Overdamped Process
Low frequency asymptotes
High frequency asymptotes
slope -2
Corner Fequency:
ωc = 1/

13. Bode Diagrams 2nd Order Process: Phase diagram

14. Bode Diagrams Second-Order Critically Damped Process
Low frequency asymptotes
High frequency asymptotes
Try construction
of Bode Diagrams
in Matlab
and/or Sisotool !
Corner Fequency:
ωc = 1/ 14

15. Bode Diagrams Second-Order Underdamped Process
Low frequency asymptotes
High frequency asymptotes
slope -2
Corner Fequency:
ωc = 1/
 The 2nd order underdamped system may exhibit a maximum in AR 15

16. Bode Diagrams Ex. of 2nd-Order underdamped Process: Asymptotes
Example 2
G(s)=N(s)/D(s)
N(s)=1
D(s)=s2+1.4s+1
D(s)= s2+2ζωns+ωn2
ωn=1rad/s
ζ=0.7
slope -2
ωc = ωn = 1/ =1rad/s

17. Bode Diagrams 2nd-Order Underdamped Process: resonance frequency
The abscissa for the max in AR:
The existence condition for the max in AR:
The ordinate for the max in AR (resonant peak):
 adapted form Pribeiro - Calvin College 17

18. Bode Plot 2nd-Order Underdamped Kp = 1
 adapted form Pribeiro - Calvin College

19. Bode Plot 2nd-Order Underdamped Kp = 1
 adapted form Pribeiro - Calvin College

20. Bode Diagrams Second-Order Undamped Process
slope -2
The phase can/must be read “modulo 360° ”
180° discontinuity jump in phase at resonance frequency 
Low frequency asymptotes
High frequency asymptotes
At the Resonance Fequency ωr = ωn = 1/
 Bode diagrams (with Kp =  = 1)
generated by SisoTool in Matlab R2007B 20

21. Bode Diagrams Second-Order Undamped Process
Low frequency asymptotes
High frequency asymptotes
The phase can/must be read “modulo 360° ”
 Bode diagrams (with Kp = τ = 1)
generated by SisoTool in Matlab R2016
At the Resonance Fequency ωr = ωn = 1/τ 21

22. Bode Diagrams Summary of Second-Order Systems22

23. Bode Diagrams Dead Time
Low frequency asymptotes
High frequency asymptotes
Corner Fequency:
none
 Bode diagrams generated by Matlab

24. Bode Diagrams PI controller
→ Try with Sisotool !

25. Bode Diagrams PD controller
slope +1
Low frequency asymptotes
High frequency asymptotes
Corner Fequency:
ωc = 1/D

26. Bode Diagrams PID controller
Low frequency asymptote
High frequency asymptote
slope -1
slope +1
Corner Fequency:
ωc = 1/D = 1/I
 Bode diagrams generated by SisoTool

27. Bode Diagrams PID controller
see risposta_in_frequenza_PID.PDF
Low frequency asymptote
High frequency asymptote
Corner Fequency:
ωc1 = 1/D
ωc2 = 1/I

28. Bode Diagrams PID controller
see risposta_in_frequenza_PID.PDF
Low frequency asymptote
High frequency asymptote
Corner Fequency:
ωc1 = 1/D
ωc2 = 1/I
The plots in Ch.17 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” are WRONG 

29. Bode Diagrams Systems in Series
 the presence of a constant >0 in the overall transfer function will move the entire AR curve vertically by a constant amount, with no effect on the phase shift
see pag , Ch.17 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice”

30. Bode diagrams in Matlab Graphical window
Example of TF:
G=1/(s^2+2*s+0.5)
Right mouse click >>> Characteristics >>> Peak Response
▼ ▼ ▼
The maximum value of the Bode magnitude plot over the specified region (possibly, the resonant frequency)
Right mouse click >>> Characteristics >>> Minimum Stability Margins
The gain crossover frequency and/or the crossover frequency
>> bode(G)

31. Bode diagrams in SisoTool Desktop window
Corner frequency of the numerator
Corner frequency of the denominator
Example of TF:
G_rat=(s+8)/(s^2+4*s+5) 
gain crossover frequency ωgc 

