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63
·*!::~~
~·1.-.
POSICAST VERSUS CONVENTIONAL TYPES
OF COMPENSATION IN A CONTROL SYSTEM
by
GERALD COOK
B.S., Virginia Polytechnic Institute
(1961)
SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June, 1962
Signature of Author
Department of Electrical Engineering, May 18, 1962
Certified by
Thesis Su rvisor
Accepted by
Chairman, Departmental Committee on Gfduate Students
__
·
POSICAST VERSUS CONVENTIONAL TYPES
OF COMPENSATION IN A CONTROL SYSTEM
by
GERALD COOK
Submitted to the Department of Electrical Engineering
on May 18, 1962 in partial fulfillment of the requirements for the degree of Master of Science.
ABSTRACT
The purpose of this thesis research was to investigate Posicast, a novel type of compensation for controlling lightly-damped systems proposes by 0. J. M. Smith in
his book, Feedback Control Systems , and to compare a
system utilizing Posicast with one utilizing conventional
types of compensation.
The research was largely experimental with all
testing being performed on a Reeves Analog Computer.
In general the results were quite favorable. The
system compensated with Posicast showed a fast response
to deterministic inputs and also was well behaved when
subjected to a random input. The transient response to
a load disturbance was not as good in the system utilizing
Posicast as it was in the system utilizing conventional
compensation, but the steady-state response was slightly
better. The frequency response exhibited no resonance
peaking at all, but did exhibit ripples at frequencies
above the half-power frequency.
As would be expected, any decision as to whether
Posicast Wm conventional compensation should be used
would depend upon the specific application. I do feel
that Posicast definitely shows promise and is worthy of
consideration where lightly-damped systems must be controlled.
Thesis Supervisor:
George C. Newton, Jr.
Associate Professor of Electrical Engineering
Title:
ACKNOWLEDGEMENT
I wish to thank Professor G. C. Newton, Jr. for
his patience and guidance in supervising this thesis.
Thanks also to Mr.
G. B. Skelton of the Electronic
Systems Laboratory who offered numerous helpful
suggestions concerning testing on the analog computer.
TABLE OF CONTENTS
Page No.
ABSTRACT
2
CHAPTER 1
6
INTRODUCTION AND SUMMARY
6
Introduction
Theory of Posicast
System Investigated
Experimental Results
Conclusions
6
6
7
10
16
39
CHAPTER 2
42
PRELIMINARY DESIGN AND TESTING
42
The Preliminary Design
Realizing the Time Delay
Testing of the Two Systems
40
48
b0
FIGURES
1
2
3
4
5
6
7
8
9
10
11
Pictorial Description of Posicast
Step Response of a Typical Lightly-Damped
System
A System Utilizing Posicast
Step Response of a Lightly-Damped System
Utilizing Posicast
Lightly Damped System to Which Posicast
Was Applied
Uncompensated System of Figure 5
Operating in a Closed-Loop Fashion
Final System Utilizing Posicast
Conventional System
Step Response of Preliminary-Design
System
Step Response of Posicast Compromise
Design
Step Response Rf Posicast Desigped for
K, = 1.09 x 10 1 BI e 1.09 x 10
7
7
9
9
12
i1
15
17
18
19
_
12
13
14
15
16
17
18
Step Response of Conventional Design
System
Ramp Response of Posicast Compromise
Design
Ramp Response of Conventional System
Load Disturbance Response of Posicast
Compromise Design
Load Disturbance Response of Conventional Design
Random Input Response of Posicast
Compromise Design
Random Input Response of Conventional
Design
19 - 24
25
26
27
28
_
U1
22
23
26
27
28
29
Frequency Response of Posicast
and Conventional Systems
Pole-Zero Plot of Uncompensated System
Pole-Zero Plot of Preliminary Design
Analog Computer Simulation of Posicast
System
Analog Computer Simulation of Conventional System
30- 5
45
49
51
52
TABLE 1
IMPORTANT RESULTS OF THE INVESTIGATION
BIBLIOGRAPHY
37
54
~1~
POSICAST VERSUS CONVENTIONAL TYPES
OF COMPENSATION IN A CONTROL SYSTEM
CHAPTER 1
INTRODUCTION AND SUMMARY
Introduction
The problem of compensating a control system which
is very lightly damped has long confronted control engineers.
Numerous solutions to this problem have been proposed and
utilized with varying degrees of success.
0. J. M. Smith
in his book, Feedback Control Systems 3 , suggests another
solution to this problem.
Smith calls his method "Posicast",*
and it is the purpose of this thesis to investigate Posicast
and compare it to a conventional type of compensation.
Theory of Posicast
Posicast can best be understood by examining an
analogous situation.
in Figure 1.
Consider the suspended weight shown
The object is to move the weight from
position 1 to position 2 without exciting oscillations.
First, the support of the weight is moved one-half the
way from position 11 to 21.
The weight swings from
position 1 to 2, one-half of a cycle, and stops before
beginning its motion back in the direction of position 1.
* Because of the analogy of his method to casting a fly,
He then
Smith first called his method "positive-cast".
shortened this to the present name, Posicast.
