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1968

A study of a model-referenced adaptive control system A study of a model-referenced adaptive control system

James William Graham

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i.,-

. j /

A STUDY OF A HODEL-REFERENCED

ADAP"TIVE CONTROL SYSTEM

BY

JAMES HILLIAM GRAHAM III )Cf~l~ )

A

THESIS

submitted to the faculty of

THE UNIVERSITY OF HISSOURI AT ROLLA

in partial fulfillment of the requirements for the

Degree of

MASTER OF SCIENCE IN ELECTRICAL ENGINEERING

Rolla, Missouri

1968

Approved

(advisor) . I

\

\ . ' •' ~~·~\~-·~~~·~~ ~\!~-·~·~·-·~~-·~·-----. -~ \. \ .\\ ..

\ .

11

A model-referenced adaptive controller for a first­

order plant with time-varying parameters is developed using

Lyapunov' s second method. The system is studied lvith the

aid of analogue computer simulation and an oscillatory mode

is encountered. A method of reducing the osci1Jat5.ons by

addition of a compensating network is proposed. The com­

pensation is obtained thr·ough the use of lincarizo.t.Lon and

root locus teclmiques. The compensated system shm·1s a much

improved error response over the original system.

iii

AC KNOvJ LEDG EivffiNTS

The author would at this time like to acknowledge the

contributions of Dr. John S. Pazdera who provided the im­

petus for this work and whose helpful suggestions and com­

ments hastened its completion.

iv

TABLE OF CONTENTS

ABSTRACT

Page

ii

AC KNO\'JLEDGEJY!ENTS

LIST OF PIGURES

LIST OF SYJ>'IBOLS

I. IN"THODUCTION

iii

vi

viii

1

A. The Adaptive Problem 1

B. History of the Design of Model-Heferenced Adaptive Control Systems 2

C. Choice of a Frob lem and Swnmary of Approach 4

II. A FIRST-ORDEH Iv10DEL-RE111Ef{l:<~NCED ADAPTIVE CONTROL SWT~ 8

A. Synthesis by Lyapunov's Second Method

1. Mathematical Development

2. Analogue Simulation

3. Results

B. Linearization of the System

1. ~~thcmatical Development

2. Analogue Verification

c. Compensation of the Linear System

1. l'v1athematical Development

2. Analogue Verification

8

9

12

14

20

21

28

33

33

43

D. Study of System Performance

1. First-Order System with Sinusoidal Variation of r and b

a. Original System

b. Compensated System

2. First-Order System with Pseudo-Random Variation of r and b and Step Changes

Page

47

48

48

49

in Ks 1~9

a. Original System

b. Compensated System

3. Comments

III. Sill·THARY AND CONCLUSIONS

IV. REC OMiv1ENDAT IONS

BIBLIOGRAPHY

APPENDIX A Analogue Simulation Drawings

49

49

58

60

63

65

67

APPENDIX B Detailed Computations - Linearized System 69

APPENDIX C Development of Pseudo-Random Signals 73

VITA 75

v

l

2

3

4

5

6

7

8

9

10

ll

12

13

14

15

16

LIST OP FIGURES

General Model-Referenced System

First-Order Model-Referenced System -Adaptive Control Unspecified

Pirst-Order Model-Referenced System -Adaptive Control Specified

System Error, e(t), and K( t) for Case l

System Error, e ( t), and K(t) for Case 2

System Error, e(t), and K(t) for Case 3

System Error, e(t), and K(t) for Case L~

Drift of Adaptive Parameters for Case 5

System Error and K(t) for Eigenvector Excitation of a Model-Referenced System

Calculated System Error for Eigenvector Excitation of the Linearized System

Root Locus of the Linearized Model­Referenced System

First-Order Model-Referenced System with Lead Compensation

Root Locus of the Linearized Compensated Model-Referenced System

System Error and K(t) for Eigenvector Excitation of the Cornpensated System

Calculated System Error for Eigenvector Excitation of the Linearized Compensat-ed System

System Error of the Original and Compen­sated Systems for Eigenvector Excitation

3

5

13

15

16

17

18

19

31

32

35

40

L~s

46

vi

Figure

17

18

19

20

21

22

23

24

Input, r, and Dynamic Plant Parameter, b, for Sj_nusoidal Variation

x (t), x (t), and e(t) for the Original S~stem wfth Sinusoidal Var·iation of r and b

x 1 (t), x 2 (t), and e(t) for the Compensated System l'nth Sinusoidal Variation of r and b

System Error for the Original and Compen­sated Systems with Sinusoidal Variation of r and b

Pseudo-Random Variation of r and b and Step Changes in Ks

System Error for the Original System with Pseudo-Random Variation of r and b and Step Changes in Ks

System Error for the Compensated System with Pseudo-Random Variation of r and b and Stop Changes in Ks

System Error for both Original and Com­pensated Systems with Pseudo-Random Variation of r and b and Step Changes in Ks

vii

50

51

52

53

54

55

56

57

a

b

r

e

q

c,d

LIST OF SYWiBOLS

model dynamic parameter

model d.c. gain parameter

plant dynamic pat'ameter

plant d.c. gain parameter

viii

controller feedback parameter for adjustment of plant dynamics

controller feedforward parameter for adjustment of plant d.c. gain

gain of the feedforward parameter adjustment loop

gain of the feedback parameter adjustment loop

model output

plant output

command input

system error, e - e m s misalignment of the model and controlled plant d. c. gain parameters, Km- KvKs

misalignment of the model and controlled plant dynamic parameters, a- (b + K8 K)

output of the lead compensator

parameters of the lead compensator

1

I. INTRODUCTION

A. The Adaptive Problem

The middle 1950s represented a remarkable upswing in

the growth or automatic control and particularly in the

development or adaptive systems. In 1958 Aseltine~ Mancini~

and Sarture1 presented a survey of the adaptive control work

dating back to 1951. The number and variety of publications

in this period alone enabled them to organize adaptive sys­

tems into five general classifications according to the

manner in which adaptation was achieved.

The general adaptive problem is to maintain a desired

response by developing within a system the ability ror self-

adjustment under the influence of changing conditions. The

problems of meeting specified performance criteria in flight

control systems is a prirne example. Of the papers cited by

Aseltine1 , et al, the majority of applications dealt with

flight systems~ and this is similarly true today. One of

the problems racing those who design flight systems is dyna-

mic parameter variation. The relatively new model-refer-

enced adaptive control system seems ideally suited to this

problem.

The general model-referenced system consists of a

model, plant, and controller as shown in Figure 1. The

plant represents a system with varying parameters and the

model represents the ideal plant from the designer's view­

point. The controller is the mechanism -llhich adapts the

plant to match the model, resulting in zero error between

plant and model outputs.

B. History of the Dest_g_n of' __ l_!lo.Qel-R~ferenced Adap_ti ve

Control Systems

2

The first model-referenced system and design criteria

\'lere proposed by Whitaker, Yamron, and Kezer2 j_n 1958.

Osburn, Whitaker, and Kezer3 sub sequent ly pub 1 i shed a paper

in which simplification of the control mechanisms and a

reduction in convergence time of adaptation were improve-

ments over the original proposal. The system was desie;ned

to minimize the integral of' the error squared. Control

signals for parameter adjustment were based on functions

of the error and time rates of change of the plant para­

meters. This method of design is referred to in later

literature as the "M. I. T. Rule."

Just prior to Osburn's paper, Kalman and Bertram4

presented their paper, "Control System Analysis and Design

Via the Second Method of Lyapunov," which marked the intro-

duction of Lyapunov's theories into Western literature.

Lyapunov's second method can be used for determination of

stability or asymptotic stability in the large, and is well

3

... ·=.-,. MODEL

' r --

-t:> r::·~ -CONTROLLER f-..::::> PLANT

[ __ _

Figure 1. General Model-Referenced System

known to be an important addition to control theory.

Parks5 used Lyapunov's second method to show that systems

developed by the "M. I. T. Rule" are unstable for certain

inputs.

6 Grayson ~ in 1961~ presented a thesis in VJhich Lya-

4

pLUrov' s second method 1:vas used to design a model- referenced

control system. Many papers quickly followed. Shackcloth

and Butchart7 presented a synthesis technique using Lya­

punov' s second method which vJas redesigned by Parks5" 8 .

The culmination of these and others came in 1967 when

Shackcloth9 presented a general design criterj_a for model-

referenced adaptive systems using Lyapunov's second method.

He has shown that stability in the large is guaranteed and

asymptotic stability is guaranteed under certain conditions.

The adaptive control loops are defined based on the error~

the input, and the plant output and its derivatives. Within

a period of months a similar design was offered by Zemlyakov

and Rutl<::ovskii 10 . The main difference j_n the papers wa.s the

manner in which a Lyapunov function was chosen, the latter

method being the same as that chosen by Monopoli8 and used

later in this paper.

C. Choice of a Problem and SWlunar;z_ of_Jll?.proach

The first-order model-referenced adaptive system of

Figure 2 was chosen for study since nowhere in the avail­

able literature was there evidence of a simulation study

in which both plant parameters, Ks and b., were asswned

5

r e

--~-~---------l~---t> 2:

Figure 2. First-Order Model-Referenced System -

Adaptive Control Unspecified

6

different from the model. Of the simulations5,9 known to

the author, only one plant parameter was assw11ed variable,

or fixed and different from the model, whlle the other

parameter was fixed and identically matched to the corre­

sponding model parameter. Since it was desired to make all

plant parameters variable, or at least fixed and different

from the model, the available analogue computer limited

simulation studies to systems not greater than first order.

Once the first-order system had been chosen, an ana-

logue simulation was under'taken as a check against the

general theory presented by Shackcloth9. Two observations

from the simulation are noted here. First, the system

contained oscillatory modes; and second, after the system

reached the steady-state for step inputs, the adaptive

control gains, K and Kv of Figure 2, tended to drift away

from their ideal values.

The use of adaptive control in flight systems and the

possibility of effectively using model-referenced adaptive

control in this area has been mentioned previously. In

this light, it was decided tlli~t tl1e oscillations previously

observed could have damaging effects of a magnitude that

would render this approach useless. However, if the oscil­

lations could be removed or at least highly damped, then

a stgnificant improvement will have been made particularly

from the standpoint of maintaining stabillty in an aircr'aft.

The basic organization for the ren~inder of the paper

will take the following form. The first-order model­

referenced adaptive system will be synthesized according

to present theory-9. Following this is a section devoted

to linearizing the system, followed by a section in which

compensation teclmiques are applied to the linearized

system. A study of several first-order systems showing

the desired improvement completes the main body of the

paper.

7

II. A FIRSrr-ORDER li10DEL-REFERENCED

ADAPTIVE CONrROL SYSTEM

In model-referenced systems~ adaptive control loops

must be determined such that the plant and model outputs

become the same. Shackcloth9 has developed a synthesis

technique for fi.ndlng the control equations of a linear~

th n -order model-referenced system in which the plant gain

and all dynamic parameters arc assumed constant or slowly

ti.me-varyi.ng. His method is used here to detennlne the

control of the first-order system of Figure 2 in which Ks

and b are assumed slowly time-varying.

8

The necessary control for a first-order linear system

is obtained from the variable gains K and Kv of Figure 2.

rrhis is seen by comparing the transfer functions~ Hm and

H ~ of the model and plant. For the model p

H (s) = ---m s+a

and for the plant

KVKS ---

s + b + KSK

(1)

(2)

9

If the plant and model are to be identical~ or matched~ it

is required that their transfer functions be equal or from

( 1 ) and ( 2) t l1a t

and

a= b+KsK

The control equations for determining K and Kv arc now

developed using Lyapunov's second method.

1. M~_tl~~ma~_~cal_Dt?_'{~l<2_pment

Equations (1) and (2) ma.y be re\·Jritten as

(s+a)em =~ Kmr

and

( s + b + K K)8 = K K r s s v s

The error between the model and plant is defined as

e = e - e m s

although an equally valid choice for error is e - 9 . s m

Now

(s+a)e = (s+a)(em- es)

( 3)

( LJ.)

(5)

Appropriate substitutions from equations (3) and (1+) yield

(s +a)e = (Km- KvKs)r -[a- (b+ KsK)Jes (6)

Let xl = K - K K and x2 = a - (b + KsK). Then m v s .

-ae + x 1r- x 2es e =

which becomes in matrix notation

C -- Ae + Bx

where

The following choice of a Lyapunov function, v, is taken

from Monopoli8 .

where P and N are positive definite matrices and in the

first-order case

N = [nl 0l

0 n 2 and

Equation (7) is a logical choice for a Lyapw1ov function,

10

(I)

in that both ter•ms are quadr•atic forms 1.·1hich 80 to ~ero

when the plant is matched to the model. Dj_fferentlation of

(7) with respect to time yields

V = eTPe + eTPe + xTNx+ xTNx

If

(9)

then (8) reduces to

V = eT (AT P + PA)~ = -eTQe

• V is negative definite if Q is chosen as a positive definite

matrix. The term ~TPe is positive definite, since P is

positive definite, and xTNx is a quadratic form ivhich is

positive definite; therefore, V is positive definite, and

asymptotic stability is assured11 . The choice of V and x result in a V which ensures that ~ will go to zero but does

not guarantee that ~will go to zero.

