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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 ExamD





Block Diagram for ExamD


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 BANDWIDTH SPECIFIEO

OUTPUT DATA

***********

FORWARD GAIN

-0.10010E 04

ACCELERATION GAIN

-0.82645E-01


-C.10010E 04 0.30000E 02 0.12373E 03 0.12000E 020.10000E 01


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



OUTPUT DATA

***********

FORWARD GAIN

0.36000E 03

ACCELERATION GAIN

0.29306E CC


0.36000E 03 0.30000E 02 0.14650E 03 0.12000E 020.10000E 01


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


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


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



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


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


0.13349E 040.27000E 02

0.26697E 040.10000E 01

0.10354E 04 0.25400E 01


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


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


Bandwidth Frequency 3.0

System roots are for gain values number one sinet Ks<

is negatl ve.

eetn

50


POLES ARE ZERO, 0.30000E 01, 0.40000E 01

NO ROOT SPECIFICATION

VELOCITY CONSTANT IS= 0.50000E 01

BANDWIDTH FREQUENCY IS= 0.30000E 01


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


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.




52


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


OUTPUT DATA

***********

SYSTEM GAIN

0.13756E 04

COMPENSATOR POLE

0.37847E 02

COMPENSATOR ZERO

0.39961E OC

53

VELOCITY CONSTANTtCOMPUTED)

0.4150GE 01


0.54972E 03 0.15081F 04 0.28735E 030.45347E 02 0.10000E 01


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


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


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


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


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.



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


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



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


0.30173E 03 0.11035E 04 0.20664E 03 0.23640E 020.10000E 01


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



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


SYSTEM POLES ARE ZERO,, 0.50000E 01, 0.80000E 01

ROOT SPECIFICATIONS ARE ZETA= 0.50000E 00,NATURAL FREQUENCY= 0.10000E 02




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


0.74145E 02 0.64829E 03 0.16483E 03 0.1f>409E 020.10000E 01


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



ROOT SPECIFICATIONS ARE ZETA» 0.50000E 00,NATURAL FREQUENCY^ 0. 14000E 02




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


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



System roots are for choice two parameters

63



ROOT SPECIFICATIONS ARE ZETA* 0.50000E 00,NATURAL FREQUENCY* 0.50000E 01

VELOCITY CONSTANT IS 0.50000E 00



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


0.29614E 03 0.10923E 04 0.24345E 030.10000E 01

0.46322E 02


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

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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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Computer techniques for implementing linear control system ... · Calhoun: The NPS Institutional...

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