Professor Walter W. Olson
Department of Mechanical, Industrial and Manufacturing Engineering
University of Toledo
Sensitivity
Outline of Today’s Lecture
 Review




Static Error Constants (Review)
Loop Shaping
Loop Shaping with the Bode Plot
Lead and Lag Compensators
 Lead design with Bode plot
 Lead design with root locus
 Lag design with Bode plot
 Important transfer functions
 Gang of Six
 Gang of Four
 Disturbance Rejection
 Noise Rejection
 Limitations
Static Error Constants
 If the system is of type 0 at low frequencies will be level.
 A type 0 system, (that is, a system without a pole at the origin,)
will have a static position error, Kp, equal to
lim G ( j )  K  K p
0
 If the system is of type 1 (a single pole at the origin) it will
have a slope of -20 dB/dec at low frequencies
 A type 1 system will have a static velocity error, Kv, equal to the
value of the -20 dB/dec line where it crosses 1 radian per second
 If the system is of type 2 ( a double pole at the origin) it will
have a slope of -40 dB/dec at low frequencies
 A type 2 system has a static acceleration error,Ka, equal to the
value of the -40 dB/dec line where it crosses 1 radian per second
Error
signal
E(s)
Input
r(s)
Loop Shaping
++
Controller
C(s)
Open Loop
Signal
B(s)
Plant
P(s)
Output
y(s)
Sensor
-1
 We have seen that the open loop transfer function,
B ( s )  C ( s ) P ( s ),
has profound influences on the closed loop response
 The key concept in loop shaping designs is that there is some ideal open
loop transfer (B(s)) that will provide the design specifications that we
require of our closed loop system
 Loop shaping is a trial and error process
 To perform loop shaping we can used either the root locus plots or the
Bode plots depending on the type of response that we wish to achieve
 We have already considered an important form of loop shaping as the
PID controller
Loop Shaping with the Bode Plot
 The open loop Bode plot is the natural design tool when designing in the
frequency domain.
 For the frequency domain, the common specifications are bandwidth, gain cross
over frequency, gain margin, resonant frequency, resonant frequency gain,
phase margin, static errors and high frequency roll off.
-3 db
Roll off Rate dB/dec
Resonant peak gain, dB
Bandwidth rps
Resonant peak frequency rps
Gain cross over
frequency rps
Loop Shaping with the Bode Plot
Increase of gain
also increases
bandwidth and
resonant gain
Poles bend the magnitude and
phase down
Zeros bend the magnitude and
the phase up
Break frequency
corresponds to the
component pole or zero
Lead and Lag Compensators
 The compensator with a transfer function
C (s)  K
sa
sb
is called a lead compensator if a<b and
a lag compensator if a>b
 The lead and the lag compensator can be used together
 Note: the compensator
sa
does add a steady state gain of
that needs to be accounted for in the final design
sb
a
b
 There are analytical methods for designing these compensators
(See Ogata or Franklin and Powell)
Lead Compensator
 The lead compensator is used to improve stability and to improve transient
characteristics.
 The lead compensator can be designed using either frequency response or root
locus methods
 Usually, the transient characteristics are better addressed using the root locus
methods
 Addressing excessive phase lag is better addressed using the frequency methods
 The pole of the system is usually limited by physical limitations of the
components use to implement the compensator
 In the lead compensator, the zero and pole are usually separated in frequency
from about .4 decades to 1.5 decades depending on the design
Lead Compensator (Frequency Design)
Note:
1) the lead compensator opens
up the high frequency region
which could cause noise
problems
2) The Lead compensator adds
phase
C (s) 
sa
sb
ab
 