A solution for P can be obtained from ATP + PA = -Q.

This solution is unique, and P will be positive definite

11

if A is a stable matrix12 . The model j_n Figure 2 is stable

provided a 2:0; hence A is stable for a stable model. Sub-

stituting for P and A

-Q = ATP + PA = -2apll ( 10)

Choosing Q = [2a]J equation (10) yields p 11 =-c: 1. rrhe

adaptive control can now be formed from equation (9).

( ll)

In the differentiat:Lon of x it has been asswncd that Ks and

b are constant or may be approxj_mated as constant.

Simplifylng (11)

The adaptive control equations for the system of Figure 2

are

(12a)

and

( l2b)

where

1 1 and

Although K may be slowly time-varying) the adaptive loop s

gains ~ 1 and ~ 2 are constant if n 1 and n 2 are chosen in-

versely proportional to Ks. This choice of n 1 and n 2 ap­

pears to add terms involving & to the differentiated

Lyapw1ov function) V, of equation (8). Since K8 has been

12

approximated as constant, so that Ks equals zero, l~ may

also be approximated as zero and equation (8) is uncl1an0ed.

The first-order system with the adaptive control specifted

in equation (12) is shown in Figure 3.

A swmnary of asstJJnptions made in this development are:

(a) the model is stable in order to ensure tlmt P is

positive definite,

(b) Ks and b, the plant parameters, are approximately

constant in order that x may be differentiated to the fonn

of equation (11), and

(c) s:Lnce N must be positive definite and J-L 1 and 1t 2

are constant, then Ks cannot change sign.

2. Analogue Simu~ation

An analogue simulation study, in which both K and b s

are not necessarily matched to the model, was undertaken to

gain a better w1derstanding of the system. Due to the

limitations of the available analogue computer only one of

the plant parameters could be varied with time. The model

parameters wore arbitrarily selected as Ks = 2 and a = 1

for use in the simulation. In Appendix A is the analogue

simulation drawing for use on an Electronic Associates, Inc.

TR-48 computer. The following cases are typlcal of many

that were simulated.

Case 1: r - 1 b = 0 K (0) = 0 v

f'-1 = 1 Ks = 1 e ( o) = 0 m

}J-2 = 1 K(O) = 0 es(o) = 0

The model-plant error and K are shown in Figure 4.

r

1

s

K v

~ K 8 m m

s+a

e

-~-------·

-K 1

s

Figure 3. First-Order Model-Referenced System -

Adaptive Control Specified

13

Case 2: Case l VJaS repeated except fl-1 -- fl-2 . - 20.

The error and K are shovm in Figure 5.

Case 3: r l b -- .5 K (0) 2 -·--v

fl-1 = l Ks - 1 e ( o) 2 m

fl-2= 1 K(O) -- l es(o) -- 2

The error and K are shown in Figure 6.

Case 4: Case 3 vms repeated except fl-1 - fl-2 -- 20.

rrhe error and K are shown in Figure 7. Cases 3 and lj. are

equivalent to a step change in b after the plant has been

adapted to the model.

Case 5: r - l b = 0 K (0) -- 2 v

f-LI = 1 Ks l e 111 (o) . - 2

fl-2 ;:::: 1 K(O) - 1 e 8 (o) - 2

In this case the plant and model are identical. A plot of

K versus Kv is shown in Figure 8.

It is noted that only fixed plant parameters were used

and that simulations involving a time-varyine; plant are

presented in a later section.

3. Results

rrhe results presented in this section are prhnarily

general observations since at this time only a general

knowledge of the adaptive process and any associated prob-

lems is being sought.

In cases one through four, there are three very notable

features which may be observed from Figures i.J. through 7.

First, there are oscillatory modes in the system as noted

+1

K 0

-1

+.5

e 0

-.5

Pigure 4.

0 5 10

TIME (sec.) 15

System Error, e(t), and K(t) for Case 1

15

+1

K 0

-1

+.2

e 0

-.2

-,--_~=~ ~-/=~~~! -±-1 ~_~_-~r-J __ ~p_~~----i--l--~~~~=-~r~r--------·-r-;-•--1--- ~

_:_ __ ~--F--~--j-----~--:--; __ ·. _____ : ____ ; ____ ;·-- ' ' . ~ ··---·-----·--·---·---.---~-----·--~ --' l I I

, I , \ . . . _ __......_ ____ ,. _ _, __ , ____ ,_ 1 ! ~ . .

i ' . ' .

! I ' t 1 l t j 1 I i

~_j~~--r-~.--':, __ ::J_\ __ ;1_· __ :~--\--~:~__j: __ __j\ __ t_\~:-t-h--~\\---;_--\-- ~----

• • I . • ' I \ i ' . • -\ ' \ \ . .\ ' . • - . I \ ' ' ' • \' \ i I i \ I I ' I ,---\-• I. I - • • ' I \ I ' \ I \ t ' \ I I \ . I I \- I \ \ i I \ I \ \ \ ' I i \ ' \ \ ' \ ' \ \ i I \ \ ' \ \ ' ' ' ' \ -+--\--, \ - l \ '. i - ,---\ • ' -\ \ \ \--~-~--\--t--\--'--\--:---~-+--'.--\-1---\-~------t--~-\--~---~-~---i-·- -

___l __ \. __ ,_U _ _j____\ __ \ __ L\_j~ .. _ _:___'L L_ll_j_1 ___ ~ -\~- \ ____ ;- i ___ '\ __ -~---~- . ___ '\_~\___L_L_ <_L_\ ___ ~\ _j __ J \ ___ \ \ __ __\ ____ _\_]_ 0 ]_ 2 3 4 5 6 7 8

TIME (sec.)

) I ) I I ) I I ' ---,--,-,.---;-,---,----;------r--.- I I -, I -rL-' ' . -r-;--· , 7-r I I , I - I I ' I I ! I ' I I I i ' ' - I I I I I ' . ! ' .

I I I I! I j i /I l l . I l j ' : L' /I / I / I I / ! i---i----i---.i--i--i---r--r-+--t--r--·--~r------r--+--+--------'-·--·---+--1-r--- ---;-- ·---~---~-_J__~-·---·---·---· -r-i , J : 1 .1 f l • : ; f ! 1 1 ; ; : ; t : 1 1 r t t i · ! ! 1 t

_;_-~-1--i--~1-'--i--;..,.,_,f--,--+i--,'-~-' ...,~· --t'--i-·-_;_l _ _:_! _ _;_! __ ,_: _ __;_' _ __;_i __ : __ : __ L_L ! : : I : ! ~ t I I ! t L.____j_: . j

l ! 1 ! f , i i , r ! I . i . ; f . .' ! : -~-At--: __ ;_ ~--~--:--·--!--· ~--.~-;--;-~·--~-~-j--r-:-----: -;~--;--:- ~-- · __ ..:

--- ~~~~~--~~-----

--.. --l--\--:. ---1- 1+\· \-~--~ \ \ '/-+-T--\-l--~-, -\-~---f---~-~---:_--~-----\+-l--\---y-~--:: --;.--- ·--·--------·--·-·--,_1., •••• \, 11 •! t 1 '\ II '

I ' I ' I \ I ...___ , ' . I I \ \ I ' ' \ ' \ , ~ • \ • • 1 ' \ \ I ' I . ·.--.--,--;--

~ \ \\ ~ 1' ~ \\ \, \\:; \ ~ \ \' l \ • \ ~ 1 -\\ ~~ \\ \ t; 1 ; ~ \1 \ ' \. \ \ \ I \ ' \ ' I I ' ~ \ \ \ I , \ \ 1 ' I \ I l. - \ -L-' I \ l • ' \ _j ' --r--·,-~,- ;1-•1--r-"l____,____.\ ·:-1· ( '1 \ 1 \-+-\ t--.. ---1-\--\' • ' ~--,~-1---l • '\----.----\""'t\--1-~~-\--\- ,-"1---~ \\,\\\'1''\\\._\ \t\\1 \\'. '\\1.\\,'\\ \\, ,\\\\\

_\..._ _1 _\___.t..--... L ~\ \.__~_l L I ...... .......___1__......__• __ L \ \_ _ _.\._ ---• . ....:. ... '...... _l I L__L L _\._- L . ______ :~ ___ \ _j_ ..._ __ ', L_

0 1 2 3 4 5 7 TIME (sec.)

Figure 5. System Error, e(t), and K(t) for Case 2 1-' m

+1

K 0

-~--~-- -· .

0

+.2

e 0

-.2

__ ; ______ !_- - ~---- T- -

L_ -- \ ------------4------------~1,--------

5 TIME

10

(sec.)

15

_( !

!

-/-- i !

-· - ·-·- -- .

0 15

1 J

/ j ' L_

'

Pigure 6. System Error~ e(t), and K(t) for Case 3

17

+l

K 0

' ; f ; r ~--., -~-;---~-:-·--.---~,----r---,------;--·-;----,--i---;---,-~,--;--~.--:----, -;--,---r--r--,-1 • I I -' I J -' ' f I ' ' I ' i ' .' f : j i i i ! I 1 : I ( i , ' I ~ i I I _L_}___;__; ____ / . I /----:_· __ : -f~f--;_---~~-l-+- ·~----/~--·-' -~: __ ,' _____ .:.---/---/ ____ i_- -/·------... 1--/-----!-----/ -----/ ___ /_ ___ 1-------/-----~--··: ____ j~-----l ____ j ___ L---~---I--

• I I 1 ! ' I J ,

0

I ! t ! I ' i

I I '

\ : ~

1 2 3

--~~-------------~--.----~----~-.-----~-·-~-

______________ , __

4 TirtiE (sec. )

5 6

i -------1 -- -~------~ ---­' I

7 8

_) / -+-!-1 _j_j'__L__j__[_~ __ L_L __ {_~r_ __ -C~L~---~--~z_-;e_~y__ !~7- ?-=-! =i=( __ ,JT_J~~7=T-~L~L __ r_=l-72 \! / 1 ! .' ! ' ' ' ' - I ' , j ' , ' ' '

+ . 051- ~-- -- _·_'i---~ . . o'i}B ___ /-!_;_:_,\}----· v_=:__~=-~~--

' -----~-----~--.-··----- -~-~-----~

e

. : ' 'I : I • : ' ' ' I j \ I ' I I '

--·-' ' ~ ' , .. ' I

\ 1• 1 ~ 1\ \ 1, 1 ', ' \ , ' \ t \ \

\ ~ ' \ \ \ , ' I \ \ ' ~ \ \ \ ' ~ '

~-~ -~----r--. -\.---t---,- .-~.----,----4-- ,~--~--·-·.----', ' ' \ \ I \ 1 I ' 1, \ \ \ 1 , ,

_\ __ ·~ _ _'-.__\~ _\_i__~\. __ ~ ___ \__~1 __ \ ____ ~ -L-'~ __ \. ___ l__\ __ \_

\--- _,, __ \~ i-'\:~t\-~;j ~r\_>:, ~-~ --\ ·"

0 l 2 3 4 5 6 7 8

TIME (sec.)

Figure 7. System Error, e(t), and K(t) for Case 4 f-1 OJ

3

K 2

1

0 ~------+-------~------~------~------~~------+------0 1 2 3

K v

4 5

Figure 8. Drift of Adaptive Parameters for Case 5 f-' \0

20

by the damped sinusoidal error response. Second, an in­

crease in the adaptive loop gains, ~, and ~2 , caused an

increase in the frequency of oscillation and a decrease in

the peak value of the error. Third, upon reaching zero

error the adaptive gains, K and Kv, were not at their ideal

values, e.g. in case one, ideal values would be K = 1 and

Kv = 2, whereas Figure 5 shows K = 0. 73 and Kv 1·ms found to

be 1.46. This apparent conflict with the theory is not

mentioned by Shackcloth9. The reasons for j_t are related

to the d.c. gains of the model and plant and the use of a

constant input; this will be discussed further in part B.

Case 5 was included to show that a drift problem exists.

Figure 8 shows that beginning vdth a perfectly adapted

system, the values of K and Kv drift in a manner that keeps

the d.c. gain of the plant equal to that of the model.

Once again the constant input was found to be a contributing

factor. The drift problem will not be discussed fu~ther

until the final section.

B. L:i.nearization of the System

The preceding investigation has shown that oscilJatory

modes do exist and that increasing the adaptive loop gains

increases the frequency of oscillation. The purpose here

is to linearize the system in an attempt to isolate the

oscillatory modes for further study. The model-referenced

system of Figure 3 is nonlinear due to multipliers employed

21

in the adaptive control; therefore, a linear analysis of

the system in its present form cannot be used. A nonlinear

system can, however, be linearized and subsequently analyzed

about a known equilibrium point. Such a point for the cho­

sen system occurs vlhen the plant and model are matched or

when x1 = x2 = e = 0. The system will be linearized about

the knO'\'In equilibriwn point and then verified by analogue

simulation.

l. l_~a tl>~~~~~ t i_ca l:__Qcye lo_r]nent

The state equations for the system of Figure 3 may be

written directly as

e = -ae + K r m m m

-be + K ( K r - Ke ) s s v s

-/I.e e '2 s = - IL e ( e -

'2 s m n r(e - e ) '' m s

e ) s

(l2a)

( 12b)

(12c)

(12d)

The method for linearizing the system may be found in

DcRusso, Roy, and Close1 3. The technique used is basically

a Taylor series expansion about the equilibrium point in

which only the linear terms are retained. The system is

then studied in terms of deviation from the equilibrium

point.

