Upload
cody-logan
View
231
Download
1
Embed Size (px)
Citation preview
Frequency Response
Time-domain vs
Frequency-domain ?
Revision 8.5 of July 17, 2015
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”
Frequency Response: definition
2
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.
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)
System Transfer Function Response to the input Asin(ωt)
1st order
purely capacitive 1st order
2nd order
dead time
PID controller
Summary of Linear Dynamic Systems
3
2
1
222
)(1
2tan
)sin()2()(1
)(
for
tAK
ty
t
P
1s
K)s(G
p
p
s
KsG p
'
)(
Let’s calculate Re[G(j and Im[G(j, then mod G(j)and arg (G(j))
Introduction to Process Control Romagnoli & Palazoglu
Frequency Response:First-Order Process
4
By comparison to AR and from the response to A sin(ωt):
Angle of a complex number in Matlab
>> phi=angle(p)
Frequency Response: a lucky circumstance !
6
for a 1st order system:•AR=B/A=mod[G1st(jω)]•Φ=arg G1st(jω)
This result is GENERAL !
1.retake G(s)2.place s = jω3.determine mod G(jω) and arg G(jω)4.AR=B/A=mod[G(jω)]5.Φ=arg G(jω) see ch.17
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
Introduction to Process Control Romagnoli & Palazoglu
Bode Diagrams(Logarithmic Plot)
Nyquist Diagram(Polar Plot)
7
10-2
10-1
100
101
102
-100
-50
0
Introduction to Process Control Romagnoli & Palazoglu
Bode Diagrams consist of a pair of plots showing:How AR varies with frequencyHowvaries with frequency
10-2
10-1
100
101
102
0
1
2AR
Bode Diagrams
8
sometimes a normalized AR/K is plotted and a normalized ω is used
10-2
10-1
100
101
10-2
10-1
100
101
10-2
10-1
100
101
-100
-80
-60
-40
-20
0
c=0.33
AR
Introduction to Process Control Romagnoli & Palazoglu
Bode Diagram
Bode DiagramsExample 1 - First-Order Process
9
Corner Fequency: ωc = 1/P
asymptotes
slope -1
Introduction to Process Control Romagnoli & Palazoglu
Bode DiagramsExample 1 - First-Order Process
10
Corner frequency:inflection point at ϕ = 45°
ωc = 1/P
Low frequency asymptote
High frequency asymptote
0)(lim
)(lim
0
0
jG
KjG P
90)(lim
0)(lim
jG
jG
Example 1
ωc = 1/3 rad/s
1st order system
Bode DiagramsFirst-Order Purely Capacitive Process
j
KjG p
'
)(
Corner Fequency:none
Low frequency asymptote
High frequency asymptote
90)(lim
)(lim
0
0
jG
jG
90)(lim
0)(lim
jG
jG
slope -1
11
Bode DiagramsSecond-Order Overdamped Process
Corner Fequency:ωc = 1/
Low frequency asymptotes
High frequency asymptotes
0)(lim
)(lim
0
0
jG
KjG p
slope -2
12
Bode DiagramsEx. of 2nd-Order Process: Corner Fequency
13
14.1
12
ss
sG
ωc = 1/ =1rad/s
G(s)=N(s)/D(s)
N(s)=1D(s)=s2+1.4s+1
D(s)= s2+2ζωns+ωn2
ωn=1rad/sζ=0.7
-80
-60
-40
-20
0
20
Mag
nitu
de (
dB)
10-2
10-1
100
101
102
-180
-135
-90
-45
0
Pha
se (
deg)
Bode Diagram
Frequency (rad/sec)
asymptotes
slope -2
Example 2
Bode Diagrams2nd Order Process: Phase diagram
180)(lim
90)(lim
0)(lim
1
0
jG
jG
jG
14
2
1
)(1
2tan
jG
0)(1 2
0)(1 2
Bode DiagramsSecond-Order Critically Damped Process
Corner Fequency:ωc = 1/
Low frequency asymptotes
High frequency asymptotes
0)(lim
)(lim
0
0
jG
KjG p
180)(lim
0)(lim
jG
jG
15
Try construction of Bode Diagramsin Matlab and/or Sisotool !