I
2
2
I
1
1
I
2
2
2
1
2
1
Figure 1.
Pictorial Description of Posicast.
Input
Output
Figure 2.
Step iesponse of a Typical Lightly-Dampea System.
-- I~
-bL-
1
At the instant in which the weight is stopped, the support
is quickly moved to position 21, relaxing the system or
removing all energy from it.
The weight now having neither
driving force nor momentum remains at position 2, the
desired position.
Half-cycle Posicast functions in a similar manner.
For example, consider the unit step response of a typical
lightly-damped uncompensated system shown in Figure 2
where T, is the natural period of this system.
Posicast
acts to subtract a portion of the input signal so that
the first output peak instead of having the value 1 + A
will have the value 1.
allow only
•
system initially.
This means that Posicast must
of the input signal to be seen by the
Then at the time Tvn/Z seconds later
when the system output has reached the value 1 and stopped
for an
instant before
swinging
back
toward
aa lesser
value,
arr
Irlp~c~rJ~
uciur·e
i~ilrri~lrr~
ucr;K
buwar'u
It=;j;st=i'
vaiut=,
aur·
Posicast must
must act
act to
to bring
bring the
the input
input seen
seen by
by the
the system
system
Posicast
ri
ry
to J.I and
and
to
thus reduce
reduce the
the actuating
actuating error
error to
to zero,
zero.
thus
The
The
system now
now having
having neither
neither driving
driving force
force nor
nor momentum
momentum
system
will remain
remain atat rest
rest atat the
the desired
desired position.
position.
will
A block
block
A
diagram
of a lightly-damped system, W(s) utilizing
diagram of a lightly-damped system, W(s),, utilizing
Posicast
is shown
shown in
Posicast is
in Figure
Figure 3.
3,
The
The input
input operated
operated upon
by Posicast
Posicast and
the corresponding
corresponding output
output are
are shown
shown in
in
by
and the
4.
Figure
Figure 4.
In the
the frequency
frequency domain
domain Posicast
Posicast can
can be
be shown
shown to
to
in
infinite number or
of zeros,
zeros.
have an bnrinite
7 ---- --
---
·-·- - -
i-
--
Figure 6.
A System Utilizing Posicast.
I
(input aiter
by Posicas
Output
,0,000/
T'in/•
Figure 4.
Step Response of Lightly-Dampeu System Utilizing Posicast.
-
~L~i~ -
·---
**A
A
,
- Sfl/
Sf/•v% ,,Z
The first pair of zeros at
S --
n
-t
cancels the pair of resonant poles of W(s) and eliminates
any resonant peaking. The higher frequency pairs of zeros
produce a pronounced ripple in the frequency response.
This effect is illustrated in Figures 19 through 24 which
show the actual frequency response of the system investigated
and is discussed in detail following these figures.
System Investigated
The lightly-damped system to which Posicast was
applied is shown in Figure 5.
This system was one under
10
study by Professor G. C. Newton, Jr., who supervised this
thesis.
The load is a tracking-radar antenna, and the
spring-friction coupling accounts for resilient load members
with damping between the drive and the load.
An ideal
torque source was assumed, and the entire system was
assumed linear over the range of operation.
A block
diagram of the uncompensated system operating in closedloop fashion is shown in Figure 6.
Here K
which has
the dimensions ft. lb./rad. takes into account the transfer
function of the transducer, amplifier, and torque source.
A large portion of the work involved in obtaining
the final system compensated by Posicast was preliminary
work, that is preparing the system so that Posicast could
be used advantageously.
From the system parameters given in Figure 5 one
sees that the spring coefficient, K1 , and the friction
coefficient, Bl, can each vary by a factor of sixteen.
Considering only the extreme values of K1 and Bl , there
are four possible combinations of these parameters and
therefore essentially four different systems which must
be compensated by a single compensation scheme.
One of
the points of most interest was the effect varying these
parameters would have upon the response of the system
utilizing Posicast.
From the introductory discussion of
Posicast it is seen that the two governing factors of
Posicast design are the natural period, Tn, and the overshoot, A, of the system without Posicast.
Therefore, if
the preliminary design, that is the final design Just
11
rori
.l
U
- .5
2
Motor inertia,Jm=1 .6xl05
it.1b.spc./rca.
Load inertia,Jl=.6xl05 ft.ib sec./rad.
Friction coefficient,Bl=1.u9Xi0 5 to
f .lb.sec./rad.
1 7 . 4XIU0
to
Spring constant,K '1,.1x10i
17 .4XiU ft.ib./rad.
Figure 5,
Lightly-Damped System to Which Posicast Was Applied.
Figure 6.
Uncompensated System of Fi6ure 5.0perating
in a Closet-Loop Fasnion,
12
-C·~-*~ I-
2
-r-C_
--
before Posicast has been added, results in a system in
which the overshoot and natural period remain constant
as K1 and B1 are varied, then one Posicast design will
fit all four systems and no changes will be exhibited in
the system response as K1 and B1 are varied.