32. Asymptotic Bode Diagrams
Hyp.: the transfer function is rational only
The Bode plots can be replaced by straight line approximations that take the name of straight line Bode plots or uncorrected Bode plots or Asymptotic Bode Diagrams.
The Asymptotic Bode Diagrams consider piecewise straight lines only.
They are useful because they can be drawn  by hand following a few simple rules, e.g.:
1. Asymptotic diagram of the Modulus
The slope of the Modulus for ω → ∞ is always equal to the difference between the degree of the numerator and the degree of the denominator.
Such a slope is
Negative for a strictly proper system
Null for a proper system

33. Asymptotic Bode Diagrams
Example a First-Order System

34. Asymptotic Bode Diagrams in MatLab® (asbode.m)
Hyp.: the transfer function is rational only
The script
asbode.m
is required from the folder  Asymptotic Bode
which is to be set as the default folder
Chiarire beta !

35. Asymptotic Bode Diagrams in MatLab® (asbode.m)
; Examination test of May 2, 2005 – Sect. 4 Part A (Esame Sez.4 Parte A)
>> Gs=tf([10 0],[1 4 8])
Transfer function: s
s^2 + 4 s + 8
>> num=[10 0]
num =
>> den=[1 4 8]
den =
>> asbode(num,den)
INDIVIDUAL CONTRIBUTIONS TO THE OVERALL ASYMPTOTIC BODE DIAGRAMS:
Guadagno (generalized gain): K = , K_db = 2 db, phi = 0 deg
Poli in origine (poles at origin): nu = -1 (No. poles exceeding zeroes at s=0)
Poli complessi (complex poles):
p,p' = /- j 2.000, omega_n = 2.828, zeta = 0.71
beta (span factor for omega_n) = 5.1, omega_s = 0.555, omega_d =
phi = da 0 a -180 deg,
Delta M_db (2*zeta (dB) = actual difference in AR at omega_n) = -3 db
Chiarire beta !

36. Asymptotic Bode Diagrams in MatLab® (asbode.m)
Examination test of May 2, 2005 – Section 4 Part A
10 s
Gp(s)=
s^2 +4s +8
overall
system
gain

37. Asymptotic Bode Diagrams in MatLab® (asbode.m)
Gp(s)=
s^2 +4s +8
Examination test of May 2, 2005 – Section 4 Part A
Line
Element
Value
Corner freq.
(omega_n)
rad/s
Left freq. (omega_s)
Right freq. (omega_d) AR ϕ
____
Gain
1.250 NA
0 deg
Zero at origin
from 0 to ∞
+90 deg
Coniugate poles
/- j 2.000
2.828
0.555
14.410
from 1 to 0
from 0 to −180 deg
Asymptotic Bode
from 0 to 0
from 90 to −90 deg
Actual Bode

38. the BODE STABILITY Criterion

39. Frequency Response Methods:
Bode Stability Criterion
Bode Stability Criterion
The closed-loop process is stable if the Amplitude Ratio (AR) of the corresponding open-loop transfer function is smaller than 1 (< 0 dB) at the crossover frequency ωco, i.e., the frequency at which the Phase Shift becomes −1800.
see Ch.18 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice”
Introduction to Process Control Romagnoli & Palazoglu

40. Gain and Phase Margin  dB ϕ -180 -180 crossover frequency
ωco
gain crossover frequency
ωgc
-180
adapted form Pribeiro - Calvin College

41. Bode Diagrams Gain and Phase Margin
We have the following system:
where K is a variable controller gain and G(s) is the plant under consideration.
The gain margin is defined as the change in open loop gain required to make the system closed-loop unstable.
The gain margin is the difference between the magnitude curve and 0dB at the point corresponding to the frequency that gives us a phase of -180 deg (the phase crossover frequency, ωco).
Systems with greater gain margins can withstand greater changes in system parameters before becoming unstable in closed loop.
Keep in mind that unity gain in magnitude is equal to a gain of zero in dB.
The phase margin is defined as the change in open loop phase shift required to make a closed loop system unstable.
The phase margin is the difference in phase between the phase curve and −180 deg at the point corresponding to the frequency that gives us a gain of 0dB (the gain crossover frequency, ωgc).
adapted form Pribeiro - Calvin College

