16.06 Principles of Automatic Control
Lecture 21
The Nyquist Stability Criterion
Can apply the argument principle to nding the stability of the closed loop system
r
+
G(s)
k
y

The closed loop transfer function is
T psq
Y psq
kGpsq
R
1 ` kGpsq
The close
16.06 Principles of Automatic Control
Lecture 24
Compensation
Compensation is the use of a dynamic controller Kpsq (as opposed to proportional control)
to improve the systems stability and error characteristics.
We have already seen compensation when we d
16.06 Principles of Automatic Control
Lecture 26
From last time, we had plant and compensator
Gpsq
1
p1 ` scfw_0.5qp1 ` sqp1 ` scfw_2q
Kpsq 9
1`s
1 ` scfw_8
The closedloop step response has 45% overshoot, when 37% expected. Why?
Look at Bode plot of H
16.06 Principles of Automatic Control
Lecture 20
Bode Plots With Complex Poles
Suppose we have a proportional feedback system:
+
G(s)
k

What values of k will lead to instability? Before we answer that, lets nd out what values
lead to neutral stability.
16.06 Principles of Automatic Control
Lecture 16
Frequency Response Design
Problems with root locus:
Works only for rational transfer functions:
Gpsq
ps zi q
ps pi q
If we start with experiment data, it may be dicult or impossible to put Gpsq in the fo
16.06 Principles of Automatic Control
Lecture 17
Frequency Response Design
Suppose we want to design a closedloop system with a specic desired response. How
might we use the FR to accomplish this?
2
n
Hpsq 2
2
s ` 2n s ` n
1
s
p n q2 ` p 2s q2 ` 1
n
(Bo
16.06 Principles of Automatic Control
Lecture 19
Bode Plots With Complex Poles
Secondorder term: Usually in denominator:
p sn q2
1
` 2 sn ` 1
This is plotted much like rstorder term, except slope in highfrequency regime is 2 (40
dB/dec).
Gpjn q
1
16.06 Principles of Automatic Control
Lecture 25
Lead Compensation
One problem with PD controller is that the gain gets large at high frequencies. So instead
use lead compensator
Kpsq k
1 ` scfw_a
1 ` scfw_b
What is the strategy? Look at Bode plot:
 
2
16.06 Principles of Automatic Control
Lecture 23
Stability Margins
Stability margins measure how close a closedloop system is to instability, that is, how large
or small a change in the system is required to make it become unstable. The two commonly
used
16.06 Principles of Automatic Control
Lecture 18
Bode Plot Construction (continued)
Note that phase of s term is
=pjq =j =j
90
To plot 1 ` scfw_a term, note that
1 ` jcfw_a p1 ` 2 cfw_a2 q1cfw_2
$
1,
!a
&
cfw_a, " a
?
%
2, a
Example:
Kpsq 1 ` scfw_20
16.06 Principles of Automatic Control
Lecture 28
The Nichols Chart
The Nichols chart may be thought of as a Nyquist plot on a log scale. A Nyquist plot is a
plot in the complex plane of
Gpjq RepGpjqq ` jImpGpjqq
looooomooooon looooomooooon
xcoordinate
y
16.06 Principles of Automatic Control
Lecture 30
Ztransform Inversion
There are 3 ways to invert a Ztransform:
1. Partial Fraction Expansion
Example
F pzq
z
pz 1cfw_2qpz 1cfw_3q
F pzq
3
z
z 1cfw_2 z 1cfw_3
Using coverup method:
But we dont know the in
16.06 Principles of Automatic Control
Lecture 29
Digital Control
At one time, most control systems were implemented using analog devices (operational am
pliers, linear circuit elements, etc). Today, most control systems are implemented using
digital devic
16.06 Principles of Automatic Control
Lecture 22
Nyquist Plot for Gpsq with jaxis poles
Consider
Gpsq
1
sps ` 1q2
Because of pole at s 0, must deform D contour pC 1 q.
Im(s)
C1
Re(s)
1
Bode:
Magnitude
50
0
50
100
150 2
10
10
1
10
0
10
1
10
2
Phase (deg)
16.06 Principles of Automatic Control
Lecture 27
Nonminimum Phase Systems
Our design rules so far are based on the bode gainphase theorem, which applies to stable,
minimum phase systems. The RHP zeros or time delays of NMP systems place fundamental
limit