For the pre-
liminary design chosen, the overshoot and natural period
did not remain constant in all four systems, but they did
remain within ten percent of each other.
(The problem
of obtaining a preliminary design which results in the
natural period and overshoot keeping approximately the
same values while the system parameters are varied is
discussed in Chapter 2.)
Since the natural period and
overshoot were not the same for all four systems, the
design of the Posicast took the form of an engineering
compromise, being designed according to the natural period
and overshoot which most closely approximated those of
all four systems.
The values chosen for the Posicast
design were Tn=.205
seconds and A=.786
.
A block
diagram of the final system compensated by use of Posicast
is shown in Figure 7.
Professor Newton in his study of this system had
designed a compensation utilizing synthetic damping.
It
is this system to which will be compared the system utilizing
Posicast.
Throughout the remainder of this thesis the
system designed by Professor Newton will be referred to
as the "conventional" system.
The block diagram of the
system is shown in Figure 8.
13
F
F
-F
A =,765
T, =.205S -C.
K,=IOx IO FT.LB.A AD.
K,=Ix IO- 2 S EC.
KS= 9.4xlO - 4 S EC.
FIGURE
7.
-~r~A
- IN IALL
T.
POS
UTILlZING
SYSTEM
s
ls~
---
--
--
-
·-
; i-~-~~-~~-~-;~'"
".""""".~"~
---
-1.
K = 4 XIO FTLB/RAD.
K2 =4.85X10-2 SEC.
K3 1.85x 10i-3 S EC?
K4=2.5X 10- 2 SEC.
FIGU RE 8.
CONVENTIONAL SYSTEM
-·
i.·
_·L-
. ·.. .--
I
N.
Experimental Results
Both the conventional system and the Posicast
system were simulated on a Reeves Analog Computer and
tested for various inputs.
(The delay required for the
Posicast system was achieved by use of a tape recorder
and is discussed in detail in Chapter 2.)
In Figure 9
is shown the response to a step input of the preliminary
system before Posicast had been added.
The natural period
and overshoot can be determined from this figure and have
values of which the averages are those stated previously,
Tn=.205
seconds and A=.786
.
Also Figure 9 con-
trasted with Figure 10 serves to point up the effect brought
about by Posicast.
Figure 10 shows the response of the
complete Posicast system to a step input.
This is the
system which was designed as a compromise for the four
different combinations of values of K1 and Bl.
The rise
times for the different parameter combinations are all
approximately 0.1 seconds, the rise time spoken of here
being defined as the time required for the system output
to rise from zero percent to one-hundred percent of its
final value.
Another Posicast compensation was designed, this
one specifically for the parameter values B1 = 1.09 x
lo5ft. lb. sec./rad. and K1 = 1.09 x 108 ft. lb./rad.
Figure 11 shows that the step response for this system
with B1 set at 1.09 x 105 ft. lb. sec./rad. and K1 set at
A.
............
%
I
\T i
Output
;V
._·___··
·
· · I_·
I_
·_ _·I ·
t (sec.)
K1 =1.09W10 8
BLI1.UO~iOb
Output
-w
B1=17
SK1=17.4w108
4
X105
A
Ai
Output
..
·
13
ofý105
Bjr·--r8=l.
Ki=17.4910
-.-q:-:~-.
-i •....
i --::i:• i
iS.1
-·I..
·L =
.~...
i .-.
--...
i .
Output
I'C-·' :
ps:r
....
.....
.. ·..
i:
.L -: ...
:
. . .:.
.le,
-..
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..
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·~i----·-··.... .....
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:i-·-!·
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i
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I~
....
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K=1.(18rqxt
=17.4XJCý~
-
Ii
.
-
I:-·-
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..
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J7
~r-····-····----,··
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·-· ·--· ·-· ··-·- ·· ·-------- -----------··-····----·---T..... i .... T
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11I
• .........
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......
I
I
I
t eC)~
4IX1.
=..C~xi
Ell
t ( . (ec.)
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Figure
10.
Step i--(spoime of' Posic%ýast (io.;ir:rl.l1
ir
iI
i
1V
.
-----
Ir
L----
·--
-
-
---
.......
.........
i.:i .. j-.:i~--i.
·:i
O'ut
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. .. .i...
. ...
. .........
-'I.
·.
....
...
,-
....
.,~. .....
.........
,..
,·
- .t
::
j
t( e c.)
7 ý.x ace,
F;
=1? 4XJC)"
······
:·;··..11··
..... . .
.il
..
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i:
k .4X1.
t
.
I:-
i;
-77 .7
F
i
imj
(I
I
:.r.....
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..i.....i.......
!..
. ;.:.
. .
II
-------------:...
K-f-#'-j r-ij±.
77.-I
:....
L
rI
iI
Hii
i
t(
)
1T
1
i;
2.
I
i::
i
g~p
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t
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:·
Pe Ponse- cf
Fe.I·
I
:
:
y e.)
I E-:ý
J ;71
C
u
xi
ICI-~-·T
__
1.09 x 108 ft. lb./rad. was quite good with a rise time of
.09 seconds.
