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Calhoun: The NPS Institutional Archive
Theses and Dissertations Thesis Collection
1968
Computer techniques for implementing linear control
system design using algebraic methods.
Walker, John Andrew
Monterey, California. Naval Postgraduate School
http://hdl.handle.net/10945/25778
NPS ARCHIVE1968WALKER, J.
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COMPUTER TECHNIQUES FOR IMPLEMENTING LINEAR
CONTROL SYSTEM DESIGN USING ALGEBRAIC METHODS
by
John Andrew Walker Jr.Lieutenant, United States Navy
B.S.M.E., University of Notre Dame, IQ61
Submitted in partial fulfillment of therequirements for the degree of
MASTER OF SCIENCE IN ELECTRICAL ENGINEERING
from the
NAVAL POSTGRADUATE SCHOOLJune I968
ABSTRACT
Design of linear control systems "is examined using
algebraic methods* Previous work indicates that the system
describing eguations are nonlinear,, Various combinations of
feedback and cascade compensated systems are analyzed
using velocity constant, bandwidth and complex root speci-
fications. Methods for I i near i 2 i ng . these system eguations
are demonstrated and examples are solved using specific
digital computer programs,, Feasibility of a general computer
program is discussed.
\
TABLE OF CONTENTS
Section Page
1
.
I ntroduct i on 9
2. Basf c TheoryI |
3. Third Order Development [4
4. Examp I es 36
5. Genera! Program 65
6. Conclusions and Recommendations 70
7. B i b I i ography 72
APPENDIX I Table of Funct i ons K (S) 73
APPENDIX II Listing of Programs 74
LIST OF ILLUSTRATIONS
Ff gure
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.3
3.9
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
Basic System Block Diagram
Third Order System with TachometerFeedback
Third Order System with AccelerationFeedback
Third Order System with Tachometerand Acceleration Feedback
Third Order System with Single CascadeCompensat i on
Third Order System with MultisectionCascade Compensat i on , Factored Form
Third Order System with MultisectionCascade Comoensat i on, Polynomial Form
Third Order System with Cascade andTachometer Feedback Compensation
Third Order System with Cascade andAcceleration Feedback Compensation
Block Diagram for Examp
Block Diagram for Examo
Block Diagram for Examp
Block Diagram for Examo
Block Diagram for ExamD
Block Diagram for Examp
Block Diagram for Examo
Block Diagram for Examo
Block Diagram for Examo
Block Diagram for ExamD
Block Diagram for Examp
e 4.1
e 4.2
e 4.3
e 4.4
e 4.5
e 4.6
e 4.7
e 4.8
e 4.9
e 4.10
e 4.11
Page
31
32
32
33
33
34
34
35
35
37
39
41
43
45
48
50
52
55
58
60
4.12 Block Diagram for Example 4.12
5.1 General Program Flowgraoh
Page
63
69
s
s
w n
w b
Ajs)
Gfc(s)
G (s)
Gc (s)
H(s)
K
K +
TABLE OF SYMBOLS
complex variable of form t+ju-
dampinq ratio
undamped natura I frequency
bandwidth frequency
M i t r o v i c function
forward transfer function
open loop transfer function
closed loop transfer function
feedback transfer function
forward system qain
tachometer qa i n
acce I erometer qain
ve I oci ty constant
7
Acknowledgements
The author is indebted to Dr G, J, Thaler of the
Nava ! Postgraduate School for the guidance and direction
provided during the preparation of this thesis,,
8
I. Introduction
Design is a word of many meanings and forms. Generally,
it means a method or procedure for determining the value of
system parameters such that the system will respond in a
prescribed manner. These methods are usually derived for
synthesis in the time domain by apolying modern state soace
techniques or in the complex frequency domain using root
locus, Bode, Nyquist, or Nichol's plot.
All the methods have their advantages and disadvantages.
Time domain analysis with state space allows the designer
to examine the entire system at one time. He can only select
one set of parameter values and investigate their effect on
the system. The advent of digital comDuters greatly aided
this process because the computer is easily adapted to a
systematic trial and error design procedure.
The complex frequency plane methods are inherently
trial and error. These methods are graphical and extremely
time consuming in their application. Present day programming
is gradually coming to the rescue but it has limitations in
the amount of useful and legible information that can be
displayed on a graph.
An outgrowth of the graphical methods was Mftrovic's
method which was generalized by the Parameter Plane technique
This procedure transforms the analytical model of the system
into a graphical representation. The benefit of this method
is that it can handle multiparameters.
9
From the parameter plane technique system design has
returned to the analytical wor ! d and developed a multi-
parameter procedure called algebraic methods. Algebraic
methods reduces the system differential equations and spec-
ifications prescribing the desired response into a set of
simultaneous algebraic equations. The simplicity of the
method is that it finds the parameter values that collect-
ively provide the required system response. The method has
developed over the years but its implementation on computers
has not been explored. It is the intention of this thesis
to investigate this area.
II. Basic Theory
The backbone of the algebraic methods is expressing the
systems characteristic eguation as a polynomial of the form
I o^S* -OK-O
where s is the comp I ex vari ab I e and is defined as
S- -St*V 4- tJ^HN/l-S*'
(2-1)
(2-2)
and the a^'s are real coefficients which are functions of
the system parameters. Two independent eguations in a ^ , S ,
and wn
are obtained if equation (2-2) is substituted in
equation (2-1 ) and the real and imaginary parts set equal
to zero. Performing additional manipulation and introducing
a new variable (1), (2), we obtain
I O^U)* &-,($)= O (2-3)K-O
£ aK u)n 0*(S) - o
where
with
0JS)=-[2S0US&+0k-z(S)]
(2-4)
(2-5)
0.(S)-O , 0,(S)^-1 MX> 0. K($)=-te (2-6)
Other values ofp^bjfor various values of S are tabulated
in Appendix I. Observe that for a real root S= - G" and
substituting in equation (2-1) gives
V\
(2-7)
Previous work provided qraphical methods for portraying
root locations as a function of any two system oarameters.
(2), (3), (4), (5). If we are willing to abandon graphical
analysis then the equations can include any number of unknown
oarameters. Proceed algebraically by substituting root
locations in the basic equations (2-3) and (2-4) giving two
equations as functions of a I I the unknowns. Requirement for
solution is the number of equations equal the number of un-
knowns. One o!: . ., s additional equations by one of two
phi I osoDhi es; first, substitution of additional root locations
into the basic equations; second, derivation of additional
algebraic relationships from system performance specifications.
Determination and solution of the proper number of equations
results in a simultaneous set of oarameter values that satisfy
stated requirements. Naturally, if the parameter values are
not satisfactory the performance specifications must be altered
or the additional equations generated changed and a new
solution affected.