Let z represent the deviation of the state variables

from the equilibrium point. The known equilibrium,

x 1 = x 2 = e ==~ 0, implies for x 1 = x 2 = 0 that a = KsK + b

and K = K K and for e = 0 that em = 9 8 • The model output -m v s

at equilibriwn is I)ur/a.

Then

z -

Kmr e ---­m a

~r e - -·-­s a

a- b K -

The linearized system is described by

z :::: J z eq --.

called the "variational equations!! of the l:Lneat'l:~.ed

2?.

( 13)

system. J is the Jacob:i.a_n matrix of the state C11uations eq

evaluated at the equ:i.librium point. rnw Jacobian is

determined13 as

-a 0 0 0

0 - (b + KsK) -K e K r J

s s s =

- J-L2 8 s J-L2 e s 0 0

J-L, r -J-Lir 0 0

At equilibrium, equation (15) becomes

-a 0 0 0

0 -a -K e K r J

s seq, s = eq

-J-L2 9 s J-L2 8 s 0 0 eq, eq,

J-Lir -J-Lir 0 0

lvhere e is the plant output at equilibrium. J is Seq, eq

used to determine the eigenvalues and corresponding

eigenvectors for ·the linearized system. The eigenvalues

( 15)

are found from 1>--I- 3 eql = 0., where

>-.+a 0 0

1>--I- 3eql 0 >--+a K e

= S Seq,

fl-2 8 s -fl-28 s >-. eq, eq,

-f-1-lr f-1-lr 0

0

-K r s

0

23

= >-.(>-.+a)[>--2 -t-a>-.+ K (f-1- r 2 + fL e: ~ S I 2 ,-;,eq,j

( 16)

In symbolic form the equations are very unwieldly, there­

fore a particular example is studied in order to shmv the

usefulness of the linearized system. Let the model para­

meters be a - 1 and Km = 2, the plant parameters be b = 0

and Ks = 1., and the adaptive loop gains be f-1- 1 = f-1- 2 == 20.

The input will be r = 1., and equilibrium values are e -- 2, Seq,

K = 1, and K = 2. Then v I>-. I- J eq I = >-. ( >-. + l) ( >-. 2 + >-. + 100) = o ( 1·-r)

Solving (17)., the eigenvalues are

AI = 0

>-. = 2 -1

>-. = 3 -.5- jlO

>-.4 = -.5 + jlO

The eigenvectors are found from Adj ~I - J eq] . The co lunm

vectors of Adj [>--I- JeJ are proportional and evaluation of

any colunm of Adj~I-Jeq] at>-.= >-. 1 gives the corresponding

eicscnvector. The matrix Adj ~I- Jeq] is given in equation

( 18) on the follmving page.

The modes of dynamic behavior in a linear system are

expressed in terms of motion along the eigenvectors, hence

A 3 + A 2 + 100 >.. 0 0 0

100A >..3+>..2 -2(A2+A) >..2+A Adj ~I - J J = \ I ( 18) L eq -40( A2 + >..) 40( A2 + >..) A.3+ 2>..2 + 21A+ 20 40(A+ 1) I

20(>.2 +>.) -20(A2 +A) 40(A+l) }+2>.2 +81>-+8~

[\) +:-

25

the matrix whose colwnns are the eigenvectors is termed the

modal matrix. Once the modal matrix is knm·Jn, time-

solutions of the state variables may be found for any set

of initial conditions chosen. Sub st it ut ion of the various

eigenvalues into (18) yields

0 1 0 0

0 1 -. 5 - j 10 -.5 + j 10 M = ( 19)

1 0 40 40

2 0 -20 -20

Solutions of -~( t) may now be calculated.

e(t)

If ~(t) is knov1n, em(t) and es(t) are knovJn and hence

= 8 ( t) - 8 ( t) may be found. The following trans-m s

formation Hill simplify calculations. Let

z - My (20a)

':Chen

z = My

Substitution of (20) into (14) yields

My = J :G1y - eq-

or

~rhe form

A = M-lJ M eq

(20b)

( 21)

where A is a diagonal matrix composed of the eigenvalues

of J is well known13. Equation (21) then becomes eq

( 22)

where

0 0

0 -1 A -

0 0

0 0

The solution of equation

0 0

0 0

-.5- jlO 0

0 -.5 + jlO

(22) isl3

yl(O)eAit

y2( O)eA2 t

y3(0)eA3t

y4(o)eA4t

26

The system response is dependent on how strongly each mode

is excited which is dependent on y(O). Since it is desired

to study the oscillatory mode, which is due to the complex

pair of eigenvalues, y(O) is chosen as

yT(o) =-c: o.o1 [o o -.625- jl.03 -.625 + jl.03] (23)

The reasons for this selection will be discussed in the

next section on analogue verification. Now using equation

( 20a), z( t) is as shmvn on the following page in equation

(24). Using equation (13) a solution for e(t) is

(25)

And

c(t) == .2L~2e-.5tcos(l0t+ 33.8°) (26)

This solution of e(t) is the result of exciting only the

last two modes of the modal matrix. The steps intermediate

to (25) and (26) may be found in Appendix B.

r 1

1 ~(t) = My(t) = o.o1\ 0

1 0

2 0

0

-.5- j10

40

-20

0 ll 0

-.5+jl01 0

40 J [( .625 + jl.03)e-( · 5 + ~lO)tl -20 -( .625- j1.03)e-( · 5 - JlO)t

(24)

1\) -..J

28

2. Analogue Verification

The analytic solution obtained for e(t) will be veri­

fied by an analogue simulation of the system of Fi8ure 3

and the values for the example being studied. The simula­

tion drawing used in conjunction with an EAI TR-48 analogue

computer is in Appendix A. First, however, the particular

choice of y(O) in equation (23) will be explained.

If the oscillatory mode is the only one to be excited,

a possible form of y(O) is

yT(o) == [o o a+ jf3 a - jf3] (27)

'l1hus there are infinitely many choices of a and f3 VJhich

should yield solutions containing only the oscillatory mode.

However, it \vas fotJnd that while the error Hent to zero for

initial conditions corresponding to (27) the model and

plant were not always perfectly matched, that is x 1 and x 2

were not zero. From (19), (20a), (27), and (13)

o em(o)-2

- a -1- 20 f3 e s ( 0) - 2 =

BOa K(O)- l ~(0) =:

(28)

-40 a Kv ( 0) - 2

This shows e (o) = 2 for any a and /3. For the constant m

input assumed, the model output equals tvro for all time

which means the dynamic model parameter, a, is not a deter­

mining factor in the adaptation process. The only require­

ment for adaption is that the controller match the plant and

model d.c. gains. From equations (l) and (2) the d.c. gains

are Km

1\n(O) = -a

and K K

v s

b+ K K s For the example chosen, (29) becomes

H (0) = 2 m

and H (0) = K /K p v

The error will be zero whenever the ratio of K to K is v

29

(29)

equal to two. Then with a constant input and initial con-

ditions corresponding to (27) the equilibrium is a set of

points represented by the line Kv = 2K. This is the same

line along which K and K drifted as shown earlier in Ftsure v 8. For each pair of values of K(O) and Kv(O) there is an

initial error, e(O) = em(o)- es(o), that causes the sy~3tcm

to end at a particular point on the equ:tlibrlum line once

the error becomes zero. There are then a set of points in

the state-space of K, K , and e tlmt represent initial con­v

ditions which will give a perfect match of the model and

plant. Using the analogue simulation, one such point was

found by trial and error to be e(O) = 0.2, K(O) = 0.5, and

Kv(O) = 2.25. Since em(O) = 2 the initial conditions on the

state variables in the simulation are

e (o) rn

e (o) s K(O)

=

2

1.8

.5

Kv(o) 2.25

From equations (13) and (30), z(O) becomes

~T(O) = [o -.2 -.5 .25]

(30)

and from equation (28) a = -.00625 and ~ = -.0103. Substi­

tution of a and ~ into equation (27) then gives y(O) of

30

equation (23). As a check, y(O) may be found from equation

(20a) as

(31)

The solution of y(O) by this method is shown in Appendix B.

The analogue simulation of the chosen example with

initial conditions of (30) was performed and a record of

e(t) and K(t) is shown in Figure 9. Equation (26), the

analytical solution is shown in Figure 10. The results of

the two error solutions of Figures 9 and 10 show a very

good correlation of peak values, and while the periods of

oscillation are not identical, they are very close.

In summary, the nonlinear adaptive system ;·ms lin-

earized about a known equilibrium point. A particular

example was chosen for study and both analytical and ana-

logue simulation results were obtained for eigenvector

excitation of the oscillatory mode. It was necessary,

however, to find appropriate initial conditions by trial

and error. The calculated system error and the simulated

system error were in very close agreement. It is concluded

then that the linear model is a good representation of the

system in the neighborhood of the equilibrium point. Fur­

ther, since the desired improvement in the system is to

remove or highly damp any oscillations, the verification of

the linear model suggests the use of linear compensation

techniques for this purpose.

K

e

+1

=~ i 0,-!-~~ l •7\ __ _{;:;_j'_[~ __ ::~~~~·-- = -,, ---~~-··-~- ···-· ' --.~--. ..---~ ;--·-

' ' ' ·-~--:

0-1~-V~-.--'v: __ }? . :q'- :~o/~= -=~=~=-~==·~ p - -_·-_-__ - __ - __ -___ - __ -_'-----------

0 ~· v . ~ . . . . . . . .

1-': ___ :____.; \ ~-' -----~-\ -~--~-----------·----------· --~----:---·--- ~--- -- ·---- ·.--~--- ----·--~-_j-,:--~-·-----;--··-

0 1 2 3 4 5 ~ 7 s TIME (sec.)

0

' jl ' ' ' . j i ' , ' i -,-Dr . ! : ' I ,---r--r- , I I / I ! / I J 1 I ! /-,-/ / / I/"'\ I I I i / ! / / l j I : I / / / l I I / / I ( I i I / I -t-l I f : / J I. ~. 1-1· ---+----, j_ -.;-+---~-r--'----·-·----"--1-J-,---+--4----·~~-·-t- !----1--~-t--t---t-f-• . : :j :/ : I : I ; ! : : ; i : ! / f / i .1 ! I : } f : ! ! I i J l ! I t ; . ! ' . I 'I\ ' . I I ' ' . ' ' ' I . I I ' ' ' I . ' . : ' ' I ' • ' ' ' 1! f '( r '4'\ : ! : 1 ! j f ,

1•

1, .' ; • :-t--,'

1;

1J • : I / ' /

1! ; / •I ,·' l 0l

' I ' I \ ' 1- ~·~. I I I ' ' ' ' i . ' ' . I I I I I I I I I I I ' I I j

~:~~·-J,~ __ j\\-T---+ I .,t-~ -1 rt~-~ _A·--~ .. ___;-·.---~-~-_:-~---~ ~--,---~-1 I :-+-~-+--~ : ' -r---f , : ; ; J ; I : 0 : ! i : ~ / ,.-..... : - 1 , : : 1 I : ! < 0 ~ I I 1 1 \ ~~ : ' ~~.... _,.--"f ' -·- ---- '--:=z. 1 cj-----...-----1 . - ;::y-.---7 ... , --~--- ='-"""'- -

' ; : ', : I ' \ I If. \ ' \ ~ • "'..__.. ; '---:- ~ ' l • ' ' : • ' • ; __ j,_:_;_:~:-\---P-' \:_L----\--1----· -· L---~-·--- --------·--' _: __ ~ ___ _;_ ___ , __ · _ _:_·_· -~---~ _ _; __ __:_;_. __ :__;__ __ : __ , '•I' ~' '')• I "-J' l:, t 1 '~: l ~: \:I I I i ~ • \' '~ i ' \ f I ' \ I I ' \ I • ' ' \ I ; I ' I ' ' I I ; ' 1 ' I \ l ' I • • l I ' • ' ' ' I . \ \I - 1 \ - I ' ' • l ' ' • I . ' • . I \ ' • ill;\ .,u ... \ . n I • ~ ••• l. I I~ I •• \, I •• I,,

> ~ \ • • • • 1 1 • • \ •, I\ •' I'\\'\\ I\ • I I • \ \ 1 \ . \ \ ', ~ I I \ ~ \ ', ' ~ \ I \ ', ~ \ ' \ L ~ \ \ \ \ , 1---L-J. ~ '- . ~:·i~ ~, -y-~.~-~ .. -r\~1 , ·• ~~.--· •. ----~-~~-:---\-\---·~--r--r----r-, 1 \ , \---\,-~\ -,_ \ ' 1\ 1

__ \~--· ~~·j _ _L__ ____ '..__ _ _L~-~_L~ _ ·._L_\_ _ _L__\_ __ ~ _ _l_ __ ~- __ ·. __ ~ _..__ ~ __ L __ _l __ \ ____ \ ___ l_ ___ ~_L __ \_ __ _l_ _ _L_L_L _ _,_ __ .:.___\_~\. __ \__~_ .. L

+.1

-.1

0 1 2 3 4 5 6 7 8 TIME (sec.)