However, when K1 and B1 were varied, the
results were not as good as they were for the system which
was a compromise.
The rise times were almost the same,
but the slight oscillations observed in Figure 10 are seen
to have increased considerably.
These oscillations are
caused by both inaccurate timing and improper step
magnitudes in the Posicast.
It should be understood that
Posicast does not increase the damping in the system, but
rather eliminates oscillations when properly designed by
not permitting them to be excited.
When mistuned Posicast
allows slight oscillations to be excited, their magnitude
depending upon the degree to which the Posicast is mistuned.
The step response of the conventional system is
shown in Figure 12.
This system is seen to be slightly
underdamped with a very small overshoot.
varied from .15 seconds to .2 seconds.
The rise times
Note that the time
scale in Figure 12 is different from that of Figures 10
and 11.
The ramp responses of the Posicast and conventional
systems are shown in Figures 13 and 14 respectively.
each case the output is well behaved.
errors differ somewhat.
In
The steady-state
The velocity constants,
f;n?37(S5)
S---O
20
~.. . ...
. .....
....
...
4 .
----
77-f-·- i .;
I
..
.......
.........
r--·
_ .T
I
~~...-i-.:
-177 7'!
Out!ju1 C
F-71:
CIO("
(-xi·
t (EC.)--I · ..
flY~'t
.
_.li-_.r'..
.
.......... . ..
7
t
--
.... .
pu~t
..
:.:
i~......
·
.......
Ali--
_j .....
13.1=1 . 4XI 0"
_
L7
..
Outtpui t
~.._!~
' -'T':-
··j-:-i-7
.
....
...
....
.
7777
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Y11
i:.
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14 J,
f
77
-:
77
_-:4 7,::
:::lL:t:
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7
77
-.4~ i i: ti :::'i I::li;l:;~
7r 77
:::::i it
!Tl:
i;
l= 1 7.4X1lO
Bf=l.Oi;X
G
I
Step Response
I
Figure 12.
of ConventiQjial Dc..-i(~ n
~:-_
_ -;
· ·
'u9r·;~I-I
es~lrr~,rcTmon
7;'33firIrci
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E,917.4X105
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i
].-.--j
:----- '---·:-.: -. i- -.:.~~~i
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i il__:j-:::jl:i
i
i
m
t (sec.)
Output
j
Kg=1.0&X1lo t
BI Z7.4'xlC'
I
'1-
·
<4...:i._l
i
--~~-.__,._....__. .
t ( e c.)
Cvtvt
KI =17.4zw1&~
1
Fizi.O~1O"
t (sec.)
Figure 14.
Ramp Response of Conventional System.
--
---L--
c-~ii~
are K2 1
(K2 + K431
100 sec.-1 for the Posicast system and
13.6 sec
for the conventional system.
For a ramp input of WC radions/second, the steady-state
error for the conventional system is .074 4.2 radians.
For the Posicast system the steady-state error between I I
of figure 3 and the output is .01COA
radians.
However,
an additional error is incurred between the input and I I
because of the delay brought about by Posicast.
is of the amount
47)N 7R~ x
This error
0Q
O4,50radians
-
and the total error becomes .055 CO radians.
Figures 15 and 16 show the responses of the systems
to step load disturbances.
The Posicast system response
is seen to be highly oscillatory.
As far as load dis-
turbances are concerned, the Posicast system is no different than the preliminary design system since Posicast as
used here does not modify load disturbances before permitting the system to see them.
Posicast can be utilized in
such a manner as to eliminate oscillations caused by load
disturbances, and the reader is referred to Feedback Control
Systems3, page 341 for further discussion of this.
The torque constants for the systems are 10 x 10 ft.
lb./rad. for the Posicast system and 4 x 108 ft. lb./rad.
for the conventional system.
Therefore, the steady-state
error of the Posicast system U x 10'
9
radians is less than
-9
that of the conventional system, U x 2.5 x 10-9 radians,
but the transient response of the Posicast system is much
24
L=-*.
Th
r
-ui-
1
less desirable than that of the conventional system.
Figures 17 and 18 show the response of the two
The input signal was achieved
systems to a "random" input.
by manually varying the amplitude and frequency of a trianglewave generator.
The frequency waz va1ied from zero to ten
cycles per second, approximately twice the bandwidth of the
systems.
No attempt was made to derive quantitative results
from the random input test, but speaking qualitatively, it
is seen that both systems are well behaved, and this was
the main point of interest, especially concerning the
Posicast system.
The frequency response curves for the systems are
As was mentioned in the
shown in figures 19 through 24.
introductory discussion on Posicast, the resonance peaking
has been entirely eliminated by Posicast.
The most striking
feature of the Posicast response curves is the ripple
exhibited at frequencies above the half-power frequency.
However, this phenomenon is directly related to the infinite
column of zeros of Posicast in the S plane demonstrated
earlier.
As the frequency of the system is varied along
the
the jW@axis,
in magnitude from
_
+
.