Inherently these equations provide a set of nonlinear
simultaneous equations to be solved. The non I i near i t i es are
generated from the system configuration or conversion of the
Derformance specifications into algebraic relationships. The
purpose of this thesis is to develop methods of solving these
nonlinear equations. Basic approach will be that of linear-
ization by substitution or manipulating the form of system
components to facilitate linearization. When linearization
12
fails it is hoped the system will lend itself to one of the
standard analytical methods for solving linear simultaneous
equat i ons (6)
.
13
III. Third Order Development
The basic system general block diagram is shown in
Figure 3-1. In this section we will use a type one, third
order plant to demonstrate the different solution technigues.
Naturally, it is reguired that some type of compensation be
employed, namely, feedback, cascade and feedback-cascade
comb i nat i on, so that the system satisfies the specifications.
The third order system allows for development of the method
while keeping the eguation length reasonable. As needed
reference will be made to higher order systems and some
higher order examples will be solved.
3.1 Feedback Compensation
Consideration will first be given to tachometer feed-
back, Figure 3-2, where the system gain (K) and the tachometer
gain (K-j-) are unknown. Solution for the unknowns reguires two
simultaneous eguations. Since the most common system speci-
fication is a pair of dominant complex roots, eguation (2-3)
and (2-4) will be implemented.
The system characteristic eguation is
sS(PitPi)Si-MPiP2+KK*)S+ MO (3-D
from which the a k coefficients and the given S and U)»\ are
substituted in the root eguations giving
Kfc,t$Wn P21- WkX&(9*(H* W)0*?jl|©)* ulM»($ = o (3-2)
|4
It can be seen these equations are nonlinear due to the
KK-t term. Linear equations are obtained by making the
following transformation of variables
X-KY-KKt
After making the substitution and rearranging we get the
following matrix form
(3-4)
T
(3-5)
Linear equations are easily solved by digital computers
with programs using matrix algebra for implementing the
standard linear equation solving techniques. It is a simple
ODeration to inverse transform the linearizing substitution
and obtain the parameter values
K~Xand
ternObserve that for this specific case the matrix equation
degenerates to give
A ~9-i cs)
and
'"fc(S)
because O (5) =
(3-6)
(3-7)
(3-8)
15
Another possible method of feedback is acceleration
feedback, Figure 3-3, which is very similar to tachometer
feedback but affects the second order coefficient of the
characteristic equation instead of the first order coeffic-
ient. This type of compensation generates a system with two
unknowns, therefore, requiring two equations which we will
obtain from the root specifications. The system character-
istic equati on i
s
sVn+«+WGSf + P1P2S+K - O (3-9)
Appropriate substitution in the root equations yields
and
^^tftKVk^+(^P^KK^^)+^^«) = o (3-11)
As in the tachometer feedback case the following linearizing
transformation of variables can be made
Making the substitution and rearranging gives the matrix
simultaneous equations
(3-12)
&($) UZpi-%- r
tx
-RLP2UXAjC^-(P1+K^(/|($-W^(S)
&(3 u£A$ Y ^m^Y^^A^Y*^)(3-13)
\6
For this type of feedback we do not get the degeneracy
due to thecortunate location of O ( S ) » but it is seen the
matrix equations for tachometer and acceleration feedback
are the same except for the second column of the coefficient
matrix. Therefore, we can exoect that when both types of
feedback are used simultaneously the matrix equation will
be very similiar to (3-5) and (3-12). At this point we will
state the results if a fourth order system is used. The
matri x equat i on i s
rj
-.«**»
3 L J L -**4<k$(3-14)
From this it is seen that the form is the same as the third
order case and all the increase in system order did was add
a term to the right hand side.
Let us now combine the types of feedback, Figure 3-4,
and solve the problem. Immediately it is seen that before
we can proceed we have to develop another equation besides
the root equations. A frequently used specification is
error constants which has different meaning depending on
the type of system. For the type one system this means we
can use the velocity constant defined as
|<v - Lvm SGoCs)(3-I5)
17
w f t h the steady state error for a ramo input, R(s) = l/s'
WThe characteristic equation for the system is
(3-16)
(3-17)
and the forward transfer function is
So(s) =
5 [ Cvv?i>(S4-?a) +*>** -«-KK^]
Aoolying the root equations and the velocity coefficient
equation we obtain the following set of simultaneous non-
linear equations in K, KK^., and KK a
(3-18)
(3-19)
qM$K+ (pim ^)\AA.4>u{§\&\n+Y&^liii) wl^Li) - o( 3_20 )
As in the seoarate type of feedback compensation we can apply
the same transformation of variables to linearize the equations
(3-22)
Making the substitution and rearranging we obtain the matrix
equation
~> <v K
LT<
>
r(3-23)
18
As predicted the above result is very similar to the
feedback cases examined separately* The coefficient matrix
second column is from the tachometer equation (3-5) and the
acceleration equation (3-13). Similarly, the right side is
not chanqed at all. From this we conclude the type of
compensation for the root location soeci f i cat i on affects terms
in the coefficient matrix while the right hand side is affected
only by the system order.
A possible specification for generating an extra equation
is bandwidth. The conventional definition of bandwidth will
be used, that is, the frequency at which the magnitude of the
output is one half the input magnitude for the closed loop
transmission with S = jWu. Because the magnitude involves
squaring the transfer function we can expect to gain a
nonlinear equation of a rather difficult form. For feedback
compensation, it is found the oroblem is not extremely difficult
Consider the tachometer feedback, Figure 3-2, where the
open loop transfer function is
GoCs)x _J£_ _( 3_24)
S(H?lfeR)H^S
and the closed loop transfer function is
C*cCS> = !S (3-25)
to which we apply the bandwidth requirement giving
-^,t
[Pl?2-W ba+KKt]+[»C-UJb
l (mP232 "
To simDlify the exoressf on make the substitution
and a (3-27)
whereby clearing gives
-K^+ZAK-^&UJ^K^-tA^Ui^^-LOb2 K2Ki * O (3-23)
Before proceding any further let us examine the second eguation
required for solution. Select the velocity constant specifi-
cation because a complex root specification generates two
equations and it is not our intention to deal with over
determined systems, hence using
Kv-^m sf" £ JS-»o L S(s-t-?l)(^tP2) t- ^<tS J
gi ves
K^V*-fc> -VU/P1V2.(3_2q)
Examination of equations (3-28) and (3-29) shows an extreme
non I i neari ty . Fortunately, if we consider KK-{- as an unknown
we can use the elimination method to reduce the system to a
quadratic equation in forward gain K. Solving eguation (3-29)
for KK t
and substituting this value in equation (3-28) gives
t-\.0-Wb/K7] X +L2A-ZW*/kv -v2Wb*?lWKvlK
20
This is easi ly solved for the system gain K yielding two
values which can be substituted in equation (3-30) to give
two va I ues of K-j.