42. Bode Stability Criterion
Example 2: 2nd order overdamped & PI 
crossover
frequency ωco
gain crossover
frequency ωgc
ARco
fgc
The distance to instability can be quantified as gain margin and phase margin.
Gain margin is calculated as
GM = 1/ARco
where ARco is the amplitude ratio at the crossover frequency ωco.
Phase margin is calculated as
PM = | fgc| = |fgc|
where fgc is the phase shift corresponding to an AR of 1.
Introduction to Process Control Romagnoli & Palazoglu

43. Limitations to the Bode Stability Criterion
The Bode stability criterion is not straightforward or not applicable in a direct way,
that is just according to its simple statement,
when:
The crossover frequency exists, but it is NOT located at –180°
the open loop gain is non-positive
the transfer function is not open-loop stable (poles with positive real part)
the phase plot is non-monotonic and exhibits more than one crossover frequency
Some textbooks state that the Bode stability criterion is not applicable in a direct way also when:
the transfer function has zeroes with positive real part
the AR plot is non-monotonic and presents more than one gain crossover (at AR=1)

44. Examination Test of July 22, 2014 (with n=6):
Limitations to the Bode Stability Criterion - I
I. The crossover frequency exists, but it is NOT located at –180°
V. the transfer function has zeroes with positive real part
Examination Test of July 22, 2014 (with n=6):
 Bode diagrams generated by Sisotool 
crossover frequency ωco
No other limitations hold for the application of the Bode Stability Criterion
However, the crossover frequency appears located at a multiple of –180° (modulo 360°),
e.g.,
–180°  k360° with kN
 As an alternative, the simplest approach is to switch to the Nyquist stability criterion
Qual è la FdT ???

45. Limitations to the Bode Stability Criterion - I
I. The crossover frequency exists, but it is NOT located at –180°
II. the open loop gain is negative
Ex.I
Transfer function: -1
s^3 + s^2 + 2 s + 1
Ex.II
Transfer function:
3 s - 1
10 s^2 + 7 s + 1
Qual è la FdT ???

46. Limitations to the Bode Stability Criterion – II
II. The open loop gain is negative
 Bode diagrams generated by Sisotool
Example II.bis:
1st Order System
new
Process Control P.C. Chau © 2002

47. Limitations to the Bode Stability Criterion - II
II. The open loop gain is negative
Transfer function: -1
-----
s + 1
Transfer function: -1
--- s
Transfer function: -1
s^2 + s + 1

48. Limitations to the Bode Stability Criterion – VI
VI. The AR plot is non-monotonic and presents more than one gain crossover
Example 3:
(Chau, 2002; Hahn et al., 2001)
 When the magnitude does not decrease monotonically and presents more than one gain crossover (at AR=1), we need to assess the stability situation at modulo 360°frequencies,
theoretically at
–180°  k360° with kN
new
Process Control P.C. Chau © 2002

49. Limitations to the Bode Stability Criterion
Sisotool reports a statement about closed loop system stability or instability even in these cases !
New 
As a safe alternative, the Nyquist stability criterion has to be used !

50. Conclusions on the Bode Stability Criterion
The Bode stability criterion presented in most process control textbooks is a sufficient, but not necessary, condition for stability of a closed-loop process.
Therefore, it is not possible to use this criterion to make definitive statements about the stability of any given process. 
A NOTE ON STABILITY ANALYSIS USING BODE PLOTS
by Juergen Hahn, Thomas Edison, Thomas F. Edgar (2001)

51. Nyquist Diagrams

52. Nyquist Diagram
The Nyquist diagram contains the same information as the Bode diagrams for the same system !!!
Introduction to Process Control Romagnoli & Palazoglu

53. Nyquist Diagram The Nyquist diagram maps
the non-negative imaginary axis from the s-plane (s=jω)
into a curve on the G-plane (G(jω))
s=jω, ω≥0
G(jω)
mapping
Introduction to Process Control Romagnoli & Palazoglu

54. Nyquist Diagram
A Nyquist diagram (polar plot) is an alternative way to represent the frequency response.
Im G (j), ordinate
Re G (j), abscissa
A specific value of frequency defines a point on this plot,
e.g., A for the point A.
Introduction to Process Control Romagnoli & Palazoglu

55. First-Order Process
Introduction to Process Control Romagnoli & Palazoglu

56. Nyquist – First-Order Process Example 1
Nyquist Diagram (for 0 ≤  < +) ω
Introduction to Process Control Romagnoli & Palazoglu