4O
--
T7/rV,
value 1 occurs at
term varies
-1A
I-A
occurs at
where
A
and
c~07/2, -
Wrs,_/2L
71
/
The value
CZ,) or CO GOrzn,')
is any integer.
h 71 or
',
The
C6p
C (CZ).
Since the remaining factor of the transfer function,
25
-.
-~
~....--
~,-
...
-----.
.:...........
I.
..........
Output
........
......
......
K l=.O9xlO
i .....
..
...
...
B=1:.Ox1l0
t se.
1,10
t(sec.)
I
I-...I
.
Output
Ki=17.4x10b
B-=17.4xl0
:
I
F -
.,..1.1.0
I10
t (sec.)
i
4j......
I
Output
I
I
...[**'
... ?..
I
. I
.·· ;:II:-···r··
.. I:·-·-··.1..-- ·: I' :·;- r1
Kl=B.U9xlO0
II...,.
Bl=17.4xI05
t (see.)
I
II
Output
17.4x10
Kl=.
Bl=1.tGxlO
....~ ...
.......
...
-
K r~
i
......
......
·
! · ··-...-........
........
iii
·
:.... .......
7......I..
FL1*' J0'~~~
n-f-,"~"~'
:.
''
20'
(sec .)
' 10
1
20
t (sec.)
Figure 15.
Load Disturoance tesponse of Posicast Compromise Design.
26
I~_
U
n-
I1
Output
I
I
.0
1
t(Lec.)
Output
8
KI=17.4x10
Bl=17.4xiu
b
t(sec.)
..
Output
.
. ...
. .
.. . .
.
. . ....
..
K1ll.Oixlio
bl=17.4xlO 5
t(sec.)
"i l...
...
.. ........
.
.':
. ..
....
.. . .
..
.....
...........
..
•I.....
I .....
....i...
..J..
Output
ii~i··.
••....-:
Kl=17.4xi0
8
Bj=1.0 ixlO0
•
...
..
......
: .:.. . ..
....
i......•
i,.....
::-+ •.....
-:•-!":':
....
•i..
: .:. i - -:. : -
,I
-.
i.,:
i ....
......
-:..:
..
-..
.....
-:i.:
..
S....
- ....
.
·
I
t (sec,)
Figure 16.
Load Disturoance Response of Conventioial
DLesign.
ý-il
cle
1/
INPUT
/`
/
·.
OUIPUT
__
·
I'z
KI=1.09x10
8
t (s c.)
Bjjl. Oxl10
INPUT
...
.....
.. . ..
..
.......
.......
.
I 1....
i-'i
......
~u
j -.... .
.. ... ....
... .....
:..l..
r^...
!....
r....
,..
.i...O
J•PU
i
Kl=17.4x10
BI=17.4xl0 5
t (sec.)
Figure 17.
Random-Input Hesponse of Posicast Compromise Lesign.
-"I
c~lc~
-i-
INPUT
OUTPUT
LI-l. UIXLU
Bl1.0 kxiU-
PUTL
L_-t.~_.:....t
... :
·- · ·--~----t·- -··
in
I
1
I
Kl=17.4xlO 8
I.
WTPU.EI
I
-
4
B1-17.4110
5
-·I
t(sec)
Figure 18.
Random-Input hesponse of Conventiunal Design.
O
-J0
0
D0I
.1
10
I
RADIANS/SEC.
FIGURE
I-"------"T---arn-aa^~
--
-3crq
-
-
-
19.
100
1000
i
O
O
i
0
O
O
O
.01
.1
1
10
100
1000
RADIAN S/S EC.
FIGURE 20.
-
~--~-~ -
-_I-. -
-
-
--
- -p~,~-i~==an~,---·En~--.1
c-c- -__
pp
lO
O ,
O
I
r
II
·
r
I
.K=I17.4x10 8 Fl EL B./RAD.
I,= I. 0•x10 5 F T. LB.S EC/R A D.
.
R.
POSICAST COMPROMlISE
...-
DESIGN
-----CONVENTIONAL
11
I·
~
DESIGN
rr
I
I
.01
I
I· I
I0
I
1000
100
RADIANS/S EC.
FIGURE 21.
-·-----------_
.~--L
---
-Y-l-LI
-L~.I
______
13--
C--..
e
-
~--.
·..
i-
--
-
-
~
·
_
I
F
it
I·_~II--I-I·_~I- _~__L
_·__I~~---II~·~--I- ---~i-~--l-l·IC------.·
-F-·-n·i.--· lr ·iS.i~·.
1
::
........
- 41I
..
...
. .....
i:
r't:
1
-·$--·-i··-i- j
·-:·-·i
··
·-LI----
:
-:--i
:
1
'I
-··1·--·--·-
i
0
0
-j
0
K-.09X108 FT. LB./RAD.
17.41AO FT.LB.SEC./ RAD.
z.
-I0
---- POSICAST C.OMPROMIS DESIG N
- ,CONVENT1IO •L DESIG N
4-
.01
10
100
I000
RADIANS/SEC.