At this point the designer has a choice of which set of gains
to use for the solution. Usually it is not a difficult choice
because one of the gain values is negative. Situations where
both gains are positive values the designer can pick the values
that suit his purpose.
Now we* I I shift our attention to the acceleration feed-
back, Figure 3-3. From previous expressions it can be expected
the feedback gain in the second order term will help the
solution. As in the tachometer case we will use bandwidth and
velocity constant specifications. The velocity coefficient is
Kv = K/PLP2.
giving one equation
The bandwidth specification gives
1 _* — (3-34)
where A^Pm-CAJt2- *«2> &-UJ&M+PI)
Rearrangement and substitution for K from (3-33) gives a
a quadratic in Ka .
21
+ [®x-c<jff)*~ *BPi P2K\, -(PI Plk»)
ZJ -0 (3-35)
Again this gives two values for the acceleration and forward
gains. As in the tachometer case the choice is left to the
designer but usually one of the choices will have an unde-
sirable negative gain.
3.2 Cascade Compensation
Cascade compensation is a widely used method of compen-
sation. In general, it is commonly refened to as a lead,
lag, or lead-lag filter. In this development we will not pre-
specify the type of filter or a criteria for rea I i zabi I i ty.
The decision of rea I i zabi I i ty is left to the system or filter
d e s i g n e r
.
The first case examined is the single section compen-
sator, Figure 3-5. In keeping with the major requirement we
need three equations to solve for the unknown system gain (K),
filter zero(Z), and the filter pole(P). If given a complex
root specification and a velocity constant we can generate
the required equations. A possible unknown is eliminated by
considering any gain associated with the filter to be included
in the overall system gain.
The system characteristic equation is
22
indicating one non I
i
near? ty , the KZ term. The velocity
constant specification yields
k, K(S+*)Cs+p)(^k2)^^J
KZ(3-37)
s^o P1PZP
which has the same nonlinear term as the characteristic
equation, making the now familiar transformation of variables
X-Kh (3-38)
provides linearity with the three unknowns P, K, X which after
affecting the root equations gives the matrix equation
O -KsPJ-PZ.
J
K
- J*>«.%- (*>*<k (3-39)
After solution we once again inverse transform the variable
for the f i Iter po I e
*- (3-40)
Unlike the feedback system a change in the system order
affects not only the right side of the matrix equation but,
also, the left hand coefficient matrix. The reason for this
effect can be seen if the reduction of the system to a forward
transfer function with unity feedback is examined. The filter
pole multiplies all poles of the basic Dlant, thus inserting
23
the filter pole in all the characteristic coefficients
except a n and a 0<>
The next logical consideration is a multisection cascaded
filter. To demonstrate the best analysis form we will use the
double section filter, Figure 3-6. The addition of each filter
section adds two more unknowns, the section zero and pole,
which reguires two extra eguations per section added. At the
risk of completely specifying the system roots the easiest
specification to use giving two eguations is a root specifi-
cation, otherwise other specifications will have to be trans-
lated into algebraic eguations. The root specification can
be very helpful since if not given explicitly it can be selected
such that it does not effect dominance of any specified roots.
Before proceding further, to ease the notation, multiply
the plant denominator giving
where
A*P/+PJ
6* P.' "Pi1
(3-41)
(3-42)
Reducing the overall system by normal methods, we achieve the
following characteristic eguation
+n(p±pz) **.)£"+ (anpit **i +*tz) s+ tint ~ o (>43)
At this point, note the transformation of the unknowns into
the seven groupings PI, P2, PIP2, KZI
, KZ2, K, and KZIZ2. This
24
double section may not be that unwieldy to solve but the
situation gets worse as the sections increase. A simple
manipulation can alleviate this undesirable condition.
Conversion of the filter to polynomial form vice factored
form prevents the generation of undesirable nonlinear terms.
Granted some remain but they can be handled by appropriate
variable transformation. The polynomial form of the filter
i s
'(3-44)
where ^~ ^' "*>*
VJ- fi-t Pl (3-45)
Using this filter form in the system reduction we achieve a
new and simpler characteristic eguation
The nonlinear terms can be linearized by
(5-46)
(3-47)
The application of the root equations twice and the velocity
constant
Kv = Ltw> 5 K(s\x^Y) - KY3 ~*o feVwfr*^(s**ifcVB*) Bit
25
yields the matrix simultaneous equation form
[*1±5K
- ffil(3-49)
where for root specification one the coefficients
A (*,0^ <£*(£)
A (ltl) « &&«.&{$)+#w£^($) y-^^jCS)
A (*A) ~ cu^^(s)
A (-*•,$)- BuJ^^t(t) -hAUJ^^s) +&H4fa($)SCO ~ ~£0Jn4i($)-At0>t fo(?)~ &*L
Sfa{$)
(3-50)
are functions of zeta one and omega one. For root specifi-
cation two the third and fourth rows have the same form as
rows one and two, ie,
(3-51)
anda($) = bo)
£U) ~ Bfz)
26
where A(3,i), A(4,i), B(3) and B(4) are functions of zeta two
and omega two. The remaining row is
A(Sj) = -/-O
(3-52)
From this development we can see the polynomial form
filter provides the transformation of variables from Z« to
Ka | _
| for i = 1, ,n. After solution of the linear equations
the filter zeros and poles can be found by factoring the numer-
ator and denominator.
3.3 Combined Compensation
Having considered feedback and cascade compensation
individually we will now examine the combination of both types
of compensation, Figure 3- 1 . Attention is given first to
tachometer-cascade compensation, Figure 3-8, which indicates
nonlinearity may be a problem. In this section the system
gain is assumed known including the filter gain. Naturally,
the three required equations are from velocity constant and
root specification.
The characteristic equation is
+ (ZL'PtP+KXk* +<)S+ k$r**0 (3-53)
from which we apply the root equations and obtain the non-
linear matrix equation
27
A 8 C o
£ P D
K o //
where
>19
i
T(3-54)
8= XKv
/f- **k# (3-55)
di = -P±Pz w*4l (f)~(pj.+Pi)tt)*k(f)-a)*h(s)
H = *>*" tfWrJ /<
Similar to the bandwidth situation this case lends itself to
the elimination method of solution such that equation (3-54)
reduces to a quadratic equation in tachometer gain K-(- from
which the compensator pole and zero may be found. As previ-
ously the two answers give a choice which should operate under
the same restrictions previously discussed.