57. Bode–Nyquist comparison First-Order Process - Example 1
Bode Diagrams
Nyquist Diagram (for 0 ≤  < +)
 Nyquist diagram generated by Sisotool
Introduction to Process Control Romagnoli & Palazoglu

58. Nyquist – Pure Integrator
Nyquist Diagram (for 0 ≤  < +)
Introduction to Process Control Romagnoli & Palazoglu

59. Nyquist - Second-Order Process
Nyquist Diagram (for 0 ≤  < +)
Introduction to Process Control Romagnoli & Palazoglu

60. Nyquist - Second-Order Process
Nyquist Diagram (for 0 ≤  < +)
Introduction to Process Control Romagnoli & Palazoglu

61. Nyquist - Second-Order Process
Nyquist Diagram (for 0 ≤  < +)
 Nyquist diagram generated by Sisotool
Introduction to Process Control Romagnoli & Palazoglu

62. Nyquist - Second-Order Process
Nyquist Diagram (for 0 ≤  < +)
unit
circle
 Nyquist diagram generated by Sisotool
Introduction to Process Control Romagnoli & Palazoglu

63. Nyquist - Second-Order Process
Undamped Process
Nyquist Diagram (for 0 ≤  < +; Kp = 1)
Actually plotted with Kp =  = n = 1
Resonance Fequency: ωn = 1/ 63

64. Nyquist - Second-Order Process
Undamped Process
extended Nyquist Diagram (for ➖ ≤  < +) 
 Actually plotted by SisoTool
(with Kp =  = 1) in Matlab R2007B 64

65. Nyquist - Second-Order Process
Undamped Process
extended Nyquist Diagram (for ➖ ≤  < +) 
 Actually plotted by SisoTool
(with Kp =  = 1) in Matlab R2018A 65

66. Nyquist - Second-Order Process
Undamped Process
extended Nyquist Diagram (for ➖ ≤  < +)
gain margin: indeterminate
phase crossover frequency: all
1st phase margin: 0 °
1st gain crossover frequency: rad/s
2nd phase margin: 180 °
2nd gain crossover frequency: 0 rad/s 
 Actually plotted (with Kp =  = 1) by 66

67.  Nyquist diagram generated by Sisotool
Nyquist – Dead Time
Nyquist Diagram (for 0 ≤  < +)
 Nyquist diagram generated by Sisotool
Introduction to Process Control Romagnoli & Palazoglu

68. Nyquist – PI controller
Nyquist Diagram (for 0 <  < +)
Kc=1
 Nyquist diagram generated by Sisotool
Introduction to Process Control Romagnoli & Palazoglu

69. Nyquist – PD controller
Nyquist Diagram (for 0 ≤  < +)
Kc=1
 Nyquist diagram generated by Sisotool
Introduction to Process Control Romagnoli & Palazoglu

70. Nyquist – PID controller
Nyquist Diagram (for 0 ≤  < +)
Kc=1
 Nyquist diagram generated by Sisotool
Introduction to Process Control Romagnoli & Palazoglu

71. Nyquist – PID controllers
§ Stephanopoulos, “Chemical process control: an Introduction to theory and practice”, Prentice Hall, 1984 
Some Nyquist plots (e.g., PID controllers) are wrong !

72. First-Order Process with Delay Example 4
Bode Diagrams
With the addition of a delay term, AR remains the same, but the Phase Shift is significantly affected.
Introduction to Process Control Romagnoli & Palazoglu

73. First-Order Process with Delay Example 4
Nyquist Diagram
The net effect of the delay is to alter the phase characteristics of the process, which results in the circling of the origin at high frequencies with a decreasing radius.
Introduction to Process Control Romagnoli & Palazoglu

74. The Nyquist Diagram in Matlab® (graphical window)
>> nyquist(G)
Right mouse click
>>> Characteristics >>> Peak Response
▼ ▼ ▼
The maximum value of the Bode magnitude plot over the specified region 

75. Nyquist Diagram: Gain and Phase MarginsPM
GM-1
unit circle
fgc
PM = | fgc| =
= |fgc|
Introduction to Process Control Romagnoli & Palazoglu

76. the Nyquist STABILITY Criterion

77. Background of the Nyquist Criterion: The Contour
A contour is a complicated mathematical construct, but luckily we only need to worry ourselves with a few points about them.
We will denote contours with the Greek letter Γ (gamma).
Contours are lines, drawn on a graph on the complex plane, that follow certain rules:
The contour must close (it must form a complete loop)
The contour may not cross directly through a pole of the system.
Contours must have a direction (clockwise, generally).
A contour is called "simple" if it has no self-intersections.
 We only consider simple contours here.