Figure 9. System Error and K(t) for Eigenvector

Excitation of a Model-Referenced System w f-1

\

=;---,,--·J_1' 1' 1 1 ,' 1 1 1 , ~~-IT ,i 1 :T;-;--,-, ---;--y-;----,r--.-r-,,-:-r---.--1-,-r -~-r-r--.' -,-r-; I f' f1 J • , 1 • 1 1 • 1 , r • • , , • 1 , , ---·· ,._ t:----'-~·--·-- 1--t--.:-----~--_J--f __ :___; __ J ____ , ___ , ______ ; __ --· . -~--: --- -- ___ ; ____ }_; _____ : ____ ..)..._ ____ ,· -- : ___ J ---1---.f--:___j_ / . ' f f 1 f , / r , 1' f / l ' ! ,' , i 1 ; J ' ' ' f ,' ! , I / 1

t ! I I ! : ; ; f : 1 ! ! ' I ! ' : : ' i ! I • I ' ! f ' ; I !1 I : I I !

+.1 t-1 ; 1 '\ : : • : ~~ : i . : : • . ! • : • · • i · · · : : i · : I .~. ! -:----~-!-\--:--· -+ :--+- ' ----.--r--~-0 ----~· __ 1 ___ ~ --~- _ _!_ __ ~ ____ _: ____ ~-;----~--:--:--~--~--~--~ -~---: _____ ~ __ ; ___ ; --~- ~---

e Ot~l_: [_j_._f-~: .L : · : \:~~:r-'\~·'-_:_0~~'"'--:---=:.""~~~-~~--~ l_jr; rl:_: \-\ ':_· ~-'--·-- J'~-· }.J_:__}.y __ ~ __ ,Y._ --·- ~----- ·----·~---:---·---·: ____ ~_: ___ : _______ : ___ ; __ .. _ --i·-· --·---; ~ 1 ~ 't \ 1 • n . 1 ' • , : • • • • . : • · . 1 • 1 \ t I

- 1 + 1• \ ~ ; I •• I ~ \ I \ ' ' ' : ~ ' ' . ~ \ i ~. !_____J_: __ ~ ', ' ,----'--• •. • I I \ .. \ ' . • .'d-'.\ I I I ' . I I . • . ·~ ' I I ' ' i ' ' I

'\ ~ \ ' • \ 1\ ·, \ ' ~ \ \ I 1 : \ \ \ \ I ~\ \ ; \ \ i \ \ \ ' 1• : \ \ ..

,-\~\-:[· \--~, i T-4, ~-\:-\~~~;~-t-, .....~,--\-"1-\\--~.---\ --- \ ___ , ___ \_\ ___ :\--\·--\ __ \_\_\ ____ \_\ ____ \_\--\-.. -y--\ --y-r-'"--\ ----.,--\ ' ~~ -~L_j__-~_.l __ L __ \ __ ..___L __ t __ L __ l, __ .__ _ _.:._ __ _1._ ----~----~---·-1--\.. __ __:.._ __ \_ ·---~---i. __ L.~--- , ___ l __ L __ ·.___l_ __ ..__ ___ L ____ L __ \_ ___ ~ __ :&.._

0 1 2

Figure 10.

3 4 TII'fili ( sec . )

5 6

Calculated System Error for Eigenvector

Excitation of the Linearized System

7 8

VJ ;\)

33

C. Compensation of the Linear System

Since the linearized model of the system h.1.s been

verified, a logical solution to the oscillatory problem is

the use of linear compensation techniques. A root locus of

the linear system suggests a specific type of compensation.

The compensation is added to the system and the syr3tem

error is calculated and verified for the particular example

studied previously.

l. Ma~pematical Development

Referring to equation (16), the only design parameters

available for which the choice is arbitrary once the model

is selected are P.t and p. 2 • A root locus may be plotted for

2 2 2 ( ) X -J-aA + Ks(p. 1r + p. 28 6 ) = 0 32

which is obtained from equation (16). It should be noted

that only the complex pair of roots, which l~cpresent the

oscillatory mode, are affected by p. 1 and p. 2 • Let P- 1 = P- 2 =c: fL'

then (32) becomes

A2 +aA+p.Ks(r2 +8~)- 0

Equation (33) can be written as

(33)

K (r2 + e2 ) 1 +p. s s == 0 ( 34)

A(A+a) 14

This is the proper form for plotting a root locus where

p. is the gain to be selected for operation at any point on

Let K = r = a = 1 and 8 = 2. These are the the locus. s Seq,

values of the example studied in the last section. Then

34

equation (34) becomes

5 1+,_,. == 0

A( A+ 1) ( 35)

The root locus of (35) is shown in Figure 11. For in­

creasing values of ,_,.~ it is seen that frequency increases

and damping decreases. It might be suggested by the root

locus that for extremely small values of f-L' there would be

no oscillation; however, as noted earlier low gains in the

adaptive loops cause an increase in peak error of the system.

Low values of ,_,. also decrease the signal from which K and

Kv are derived. Correspondingly, there is an increase in

the time required for adaptation of plant to model. The

most lmportant observation is that large values of fL are

necessary for adequate tracking of the model by the plant

but the result is high frequency oscillations in the system.

Under an assumption of large values of JL' the root locus

in Pigure 11 shows that j_mproved damping can only be ob­

tained by drawing the complex roots further into the left

half plane. 1J.lhis can be done by addition of appropriate

pole:3 and zeros.

A passive lead compensator adds one pole and zero, the

zero being closer to the origin. The transfer function of

~he lead network is

ds + c where d > 1. G(s) ==

s+c

Because oscillations occur in both adaptive loops the input

to the lead compensator was chosen as the system error and

Imaginary

7.25 +j6

3.25 +j4

.85 +j2

fL =. 05 ---------+---*-·~~~~-R_eal

-2 -1 +1

.85 -j2

3.25 -j4

7.25 -j6

Figure 11. Root Locus of the Linearized

Model-Referenced System

35

36

the output was then used as the error signal for deriving

K and Kv. The output of the lead is denoted by "q."

Figure 12 shows the original system of Figure 3 with the

lead compensation added. The procedure from this point is

very s:tmilar to the preceding section.

• K

• K v

The state equations of the new system are

= -ae +K r m m

= -be + K ( K r - Ke ) s s v s

= -,u.2eSq

= fl-1 rq

q = - cq + ce + de

__ -cq+(c-da)em- [c-d(b+KsK)]es+d(Km-KvKs)r

(36a)

( 36b)

(36c)

(36d)

(35e)

':Phc compensated system is now linearized about the equili-

b r :i.L:nn x 1 = x 2 ::-.:: e = q = 0 . The variational equations are

where

w = J w - eq-

w =

Ktrtr e -­

m a

a-b K---

q

d at equilibriwn for the and the Jacobian evaluate

( 37)

(38)

r

K m

s+ a

-K l s

-----~-->

s+b

e m

e s

e

Figure 12. First-Order Model-Referenced System

with Lead Compensation

31

q

38

compensated system is

-a 0 0 0 0

0 -a -K e K r 0 S Se~ s

J -- 0 0 0 0 -fL e eq 2 Seq,

0 0 0 0 fl-1 r

c- da da- c dK e -dK r -c S Seq, s

Eigenvalues are then determined from

S I 2 Seq_'j 1>..1-Jeql= >..(>..+a){(>..+c)[>..2+a'X.+K (fLr2+tL 92 ~ +(d-l)'AK (fL r 2 +p. e )} :~ o (39)

S I 2 Seq,

The example studied for the original adaptive system is

used in studying the compensated system in order to facj_lt­

tate a comparison of the two systems. The values to be

used are

K -- 2 a = 1 e = 2 fl-1 = 20 m Seq,

K - 1 b = 0 r = 1 fL2 = 20 s

The choice of c and d can be seen from a root locus of

the linearized compensated system. Let fLI = fL == fL• rrhc 2

substitution of values, except p., into equation (39) yields

>..('A+l){(>..+c)[>..2 +>..+5fL]+5J-L(d-l)>--} = 0 (~-0) Note that a designer now has three parameters he may select,

these being fL, c, and d. Now let p. = 20 as in the original

system. A root locus may be obtained from

( >-. + c) [ >-.2 + >.. + 100] + 100 ( d - 1 )>-. = 0 ('+1)

and will be plotted assuming d as the variable. rche proper

root locus equation is 100>..

l+(d-l)(>..+c)(>..2+>..+100) ~ 0 (42)

39

For values of c = 10, 15, and 20 the loci are shown in

Figure 13. A choice of c ~20 was made to eliminate scallng

Problems in the analogue s· 1 t · lmU a J_on. From Figure 13 it ls

seen that for c = 20 the best damping can be achieved. rchc

root locus shows that increasing d causes the pole at c to

move toward the origin which incr·eases the tlme constant of

the mode due to c; therefore, the response time of this mode

increases. The selection of d was based on obtaining a good

damping ratio while maintaining a response time for the mode

due to c which is lower than that for the mode due to the

complex roots. For the example under consideration, d == ?..9

appears to satisfy both of these requirements, and the root

locus predicts the oscillatory mode will have a time con­

stant of l/5.5 and a frequency of 13.1 radians/second. As

seen from the root locus the time constant of the oscilla­

tory mode is a function of fL, a, c, and d whereas the time

constant of the oscillatory mode in the original system is

found as 2/a. A compensator has been specified and a solution of the

system error will be pursued as a check against the root

Of frequency and time constant. Substi­locus predictions

tution of values into (39) yields

~c A+ 1){CA+ 20) [ A2 +A+ 100] + 19oA} - o

>..( >,. + 1) {>..3 + 21>..2 + 310>..+ 2000} ~ 0

~ ( A + 1 )( A + 10 )( A 2 + llA + 200) = 0 ( i~ 3)

The eigenvalues from equation (43) are

Root Locus for Ima[:;1nary -----------

-20

Pigure 13.

C=20 C=15 C=10 +jl5

+j10

+j5

15\ 10\ ; .~--- ·~-----:!~.~-:::_:=:...::::: __ -=--:---=[ ....=...::: .:.: ::...:-::- :t _ _l~~ a 1

-15 -10 -5

-j5

"//~ -j10 "I

/I I

I I

II I -jl5

I I

I I I

Root Locus of the Linearized Compensated

Model-Referenced System

XI -

x2 =

)..3 =

0

-1

-10

)..4 = -5.5- jl3.03

)..5- -5.5+ jl3.03

The modal matrix is found from Adj [AI- Jeq] which is shmvn

in Appendix B. The ith colwnn of M is the eigenvector

corresponding to >-.i.

0 1 0 0 0

0 1 190 9(4.5- jl3.03) 9(4.5 + jl3.03)

M = 1 0 68'-+ 684 68l~ ( }~}~)

2 0 -342 -342 -342

0 0 171 17 .1(5 .5 + jl3.03) 17.1(5.5- jl3.03)

To simplify calculations the follovJing transfonr.atton is

made

w = Mv ( 45)

Then, as shown earlier

v = Av (lJ-6)

where

0 0 0 0 0

0 -1 0 0 0

A. == 0 0 -10 0 0 (47)

0 0 0 -(5.5 + jl3.03) 0

0 0 0 0 -(5.5- jl3.03)

The solution of equation (46) is

v(t) == e A.tv(O) (lt8)

As before, trial and error was used to find the following

initial conditions for eigenvector excitation of the oscil­

latory mode in which the system reached the assumed

equilibrium point.

e (o) m

es(o)

K(O)

Kv(o)

q(O)

Using these values equation

w(O)

==

(38)

--

2

1.8

.5

2.25

.26

becomes

0

-.2

-.5

.25

.26

Equation (1~5) may be rewritten and solved as

0

0

.000018

-. 00037 - j. 000736

-. 00037 + j. 000736

lt2

( 1+9)

(50)

-l M may be found in Appendix B. The solutlon for v(t) is

found from equation (48) as

0

0

v ( t) == • 000018 e -lOt (51)

(- .00037- j .000'736)e-( 5 · 5 + jl3 .o3)t

(- .00037 + j .000736)e-( 5 · 5 - jl3 .o3)t

The system error is the difference of model and plant

outputs or from equation (38)

e(t) == w1(t)- w2(t) (5~)

Appendix B contains the solution of e(t) from equations

( 1+4), (4-5), and (51). The error due to eigenvector excita­

tion of the oscillatory mode only is

e(t) == .206e-5 · 5tcos(l3.03t+7.7°) (53)

It is noted that the time constant and frequency of oscil­

lation for this solution were correctly predicted by the

root locus of Figure 13.

2. A~alogue Verification

As a check for accuracy of calculations based on the

linearized model of the compensated system, a simulation of

the system of Figure 12 was made using the values of the

example being studied and the initial conditions of equation

( 1+9). rrhe simulation drawing used in conjunction vJith an

EAI TR-1~8 analogue computer may be found in Appendix A. 'l,he

system error, e(t), and K(t) were recorded as shown in

Figure 14. To facilitate comparisons the calculated system

error, equation (53), was simulated and its time solution

is shown in Figure 15.