Finally, we shall consider the combination of accelera-
tion feedback-cascade compensation. Experience indicates the
28
acceleration gain wilt simolify the equations, but for this
case we find the compensator zero prevents this complete
simplification. Using the root and velocity constant specifi-
cations with the characteristic equationS«+ (PJ+-PZ JP+ W*)S^ {P1P2 + PP1-+ PP2+Hk.<JLt)s
;L
+ ( PJLPZPffC)S V- <* ^^ (3-56)
we obtain the nonlinear matrix equation
where
C
1
a
D
O
O£
o
T
[7*1
* I
1K*i
(3-57)
C = (Pl*pz)u)*?'d>,(f) + iv*<fiJS)
D- - K fa ( S)
6 = - pipzcu£ &(?) - (pi+ pi) to<c,
4dS)-CJU *<£*(?)
H = - KU)n.h($) - MPl totfatQ -{&tPi)a)Jfaff-Mf4ii f)
The system of equations can be reduced by elimination (like
the tachometer case) to a quadratic in acceleration gain K a
from which the compensator pole and zero can be found.
In this section we see the simultaneous equations
(3-58)
29
easily lend themselves to solution by elimination but appear
to be close to !inear This is true if a fourth specification
is given that doesn't create non I
i
neari t i es (like bandwidth).
The linearization method used throughout this thesis can be
used for the K-^Z or K aZ term. The situation is somewhat
easier if a multisection filter is used. There is a nonlinear
product for each section of filter. Therefore, if the number
of filter sections is even then specify an additional k/2
complex roots, linearize and solve.
30
/>
>(M)
1+
XIc-I
I
CD0)
CO
CO
CO
r+fD
3
CD
OOX"
0)
X)-I
0)
3
^r
I
1
o oo Q)
3 co QT3 O —0> 0) --N
3 a. coco a> •
—
oo r+3 "n
—'•
{r> ft> o
1fD <v 3
i 3 a Ico cr -—•*
0)0) CO
!r+ O >—
'
i — X*
1 o
I
3
!
I
i |
1 /
i
t) CO— o0) ^a coH- —
.
O
31
K
Third Order System With Tachometer Feedback
Figure 3-2
h *<*Pi K C
I — 1
nS(S+Pl)(S+P2)
KaS 2
.
Third Order System With Acceleration Feedback
Fi gure 3-3
32
Third Order System With Tachometer AndAcceleration Feedback
Figure 3-4
z s±z.S+P
JtS(S+PI )(S+P2)
Third Order System With Sinn I e Cascade Compensation
Figure "5-^
33
R4-
-V
\ r (S+Z|)($+Z.2)(S+P|)(S+P 2 )
)
w Ks(s+p|Xs+p»
)
Third Order System With Multisection Cascade CompensationFactored Form
Figure 3-6
.a ;
i\
S^+XS+Y.s2+ws+z
JLS(S+P»)(S+P£)
Third Order System ^'ith Multisection Cascade CompensationPo
Iynomi a I Form
Figure 3-7
34
R L z-J
s+zS+P S(S+P| ) (S+P2)
Third Order System With CascadeAnd Tachometer Feedback Compensation
Figure 3-8
R T+
<? S(S+P| )(S+P2)
Third Order System With CascadeAnd Acceleration Feedback Compensation
Figure 3-9
35
IV. Examp I es
Examples in this section were solved on an IBM 360
computer using subroutine programs furnished with the
computer. The programs used are specific in nature and
appear in Appendix II. The following points apply to all
examp I es
:
a) negative feedback is desired so that negative
feedback constants are not acceptable since this implies
positive feedback.
b) when a cascade pole or zero is found positive
sign indicates left half plane and neqative sign indicates
ri ght ha If p lane.
c) for bandwidth and feedback-cascade combination
examples, there are no logic statements in programs to select
the proper set of parameter values for determining the system
roots. This part of the program was written with a priori
knowledge. For roots the sign convention mentioned above does
not apply, use the standard sign convention.
36
Examp I e 4.
I
Fi gure 4.
I
Using feedback compensation design a system with
open loop transfer function in Figure 4,1.
Root S pecifi cation: Zeta = 0.5, Natural Frequency = 10.0
No Velocity Constant or Bandwidth Specification
37
INPUT DATA
**********
POLES ARE ZERO, 0.500000 01, O.fcOOOOD 01
ROOT SPECIFICATIONS ARE ZETA= 0.50000D 00,NATURAL FREQUENCY^ 0.100000 02
NO VELOCITY CONSTANT SPECIFIED
NO BANDWIDTH SPECIFIED
OUTPUT DATA
***********
FORWARD GAIN
C.10000E 03
TACHOMETER GAIN
C.80000E OC
COEFFICIENTS OF CHARACTERISTIC EQUATION(ASCENDING ORDER)
C.10000E 03 0.11CO0E 03 0.11000F 02 0.10000F 01
THE SYSTEM ROCTS ARE
REAL PART
-0.10000E 01 -0.50000E 01 -0.50000E 01
IMAGINARY PART
0.0 -0.86603E 01 0.86603E 01
38
Exarno le 4.2
Figure 4.2
Design an acceleration feedback system with the open
Iood transfer function shown in Figure 4.2.
Root specification: Zeta = 0.5, natural frequency = I 1.0
No velocity constant and bandwidth specification.
This example indicates an unattainable solution since
solution gives negative K and Ka and a right half Diane
root.
39
INPUT DATA
**********
POLES ARE ZERO, 0.10000D 01, 0.50000D 01, 0.60000D 01
ROOT SPECIFICATIONS ARE ZETA= 0.50000D 00,NATURAL FREOUENCY= 0.110000 02
NO VELOCITY CONSTANT SPECIFIED
NO BANDWIDTH SPECIFIEO
OUTPUT DATA
***********
FORWARD GAIN
-0.10010E 04
ACCELERATION GAIN
-0.82645E-01
COEFFICIENTS OF CHARACTERISTIC EQUATION(ASCENDING ORDER)
-C.10010E 04 0.30000E 02 0.12373E 03 0.12000E 020.10000E 01
THE SYSTEM ROCTS ARE
REAL PART
-0.34194E 01 0.24194E 01 -0.55000E 01 -0.55000E 01
IMAGINARY PART
0.0 0.0 -0.95263E 01 0.95263E 01
40
Examp I e 4.3
Figure 4.3
Redesign of Example 4.2.