78. Background of the Nyquist Criterion: The Nyquist Contour
we’re interested in whether there are any right half-plane roots of the closed loop transfer function
we choose the Nyquist contour encircling the entire unstable region of the complex s-plane
The Nyquist contour is an infinite semi-circle that encircles the entire right-half of the s-plane:
The semicircle travels up the imaginary axis from negative infinity to positive infinity.
From positive infinity, the contour breaks away from the imaginary axis, in the clockwise direction.
Finally, it forms a giant semicircle.
s = jω
s = -jω
s-plane
a pole to the closed-loop transfer function (or equivalently a zero of the characteristic equation)

79. Background of the Nyquist Criterion: The Contour Mapping
When we have our contour, Γ, we transform it into another curve (i.e., the transformed contour) in the C-plane, ΓF(s), by
plugging every point of the contour Γ into the function F(s)
and taking the resultant value to be a point on the transformed contour ↓
mapping

80. Background of the Nyquist Criterion: The Cauchy's argument principle
If we have a contour, Γ, drawn in one plane (say the complex Laplace plane), we can map that contour into another complex plane (say the F(s) plane) by transforming the contour with the rational function F(s).
The resultant contour will circle the origin of the F(s) plane N times, where N is equal to the difference between Z and P (the number of zeros and poles of the rational function F(s), respectively).
 Cauchy’s theorem thus tells us that there is a relationship between the value of a contour integral and the poles that reside within the contour.

81. Background of the Nyquist Criterion: The extended Nyquist Diagram
The extended Nyquist plot is the “mapping” of the Nyquist contour from the s-plane into a curve on the G-plane using a transfer function, G(s), as the mapping function !!!
extended Nyquist plot
s=jω
s=-jω
s-plane
G(jω)
G-plane
mapping
 the resulting map is symmetric about the real axis
Introduction to Process Control Romagnoli & Palazoglu

82. Extended Nyquist Diagram. 1st case: the extended Nyquist plot is closed
Nyquist Diagram extension to - <  < + for a 1st-Order Process
 Mirror image with respect to the real axis
 Obtain a unique contour (polar plot) on the complex plane
 Follow the G(s) contour for - <  < +
extended Nyquist plot
  -
 = 0
Introduction to Process Control Romagnoli & Palazoglu

83. Extended Nyquist Diagram. Example 1ter: a First-Order Process
Extended Nyquist Diagram (for − <  < +)
Introduction to Process Control Romagnoli & Palazoglu

84. Example 6: Kc = 1 a 4th order Transfer Function
with:
Kc = 1
Nyquist plot initial point ( = 0):
Nyquist plot final point (  ):
Extended Nyquist plot initial point (negative frequency   ):
→ Try using
>> nyquist(G4)

85. Background of the Nyquist Criterion: The critical point
Important property of the Nyquist plot
The (-1, 0) point is so important in the Nyquist plot.
The reason can be deduced from the characteristic equation
1 + GOL(s) = 0
This equation can also be written as GOL(s) = -1, which implies that at the (-1, 0) point:
AROL = 1 and ϕOL = -180°
The (-1, 0) point is referred to as the critical point.
An encirclement of the origin in 1 + GOL(s) = 0 becomes an encirclement of -1 in GOL(s) = -1

86. Nyquist Stability Criterion: simplified form I: for open-loop stable systems
Stephanopoulos’ formulation
Most process control problems are open-loop stable. Consequently,
the closed-loop system is unstable if the extended Nyquist plot for GOL(jω) encircles the -1 (critical point), one or more times.
see:
§ Stephanopoulos, “Chemical process control: an Introduction to theory and practice”, Prentice Hall, 1984

87. Nyquist Stability Criterion: counting the encirclements
 The Nyquist diagr. G(j) here is without the “mirrored” part!
 The unstable one derives from the same transfer function of the stable one after multiplication by a constant K>1