In comparing Figures 14 and 15 it is seen that once

again there is an excellent correlation between calculations

based on a linear representation of the system and results

obtained from an analogue simulation of the actual system.

The most important observations come from comparing the

system error of the original and compensated systems which

K

e

- i ---,------ --,, ---T

/ ! i / /- __ ,. ____ / ___ ,~ ___ .: ______ L__ L~ ~~ ' ' ! /--~.'--,~· ----:--/·-.~---·'----:--- '-~-.---

+: ~~-· ,·_:=_i -L_'_~=· "'" __ :_-__ -_-~-------~--

l·-: -, ':-.:~ --\ ! : ·-·-~·\-~-- ----.------------- ---:--.-~-- -~:------

,.,_"'::.- --"=_:_-; "::":::;::::::"" .:::.:..=:..::-..:..=.:..;-_ ·=::..=:::::-:-_--:::::::::-'-:::~---==- -_--:==._-::=.=-.::::-:-:-:=_-::.-.__:"::=._ ~

+.1

0

-.1

0

.1\: / ,'

!\

.5 TII'JIE (sec. )

1

I ! r' I ! I ' 1 1• , ' I I j ' I I I I , ' -t-- · . ~ J • · • I I , I , I I i , r~~-~~~ ' I i I I· . ! I i .• ' .. -I I i I ' ; ; I • ' I ! ' i ,____, I I I ! ' I

i : ' : I : : ! ! I I i 1 ; I i . . . ' -------'- I t ' _ _i__' ~ I i i :

~.--:

! ! j / r / J f • • , / : I I i / i ! : ; ! ~~--~-~---·---,-· --t-~:---: __ t -~-,~1 __ _j _ __,_ _ _; __ . . .

! "--·--:---~-------;--: --...--------'~--~- -~--~-- ___:, ___ , __ , __ i _ ____:.__~--~---', --- -~------;-- -r··--

\ \ \ \ \ \ \ \ . I \_ , \ 11 \ i, . '