Root Specifications: Zeta = 0.5, Natural Frequency = 12.0
No Velocity Constant and Bandwidth Specified
41
INPUT DATA
**********
POLES ARE ZERO, O.IOOOOO 01, 0.50000D 01, 0.60000D 01
ROOT SPECIFICATIONS ARE ZETA= 0.50000D 00,NATURAL FREQUENCY= 0.120000 02
NO VELOCITY CONSTANT SPECIFIED
NO BANDWIDTH SPECIFIED
OUTPUT DATA
***********
FORWARD GAIN
0.36000E 03
ACCELERATION GAIN
0.29306E CC
COEFFICIENTS OF CHARACTERISTIC EQUATION(ASCENDING ORDER)
0.36000E 03 0.30000E 02 0.14650E 03 0.12000E 020.10000E 01
THE SYSTEM ROCTS ARE
REAL PART
-0.60000E 01 -0.60000E 01 -0.37068E-07 -0.37068E-07
IMAGINARY PART
-0.10392E 02 0.10392E 02 -0.15811E 01 0.15811E 01
42
Examo I e 4.4
+ K
S(S+I) (S+2)
KaS^ + K T S
Figure 4.4
Design a third order system with acceleration and
tachometer feedback for the ooen loop system in Figure 4.4.
Root Specification: Zeta = 0.7, Natural Frequency = 10.0
Velocity Constant = 6.0
No Bandwidth Specification
43
INPUT DATA**********
POLES ARE ZERO, O.IOOOOO 01, 0.20000D 01
ROOT SPECIFICATIONS ARE ZETA* 0.700000 00,NATURAL FREQUENCY= 0.10000D 02
VELOCITY CONSTANT IS 0.600000 01
NO BANDWIDTH SPECIFIED
OUTPUT DATA
***********
FORWARD GAIN
0.37500E 04
TACHOMETER GAIN
0.16613E OC
ACCELERATION GAIN
0.12933E-01
COMPUTED VELOCITY CONSTANT IS = 0.60000E 01
COEFFICIENTS OF CHARACTERISTIC EQUATION,(ASCENDING ORDER I
0.37500E 04 0.62500E 03 0.51500E 02 0.10000E 01
THE SYSTEM ROCTS ARE
REAL PART
-0.70000E 01 -0.70000E 01 -0.37500E 02
IMAGINARY PART
0.71414E 01 -0.71414E 01 0.0
44
Examp I e 4.5
:®-JKD- K
S(S+2)(S+6)(S+9){3+!0)
KaSd+K TS
Figure 4.5
Design a fifth order system with acceleration and
tachometer feedback for the open loop system Figure 4.5.
Root Specification: Zeta = 0.5, Natural Freguency = 4.0
Velocity Constant = 0.5
No Bandwidth Specification
45
INPUT DATA
**********
POLES ARE ZERO, 0.20000D 01, 0.60000D 01C.90000D 01, 0.100000 02
ROOT SPECIFICATIONS ARE ZETA= 0.500000 00,NATURAL FREQUENCY= 0.40000D 01
VELOCITY CONSTANT IS 0.50000D 00
NO BANDWIDTH SPECIFIED
OUTPUT DATA
***********
FORWARD GAIN
0.13349E 04
TACHOMETER GAIN
0.11909E 01
ACCELERATION GAIN
0.65497E-01
COMPUTED VELOCITY CONSTANT IS = 0.50000E 00
46
COEFFICIENTS OF CHARACTERISTIC EQUATION(ASCENDING ORDER)
0.13349E 040.27000E 02
0.26697E 040.10000E 01
0.10354E 04 0.25400E 01
THE SYSTEM ROCTS ARE
REAL PART
C.63277E OC0.20000E 01
-0.11184E 02 -0.1U84E 02 -0.20000E 01
IMAGINARY PART
0.00.34641E 01
-0.26026E 01 0.26026E 01 -0.34641F 01
47
Examp !e 4.6
Figure 4.6
Design a tachometer feedback system for the open
looo system in Fi'gure 4.6.
No Root Specification
Velocity Constant = 0.91
Bandwidth Freguency = 4.0
Systems roots are for number two aain values since system
gain one is n e o a t i v e .
48
INPUT OATA**********
POLES ARE ZERO, 0.50000E 01, 0.60000E 01
NO ROOT SPECIFICATION
VELOCITY CONSTANT I S= 0.91000E 00
BANOWIDTH FREQUENCY IS= 0.40000E 01
OUTPUT DATA***********
SYSTEM GAIN ONE = -0.31554E 02TACHOMETER GAIN ONE= 0.20496E 01
SYSTEM GAIN TWC= 0.41920E 02TACHOMETER GAIN TWO= 0.38325E 00
COEFFICIENTS OF CHARACTERISTIC EQUATION(ASCENOING ORDER)
0.41920E 02 0.A6066E 02 0.11000E 02 0.10000E 01
THE SYSTEM ROOTS ARE
REAL PART
-0.12317E 01 -0.48841E 01 -0.48841E 01
IMAGINARY PART
0.0 -0.31905E 01 0.31905E 01
49
Examp I e 4.7
R ^<p-^> h8 (3+3) (8+4)-m«u>m i»»»i»iiimiii> ii ii
Figure 4.7
Design an acce lerometer feedback tyitem far the
loop system Figure 4.7.
No Root Specification
Velocity Constant = 5.0
Bandwidth Frequency 3.0
System roots are for gain values number one sinet Ks<
is negatl ve.
eetn
50
INPUT DATA**********
POLES ARE ZERO, 0.30000E 01, 0.40000E 01
NO ROOT SPECIFICATION
VELOCITY CONSTANT IS= 0.50000E 01
BANDWIDTH FREQUENCY IS= 0.30000E 01
OUTPUT DATA***********
SYSTEM GAIN CNE= C.feOOOOE 02ACCELERATION GAIN GNE= 0.49388E-01
SYSTEM GAIN TWC= 0.60000E 02ACCELERATION GAIN TWO= -0.50494E 00
COEFFICIENTS CF CHARACTERISTIC EQUATION(ASCENDING ORDER)
C.60000E C2 0.12000F 02 0.99633E 01 O.IOOOOE 01
THE SYSTEM ROOTS ARE
REAL PART
-C.29863E OC -0.29863E 00 -0.93660E 01
IMAGINARY PART
0.2513AE 01 -0.25134E 01 0.0
51
Examp I e 4.8
R ^ £±Z_S+P S(S+5)(S+7)
Figure 4.8
Compensate the open loop system in Figure 4.8 using
a single section cascade compensator.