88. Nyquist Stability Criterion: counting the encirclements
Stephanopoulos’ practical procedure
Have your extended Nyquist plot on a piece of paper
"Place a pencil at the critical point (-1,0). Attach one end of a thread at the pencil and with the other end trace the whole length of the extended Nyquist plot. If the thread has wrapped around the pencil then we can say that the point (-1,0) is encircled by the Nyquist plot."
see:
§ Stephanopoulos, “Chemical process control: an Introduction to theory and practice”, Prentice Hall, 1984

89. complete Nyquist plot with closure at infinity
Nyquist Stability Criterion:
counting the encirclements
complete Nyquist plot with closure at infinity -1
Wikipedia’s practical procedure
An easy way to remember how to count N in the clockwise direction:
 draw a 2-color oblique half-straight line from the critical point
crossing RED gives +1
crossing BLUE gives -1
 3 cases in the figure!
Wikipedia

90. Extended Nyquist Diagram
Extended Nyquist Diagram. 2nd case: the extended Nyquist plot is not closed
Nyquist Diagram extension to - <  < +
In order to obtain a closed polar plot, we introduce closure at infinity
closure at infinity consists in rotating clockwise of π angle with an infinite radius for every pole with Re(.)=0
Nyquist4(pole at origin).swf
see pag
prof. Lanari
“Stability – Nyquist”

91. Nyquist Contour. case 2a: the transfer function has a pole at origin
Nyquist contour construction
s-plane
⎯▶︎ the extended Nyquist plot
will be closed with closure at infinity in the G-plane

92. Nyquist Diagram extension to - <  < +
Extended Nyquist Diagram. case 2a: the extended Nyquist plot is not closed
Nyquist Diagram extension to - <  < +
s + 2
G(s) = -----
s^2
Example 5:
  -
 = 0+
  +
 = 0−
mirroring
The Nyquist diagram must be closed:
Connect manually the point corresponding to “ω=0−” to “ω=0+” on the G-plane
The connecting curve must be drawn clockwise
 In this example’s case the clockwise path is not the shortest!

93. Nyquist contour. case 2b: G(s) has two or more imaginary (resonant) poles
Nyquist contour construction
s-plane
⎯▶︎ the extended Nyquist plot
will be closed with closure at infinity in the G-plane

94. Nyquist Stability Criterion:
general form
Nyquist Stability Criterion
Given the closed-loop characteristic equation
1 + GOL(s) = 0,
if N is the number of times that the extended polar plot of GOL(jω) encircles the (–1,0) point in the clockwise direction as ω is varied from –∞ to +∞,
and if P≥0 is the number of the poles of GOL(s) in the right half plane (RHP),
then Z = N + P is the number of unstable roots of the closed-loop characteristic equation.
Notes:
N  Z (NB: a counterclockwise encirclement has a negative sign)
 GOL(jω) must not intercept the (–1,0) critical point
Chemical Process Control - A First Course with MATLAB Chau p.161

95. Extended Nyquist Diagram
Extended Nyquist Diagram. 3rd case: the Extended Nyquist Diagram passes through the critical point
The closed loop TF GCL(s) has poles with a null real part if and only if the Nyquist plot of the open loop TF GOL(s) passes through the critical point (-1, 0) 
marginal stability or
BIBO instability
at closed loop
depending on the multiplicity of poles with a null real part.
 Correspondingly, the phase margin is always zero under this case.
see pag. 6
prof. Lanari
“Stability – Nyquist”

96. Nyquist stability criterion: Important features
It provides a necessary and sufficient condition for closed-loop stability based on the open-loop transfer function.
The Nyquist stability criterion allows stability to be determined without computing the closed-loop poles.
A negative value of N indicates that the -1 point is encircled in the opposite direction (counter-clockwise). This situation implies that each countercurrent encirclement can stabilize one unstable pole of the open-loop system.
Unlike the Bode stability criterion, the Nyquist stability criterion is applicable to open-loop unstable processes.
Unlike the Bode stability criterion, the Nyquist stability criterion can be applied when multiple values of co or gc occur.

97. The Extended Nyquist Diagram in MatLab®
Right mouse click
>>> Characteristics >>> Minimum Stability margins
▼ ▼ ▼
unit circle 
>>> Show >>> Negative Frequencies
Extended Nyquist plot
>> nyquist(G)

98. The Extended Nyquist Diagram in MatLab®
>> nyquist(G) 
the Matlab nyquist command does not take poles or zeros on the jw axis into account and therefore produces an incorrect plot