· ' · ' · 11 ·, - · ·1 • . 1 · • , • · · · • . • , , • \ :---<1--.. -:-\ \ \ \, \ ·.- \ \ \ ' \ \ ~ \ \ \ \ \ ; -~ \ \ ~ \ \ \ \ ' \ \

~~~s.-r-i=~LU~Lcc=c\0·~ ... --~,c~~=I~\~-·cy-:\=r~~=:r=r-~~"~\"-~~~ 0 .5 1

TIIVJE (sec . )

Figure 14. System Error and ~(t) for Eigenvector

Excitation of the Co~pensated System ,._ ...,_ ..;:::-

e

+.1

0

-.1

i 1 / l 1 1 : ~--.,--~_ --;--r--·r---r----,- , . ---r-1 / , I ! I t / .1 _; . : / ,' f I I i I ; / ! I I

-+--t--;-\--;---.--i--~Tt!--i:--t---:--~. ---~~--t----~·-; -!--t·-r--r ~([=~---!--- / /~--~-- i-

----------------~-- __ ; ____ ~------.-------------'--!-~·---1----+----~----

~----~----~~~--~~-----

r-\ . : ~': ·:--L_, __ : __ -1 _ _,_ __ --~---: __ J _ _: __ :-----\--;·--+-~---i--.--;--\ ~ \ ; I \ ; 1. I \ \ • \

. : . I . \ \ -. : I \ : ' \ ' ' I \ \' ; \ \ 'I \ \ I '· i \ ' \· 1 \ , ' I. l \ \ \ \ \ \ \ \ , \ , '. \ '1 , 1 I I ' \ _l ~ 1 _

~\~~ \ \Ll. _\_i_C\\~-\\-r-\·--\-=·:·-·-;--\---\--~~-r~\---\~---\ __ \ \ r~~=-\.-~\ __ 0 .5

Til'wlE (sec. )

1

Figure 15. Calculated System Error for Eigenvector Excitation

of the Linearized Compensated System

~ -.-v.

+.2 -!r - - - Original System

- Compensated System / \

I \ I \

I \ I +.1 + \ I \ I

I \ I I \ I

I \ I \ I

0 ·l . \ I 7' If ~ +-= ~ I I I .\ . I \

I I \

I I \

I \ I

- .l i" \ I \ I

\ I \ I

\ I \ I

' / \ I

\ I -.2 t \ I -I I . !..

I • 0 .2 .4 /" .8 1.0 .o

r:"lr ~~ ( \ 1 1¥;.t. sec . ;

Figure 16. Syste~ E~ror of t~e O~ig:~al a~d Co~pensated

Sys~e;-;,s for Eit.,;envec'vo~ 2xcita~:i.on

/

1.2

..__ _,__ 0\

are shown in Figure 16. The systems \-Jere identical, inclll­

ding initial conditions, except for the additlon of co!npcn­

sation. The compensated system b..as a heavily da1npcd error

response. The peak overshoot has been significantly 1·ed'J'·cd

and the response time has been decreased by a factor of

approximately ten as compared to the original system. 'rho

same conclusions are found by comparing the system en·or of

equations (26) and (53).

In summary, it bas been found t:b..at J.are;e adaptl·vc loop

gains, fL 1 and f'- 2 , are necessary for good paraP1cter t t :.1.ck 1 n 1 ~

but this creates higher frequencies of the osct 11atory 1:1udc!.

A root locus of' the original linearized system ba~:; :;hovm

that the system may be improved by drawing the cowp1cx roots

further into the left half plane. Compensatlon 1·ms added

in the f'orm of a lead net·work. The study of an ex:-unple

showed that the compensation significantly improved the

oscillation problem. An increase in damping 1·:as predicted

by the root locus of the compensated system and verified

both by an analytic solution and an analogue simulation.

D. St~~ of System Performance

The example studied for the linearized system and the

1 d to simp 1 i f y the linearized compensated system was selecce

t . y~tem error. This mathematics involved in calcula ·lng s •=>

Systelns in which parameter variation section provides two

Due to t he number of multipliers on the was more general.

available analogue computer only one of the plant p:ll<lmctcr:3,

Ks and b, could be continuously time-varying. 'l'he pl:1nt

dynamic parameter, b, was arbitrarily chosen as the p:lra­

meter to be varied continuously. In the first system the

plant has a sinusoidal variation of b and of the input r.

The second system has pseudo-random variation of b and r

and step changes in Ks.

Although no restrictions are placed on the input one

of the assumptions in the development of the model-

referenced adaptive control ·was that plant var:tat :tons 1:oq ld

be "slowly" time-varying. It was noted that for hic;llcr

adaptive gains, ~ 1 and ~2 , the system adapts more rapidly.

In view of' these two points the value of J-1- 1 and J-L2 and the

Chosen to delnonstrate the 1m­rate at which b varies were

provement in system error response with the addition of

compensation. For the two types of variation selected both

th t d Systems were simulated on e original and the compensa e

an analogue computer.

1 ~.r-tth Sl" nusoidal Variation of l' and b · Fir s t - 0 rd e r S y~s~t~e~r~n~v·~ .... ~:.:_::::.::..:..:..:.::.:.:...:_;_;_---

a. Original Syst~

used in the system of Figure 3 'dere: The values

b(t) .5 +sin. lt a === 1

K 2 20 -- ~I -m

Ks 1 20 r(t) === sin .45t - 1-'-2 -

The initial conditions were:

e ( o) 0 es(o) === 0

Kv(o) K(O) .5 -- 2 - m

l~g

The time plots of b and r are shown in Figure 17. Plots of

x 1 , x 2 , and e are shown in Figure 18.

b. ~o1~pensated System

The values and initial conditions used in the

system of Figure 12 were the same as the original system.

The compensator lmd parameter values of c = 20 and d = 2.9;

and q(O) was zero. The time plots of x 1 , x 2 , and e are

shown in Pigurc 19. An expanded portion of the error is

shovJn in I~,i£-Su.rc 20 for both the original and compensated

systems.

2. First-Org~£-~?tGEI!_~vith Pseudo-Random Variation of r

anq_"!_J_ and S~_e_p Changes in Ks

a. Or·ivtn8.l System ----- _,;. _____________ _ The values for the model and adaptive loop gains

used in the system of Figure 3 were:

a 1 ~= 2 J-L, - 1-'-2 = 20 -

The variation of r, b, and Ks are shown in Figure 21. The

P C" d d · 1 r and b. were generated using a ueU 0-l'D.Tl om s:tgna S, .,

digital computer. The method used is presented in Appendix

C. The initial conditions were:

K (o) "= 1 K(O) = .5 v

r ;s shown in Figure 22. and the resulting system erro ~

b. _9o_!_lpcns~l ted System conditions for the system

The values and initial

of r:~· wc.,...,c tl'lc same as the original 1 lgurc 12 .L _

compensator had parameter values of c = 20

system. The

and d = 2.9; and

r 0

7 1 . ,' ~-~-~~-/--,--,-, --,-. --1 - --.' ~-r--,-,--7-- ---,--~:---r· -:--r·--·-·:·-~;--~-~--r-·-·;--~~--r-;--r--r---r-·-:-;--, ---T---7·-:----r-~r---r-,-1 ·' i ; I I ' t I I ' i J i i ! : ; ! i I I ,I I ! ! ! i I I : I i I ! , ; I I I '

O]J. . . : .~,.. ~· ~ .. - . . .' ,. r . I ,· • ,' ~7'1 ___ ,. -·-r.---.--·-7~-.·-.--;z;:.· .. :-.. ,.-' -:--?<~,-' -, ---~ I Z' ,~ Lt' ~-.:-. . ·!1 . n. , I • I , . .. , . f', , I • , . / \ , ~ ·' . ·; , • ,~,., . • .. \ . . ,. ! . \ • • . • \ ; ' ~ ; /: . ; : I I <; : ,: f i : i \: <I ' :' i / : ~ i ~ ! ! .I ', ; l • ; ; \ ; i ·' ~ \~ : I _ -----t'--i--•---l~C·--· --t--"'~---r----·;--;---'--:-----<.--··-·-,-.1 .. ____ ... , .... :...\ --+----·r---·- .. ·-·---- , ____ , --~-~··-\--·-(---\ -·---·=·t------- ---·~-' : ., : \ : !' ' : ;: \ : I : \ ; ' ' ; 'I ; 1 ~ ; I i : I ; : . ! : ; : . .'; ; i l : i : I I ; ' ' I 1

\ ; ,. ., .: L ' . . ' '\ ' ' ·.· ' I ' \ : I : : I ''I • . : f ' .( ' I I I '' \ ' I • I ' I : J ' I . . ' . ) '--t ' : I. t · . I, -~+.---t ' --~-·--~--· .1___',. '.·--·:-~+--·} __ . .J __ l.__;----1---~~~-'~·-'--i \ , '----~-. ' ' \ : I : \ ·LJ· \ I I ' \ : : I ; I-:\· ' ' : ·. ! ' \. 1 : I i ! ' I ' I' ·\ : I ' \ I ' I I i •I I ' ; ' ' I. : ' ' ' 'I I ' I ' • ' ,. ' ' I ' \ ' : ' I • \ • I ' i ' ' , ' : . \ I •, I I I ' ' : . I' I ' ' ' ., • • • • I • . . i I ; • I l ' . I I\ 'I I i . ' • t 'f

-1

-. -~~.-+-:-.-;-. , .. 1 . . ,-t-;---1--r--. -\,--~-~--~,-'-,~ .. -. \--;---· ---·· ---! --..,-·, ---: --:--·,-- --~---·-.-\-·f·-~--\--1- .. --~---· .. --~· .. -----\\----r~ --; , .... ,-• •. , . I • . I . \ J ·. . • . I • • , . • , • • • • • , • . • ·u , ' . . . . , 1+. I \I ; I 'i ' j I '\ ·'. \ i ' \' ' ' I \ I • \. ' ', .. I ' : : '. I I ' \ I : I \ •I + I ' ' ''; 1 >« ·~ · ·.·' \.J __ ;---~--~~---""""-:--·--'v..--~_; .. )~ --·-~·\..L ~ · ' L.M--

t \ ' I I ~

+1

' I ' I ' . I ' ' ~ \ ~ \ ' I ' ' ' ' ~ \ '. ' , 1 • \ I 1. \ \ ____..__..... __ _..__..._--....,..____._ __ .. __ ---••- -----4----·- -•••' •r•o•• ,,,...,._,, •-•••<••- t,,,.

0 25 50 75 100

Tirvffi (sec. )

125 150 175 200

r -~ --~----r: , J • _ , .! I i ! ! I I i I . I , . I • I ;--,.-- ---· i r·:--i-+--,__; _ _! : I / t' / I ! _:" ____ ! -' t- --r----- __,. __ . ; l I I : . __ ___,__ . I . I I i I ! r-· - -, " . , . I . ~zm· ! • . ,__, __ ._ .... _ .... : .. , I I • • I ' r-~ :--r·---~--T--" j I ' I t t' I I I I I ' ---.. ·----·--· ' ' I I I I J ' I I .--T-' --

. ' · • , ' • . '' ' 1 , , i I ! -,.--- '-'---" ; ' i . • j ,.,. . . .~ I ' ' ' ' . ' • I ' ' --·-- --'- -- . ' . ' ' , , i i \_ I ; ' . I I / j ! I ~--~ . --· -+-__:_

, , I i ~-· --:---~~----! ___ 1 7A' • 1 : :...___' , /--:---f--/ : ~ I • • • ·-··- _L ___ I t ' : ' • ; ~-t-"\ I .....;_· i i ' ' J; . I • : ~~ . • ______ '\: __ : -: '---~- ------Hu , , , 1

, \ , '· , -,- , -!-----L-;____L__j ___ ;_~7 · .--. · : :----~----;--·--1 I . . ' ' ; . \ ', \ \ . Tu 'i :-'. -\- u-:_:_,-;~.-;-c-T-, ! \ ,--·rc-;--'----i--,--~-

1 t I · 1 :

' I , ~---'-

b 0

0 25

Figure 17.

50 75 100 TIME (sec.)

125 150 175

Input, r, and Dynamic Plant Parameter, b, for Sinusoidal Variation

200

~v-.

0

Lfl 0 0 . 1.!"'\

0 •

0 0 rl

~ . C)

Q) rn -~ 1--1 8

mom C\J (\j . (\] .

f L..A t

\-=!I I ----~~· . ; :1 g . . 1 . C\J ---: . -. !I. _-· ~-! ·--... j· 7.: :·~. 1~7:

I ·r·. ~~J · ... ·.· l

I I • - - I J

I I - . ' ·- .. ·1 -T-··r . i o ·- \,~ .. ·-;··· I ~

.... =~~.J .I .. ~ .. I . . . t'-.1

- . ;. 'j

. . . l 'i : .. J

. . ! '1' ··- "J . j .... / I - ... -7 1 . I . . 1 ... ·; .. T'"-:7 .

: I<_~; ( y. •,.

Lf'\OU'\ (\J C\1

• r-1 • + ;:.< I

() Q)

0 (/) 0 ......_,. rl

(:)

<.) I •

l)

' 0

. ..; )

'--- _..., ()

,0

. ) ('j

•: ..

r -1 (~ '(j •. -1

0 t') ,., ·" ~--: . .;

U)

(."\j . '·-: >< ~)

. C) rl

C) I:..; :_j :_.Q

......--1 r~

., -1 ·-· . -

..L. 05 I i I i ! ! i / ! r--r--; r·-; I / I H ! :' i i ! -I -,-----; lj_f ! ! ! I I i I ,Cl_! -r • j • 1 r j $, I • I • I j 1 j. t t 1 r ~. 1 I I • j ~ ' • '• J ' '1 ' I ' , ' , ' J . ' ' . ' I . , ' r ' .. i ,., ' ' , 't. I , 'I J I'' ' •j, • I ; • • II, , l I ' I I I I ' ·i ' I ' I : •I I • ' I •.' I I I . ~~- 'ill' ' . . . I '.I ' . ' ' ' ' ' ' ' 'I' ' ' I! I • J· . ' ' --- 4-- ~~,+-- fJ--··------p----·-·--r--,1------ -·~- 1-~--~---'-j•-- ...... -..- } ··-----;_--·-·t . I' . ' J ,. • • I • I ,. l I '~ ' '. ' ., I ' .. ' . . ,, 'I . ' 'j • '·"! . • ' • ' . . - ; t I ! I • i, • •I • ... . I ' .. .. • ,. ~ l • t I I I I ! I ': ' I -: l . ~ ,,, . : ·~ .. __ : ;~.: ~ ';: Jt ·,; ·: I ~-..;: .. _ . J': ',L./1' 'ft._~' i_.-Ji· '· I\...: e 0 f. . ,....)!.......,~ 'q;~r-,r·•~:r:-;··---=-;•·"'"'~-_,.""r--c~.···~~-•:T-.·- ..F--l'"",··-''1-c:l-4

~--:-J~--:~~- H--'.-+J._:_;,_j; i • : 'J-~~-~-,--':- :l-~ : ___ , __ \ :_~--~-:__ -'- '!---'- __;.~ '--'---.05

_\ __ \ \ L \ \ \ -\ \ \ ~_\__l__\ \ __ U--:__1_\_j ____ \ ____ \ ____ \ ___ \, ___ \ __ '. ___ L_ _____ ,_ '---· __ j __ ~_i __ , ___ , _____ _

+.25 X2 0 -.25

+.25 Xl 0 -. 25

0 50 100 150 200

TifiiE (sec. )

• l I • I I 1 , 1 • I • ; 1 r--;--,--~-----------r---1 --. 1 ; l

~ ! : I j I 1 ! I ! r J i I I i i I i • r r ,I . I i . : ! I ;

; : : 1 . : ~---: --i . ~ -.-· --!--:--£_-, ---'--:,-/\; ·~ !-~--:--!--~- ~--~~--: ----.---·-rc:,-~----;--~--1 --r--:--7 i , :~--; --. -~~: !j~,--...._ . : ! . ~''='t· '-' "'~-- . q- .. ,......-.,._r'\) ·\~~· .. : : .. _..,J4~ -- -~ ·-- ----~----- __ ._ +--_....., "·---- --- \ -. . I . . : ' "'~ . ~- :""i'i...;_/"': ~ . ,"----:-\: . ·'.r..J ---./'-:;::./'• ' . . . \1 ' • , ~· _..! ,r-r . , , _,.. ~ i. ·, ' 1

\ T ~ : l ~ ~ i ~ : ';~---~--;--L t ; i __ ._i __ ··---~~__:\ __ ~ _ _j __ i __ l __ ~--~--LJ_j. __ ~_J~-l~-J~j__J __ \ • \ ~ \ \ \ \ \ \ ~ I 1 . : ' ' ;' . . ' ~ : I ; \ : ! t \ I I ~ : ~ \ ~ ' \ ~ ! I I ~ \ I \ \ ~ ~ \ • L ___ ~~ -.L_~: ___ j_ __ l_._ ~- \ _ _j__

0 50 100

TIME (sec.) 150 200