FIGU RE 22,
----
-~`-----
LI
I
c'
-
III
I
I
~'
"'~"
`"~~`~~~"~'~
~`
---- ------- ·--~ -----·---- ;----·-
L
I,'
I
W
i-
--- Kt 17A XlO8 FT. LB/RAD.,, = 17.4 KI OFT.LB.SEC./RAD.
---- K=
1O9XlO8 F LB/RA D., B,= 1.09 105 FT. LB.SEC/RAD.
8
to
POSICAST DESI GNED FOR K,= .O9xIO,
1
-20 1.O
__.
B,=LO9X1O 5
~~____
__
100
10
1000
RAD IANS/SEC.
FIGU RE 23.
L~---
i
.. _~
-
_
..· ~
*I
~~~Y
..~._i_.~.
i....
__
_
-
IU
O
O
-10
-20
-20
.01
J
I
10
RADIAN S/S EC.
R
(y
~IGURt
24,
100
1000
W(s) as shown in Figure 3, is generally a monotonically
decreasing function at frequencies above Qn, it is seen
that depending upon the rate at which W(s) drops off with
increasing frequency and the ratio of the extreme values
which the 1
4- A
term may take on,
various degrees of rippling may be exhibited at the higher
frequencies.
The important results of the investigation are
listed in Table 1.
36
'I
i
I
TABLE 1
Conventional
System
Posicast
Compromise
System
Posicast
Designed
For :
/(,:- ,x/o
I(Ir 4o 0FX /O0
Average rise time for step input.
.17 sec.
.1 sec.
.08 sec.
Average overshoot for step input
of magnitude N radians.
.012 N rad.
.03 N rad.
.033 N rad.
.074 4rad .
.055 Arad.
.056olJrad.
Average steady-state error for velocity input of magnitude cc rad/sec.
Average error overshoot for step
load disturbance of magnitude U ft.lbs.
0
Average settling time for step load
disturbance (time to settle within
5% of final value)
.11 sec.
Average steady-state error to step
load disturbance of Magnitude U.
2.5x0l9 Urad.
.5 U rad.
.3 sec.
.5 U rad.
3 sec
ixlO-9Urad. ixlO' 9 U rad.
TABLE 1 CONTINUED
Conventional
System
Posicast
Compromise
System
Posicast
Designed
For
9/ =-oTx/0
-
Average half-power frequency
(frequency where
WJI -.707).
Average Resonant Peak
(arithmetic scale).
27.25 rad./sec.
29.5 rad./sec. 29 rad./sec.
1.17
1.0
1.0
Conclusions
From Table 1, we see which factors have been improved and which ones have suffered adverse effects
through the use of Posicast.
Perhaps the most important
improvement is the reduction of the step response rise
time.
The bandwidth has been increased but only slightly.
The resonant peaking has been eliminated entirely, but
in the conventional system the peaking was not enough to
create any serious problems.
The steady-state error for
velocity inputs favors Posicast.
By far the factor which
suffered most through the use of Posicast was the transient
error response to a load disturbance.
However, even with
its fifty percent overshoot the error of the Posicast
system is at no time as large as the steady-state error
of the conventional system.
Because Posicast eliminates
resonant peaking and overshoot in the response to input
signals, higher gain and less damping can be used in a
system, thus bringing about a reduction in steady-state
errors.
One requirement imposed by the use of Posicast is
that the input signal must be explicit.
In most control
systems the only signal which is of importance as far as
the system is concerned is the error signal, and this is
the signal measured by the system when in operation.
For
the use of Posicast, the input can be made explicit by
39
__
measuring the output signal and adding this to the error
signal through a summing amplifier.
However, these
measurements as well as the Posicast network must be very
accurate since the output of the system can be no more
accurate than the input regardless of how good the system
is. Also the input in the case of the tracking-radar
antenna may vary from zero to 360 degrees.
The net requirement then becomes an accurate
measurement of angles up to 360 degrees, an accurate
summing device to add the error and output signals, and
a Posicast network capable of handling a wide range of
inputs.
If a magnetic tape or similar device is used to
achieve the time delay, the range of the input signal may
be quite significant.
One possible means of alleviating
the situation would be to shift the output reference
periodically to the operating point at that time so
that the magnitude of the output signal measured from
the reference could be maintained reasonably small.
More
work on this problem would be required in any system in
which the input is not explicit.
An analysis of how Posicast compares to the conventional system depends upon which characteristics of
system behavior are most important in the specific
application.
If the problem Just mentioned can be over-
come, I believe that Posicast should be considered for
compensating lightly-damped systems since it has demonstrated several favorable characteristics.
The remainder
40
of this thesis is concerned with the problem of achieving
the preliminary design and with simulating and testing the
systems.
Illl)irC
--c_~--
2
CHAPTER 2
PRELIMINARY DESIGN AND TESTING
The Preliminary Design
This section is concerned with the preliminary
design, that is the design required in modifying the
original system shown in Figure 6, to a form to which
Posicast could be applied advantageously.
The main objective of the preliminary design was
to obtain a stable system with a short natural period,
a large velocity constant, and a large torque constant.
No effort was made to limit the overshoot since theoretically this could be eliminated by Posicast.