Bode DiagramsSecond-Order Underdamped Process
Corner Fequency:ωc = 1/
Low frequency asymptotes
High frequency asymptotes
0)(lim
)(lim
0
0
jG
KjG p
slope -2
16
5.0 with
12)(
22
ss
KsG p
Bode Diagrams2nd-Order Underdamped Process: resonance frequency
r n 1 2 2
0.707
Mp G r 1
2 1 2
0.707
adapted form Pribeiro - Calvin College 17
The abscissa of the max:
The existence condition:
The ordinate of the max:
Bode Plot 2nd-Order Underdamped
adapted form Pribeiro - Calvin College18
Bode Plot 2nd-Order Underdamped
adapted form Pribeiro - Calvin College 19
slope -2
!
!
Bode DiagramsSecond-Order Undamped Process
Low frequency asymptotes
High frequency asymptotes
0)(lim
)(lim
0
0
jG
KjG p
20
1 with 1s
K)s(G
22
p
)j(Glimn
At the Resonance Fequency ωn = 1/ Bode diagrams generated by SisoTool
(with Kp = = 1)
Bode DiagramsSummary of Second-Order Systems
21
Bode DiagramsDead Time
Corner Fequency:none
Low frequency asymptotes
High frequency asymptotes
)(lim
1)(lim
jG
jG
22
Bode diagrams generated by Matlab
Bode DiagramsPI controller
23
Try with Sisotool !
Bode DiagramsPD controller
Corner Fequency:ωc = 1/D
Low frequency asymptotes
High frequency asymptotes
24
slope +1
Bode DiagramsPID controller
1 with
11
ID
IDcPID s
sG
Corner Fequency:ωc = 1/D = 1/I
Low frequency asymptote
High frequency asymptote
slope +1slope -1
25 Bode diagrams generated by SisoTool
Bode DiagramsPID controller
Corner Fequency:ωc1 = 1/D
ωc2 = 1/I
Low frequency asymptote
High frequency asymptote
26
see risposta_in_frequenza_PID.PDF
Bode DiagramsPID controller
Low frequency asymptote
High frequency asymptote
27
Corner Fequency:ωc1 = 1/D
ωc2 = 1/I
see risposta_in_frequenza_PID.PDF
The plots in Ch.17 - Stephanopoulos, “Chemical process control: an Introduction to theory and
practice”
are WRONG !
Bode DiagramsSystems 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. 330-331, Ch.17 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice”
28
Asymptotic Bode Diagrams
29
If the transfer function is rational, 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. For example a First-Order System:
1s
K)s(G
P
P
http://lpsa.swarthmore.edu/Bode/BodeHow.html
Asymptotic Bode Diagramsin MatLab® (asbode.m)
30
The script asbode.m
is required in the folder Asymptotic Bode
http://www.diee.unica.it/giua/ANSIS/
Asymptotic Bode Diagramsin MatLab® (asbode.m)
31
; Examination problem of May 2, 2005 - Section 4 Part A; (Problema d’esame del 2.05.05 - Sez. 4 Parte A)
>> Gs=tf([10 0],[1 4 8]) Transfer function: 10 s-------------s^2 + 4 s + 8
>> num=[10 0]num = 10 0
>> den=[1 4 8]den = 1 4 8
>> asbode(num,den)
Guadagno (gain): K = 1.250, K_db = 2 db, phi = 0 degPoli in origine (poles at origin): nu = -1Poli complessi: p,p' = -2.000 +/- j 2.000, omega_n = 2.828, zeta = 0.71(complex poles) beta = 5.1, omega_s = 0.555, omega_d = 14.410 phi = da 0 a -180 deg, Delta M_db = -3 db
Asymptotic Bode Diagramsin MatLab® (asbode.m)
32
Examination test of May 2, 2005 – Section 4 Part A
overall system
Asymptotic Bode Diagramsin MatLab® (asbode.m)
33
Examination test of May 2, 2005 – Section 4 Part A
Line Element Value Corner freq.(omega_n)rad/s
Left freq. (omega_s)rad/s
Right freq. (omega_d)rad/s
AR ϕ
____ Gain 1.250 NA NA NA 1.250 0 deg
____ Zero at origin
0 NA NA NA from 0 to ∞
+90 deg
____ Coniugate poles
-2.000 +/- j 2.000
2.828 0.555 14.410 from 1 to 0
from 0 to −180 deg
____ Asymptotic Bode
2.828 0.555 14.410 from 0 to 0
from 90 to −90 deg
- - - - Actual Bode
2.828 from 0 to 0
from 90 to −90 deg
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.