I ' I ! t • t I • t I I I I t I I I I • i ' I I • j I • I ' ' I ' ! . j I I I : I I ' I . ;__!___!.__ , _ _L __ I __ : -~ l_____L__L ~ I : i i __ ;, ' I i ' : : I Y1 i i ! I i i : i ~· I ' ; : ' 1 : : /l ,- ; : i -;-;- : :~-/ :r-7.-~-·----:--- ---~----~----/-17 -t--,--;------,---------t--:--~ -- l-: '>J ',~· I . ' . . • • . , __... '· t I -·~----·'---·--~ ·~ . !_ : I___.:~---~~

i : I I : • ' I I I .. ~~ . . . --,-- ·- - '0 "'--: - - ·------ ~-""-J~. ----- _,_ ; : ~ ll: ij : ! ' i ; ; ; , 1'\, ; 1' ~ "-.-'....._ I ' •....._ _ ! ' '' ~ < .__ ---v ~.-:_ .. ···'--~.--~·-·-~--~--· ~---·---~~-!_~ ··-~-- ··-.. ··-------·--. . ' I I~ I I • ~ '! 1 I i ! .

\ \ \ \ ', i \ I . ~ . i \ I \ ' I ~ ' • I \ 1 ; • I I • ' I --~ I I t : __ ~ -._. __ ; __ · ____ . _______ . __ _:__, __ ~~--· l

0

Figure 19.

50 100

TIME (sec.) 150

x 1(t), x2(t), and e(t) for ~~e Coopcn3ated System

\·:it~1 Sinusoidal ·,.rar'~at ::.o~;. of r ar.d b

200

\....­:\)

Original System

+.05 r -:----;-" ---,--T------r---- ---;---;--·;----r-----1 ---- T----r------1--r--

; i i I I . I ! f I ; I • I ' ' ,I j • l 1 ,1 I i :

f ': / ~~ ' ,' ' ! ~ / I -- ; ;'! ! I t ;_;--. ~' I ; ' ! ' l_ . . ' . I' ' . '

::::. '. i~--11,, ll:~·--------------)"\-~.~--------~.~---.- \--~,-~~--.~--·-------~--r\-7\~.--~·-:·----_;, : /"'\ ;I ,., · ·.· · -~- l /1~_1 '-1~-~~- :--',_/-'+-.p.:..~-,..._""' __ r~:·-·,_;_\J:_v·~,!f:.._.:..·-"---:::::.__-r -U1 r :.-: ~V-.=.~---~f- -\ 1~-'i :-~~~'-l.r--- -~. __, ..f---t.-;~. I I. I' . \i '-'~ I . 'I \• \J v ........ "-' ' ' I/ ' 1.1 \; : V · ' ' 1·;.:\J '. \ • •:'l,;v v '' dv I,..;

05 + ~-:-. • . . ' t H-:v~ --:-. _:_---~:~--v--~,--:-----;.- ---~~~-=:-=--~ ---,_-· \(;,---:·--:------·.-\,}- -~=~=~---·--,-- • 1 ~ , l ' 0 \.1 \ • \, 1 ' 1 I I -\ \__ t ~_j_ \ -~ \ -~ ~ ~- ___ ;_ ~-- L \ '• -- \ ·-- \ ., :. '------~-- ___ I"---" \ ...... L ·.- - \ ____ L

e 0

0 10 20 30 40 TIME (sec.)

Compensated System

+.05 ---------- --------------r-____,...--·, I ,--,~- I ' '

! : i ! ! ! I ! I ( ! ! I I I ! ( ' ! ,.. I ! I I I / I I I I I ; : ' i t· I I ! } __ , ; I ! ! J t . I . ' f I ...

.l., _ _;__+-1 _ _._,_..__ ... I -+--_;_-./ql ' I . . ' i ' i ; : : ' : : I ! : I i : I ; i I : : ! ' .

I j I I i : i I I ; ' j I I : ; i , 't\ : I I ' I ; I ; I I , I ! 1/\ • , 1 ~ \ , 1

; • ; I f--~---~:--:-+--~--; __ : __ ;--;~~---~~--~-~-:---f-i--t--;--;--'-~.· -c:-\ ~~:- '=,- _!_~ :J:~-Tj=_j\ t~ -C- ~-~- _':"/'vc -:~, i- -~ ;_ \£ _ L \~:~\JJ,;: __ ---, [~/ __

\ ' . \ . ~ \J, I l ' l \ • ' - • ' I \ • . I \ • ' l I . . 1 \ ' ; ' 1 \ • \ I I , 1 • I < I I 1 ' l • > • I '

\ ; • ' ' ' ' • l \ ', ' \ \ I ' : ~ ~ \ ~ I ; \

', \ \ I \ \ \ \ \ \ \ \ 1 I '; 11 \ \ ~ \ \ \ 11

: \ \ \ .__ __ \ ___ J___ ~--- ~--· \ _..__ --~ ______ ••• \ -~-- _ -~- : __ -~- __ •h ... \ ____ .._ __ ---~ L '--- ;· -~ __ \ _____ ..,.. __ ____t ______ ___.._..._ ____ ~.---l.--\' -~____.. ___ .l_-...__

3o 4o

·----- --·· ~-----·

e 0

.05

0 10 20

TIME (sec.)

Figure 20. Syste~ Error :or the OriGi~a~ a~d Compensated

Syste:ns ·.·:ith Sinusoidal Var::.a~ion o.i." r and b v~

UJ

-1

TT~f ·l··-r-I'-T'··r-r-r-r--r-r-7---~-~~~----r--:-:-T--T{-r--r--r--r-r-~---r-;-T7---;~r;--~--~--r~TT-r --:--/--l-,-. -r-+-+--+--...:--i---lf-i--:'-!--t--i--r---r-r----:--rr.-.:--i---r-!---;-1---l---f---:r--f---1-1--,,--t--;--+--t---;-- ~A:-, :' f i ' l ! ; : 1 ; ':' ! , i ! f 1

( :' • • ~._; !_j_j__! ! 1 L_,/1 f ( ! { I f ! ; ' i L_ij_j_ ~ • \.../\' ..-. .. 1 i ' I j/ I

I j ! """' • ·• / ' ' ' • I I

r 0 -----(',-,--.,----~~~-.-.-·-_--,---;-- \ ;'\ r-.-.-~~~---.--.---.;--~-- ·,--- ' :' \ ;.--·i---,..r\ -: --~---:--;,·-;, ---.--;~\~~-;-r(

1 , / ~ J 1 , - • 1"1 I · \ 1 ,'• 1 1; • , . { • f ' I J \ r • I ' I \ / • r- • 1 ,.._...__.___r.,. . ' .c , _.,._,--1..--v--t-_----.. ~-~-- -~---____ .... ___ - , _____ --- ---.......... -t-·-·- ... __ _,__,.. -~...,.- \--·--'"-~~-.. --+-----'-• . . i I , . \ . r,J, ,. . .... , " \ /".. , ·. r. ·. ' . ~ , , ' ·v . , ' , , ' ' I \ ' ' ,

. :_ \ / \ . ' ; ·. . :! ~ \ :. : ·\ :; I J ! ' I "'-. ' ! ' ' ' 'v' ·, I ' . \ ' I : \ . 'v ~ '' '

+1

+1

b 0

-1

+2

K 8 0

~ J ! I . ' \ . • \ ; I . : . \ -i I '· I ' \.t I ~ ' \\.' : ~ I : ~ :------\.}·--·--•--·-------'>-··---, -·•-·------- v-·- ,.->-------"'."'· -----~----; ·--------- --- - ____ , ,_ - \ ·--~-. ,______ ~ :- f--~ ··--- -\ 'i --------~\ -: --- _, '~ ~ i \ 1 : \j . \ : . ' \ i : : 'j \ / . \ I 1 I \.; i .... ../ ' : \ : v

. l . ', i : . ·~___~· --------1·~-· 'V \ ;

. ' \ ~ \---~-------·--~--

1, \ \ \

~'-\ \ '\i \ \-~-\l..:.--~·,\ __ ,: ___ \-~\--1~. -~-·~~--~-----~.--'--; -\ .---~=-\.----;--~-:; __ .-+---~-----;- ~ -~--__, ___ , \ '\ \ I i I ' ; I \ \ \ \ \ -. \\ ' . \ ' . \ \ ' \

:_:_l _i_ __ --~--___},.._______~l.--~--·~~·--'--'--__;_. ____ _L __ :.._ __ L__:_ ____ _',. ____ ~_l _____ \ ___ -~ ____ , ··' ___ \ ___ . _____ '._ ___ I \ -~---- L_ \ _J...... __ L

0 50 100 150 200

TIIVill (sec. )

! I ~ I ' ·___1___;_~_J_!-~f-:--i--f:-' ~· -:-: :-:· · .' · 1--.~. ; i . : : i · ~· : ~ ~i---: _. -· --~-~-! ~r=-~· ____ :~;---r---~---~ __ ,_\_; ___ ·:/~.Lz:~--ucJ.=rr- :-:=-\ ; ; : ~ . : ~----r--,-. -:-:'C/ ' . ' : :·';fjj=~l ! :

-r \ l-= . . G_ ~-:-, - J~'-=' ,--\---.-~,-;-_:--; j L_! __ . __ ~. ;----·~l-- -\_L 0 50 100 150 200

TltrlE (sec. )

--r-~-:-,--1 ;-:---1: ;;-;-:--lr-;-; r i r:--;-;---, I -,---.-, -- I -, l I i I 1-----r--.~ i i ! 1 : : ,i ! j ; : : , ! ; I 1 •1 ~ ,: . : ' ! I I , j , ! 1

It ,/ fl 1 I . 1 • ' I 11 '---;-: '/ ' '

---'-- •: :r I ____ '' i --·· -·---~- , ___ · _L ·--· '_: ___ , _:-

- _i -----·---

0

Figure 21.

-~---

50 100

TDIIE (sec. )

150

Pseudo-Random Variation of r and b ar..d Step Changes in K s

200

\...~ .;::-

+.2

e 0

-.2

1-7-~--,--s J , r , ~ , • --t--1·-r ~ , ;--,---,---r-- -r·--·-,-. ---;--,--~---r---~- ·:--·-,r--r·--,---l--;------r--r--,--,---,-----r--r----, I / ' I I f I I j I ; I I I ; ! , . I ' . ' I . I I I ' , I , j I I f I I / I I j I • I ! i I I l f I ; ' ' i I ' : . . i J i .I 1 I I I I I I ' j I i I I I I ~ 1 • _L_' , .1 ~t--:--L-1--.--/ • I .~ t I ] ___ .' __ : ___ ]_ __ ~· --L-~.--L___j. __ ; ___ :__ __ :.___,, __ .;___;__t__[ _ _j ___ l ___ L--f~ I I I f ·' t t ' I • I I I I I I f .' t I I I I ' I ! • : ! f I t f I I I , / J I I : ! ! ·' i j i ! i ;. ! ! / • ; 1 t I I ! . ~ I l I i ,1 / i I f f : I f ! ! j ' I

: t J ! ' -f,___! ' - ,"' I I I . I I ' I _J_ 1 ' , J • , • • f I , • . : . • J • •! , ' I . ' , r' ' I ' I r ! I ~ I ; : I ,"; ! ' : ! l :. I ! : j : I I I ' . j] J

• " I (i • I ,! I ··f --;-:--7t--.---~·~-.-ll\ _____ :_ --,--,--,--.--~-;-----:----J....----r-- ·~--: ----:----· -----:1------:----J--, :r·---~-- -----:----· ,;----.. :~ ---,t-

~' :, 1 ' " ·• .1 •• , • " • , 1, .r ,, >!!I ·f'•

. ' i . ~ I ' ~.... ' ,, " ' • '- '"'" i : I ' ' • ' ' (1 . I'! ' I. ' ~ ' A ~ I 1 I ·' .. ' II •• '.~ , .. , •. 'l '1 ., Jr, I' • '"· ,, I 1. • • ~ ,. --.J••- ,,,,, I I . -J\~· ·rJ·~ )- _JJ_.___.rv-...... ~~i'.-.:A~-~- -·- --·· .. -----~-·- ·,.-.:..t:=·-~-tr-· ·'--R--'·-'-4i~~- ·4 " I ''·\J ~ f ,, •. II• ' ••• - •• • 'I li'·· \,.... ,, 'I . lll'lr-"'\J •. v I• • ' ij ' I'¥ I I \,1 • • ' •• " ,, I li ' I 'I ' :• I • .I ' I I ~ ' ;! v I I ' ' \I I .

t , J ; I ~ ; ; • ; : ( ; : f \ 1{ ~ .: " II ; l 1 1

• :,• ~ , I \ : ~ \j f : 1 \ \ .'J \ ~ • • ; -~~~:-~----.·\t-·-.--~-! -,~--~-- I • ~~-~--i--•--,j---j;i·-t-~---:---,l---~;i--, --;-- t-;--,-----;~---i--_.~--i-·-----'j--\J--1~'----• : • I • ~ \ ' ' • ' ' ' : \ I • ' I • I ~ ' '.i ' ' ' I ' I . ' ' 'I I \ 1 1 I I \ ' ll, ~ • ' • 1 1 I • 1 I 1 1 , 1 ' 't! t \ 1 I l \ • 1 \ \ ~ \

_ , • ' • • ' • ' \ \ ' • \ I I ' ' ' o I ' • ' ' ' ' I , • ' • •---

\ • \ I I I ' \ ' I I I ' \ I I I ' I ' I ' . \ ' I ' ' I I \ I I I ' I \ I I 1 \ ' • \ • \ 1 \ • 1 • \ I • ' \ 1 I < 4 I I I l 1 , 1 _j >

1 \ ' ~ \ \ ~ \ I 1 \ \ ~ \ \ i 1, 1 ~ \ I 1

', \ 11 ~ ' ' \ ; \ ' \ \ -l-~-\----.1-..\---.--~~1----.---~ , \--\-···~----,---~----· ·-----~ -~-J --,-~-~-~---~--" _\_._L---\..-_, __ L_.__,____._ _,_

\ I\\~\ I ~' \\' \\ \' I I\\\\\\\\ Ill I I I\ \, \_ \_ _\ __ \ __ \._ __ ~ ____ \__~ __ j_._\ __ j ___ ~----~--1_1.. _\__\ _\ \,\ \ .~\ __ tj ___ \ __ j ___ \ ____ \ ___ ~--. .\ ____ . __ 1_L __ ,_\ __ j ____ L\_~,___\ __ L

' ' \ i ~ 0

Figure 22.

50 100

TIME (sec.)

150

System Error for the Original System with Pseudo-Random

Variation of r and b and Step Changes in K s

200

\...~ \..-"';

+.2

e 0

-.2

' I I ___;_~'--.1-....l

~-~--,-- .I i • I • i ~.,-------~ ---;~-r--j.--r----r----y--;·--r--; I . T-i__,---·~ ) r-,-! I I I I ·I I ' I I ' I ' I ' I 1 I I ' ' I I ' ' I ! I I 'I I i ,' I , _, ·_I .' • _. / I 1 1 , r I 1 ,: , / , .1 , ! / I ! / . .' . :

;----:---r-.-i~-.~--_;---r--~-· --:----:'--/~ .. --:---::---.~-t--_~----/-a--r----r-~;--:--7--.:---_r--~r--_--;;-- /--~--;--;t-r--!~-r--:--: : i ' i ' , ! f -' ~ fl : ,

+-~----~~r-i~~~.---_~:~1;~~~·~~-~.._~---.-n-;~~-.-, ~~---!1-'~ ~ · : ' - f -: ~} 1- '-'- ~--· --:--~~--7-- -"-k !_J_H--1 !~ ~J \ ~ J+--~· -~J!......;....D._ :-~.._~; t_ ~t-~~~ =-- C: ~. _ _,_: ~-0-=J-,_.J .rv:=;..c-- ~____...-:, -r~~i;_.,.;~ . ._•--I~.~·~J

I v:-: r· ,. , -1 .. \ 'i \I v .\ . ~ i· \' . \i - ~- ~ v- i .r-;t :, ~ ~J ~- . v ·'t, : . v \ 1 ~ ~, ~ : -'-. --·.----~~--~ --~- ,\'. __ , __ ~---- \ ·- \i ___ L-~:-~ _ _j ____ ', _: ____ _: ____ -"---~; __ : ;,! . ---·---)L-, __ j; -:·"---~1- ~-~--~--~-~+--; JJ:---~----:\j--\j-1--;

.,J ~ 1 11 l 1 \ ~ • \ ! ~ \ --',--:--~-'---r--,---,---,------' ' ·----.--------- ' '

\ i . ' \ \ \ ·~ \ \ • \ I ·.• i ·,_ ', '. \ ', \ I -~ ~ \ \ \ \ - \ 1,- ~ - !l l \

\ ' \ ' I ~ I \ • ' l \ \ I I I ' \ I ' \ 'I - ', \ \ '\ I i . ; I \ . ' ' ' \ \ \ ' ' ' ' ' l ' \ ' ' • . . . l l ' ' ' ' ' \ • . ' \ \

\\\-r-0--~-, --\\\ ~-.-~;--\--;--\-\_\_\ -\--\:- ···.,--- ' -- \---,_--\-, --~~\--- \--\-. -\-r-T\--Y~\,_\_\\_\_\_\-~\\--' __ \ _L_ _ __1 ____ \ ___ __1__~---l..--~)._ ___ )._~:~_\_ ___ .l____L_ __ ~__L_____j,_____L__l__J__~----- ')._ _____ _'._ _____ .. \ ___1 ____ \__ _____ :.._ ____ \..__ ____ _\__ __ L__\_ __ -L- ___ l_ ___ \. ___ ,__ __________ ~------\__ ___ , ______ l ___ \_ ____ _._ _ _______l_ __

0

Figure 23.

50 100

TIME (sec.)

150

System Error for the Compensated System with Pseudo-Random

Variation of r and b and Step Changes in K s

200

\_.,"";

0\

+.2

e 0

-.2

+.2

e 0

-.2

Original System -;------,..-------T~7-~- -,---~---,--r·--,r---~-----~-------T _____ T ____ r--::-·-;----;·-- ·i r' 1 · . 1 • , · ' ; 1 •· · ,. 1 1

I ' I ' l I ' I I .' ' 'i It 'I I I ' I .' I ' I I I ' I I ·' I I I I I I ' ' .' ; 1 r-r-r J .' I l .J I i I I I I :

I ! I / I I I ! : _: li ; ! / 1 ;' I / :' ,' i / / : ! f J ;' i ; / i !

~r--- ~-~~-~ c=~~~};.sr -n-:-, --r-r-;.--:---_, ---r-~·---:--,~~::-r~;---.: _r-t-G / 1 I ' ' I ;__; __ ~

' I

~. I : I , .· ' '',,,,· _ _/ :i: •- -~.-1 !!'"'-j-v-~-<i-r. . , .... ':,·~ . ·'•"T·

---~_j __ ~: _ _j__j_~;--~-·-·. ___ · __ ·_ .. --· ~ \_~ ~: -~ ·J

:--~--~--------~;--- ----;..:~~-~---- J_ __ : ----------':, •• --~--·--------~7~.-.~;;-~-' ' - ~ • \ ' \ ""'' . --- ~, I I ' . I '"' ,., --· ' J • I ' '"'" I -··~~-------------'1---------t------~, ~r··-..~----...,.__j___,.!,j'- ...., "' - \..._f ' v ; ~ ,! ·) ' ' "-.,_/ . \ J

' :\ \,. ' I

\H I ~--. --__ , ____ ____.. __ -'-----

' \

'.----~,____: ___ ... ,-~,--\_ll __ \--\-~ -~~. -~--~_j _ _:t--\-\-..;. ' \ ' I I ' I ' ' I \ I \ \

\ _____ \_ ___ \ __ L_ __ ~----~--~L-~~-_l___l ____ ~~l-----L--j_ __ \ __ L_l

\1

-;----\-~--~--...,1 \ \---,-~---\~-~---~.-__'.. __ ·\ ---·~- --\\ ,·,,,\~\\'.'._\\\\

__ \ ___ _j_ -~·~ _____ l ___ , ___ J __ ~L___.:.. __ '...__ __ ~-~~~__:, __ _:, _____ \

40 50 60 70 TIVJ.E (sec.)

Comoensated System