It was desired
that the overshoot and also the natural period vary only
slightly as the system parameters K] and B1 were varied
since the design of the Posicast depended upon these
two factors, natural period and overshoot, and was to
remain unchanged for all values of K1 and B1.
An examination of the closed loop transfer function
of the original system shown in Figure 6 reveals that it
is unstable for all real values of Km.
Km(BlS + K1 )
eerror
S2 [JmJL S2 +Bl(J
+ JL) S + KI(Jm + JL)I
__
Km(BIS
__m
4 K1)
2
SJmJLS +B (JmJL )S34K (Jm4JL)S
mBlI
S
+KIK1
The corresponding array for the Routh Criterion is:
JmJL
Bl(Jm 4 JL)
KIB1(Jm 4 JL)2
-
KmB1 [KlBl(Jm4JL)
JmJLKmBl
2
- JmJLLmBIB
- B12 (Jm + JL) 2 KnK1
Bl(m + JL)
K1 (Jm 4 JL)
KmK1
KmBl
KmK1
By restricting all the terms in the first group to being
greater than zero, we arrive at the condition Km2 < 0
which cannot be satisfied by a real number.
For Km = 0,
the system is marginally stable.
By examining the open-loop transfer function of
the system and noting the second order pole at the origin,
it is realized that phase lead is very desirable.
Based
upon this observation it was decided that a lead network
in the forward loop should be given some consideration.
Examining the pole zero sketch of this transfer
function shown in Figure 25 and realizing the desirability
of moving the poles to the left and away from the imaginary
43
From
axis, one recognizes the need of increased damping.
this observation it was decided to consider synthetic damping
through the use of rate and acceleration feedback subtracted
at the input summing point.
Once the types of compensation to be considered
had been selected, the design was largely trial and error
since the objectives of the preliminary design did not
correspond to those of any known optimizing techniques.
Much of the work was performed on the analog computer.
First a second order lead network of the form
( ~i..I)
was designed for the system.
frequency of the term JmJLS2+ Bl
( Jm J L )S
4 K
The resonant
(J m J L
)
of
the open loop transfer function was found to be fifty
radians per second.
The lead network was designed to
give its maximum lead effect at this frequency.
0C was
chosen to be 20 since this is generally the largest practical
value it can be given.
Tc was .0045 seconds, calculated
from the relation
where Q. is the frequency
where maximum lead is desired.
The lead network achieved
a phase lead of 1300 at 6~ and stabilized the system for
a limited range of values of Km.
However, when this
system was simulated on the analog computer and tested
for a step input, the shortest natural period obtainable
was two seconds.
It was not known how short a natural
period it would be possible to achieve, but it was known
that the average rise time of the conventional system was
.17 seconds.
Since in the Posicast system the rise time
-1.15+j48
.--
I
_ Ipp__
e
le
|
I
i
C,ý
IN
-1.i-j46
Figure 25.
Pole-Zero Plot of Uncompensated SsStem.
45
_ ~__
__
is one half the natural period, it was felt that a natural
period considerably less than two seconds should be striven
for.
Next rate feedback from the load was attempted.
This compensation added damping and stabilized the system
for a limited range of values of Km.
The system was
simulated on the computer and tested for a step input,
varying the amount of rate feedback and in each case
using the largest value of Km permitted by stability
considerations.
Still the shortest natural period obtain-
able was approximately two seconds.
At this point load-
acceleration feedback was introduced, but no significant
improvement resulted.
Rate feedback from the torque source was then
attempted and the natural period was decreased to .26
seconds.
The relatively short natural period was encourag-
ing, but neither it nor the overshoot remained even close
to constant when K1 and Bl were varied.
However, since
this scheme had given the most favorable results up to
that time, it was decided to give it further consideration
and to attempt to improve its performance by modifications.
Load-rate feedback in conjunction with the torque-sourcerate feedback was then attempted but with no improvement.
Finally it was found that best results could be
achieved by using load-acceleration feedback along with
torque-source-rate feedback.
The natural periods for the
four combinations of values of K1 and B1 averaged .205
46
_i ;~_ _~I~~
_
__I_*I_·
___
__p___
seconds and the overshoots averaged .786.
All values
were within five percent of these average values.
step responses are shown in Figure 9.
i
The
The torque constant
was 10 x 10
ft. lb./rad. and the velocity constant was
100/second.
The reason for desiring large torque and
velocity constants obviously was to decrease the steadystate errors to load disturbances and velocity inputs
respectively.
One may wonder at the preliminary design being
terminated at this point since better results could
conceivably be achievable.
In a trial and error design
such as this in which there is no analytical method which
can be used to establish guide figures, there is no way
of knowing when an optimum or near optimum system has
been achieved.
Therefore, recognizing that the torque
constant, velocity constant and rise time compared
favorable with those of the conventional system, and
realizing that a limited amount of time was available
for
hlis phase of the project,
it was decided that this
preliminary design would be used.
A block diagram of
this design is shown in Figure 7 where the Posicast
has been included also.