Root Specification: Zeta = 0.5, Natural Frequency = 6.0
Velocity Constant = 4.15
No Bandwidth Specification
52
INPUT DATA**********
POLES ARE ZERO, 0.50000D 00, 0.700000 01
ROOT SPECIFICATIONS ARE ZETA= 0.50000D 00,NATURAL FREQUENCY^ 0.600000 01
VELOCITY CONSTANT IS 0.41500D 01
NO BANDWIDTH SPECIFIED
OUTPUT DATA
***********
SYSTEM GAIN
0.13756E 04
COMPENSATOR POLE
0.37847E 02
COMPENSATOR ZERO
0.39961E OC
53
VELOCITY CONSTANTtCOMPUTED)
0.4150GE 01
COEFFICIENTS OF CHARACTERISTIC EQUATION(ASCENDING ORDER)
0.54972E 03 0.15081F 04 0.28735E 030.45347E 02 0.10000E 01
THE SYSTEM ROCTS ARE
REAL PART
-0.39200E CC -0.30000E 01 -0.30000E 01 -0.38955E 02
IMAGINARY PART
0.0 -0.51962E 01 0.51962E 01 0.0
54
INPUT DATA**********
POLES ARE ZERO, O.IOOOOD 01, 0.150000 02
ROOT SPECIFICATIONS ARE ZETA* 0.665000 00,NATURAL FREQUENCY* 0.596000 01
VELOCITY CONSTANT IS* 0.280000 02
NO BANDWIDTH SPECIFIEO
ROOT SPECIFICATIONS ARE ZETA* 0.50600D 00,NATURAL FREQUENCY* 0.14660D 02
OUTPUT DATA***********
FORWARD GAIN
0.34637E 04
FILTER ZEROS ARE
REAL PART
-0.51508E 01 -0.91788E 01
IMAGINARY PART
0.0 0.0
56
Ex amp ! e 4.9
R4-
- A
H»"S2+X+Y
"s"2+fs+F S(S+I)(S+I5)
Figure 4 . Q
Design a double section comoensator for the ooen
loop system Figure 4.9.
Root Specifications: Zeta I = 0.665, Natural Frequency I
5.96; Zeta 2 = 0.506, Natural Frequency 2 = 14.66
Velocity Constant = 28,0
No Bandwidth Specification
FILTER POLES ARE
REAL PART
-0.14107E 02 -0.14107E 02
IMAGINARY PART
-0.13817E 02 0.13817E 02
COEFFICIENTS CF CHARACTERISTIC EQUATION(ASCENDING ORDER)
0.16376E 06 0.55482E 05 0.10125E 050.85632E 03 0.44213E 02 0.10000E 01
THE SYSTEM ROCTS ARE
REAL PART
-0.39634E 01 -0.39634E 01 -0.21451E 02 -0.74180E 01-C.74180E 01
IMAGINARY PART
-0.44512E 01 0.44512E 01 0.0 -0.12645E 020.12645E 02
57
ExamD I e 4.10
L _a±z.
S+P500
S(S+5)(S+3)
Figure 4. 10
Design a single section cascade compensator with
tachometer feedback for open loop system Figure 4.10.
Root Specification: Zeta = 0.5, Natural Frequency = 7.0
Velocity Constant = 0.5
No Bandwidth Specified
These specifications can not be satisfied. Choice one
gives negative feedback gain and roots indicated. Choice
two gives undesirable right half plane compensator.
58
INPUT DATA**********
SYSTEM POLES ARE ZERO,, C.50000E Olt 0.80000E 01
ROOT SPECIFICATIONS ARE ZETA= 0.50000E 00,NATURAL FREQUENCY= 0. 70000E 01
VELOCITY CONSTANT IS = 0.50000E 00
NO BANDWIDTH SPECIFIED
OUTPUT DATA***********
CHOICE ONE
TACHOMETER GAIN= -0.73364E-01COMPENSATOR POLE= 0.15640E 02COMPENSATOR ZERO = 0.60346E 00
CHOICE TWO
TACHOMETER GAIN= 0.60888E 00COMPENSATOR POLE= -0.46819E 02COMPENSATOR ZERO= -0.26924E 01
COEFFICIENTS OF CHARACTERISTIC EQUATION(ASCENDING ORDER)
0.30173E 03 0.11035E 04 0.20664E 03 0.23640E 020.10000E 01
THE SYSTEM ROOTS ARE
REAL PART
-0.28840E 00 -0.35000E 01 -0.35000E 01 -0.21352F 02
IMAGINARY PART
0.0 -0.60622E 01 0.60622E 01 0.0
59
Examp I e 4.11
S+ZS+P
500S(S+5)(S+8)
Fi gure 4. I I
Redesign Example 4.10 for single section cascade
compensator with tachometer feedback for open loop system
Fi gure 4.11.
Root Specifications: Zeta = 0.5, Natural Freguency = 10.0
Velocity Constant = 0.5
No Bandwidth Specified
Data on page 61 shows results for Wn = 10.0 and system
roots for parameter vaues choice one. I f Wn is changed
to Wn = 14.0 data on page 62 shows both choices of
parameter values are satisfactory.
60
INPUT DATA**********
SYSTEM POLES ARE ZERO,, 0.50000E 01, 0.80000E 01
ROOT SPECIFICATIONS ARE ZETA= 0.50000E 00,NATURAL FREQUENCY= 0.10000E 02
VELOCITY CONSTANT IS = 0.50000E 00
NO BANDWIDTH SPECIFIED
OUTPUT DATA***********
CHOICE ONE
TACHOMETER GAIN= 0.16103E 00COMPENSATOR POLE= 0.34088E 01COMPENSATOR ZEFO = 0.14829E 00
CHOICE TWO
TACHOMETER GAIN= 0.12470E 01COMPENSATOR POLE= -0.21609E 03COMPENSATOR ZERQ = -0.22957E 02
COEFFICIENTS OF CHARACTERISTIC EQUATION(ASCENDING ORDER)
0.74145E 02 0.64829E 03 0.16483E 03 0.1f>409E 020.10000E 01
THE SYSTEM ROOTS ARE
REAL PART
-0.11786E 00 -0.62909E 01 -0.50000E 01 -0.50000E 01
IMAGINARY PART
0.0 0.0 -0.86603E 01 0.86603E 01
61
INPUT DATA**********
SYSTEM POLES ARE ZERO,, 0.50000E 01, 0.80000E 01
ROOT SPECIFICATIONS ARE ZETA» 0.50000E 00,NATURAL FREQUENCY^ 0. 14000E 02
VELOCITY CONSTANT IS = 0.50000E 00
NO BANDWIDTH SPECIFIED
OUTPUT DATA***********
CHOICE ONE
TACHOMETER GAIN= 0.29434E 00COMPENSATOR POLE= 0.46165E 01COMPENSATOR ZERO= 0.21652E 00
CHOICE TWO
TACHOMETER GAIN= 0.16032E 01COMPENSATOR POLE= 0.43558E 03COMPENSATOR ZERO= 0.87812E 02
COEFFICIENTS OF CHARACTERISTIC EOUATION(ASCENDING ORDER)
0.10826E 03 0.71652E 03 0.24718E 03 0.1T616E 020.10000E 01
THE SYSTEM ROOTS ARE
REAL PART
-0.15980E 00 -0.34564E 01 -0.70001E 01 -0.70001E 01
IMAGINARY PART
0.0 0.0 -0.12124E 02 0.12124E 02
62
Examp le 4. 12
Fi gure 4. 12
Design single section cascade compensator with
acceleration feedback for open loop system Figure 4.12.