Introduction to Process Control Romagnoli & Palazoglu34
see Ch.18 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice”
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 cross over frequency, ωgc).
Bode DiagramsGain and Phase Margin
We have the following system: where K is a variable (constant) gain and G(s) is the plant under consideration.
adapted form Pribeiro - Calvin College35
-180
Gain and Phase Margin
adapted form Pribeiro - Calvin College36
dB
ϕ
-180
Bode Stability CriterionExample: 2nd order overdamped & PI
10-2
10-1
100
101
102
10-5
100
105
10-2
10-1
100
101
102
-200
-180
-160
-140
-120
-100
-80
AR
crossover
frequency ωco
k 0.5, I 0.5
This closed-loop system is stable as the AR is less than 1 at the crossover frequency.
The distance to instability can be quantified as gain and phase margins.
Gain margin is defined as GM=1/ARc, where ARc is the
amplitude ratio at the crossover frequency.
Phase margin is defined as PM=|-1800 - c|=1800-|c|,
where c is the phase shift
corresponding to an AR of 1.
Introduction to Process Control Romagnoli & Palazoglu37
Limitations to the Bode Stability Criterion
The Bode stability criterion is not straightforward when there are multiple crossover points or at least one of the Bode plots is non-monotone.
When the magnitude does not decrease monotonously (see this example!), we need to assess the stability situation at higher frequencies, or in other words, crossover at phase angles larger than –180°.
Theoretically, we need to look at all multiples: (–180° k360) with kN
Process Control P.C. Chau © 2002 38
Limitations to the Bode Stability Criterion
The Bode stability criterion is not applicable in a direct way, that is just according to its simple statement, when:
•the gain is negative
•the transfer function is not open-loop stable (poles with positive real part) •the transfer function has zeroes with positive real part•the phase plot exhibits more than one crossover frequency (at ϕ = –180°)•the AR plot presents more than one crossover (at AR=1).
39
Sisotool reports a statement about closed loop system stability or instability even in these cases !
Bode Stability CriterionExample: crossover frequency NOT located at –180°
When no limitations hold for the application of the Bode Stability Criterion, the crossover frequency is to be located at a multiple of –180°(–180° k360°with kN)
40
Bode diagrams generated by Sisotool
The Nyquist diagram contains the same information as the Bode diagrams for the same
system !!!
Introduction to Process Control Romagnoli & Palazoglu
Nyquist Diagram
41
The Nyquist diagram maps the positive imaginary axis from the s-plane (s=jω) into a curve on the G-plane (G(jω)) !!!
Introduction to Process Control Romagnoli & Palazoglu
Nyquist Diagram
42
s=jω, ω≥0 G(jω)
A Nyquist diagram (polar plot) is an alternative way to represent the frequency response.
A specific value of frequency defines a point on this plot,
e.g., for the point A.