The open loop transfer function of the preliminary
design system chosen is given below.
4,
-9error
Km(B 1 S + K 1
)
3
Jm LS4+ [B
I (Jm-JL)-Km(K B-K2L) S
+ [Kl(Jm+JL)+Km(KlK 3'eBlK 2) S2+KK2KlS
47
Figure 26 shows a pole zero sketch of this system and
exhibits the increased damping achieved through the use
of the rate and acceleration feedback.
The damping could
be increased further by increasing the quantities of rate
and acceleration feedback or decreasing Km, but increasing
the quantity of rate feedback decreases the velocity
constant and decreasing Km decreases the torque constant.
Doing either of the above increases the natural period.
Damping was required to stabilize the system and permit
use of a sufficiently large value of Km to make the system
reasonably fast.
Beyond this it was felt that damping
was more detrimental than helpful toward the system
response since Posicast prevents oscillations due to
input signals from being excited.
Therefore, higher
damping was not sought.
Realizing The Time Delay
After the preliminary design was obtained, the next
problem confronted was that of achieving a time delay for
the Posicast network.
transmission line.
Smith 3 had suggested the use of a
An effort was made to approximate this
by means of lumped elements, but when simulated on the
analog computer this scheme gave very poor results.
The
possibility of approximating the Taylor expansion for
e-STn/2 with a ratio of two polynomials and simulating
this on ~he analog computer was also considered, but
48
7 im
j6o
v
-37+-3102
Ihi
-86
Figure 26.
Pole-Zero Plot of Preliminary System.
49
J
_·
.___
published literature on simulation techniques reported
this method to be stable only for frequencies from zero
to fifty cycles per second which would rule out step
inputs.
Finally an Ampex F. M. tape recorder with separate
record and play-back heads was obtained.
The spacing of
the heads was 1.5 inches and the only tape speed at which
the recorder would both record and play back was thirty
inches per second.
This meant that a delay of .05 seconds
could be achieved through the use of this tape recorder.
Since a delay of Tn/2 or .103 seconds was required,
time scaling had to be introduced into the computer simulation to make the system operate at .103/.05 or 2.06 times
real speed.
This was achieved without difficulty by
increasing all integrator gains in the simulation by a
factor of 2.06.
The system was now ready to be tested.
Simulations of the Posicast system and the conventional
system appear in Figures 27 and 28 respectively.
Testing of the Two Systems
Throughout the testing of the Posicast and conventional systems and the interpretation of the results,
an attempt was made to maintain an objective approach.
The tests used were those which it was believed would
give the most information about the behavior of the
systems.
The same tests were performed several times
50
~...__~dt
0= 1/
I+A
= A /I+A
0)= Km /40x
08
.4x•10I/,~
K, /8 x 108
0:: 8K(Q/J;
(9=B(/4 x(0 5
- .1
(5-=8X 10 /J
.2
@- K,xlo7/LT
-= so K
Figure 27
Analog Computer Simulation of Posicast System at
1/ 5 0th
Real Speed
(Computer was switched to 100 times real speed to give a final speed of 2 times real speed)
I -- i -Y --
_-
·. I i 1 _ _
---~s~--L----~CI~·-
-i-·1~~-_ - - ·1--
;· ;e Li_·ii -~pi'i---_it
._-_c.rzFi=Lt;;n;=_?
·
U
.4
K, //40,I10"
SlOS/ T
K, /4 x 10'
4J;.
- 10O6/
Ios/4 JL
(D= 1,/4 x 0
= K, x 10 '/J,
-= .4
I=
.s/47
O= 160 K-5
FIGURE 28
Analog Computer Simulation of Conventional System at
1/ 1 0th
Real Speed
I____
on the systems to make sure that the results were
consistent.
Also, each test was performed with various
magnitudes of input to ensure that the response was independent of magnitude.
The step input and the step load disturbance were
achieved simply by actuating a contactor connected to a
D. C. voltage source.
The velocity input was obtained
by passing a step function through an integrator on the
analog computer.
A sine-wave generator was used for
determining the frequency response.
For the random input,
a triangle-wave generator was utilized with the frequency
and magnitude of the signal being varied manually.
It is
recognized that this type of input is not truly random,
but it is believed that it is somewhat representative of
the type input to which one might expect a trackingradar antenna to be subjected.
It was hoped that the
systems could be tested with a noise generator in cascade
with a low-pass filter, but no noise generator was readily
available of which the lower half-power frequency was less
than the upper half-power frequency of the two systems
to be tested.
All important results and conclusions of this investigation have been listed and discussed at the end of
Chapter 1.
53
I_I__
_
_
BIBLIOGRAPHY
1.
Newton, G. C. Jr., Gould, L. A., and Kaiser, J. F.
Analytical Design of Linear Feedback Controls,
John Wiley and Sons, Inc. New York, 1957
2.
Savant, C. J. Jr., Basic Feedback Control System
Design, McGraw-Hill Book Company, Inc. New York, 1958
3.
Smith, 0. J. M., Feedback Control Systems,
McGraw-Hill Book Company, Inc. New York, 1958
_