Root Specifications: Zeta = 0.5, Natural Freguency = 5.0
Velocity Constant = 0.5
No Bandwidth Specified
System roots are for choice two parameters
63
INPUT DATA**********
SYSTEM POLES ARE ZERO,, 0.50000E 01, 0.80000E 01
ROOT SPECIFICATIONS ARE ZETA* 0.50000E 00,NATURAL FREQUENCY* 0.50000E 01
VELOCITY CONSTANT IS 0.50000E 00
NO BANDWIDTH SPECIFIED
OUTPUT DATA***********
CHOICE ONE
ACCELERATION GAIN'COMPENSATOR POLE*COMPENSATOR ZERO*
-0.26303E 000.20500E-44
-0.13047E 02
CHOICE TWO
ACCELERATION GAIN=COMPENSATOR POLE-COMPENSATOR ZERO*
0.37030E-010.14807E 020.59227E 00
COEFFICIENTS OF CHARACTERISTIC EQUATION(ASCENDING ORDER)
0.29614E 03 0.10923E 04 0.24345E 030.10000E 01
0.46322E 02
THE SYSTEM ROOTS ARE
REAL PART
-0.28868E 00 -0.25000E 01 -0.25000E 01 -0.41033E 02
IMAGINARY PART
0.0 -0.43301E 01 0.43301E 01 0.0
64
V. Genera! Proa ram
The problems solver) in this thesis were done on an
individual basis with no attempt made to write a Genera!
oroqram. In retrosoect it is seen that a qeneral oroaram
could be written provided the type of specifications to
be satisfied are selected, the maximum system order Includina
any compensation effects is stated, the desired output
information and format is known, anH the system configuration
is cons i dere^o
Our prime specification was a complex root specifi-
cation which is dependent on the form of the characteristic
equation coefficients. In all the cases considered the riaht
hand side of the matrix equation had the same fcrm The form
is a function of system tyoe and order, and has two general
formsv\ -
LK » C
fllKU)*^,, it)(5~!)
from equation (2-3) and
(5-2)
from equation (2-4) where
Glv\-\ s SuiM Of At i \'
•. .
U.vi~l - Sum -s
\ \
yi-'"f\, s< l rou
, (5-3)
The left hand si^e coefficient matrix also reduces to a
J(5-4)
convenient form
where R is an unknown aooearing in the j coefficient
of the characteristic equation. If a Darticular coefficient
of the characteristic equation has no unknowns then the
aopropri ate olace in the coefficient matrix above is reDlaced
by a zero. As noticed for the cascade comoensation system
terms similar to the right hand si^e are aenerated in the
left hand coefficient matrix multiDlyina the unknown filter
oole or oolynomial coefficient exceot the coefficients are
(5-5)
andVI
y.
K-Vl (5-6)
where|< =. v\ + oRc€>e. of Filter.
fc-fl
tin-/ -L -.i~
Pc
tfn-t* = 5c>/w of rHB Pzopoer of c of- the. FtusG. Potej
The root equations can be generalized by aoolying them
to a characteristic equation with coefficients
A< - &<»< + &«£ + £ K (5-7)
nd then the result can be reduced to a qiven situation by
66
reading in the nrooer value f or the coef f i c i ents ^K > Sk and f-y..
The velocity constant also has certain Drooerties that
make it con H ucive to Drogramming The basic form is
m
/fn-(5-8)
where n is the number of system ooles and rr is the number of
system zeros. The form above aoolies for systems with unity
feedback and acceleration feedback,, If tachometer feedback
is used the velocity constant becomes
m
ft(5-9)
TTPt-t kkt /T£l
The remaining situations are for cascaded systems an
the form is now
mK, = K TT-ic 7/ </
~7TP<: jfp-
(5-10)
where k is the number of cascade filter sections. The cascade
acceleration combination is as above but the tachometer
cascade combination becomes
(5-11)
In all the above situations if any Dole or zero is not in the
system it is reo laced by a one where as removal of tachometer
feedback i mo lies K. = o
The final soeci f i cat i on presents a rather te H ious oroblem
as the genera I form shows
67
/F
/n-i
£*)
SB4 u)u(5-l2)
where the a^'s are a function of the unknowns. Since the terms
are squared it is recommended that bandwidth be set up for the
most difficult case exoected to be encountered.
Once the soeci f i cat i ons have been oreset in the program
they can be imDlemented by reading in aooropriate constants,
then using logic statements the simultaneous matrix eguations
can be formed. Again as in the bandwidth case the nonlinear
cases using the elimination method for solution should only be
set up for the maximum system configuation and order expected
to be encountered. This is because the computer can process
numbers but is unable to process unknowns specified by alpha-
betic characters. Designers may find that for cases of this
nature it may be beneficial to use an aooroDriate iteration
routine mentioned in the next section. A rough aenera I flow
chart of major ODerations is shown in Figure 5-1.
68
r- *"- Read Data
rPrint Input Data!
IL o q i c For
Selection And
Formu I at i on
of
Equat i ons
RootSpecifications
Ve I oci tyConstant
Bandwi dth
Solution ofEquati ons
Formulation of OutputSpecification Check
Other Desired Computation
iPrint Output I
?
General Program Flowgraphr i gure F-
I
69
VI. Conclusions and Recommendat i ons
The purpose of this paper was to investigate the feasi-
bility of using a computer for designing with algebraic methods.
It has been shown that this can be accomplished, in most cases,
by the linearizing transformation of variables or manipula-
tion of system analytic form. This was not true for the
feedback-cascade compensation cases or when bandwidth speci-
fication was to be satisfied but these problems could be
solved by application of a straight elimination process.
The important advantage of this method is it only restrains
the system sufficiently to solve for the unknowns. Other
design attempts tended to specify all root locations in imple-
menting a solution.