Im G (j), ordinateRe G (j), abscissa
Introduction to Process Control Romagnoli & Palazoglu
Nyquist Diagram
0
0
AR
Im
Re
A
43
First-Order Process
Introduction to Process Control Romagnoli & Palazoglu
bja1
k)(j
1
k =
1
)1j(k
)1j(
)1j(
1j
k)j(g
2222
22
bja1
k)(j
1
k =
1
)1j(k
)1j(
)1j(
1j
k)j(g
2222
22
g(s) k
s 1 g(j)
k
j 1
1
k)()j(GImb ;
1
k)j(GRea
2222
1
k)()j(GImb ;
1
k)j(GRea
2222
44
Introduction to Process Control Romagnoli & Palazoglu
Nyquist Diagram (for 0 ≤ < +)
Example - First-Order Process
1
)()(Im
1)(Re
22
22
kjG
kjG
1
)()(Im
1)(Re
22
22
kjG
kjG
45
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
Real
Imag
inar
y
ω
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
RealIm
agin
ary
Introduction to Process Control Romagnoli & Palazoglu
Bode – Nyquist comparisonExample - First-Order Process
1
)()(Im
1)(Re
22
22
kjG
kjG
1
)()(Im
1)(Re
22
22
kjG
kjG
10-2
10-1
100
101
10-2
10-1
100
101
10-2
10-1
100
101
-100
-80
-60
-40
-20
0
c=0.33
AR
arg(g(j)) tan 1( 3)
Bode Diagram
AR g(j) 2
92 1
46
Nyquist Diagram (for 0 ≤ < +)
Introduction to Process Control Romagnoli & Palazoglu
))j(Garg(
)j(GAR
Nyquist – Pure Integrator
G(s) 1
s
47
Nyquist Diagram (for 0 ≤ < +)
Introduction to Process Control Romagnoli & Palazoglu
))j(Garg(
)j(GAR
Nyquist - Second-Order Process
G(s) Kp
2s2 2s 1
overdamped
48
Nyquist Diagram (for 0 ≤ < +)
Introduction to Process Control Romagnoli & Palazoglu
))j(Garg(
)j(GAR
Nyquist - Second-Order Process
G(s) Kp
2s2 2s 1
critically damped
49
Nyquist Diagram (for 0 ≤ < +)
Introduction to Process Control Romagnoli & Palazoglu
))j(Garg(
)j(GAR
Nyquist - Second-Order Process
50
Nyquist Diagram (for 0 ≤ < +)
unitcircle
Introduction to Process Control Romagnoli & Palazoglu
))j(Garg(
)j(GAR
Nyquist - Second-Order Process
51
Nyquist Diagram (for 0 ≤ < +)
Nyquist - Second-Order Process
)j(Glim
)j(Glim
0)j(Glim
K)j(Glim
1
1
p0
52
1s
K)s(G
22
p
Resonance Fequency: ωn = 1/
Undamped ProcessNyquist Diagram (for 0 ≤ < +)
! real always
1
K
1j
K
)j(G
22
p
22
p
Nyquist - Second-Order Process
53
1s
K)s(G
22
p
Actually plotted by SisoTool
(with Kp = = 1)
Undamped ProcessNyquist Diagram (for 0 ≤ < +)
!
!
! real always
1
K
1j
K
)j(G
22
p
22
p
Introduction to Process Control Romagnoli & Palazoglu
))j(Garg(
)j(GAR
Nyquist – Dead Time
54
Nyquist Diagram (for 0 ≤ < +)
Introduction to Process Control Romagnoli & Palazoglu
Nyquist Diagram (for 0 < < +)
))j(Garg(
)j(GAR
Nyquist – PI controller
G(s) Kc 11
Is
Kc=1
55
Introduction to Process Control Romagnoli & Palazoglu
))j(Garg(
)j(GAR
Nyquist – PD controller
G(s) Kc 1 Ds
Kc=1
56
Nyquist Diagram (for 0 ≤ < +)
Introduction to Process Control Romagnoli & Palazoglu
))j(Garg(
)j(GAR
Nyquist – PID controller
G(s) Kc 1 Ds 1
Is
Kc=1
57
Nyquist Diagram (for 0 ≤ < +)
Some Nyquist plots (e.g., PID controllers) are wrong !
