NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Nyquist Stability Criteria
D Bishakh
Dr.
Bi h kh Bhattacharya
Bh tt h
Professor, Department of Mechanical Engineering
IIT Kanpur
Joint Initiative of IITs and IISc - Funded by MHRD
Module 2- Lecture 14
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
This Lecture Contains
 Introduction to Geometric Technique for Stability Analysis
Frequency response of two second order systems
Nyquist Criteria
Gain and Phase Margin of a system
Joint Initiative of IITs and IISc - Funded by MHRD
Module 2- Lecture 14
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 14
Introduction
In the last two lectures we have considered the evaluation of stability by
mathematical
th
ti l evaluation
l ti
off the
th characteristic
h
t i ti equation.
ti
R th’ test
Routh’s
t t and
d
Kharitonov’s polynomials are used for this purpose.
There are several geometric procedures to find out the stability of a system. These
are based on:
 Nyquist Plot
 Root Locus Plot and
 Bode plot
The advantage of these geometric techniques is that they not only help in
checking the stability of a system, they also help in designing controller for the
y
systems.
Joint Initiative of IITs and IISc ‐ Funded by MHRD
3
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 14
Nyquist Plot is based on Frequency Response of a Transfer Function. Consider N
i t Pl t i b d
F
R
f T
f F ti
C id
two transfer functions as follows:
T1 ( s ) 
s5
s5
;
T
(
s
)

2
s 2  3s  2
s2  s  2
The two functions have identical zero. While for function 1, the poles are at ‐1 and 2 respectively; for function 2 the poles are at +1 and 2 Let us excite both
and ‐2 respectively; for function 2, the poles are at +1 and ‐2. Let us excite both the systems by using a harmonic excitation of frequency 5 rad/sec. The responses of the two systems are plotted below:
Stable Frequency Response of T1
Unstable Frequency Response of T2
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 14
Frequency Domain Issues
Consider a closed loop system of plant transfer function G(s) and
Feedback transfer function H(s) respectively.
respectively
The closed loop transfer function corresponding to negative feedback
may be written as:
G ((ss )
T (s) 
1  G (s) H (s)
 Poles of 1+G(s)H(s) are identical to the poles of G(s)H(s)
 Zeroes of 1+G(s)H(s) are the Closed Loop Poles of the Transfer
Function
 If we
e ta
take
e a Co
Complex
p e Number
u be in tthe
e ss-plane
pa ea
and
d subst
substitute
tute itt into
to a
Function F(s), it results in another Complex Number which could be
plotted in the F(s) Plane.
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 14
Cauchy Criteria
Mapping:
A Clockwise contour in the s-plane results in
Clockwise contour in the F(s) plane if it
contains only zeros
A Clockwise contour in the s-plane results in
anti-Clockwise contour in the F(s) plane if it
contains only poles
If the contour in the s
s-plane
plane encloses a pole or a
zero, it results in enclosing of the origin in the F(s)
plane
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 14
Example:
p A Clockwise Contour in the s-plane
p
for G(s)
( ) = s-z1
Reference: Nise: Control Systems Engineering
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 14
Example: A clockwise contour around a Right-half Plane Pole for a
function G(s) = 1/(s-p
1/(s p1 )
Reference: Nise: Control Systems Engineering
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 14
Nyquist
yq
Stability
y Criteria
– Number
Number of Counterclockwise (CCW) rotation N of Counterclockwise (CCW) rotation N = P
Pc – Zc (Pc –
no. of enclosed poles of 1+ G(s)H(s) and Zc – no. of enclosed zeroes)
– For a Contour in s‐plane mapped through the entire right half o a o ou
s p a e apped oug
ee e g
a
plane of open loop transfer function G(s) H(s), the number of closed loop poles Zc (same as the open loop zeros) in the right half plane equals the number of open loop poles Pc in the right half plane minus the number of counterclockwise revolution N around the point ‐1 of the mapping. – Z c= Pc ‐ N NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Consider a plant transfer function G(s) as follows:
2
s  12s  24
G(s)  2
s  8s  15
For a unity
y feedback closed loop,
p, find using
g Nyquist
yq
Criteria
whether the system will be unstable at some values of K.
(Vary K from 0.5 to 10)
Module 2- Lecture 14
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 14
The Nyquist diagram corresponding to unity Gain and the root locus are shown below for your reference.
Nyquist Diagram
Root Locus
0.4
0.3
0.3
0.2
0.2
Imaginary Axis (seconds-1)
0.4
Imaginary Axis
0.1
0
-0 1
-0.1
System: tf1
Gain: 0
0.51
51
Pole: -6.63
Damping: 1
Overshoot (%): 0
Frequency (rad/s): 6.63
0.1
0
-0.1
-0.2
-0.2
-0.3
-0.3
-0.4
-10
-0.4
04
-1
-0.5
0
0.5
1
1.5
-8
2
-6
-4
-2
0
2
Real Axis (seconds -11)
Real Axis
Joint Initiative of IITs and IISc ‐ Funded by MHRD
11
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Gain Margin
g
The gain margin is the factor by which the gain can be raised such that the contour encompassed the unity point resulting in instability of the system. Following the figure below, gain margin is the inverse of the distance shown in the figure.
shown in the figure.
Module 2- Lecture 14
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Phase Margin
g
The Phase Margin is the amount of phase that needs to be g
p
added to a system such that the magnitude will be just unity while the phase is 1800 . The figure below is showing ‘theta’ to be the phase margin.
Often control engineers consider a system to be 0 q
y
p
g
adequately stable if it has a phase margin of at least 30
. Module 2- Lecture 14
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 14
Special References for this lecture
 Control Engineering and introductory course, Wilkie, Johnson and Katebi,
PALGRAVE
 Control Systems Engineering – Norman S Nise, John Wiley & Sons
 Modern Control Engineering – K. Ogata, Prentice Hall
Joint Initiative of IITs and IISc ‐ Funded by MHRD
14