An area of possible future study is converting other
system specifications into algebraic equations and analyzinq
techniques of solution required. We have dealt with only
three specifications in this paper. Can we effectively use
peak overshoot, steady state error, settling time, resonance
frequency, or rise time? In analyzing these specifications
nonlinearities may aooear that defy solution by elimination,
therefore, it may require using Herat ion methods for
solution like Newton's method or steepest decent, or a
combination of these. The results obtained by these methods
are dependent heavily on the initial value guess and to a
lesser degree the quality of convergence to the final value.
70
An interesting ooint for further investigation would
be the possibility of deriving another definition for band-
width base H on a criteria other than magnitude. If band-
width can be specified as a comolex number then the non-
linearities are not compounder) by sguaring and the specifi-
cation will reduce to a two equations similar to the root
equat i ons
.
71
BIBLIOGRAPHY
I. Siljak, D. D , Analysis and Synthesis of Feedback ControlSystems in the Parameter Plane, I-Linear ContinuousSystems. IEEE Transactions on Application and Industry ,
vol. 83, No. 75, November, IQ64, pp. 449-458.
2. Thaler, G. J., Siljak, D. D. and Dorf, R. C. AlgebraicMethods for Dynamic Systems. Interim Technical ReportPrepared for NASA Under Contract NAS2-26Q9 . 1966.
3. Nutting, R. M., Parameter Plane Techniques for FeedbackControl Systems. M.S. in EE Thesis. Naval PostgraduateSchool . 1965.
4. Siljak, D. D., Generalization of the Parameter Plane Method.IEEE Transactions on Automatic Control , vol. II, No. I,
January, 1966, pp. 65-70.
5. Thaler, G. J., Elliot, D. W. and Heseltine, J. C. W.Feedback Compensation Using Derivative Signals, II-Mitrovic's Method. AIEE Transactions on Application andIndustry . September, 1963.
6. Zaguskin, V. L., Handbook of Numerical Methods for theSolution of Algebraic and Transcendental Equations .
Pergamon Press, 1961.
72
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73
APPENDIX II
LISTING OF PROGRAMS
74
z z a Uj H CTfr
o c •h * o <T **•—olu o ** z u Z #*Ht-X o_j ITi o <i * <1<I<C>- *clu «—
•
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1
O 0£CO<ILU «-» >- < » XX••exl* X in Xo - » O.CL
*XLU 0< w l-> m • 1 1
o»- >Z - 3 -•CO cm •b. »-4f\J
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cc »—>>-rwt- —
.
LU K am • mm XXDjt-iiJDli.' LO LOZ •V in >v Q-CLIUDU2 M •^ LUC *: m r-> m **ujZOOa: - u —1 U IT* o m CM CMLL<I_JLU3LU * ClLU < • » o **coujceo •> —
»
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94
INITIAL DISTRIBUTION LIST
No. Cod i es
I. Defense Documentation Center 20Cameron StationAlexandria, Virginia 223 14
2. Library 2Naval Postgraduate SchoolMonterey, California 9394-0
3. Naval Shio Systems Command (Code 2052) I
Department of the NavyWashington, D. C. 20360
4. Prof. G. J. Thaler 5Department of Electrical EngineeringNaval Postgraduate SchoolMonterey, California 93940
5. Prof. D. D. Si I jak I
Department of Electrical EngineeringUniversity of Santa ClaraSanta Clara, Calif. 95Q53
6. Prof. Toyomi Ohta I
Electrical Department, Defense AcademyObaradai, Yokosuka, Japan
7. LT H. H. Choe I
Division of Control EngineeringUniversity of SaskatchewanSaskatoon, Saskatchewan, Canada
8. Dr. K. W. HanI
Institute of ElectronicsNational Chiao-Tung UniversityHsinchu, Taiwan
9. Dr. S. H. Kiu I
P. 0. Box I
Shi h-y i-FenLungtan, Taiwan
10. Dr. S. M. Seltzer I
D/5I/-0I B/102Lockheed Missiles & Space Co.P. 0. Box 504Sunnyvale, Calif. Q4088
95
11. IVY. D. R. Towi ! I
Welsh College of Advanced TechnologyCathays Park, Cardiff, Wales
12. Dr. A. G. ThompsonDepartment of Mechanical EngineeringUniversity of AdelaideAde ! ai de, Austra ! i a
13. Mr. James CoxBldg. 151, Dept. 55-35Lockheed Missile & Space Co.Sunnyvale, Calif. 94088
1 4. LT John A. Wa Iker, Jr a
Philadelphia Naval ShipyardPhi lade Iphia, Pa. 19! 12
96
Unci assi f iedSecurity Classification
DOCUMENT CONTROL DATA -R&Dintv classi titration of title-, hotly of abstract and indexing annotation must be entered when the overall report is classified)
1 originating ACTIVITY (Corporate author)
Naval Postgraduate School
Monterey, California ^3940
Za. REPORT SECURITY CLASSIFICATION
Unci assi f ied26. GROUP
J REPOR T TITLE
Computer Techniques for Implementing Linear Control System Design Using
Algebraic Methods
4 DESCT'PTivE NOTES (Type of report and, inclusive dates)
M.S. Thesis - Electrical Engineering - 19685 AUTHORISI (First name, middle initial, last name)
Walker, John A. , Jr.
Lieutenant, United States Navy
6 REPOR T D A TE
June 1968
7a. TOTAL NO. OF PAGES
97
7b. NO. OF REFS
8a. CONTRACT OR GRANT NO.
b. PROJEC T NO.
9a. ORIGINATOR'S REPORT NUMBER(S)
9b. OTHER REPORT NO(S) (Any other numbers that may be assignedthis report)
10. DISTRIBUTION STATEMENT
special export, controls and each
;nts or foreign nationals may be made- only with p
mmmmwmII SUPPLEMENTARY NOTES 12. SPONSO RING Ml LI T AR Y ACTIVITY
Naval Postgraduate School
Monterey, California 93940
I 3. ABSTR AC T
Design of linear control systems is examined using algebraic methods.
Previous work indicates that the system describing equations are nonlinear.
Various combinations of feedback and cascade compensated systems are analyzed
using velocity constant, bandwidth and complex root specifications. Methods
for linearizing these system equations are demonstrated and examples are solved
using specific digital computer programs. Feasibility of a general computer
program is discussed.
DD "** 1473I NOV 89 I "T / WS/N 0101 -807-681
1
(PAGE 1)Unclassi f ied
Security ClassificationA-31408
Unc I assi f i edSecurity Classification
KEY WORDS
Algebraic Methods
Linear Control Systems
DD ,
F°"v
H..1473 back,
S/N 0101-807-6821Unclassified
Security Classification
98
3 2768 0041 5866 7
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