a
b

1  sin f
f
m
b1
1  sin f
a  m 
b
Mechanical
Lead
Compensator
b2
a
xi
x0

k
y
Lag Compensator
 Lag compensators are used to improve steady state
characteristics where the transient characteristics are
adequate and to attenuate high frequency noise
 In order to not change the transient characteristics, the zero
and pole are located near the origin on the root locus plot
 The starting point for the design on a root locus is to start with
a pole location at about s = -0.001 and then locate the pole as
needed for the desired effect
 In order to not give up too much phase, the zero and pole are
located away from the phase margin frequency
Lag Compensator
Mechanical
Lag
Compensator
b2
xi
k
b1
b
a
x0
Note that the lag compensator causes
a drop in the magnitude and phase
This could be useful in reducing
bandwidth, and improving gain
margin; however it might reduce
phase margin
Sensitivity
 Sensitivity is an evaluation of how the system responds to
various signals compared to the design signal
 In general, we want the system to respond to the reference
input
 We do not want the system to respond to noises and other
signals that do not contribute to the accuracy of the desired
output
Several Transfer Functions
N Measurement
Noise
Disturbances D
R
++
F (s)
E
C (s)
Controller
h (s) 
 (s) 
U (s) 
E (s) 
PCF
1  PC
PCF
1  PC
CF
1  PC
CF
1  PC
F
1  PC
R(s) 
R(s) 
R(s) 
R(s) 
R(s) 
P
1  PC
P
1  PC
1
1  PC
PC
1  PC
P
1  PC
u
++
h
Y
++
P(s)
Y
-1
Y (s) 
U
Process
D (s) 
D (s) 
D (s) 
D (s) 
D (s) 
1
1  PC
PC
1  PC
C
1  PC
C
1  PC
1
1  PC
N (s)
N (s)
N (s)
N (s)
N (s)
E CL  R  h  R 
PCF
1  PC
R(s) 
P
1  PC
D (s) 
PC
1  PC
N (s)
The Model
R
++
F (s)
U
E
C (s)
-1
Controller
u
++
h
P(s)
++
Y
Y
Process
P
1
1  PC
1  PC
P
 PC
1  PC
1  PC
1
C
1  PC
1  PC
 PC
C
1  PC
1  PC
P
1
1  PC
1  PC
“Gang of Six”

Complementary
Load

Sensitivity
Sensitivity

Function
Function


PCF
PC
P
R
T
F

T

P
S


1  PC
1  PC
1  PC
 
 D 
 
CF
C
1

CFS 
CS 
S 
 N 

1  PC
1  PC
1  PC

Noise
Sensitivity

Sensitivity
Function


Function
“Gang of Four”
 PCF
1  PC

Y
   PCF
  1  PC
h
   CF
u   
  1  PC
U
   CF
 E  
1  PC
 F
 1  P C
N
D
Example
R
E
++
1
K (s  z)
U
T 
PS 
CS 
S 
1
s
s p

P
1  PC
C
1  PC
1
1  PC


K (s  z)
Y
s p
s  p 1
T 
K
sK
If p is in the rhs, this is unstable
PS 
K (s  z)  s  p 
s  s  p   K (s  z)
CS 
s
s  ps  K 
K (s  p)
sK
s p
s  s  p   K (s  z)
K
s
s  s  p   K (s  z)

++
s p
If p  z then P C 
s s  p
K (s  z)

++
Process
K (s  z)
1  PC
h
1
Y
Controller
PC
u
s
-1
PC 
N
D
S 
s
sK
T his is a additional reason w hy
pole/zero cancellation is a bad idea
Disturbance Rejection
 We want our system designed such that the disturbances to the system are
attenuated
 Harold S. Black gave us the answer: negative feedback
Y (s) 
PCF
1  PC
R(s) 
P
D (s) 
1  PC
1
1  PC
N (s)
C onsider the open loop case w here R ( s )  0 an d the plant effect is P  1: Y ( s )  D ( s )
In the feedback case, Y 
T he sensitivity function
1
1 C
S 
D 
1
1  PC
A s long as C is positive, the disturbance is reduced
is the tra nsfer function of the output to disturba nce
N
D
R
F (s)
E
++
U
C (s)
-1
Controller
u
++
Y
Process
h
P(s)
++
Y
Disturbance Rejection
Disturbances with
a frequency less
than the cross over
frequency are
attenuated while
frequencies higher
would be passed
Fortunately, most load
disturbances are low
frequency and often
can be treated as step
inputs
Noise Rejection
 We would also like noise rejection
 Noise is most often high frequency signals caused by the
sensors used to measure
 Noise is presented as a result of the feedback terms
 We do not have noise as defined here in an open system
 In the closed loop error, noise is multiplied by T, the
complementary sensitivity function,
T 
PC
1  PC
 In a system without a pre-filter, this is the transfer function
 For this reason high frequency roll-off is important
Limitations
 Systems with right hand side poles and zeros are inherently
hard to control
 For a system with right hand side poles, pk, Bode showed that


0
log S  i  d   

pk
 Improvements in one frequency region are met with
deteriorations in another frequency region
 Sometimes called the waterbed effect
Summary
 Important transfer functions
 Gang of Six
 Gang of Four
 Disturbance Rejection
 Noise Rejection
 Limitations
Next: Robustness