Nyquist – PID controllers
§ 17.4 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice”, Prentice Hall, 1984
58
10-2
10-1
100
101
10-2
10-1
100
101
10-2
10-1
100
101
-400
-300
-200
-100
0
AR
)5.0()3(tan))(arg(
19
2)(
1
2
jg
jgAR
With the addition of a delay term, AR remains the same, but the Phase Shift is significantly affected.
Introduction to Process Control Romagnoli & Palazoglu
13
2)(
5.0
s
esg
sBode Diagram
Example - First-Order Process with Delay
59
-0.5 0 0.5 1 1.5 2-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
Real
Imag
inar
y
g(s) 2 e 0.5s
3s 1
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
Nyquist Diagram
Example - First-Order Process with Delay
60
Introduction to Process Control Romagnoli & Palazoglu
Nyquist Diagram:Gain and Phase Margins
61
PM
GM-1
unit circle
PM = |-1800 - c| =
= 1800 - |c|
Background of the Nyquist Criterion:The Contour
62
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:
1.The contour must close (it must form a complete loop)2.The contour may not cross directly through a pole of
the system.3.Contours must have a direction (clockwise, generally).
A contour is called "simple" if it has no self-intersections. We only consider simple contours here.
Background of the Nyquist Criterion:The Cauchy's argument principle
63
If we have a contour, Γ, drawn in one plane (say the complex Laplace plane), we can map that contour into another plane, the F(s) plane, by transforming the contour with the function F(s).
The resultant contour will circle the origin point 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 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.
Background of the Nyquist Criterion:The Nyquist Contour
64
Since we’re interested in whether there are any right half-plane roots of the closed loop tranfer function, we choose the Nyquist contour encircling the entire unstable region, that is the right half of the complex s plane.
The Nyquist contour is an infinite semi-circle that encircles the entire right-half of the s plane:1.The semicircle travels up the imaginary axis from negative infinity to positive infinity. 2.From positive infinity, the contour breaks away from the imaginary axis, in the clockwise direction.3.Finally, it forms a giant semicircle.
s=jω
s=-jω
s-plane
Introduction to Process Control Romagnoli & Palazoglu
Background of the Nyquist Criterion:The extended Nyquist Diagram
65
G(jω)s=jω
s=-jω
s-plane
the resulting map is symmetric about the real axis
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(jω), as the mapping function !!!
Introduction to Process Control Romagnoli & Palazoglu
Nyquist Diagram extension to - < < + for a 1st-Order Process
Extended Nyquist Diagram. 1st case: the Nyquist plot is closed
Mirror image with respect to the real axis
- = 0
Obtain a unique contour (polar plot) on the complex
plane
Follow the G(s) contour for - < < +
66
Introduction to Process Control Romagnoli & Palazoglu
Nyquist Diagram (for − < < +)
1s3
2)s(G
Extended Nyquist Diagram. Example 1: a First-Order Process
1
)()(Im
1)(Re
22
22
kjG
kjG
1
)()(Im
1)(Re
22
22
kjG
kjG
67
Nyquist Diagram extension to - < < + for the following transfer function:
Extended Nyquist Diagram. 2nd case: the Nyquist plot is not closed
68
s + 2 G(s) = -----
s^2
-
= 0+
+
= 0-
The Nyquist diagram must be closed. Connect manually the “ω=0-” to “ω=0+”. This should be done clockwise. In this example’s case the clockwise path is not the shortest.
Nyquist Diagram extension to - < < +
Extended Nyquist Diagram. 2nd case: the Nyquist plot is not closed
69
In order to obtain a closed polar plot, we introduce closure at infinity, which consists in rotating clockwise of π angle with an infinite radius for every pole with Re(.)=0
see pag. 14-20prof. Lanari
“Stability – Nyquist”
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.
Background of the Nyquist Criterion:The critical point
70
Important property of the Nyquist plot
Extended Nyquist Diagram. 3rd case: the Nyquist contour passes through the critical point
71
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
orBIBO instability
depending on the multiplicity of poles with a null real part
see pag. 6prof. Lanari
“Stability – Nyquist”
The Extended Nyquist Diagramin MatLab®
72
unit circle
>> nyquist(G)
Right mouse click >>> Characteristics >>> Minimum Stability margins
the Matlab nyquist command does not take poles or zeros on the jw axis into account and therefore produces an incorrect plot.
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.
Nyquist Stability Criterion:simplified form I: open-loop stable systems
see:§ 18.4 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice”, Prentice Hall, 1984
73
Nyquist Stability Criterion:simplified form I
Real Axis
Imag
inar
y A
xis
0
0
2
2-2
-2
(-1,0).
unstable
stable
74
The Nyquist diagr. 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
Nyquist Stability Criterion:simplified form I
see:§ 18.4 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice”, Prentice Hall, 1984
75
Stephanopoulos’ practical procedure:1.Have your Nyquist plot on a piece of paper2.
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
Nyquist Stability Criterion:general form
Chemical Process Control - A First Course with MATLAB Chau p.161 76
Wikipedia’s practical procedure:An easy way to remember how to count N in the clockwise direction in the equation:
Z = N + P
draw a 2-color oblique half-straight line from the critical point crossing RED gives +1 crossing BLUE gives -1
Nyquist Stability Criterion
Wikipedia http://commons.wikimedia.org/wiki/File:Nyquist_Criteria.svg
completeNyquist
plot
77
1. It provides a necessary and sufficient condition for closed-loop stability based on the open-loop transfer function.
2. The Nyquist stability criterion allows stability to be determined without computing the closed-loop poles.
3. 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.
4. Unlike the Bode stability criterion, the Nyquist stability criterion is applicable to open-loop unstable processes.
5. Unlike the Bode stability criterion, the Nyquist stability criterion can be applied when multiple values of c or gc occur.
Nyquist stability criterion:Important features
78
Example 14.6
Evaluate the stability of the closed-loop system in Fig. 14.1 (see below) for:
4( )5 1
s
pesGs
(the time constants and delay have units of minutes)
Gv = 2, Gm = 0.25, Gc = Kc
1.Obtain ωc and Kcu from a Bode plot. 2.Let Kc =1.5Kcu and draw the Nyquist plot for the resulting open-loop system.
79
Example 14.6
Figure 14.1 Block diagram with a disturbance D and measurement noise N.
80
Figure 14.7 Bode plot for Example 14.6, Kc = 1.
10
The Bode plot for GOL and Kc = 1 is shown in Figure 14.7.
For ωc = 1.69 rad/min, OL = -180° and AROL = 0.235. For Kc = 1, AROL = ARG and Kcu
can be calculated from Eq. 14-10. Thus, Kcu = 1/0.235 = 4.25. Setting Kc = 1.5Kcu gives Kc = 6.38.
SolutionExample 14.6
81
Figure 14.8 Nyquist plot for Example 14.6, Kc = 1.5Kcu = 6.38.
SolutionExample 14.6
82
The Nyquist diagr. here is without the “mirrored” part!
Example 2a 4th order Transfer Function
83
1s3s2s5.0s
K)s(G 2c1
with: initial value:
Kc = 1P = 0
08.0jGlim)0(G0
1
Nyquist plot initial point ( = 0):
0jGlim)(G1
Nyquist plot final point ( ):
Extended Nyquist plot initial point (negative frequency ):
0jGlim)(G1