:-:--:,~:-.~, , .~ ~-n~F~· -r-~-~-r-~-T!-~-~~--~~ ~~ --~---~ -.IT!' , I I , I ~~-.-1 I I 1-, r : I I i I l f i I I '1 r ( ! I i ' :' j ; -~ ;' . .' / ; ' ! I .' I : I / : f ' ____ j_____._.._ ___ L _ _J___(__ __ _.___ __ r------;----, _ f.--,._..._-.-~--·~--..~. __ ,._ ____ ,·---,-- ~-~--+--t--.~-~---1~-~---+------f--. --t-- -t-----~

I ' I I f I I I i I 1 ' I ' f ' I I ' ' • / ! , I ' , I ; . I i ,; i / j I ! i i 1 · ! i ! f / i ,• , 1 ' i ' i ; ! / I ; f f I r I / ,J ,1 ; I ' . • • I ~' .- TTl --· __ . __ :...__j__ f I :: ' . • I ' • ! ' I I ' ~:___:_ ~!_.1_

' • I I·'''.''.' ' :'.' I.'' t· '1. ; ~ ~ ! ~ ; ; ; i 11 : ; ; ,' I ; : : I i I I : : t J ~ :.---1--:---~- -: -~--~---·------ '7:J-~..;_~--·-· -1--~~--~-------7 ---- -~--·-- :--- _ ___; ___ ~ "---L---~-- J ____ {~------~-~ ~.. :---+-~1 -\ ___ ; __ ~ __ 1.__

~ : : : : I : ; II: . ,--\· I ~------ ' ' : ' I ' 1 : ' I ---...___ ' I - ' ,- I ' I .-...=:::::::: - t"l "\-1·---'-~-~-----=-·-c~-•--==..~

. ----\-

. , . -~· : . \ . . . . : . \: ' , . . . . \ _,...-• I 1 i • < , ' • I I 1 - 1 '\.../, : :

-•-• ._ ' - • --\1· •-• ----·-- c-.\·--;-.- I \J • -=·-' ·-~ _ _j ~---:-·-j-·--t--~;--~-\--~---;--··~-~---~-:-

- \-~--~-"

I ,

\ I ' '. -- \---~--

\ \ \

40

___ .. ___ ____.._._. '

50

\ I

\ - \

60 TI:·:E (sec.)

1 \ __ , ' \

' 1. I \

~- --'- ___ .._ --\ --\---\-----1 \ \ \ \ \ .

I

\ I

70

F1~ure 24. Sy ,..,~·em T.'·,-,Y1oY1 ·"or "o· ot:1 oY1~ ..,..i"'a' and ,..,0'""'8"''""'.1-Cd Syc-""ems ...• .c..·~ 0 V .i.l .L..J.. ~ .i... .1. ' ! L .1-t) ! 4 .l. V .l.I.:._J - J. •UU V tJ V .il ~·, J.. V l •

P;:;cudo-Hand.o:--:1 Variat::.on o:'"' Cl~~ J ~'--.n C~J. .. U S~CL} c;un;:;e:::; ·~ .Ld :<:~ ._,

\...~ -....;

58

q( 0) was zero. The system error is shown in P:tgu.rc 23 and

an expanded portion of system error for both the oriGinal

and compensated systems is shown in Figure 24.

3. Comments

For the original system with sinusoidal variations the

error response is very oscillatory as noted from Fie;urc 18.

The tracking of K and a, denoted by x 1 and x2 in Flgure 18, m

are also oscillatory. In Figure 19 the corresponding c, xl,

and x 2 for the compensated system show a definite improve­

ment in that oscillations are heavily damped without an

:i.ncrease in peak values. The error for both systems may be

compared directly rrom Figure 20 which again points out the

improvement obtained from compensation.

Although the tendency to oscillate is not as prevalent

in the system with pseudo-random variations, it can be Gccn

from Figure 22 that hlgh frequency oscillations do occut' in

the original system. These oscillations have been hcavlly

damped in the compensated system as noted from Fic;ure 2 3·

The error peaks ror both systems are very large. In the

original system the maximum error is approximately 12J percent of the maximum model output while for the compen­

s t Errors of thiS magnitude a ed system it is 14t percent.

are undesirable but previous analysis and results have . 1 p gains will decrease

Shown that increasing the adaptlve 00

for both original the peak error. Figure 24 shows the error

the improved control and compensated systems and indicates

Of OSCillationS.

It is worthwhile to note that b, the dynamic plant

parameter, ror both sinusoidal and pseudo-random variation

is allowed to vary quite severely and to even be nesatlvc.

1'1ithout the adaptive control, compensated or uncompcn~3atcd_,

a negative value of b would indicate positive feedback in

the plant. 'rhis is the same as operating in the rie;ht half

plane on a root locus diagram which means the plant has be­

come unstable. The model-referenced adaptive control has

thus brought stability to an otherHise unstable plant.

60

III. SIDvTI\1ARY AND CONCLUSIONS

The first-order model-referenced system was selected

for study and the necessary adaptive control was developed

using Lyapunov' s second method. Several first-order :::.ys'Lcr1s

were simulated on an analogue computer. This revealed an

oscillatory mode in the system. In the light of past and

present usage of model-referenced systems there 'i'Jas a need

to obtain more information about the oscillatory mode and

the possibilities of controlling it.

The model-referenced system as shown in Flgure 3 is

nonlinear. By linearizing the system about a J.mm·1n equi­

librium point the eigenvalues and eigenvectors were found.

The oscillatory mode was indicated by the complex pair of

ei t d and System error was genvalues. An example was sclec e

calculated f'rom the linearized system and also found from

a S. 1 h · ~l·gure 3. A comparison ~mu.ation or the systems own ln L

of the error from these two methods using eigenvector ex-

citatl.· on o+> t mode was made,. and it is con-J. the oscilla-ory

eluded that the linearized system is a very good approxi-t ·n the neiehborhood

mation o.f the nonlinear adaptive sys em l

of the equilibriwn point.

61

A root locus involving the complex cie;envalues of' the

linearized system, and also slmulation studies, sho'l'lcd that

an inc.J:.~ease in adaptive loop e;ains, fL, and fl-2 , caused the

f'requency of' oscillations to increase. However it ·\\'as

f'ound from simulations that large adaptive loop Gains are

necessary f'or good adaptation of' the plant parameters to

the model parameters. The cost of' good adaptation qualities

is then a substantial increase in the freqLlency of' the

oscillatory mode.

Having studied the root locus and lmowints that the

linearized system was a valid rep:cescntat:ion, linear com-

pensation techniques were thought to be a possible :3olut1.on

to the oscillatory problem. A general 1ead compensation

network was added to the origina1 adaptive system, and in

order to study the new compensated system linearization

tec~1iques were applied. The eigenvalues were round and

again an oscillatory mode was indicated. The same example

used f'or a study of' the original system was chosen for a

study of' the compensated system. A root locus of' the com­

pensated system was then used to determine values f'or the

parameters of' the compensator which would give good damping

of' the oscillations. Initial conditions were chosen tlmt

would excite only the oscillatory mode, and once again good

correlation was f'ound f'or results obtained by simulation of

the compensated system of' Figure 12 and by calcu1atj_on .fPom

the linearized compensated system.

62

A comparison of the system error for both ori~inal and

compensated systems sho\·Icd that an outstand1ng :impl'ovcmcnt

in the reduction of oscillations v1as m3.de by the addition

of compensation. This was also shown from the study of two

systems for which plant parameter and input variations were

more general.

It is concluded tb~t model-referenced adaptive control

systems designed by Lyaplillov's second method should begin

with an assurnption of large adaptive loop gaLns Nhich arc

necessary for good tracking. 'This assumption j mpl_ics h:i.ghly

oscillatory modes. It is further concluded then that com­

pensation which can be designed using linearization and root

locus tectmiques should be added to the system for the pur­

pose of controlling the oscillations.

The following recommendations of' f'uture work represent

problems and items of' interest which were not covered in

this paper.

1. The problem of' the adaptive control drif'tlng and/or

not reaching ideal values f'or constant inputs Darrants fur­

ther study. It is suggested that a small amplitude noise

signal at the input might be helpful.

2. It should be shown whether or not the compensated

system is stable ln the large.

3. Compensation has been added to both adaptlve loops

of' the f'irst-order system. For the exa.mple studied in sec­

tion II-C if' the compensation is added only to the adaptive

loop involving K the following equation is obtained f'or

plotting a root locus.

80( d- 1) A 0 = 1+

( A.+ c) ( A 2 + :\ + 100)

This is identical to equation (1~2) except that the number

100 in the numerator of' ( 1~2) is now 80. r:ehe t•do loci are

identical then except f'or values of' d at each point on the

loci. Any advantages 3 disadvantages, or redundancies that

6h

occur :from compensating various combination:J or inuiv:tdual

adaptive loops would be a worthwhile and intcrcst:i.ng :.>tu.dy.

4. If the compensated model-referenced adapt i vc ::;y~3tcrn

is to be fully exploited, worlc should be done on opt:Lrnl:.;:a­

tion or the compensator.

5. It is not necessary that N on page 10 be a d:iac;onal

n~trix. Satisf'actory pcrfornmnce might be achieved by a

di.ff'erent choice of N and at the so.mc time cllJnlnatc the

need .for a compensator.

BIBLIOGRAPHY

1. Aseltine, J. A., r1Iancini, A. R., Sarture, C. W., "A Survey of' Adaptive Control Systems," IRE Trans. on .:f:.~~omatic Control, no. PGAC-6_, pp. 10-2-los;­December_, 19?8.

2.

3.

5.

6.

~r .

8.

9.

Whitaker, H. P., Yamron, J., Kezer, A., "Design of' Model Reference Adaptive Control Systems f'or Aircraft," Heport R-164, Instrumentation Lab., J\1assachusetts Institute of' Technology, Cambridge; September_, 1958.

Osburn_, P. V., Whitaker_, H. P., Kezer, A., "New Developments in the Design of' Model Reference Adaptive Control Systems, 11 Institute of' Aer•onautical Sciences_, paper no. 61-39.

Kalman, R. E., Bertram, J. E., "Control System Analysis and Design Via the Second r1ethod of' Lyapunov: Part I, Continuous-Time Systems," Trans. ASME_, vol. 82_, pp. 371-393; June, 1960. -----------

Parks, P. C., 11 Lj_apunov Redesign of' Model Reference Adaptive Control Systems, II IEEE rrrans. on Automatic Control, vol. AC-11, pp. 3~~30(; July, 1966.

Grayson, L. P., "The Design of' Nonlinear and Adaptive Systems Via :r:...yapunov's Second fv1ethod," Polytechnic Institute of' Brooklyn, Report PIBMRI-937-61; ,July, 1961.

Sha.ckcloth, B., Butchart, R. L., 11Synthcsis of' l\1odel Reference Adaptive Systems by I.Jyapw1ov' s Second Method," Theory of' Self-Adaptive Control Systems ( Proc. Sceona I:biJfc-syrnposlwn-y-:-New~rork_, N. Y.: Plenmn Press, 1966_, pp. 11~5-152.

l\1onopo li, R. V. , "Lyapunov 1 s Net hod for Adaptive Control-System Design," IEEE Trans. on Automatic Control, vol. AC-12_, pp.-3~55S; June, 1967.

Slk'l.ckcloth_, B., "Design of' Model Reference Control Systems Using a Lyapunov Synthesis Technique," Proc. IEE, vol. 11L~_, pp. 299-302; February, 1967.

10.

11.

12.

13.

14.

Zemlyakov, S. D., Rutkovskii, V. Yu., "Synthesis of' Algorithms f'or Changing Adjustable-Coef'i'ic:tents in Self'-Adaptive Control Systems v1ith a I·1odel," D~klady, vol. 12, pp. 1~18-419; November, 1967.

66

Hahn, W., Theory and Application of' Liapunov's Direct Method-:--Engievloocr--CTlf'-:fs, -N .-J-:!-Prentice-Hall; 19-o-3-:;_p • 9 s .

Ogata, K., State Space Analysis of' Control Systems. Englevvoocrciif'fs, N. :r:--: --Prentice-Hall, 1967, pp. 461-l-1-63.

DeRusso, P. M., Roy, R. J., Close, C. M., State Variables f'or Engineers. Nm'l York, N. --y-:-=- Wiley, -----Ci=------Ic:·-··or:g-··--19u5, pp. ~79- -1- 3.

Savant, C. J., Control System Design. New York, N.Y.: McGraw-Hi l1-;-l9-64--:;--2nd.ed.-=-;-pp--. -218-220.

APPENDIX A Analogue Simulation Drawings

A-1. Analogue Simulation Drawing of the Original System

Any scaling necessary for actual use with an analogue

computer has not been sho"Vrn.

r

-9 m

-K(O)

e

68

A-2. A:_~alogue Sj_mulation Drawing o:f the Compcns~_ted __ S:[>~tcm

Any scaling necessary :for actual use with an analogue

computer has not been shown.

~----i z

p(O) - z(O)- de(O)

APPENDIX B Detailed Computations - Linearized System

B-1. Solution of e(t) for the Linearized System from

Equation (25)

From equation (24)

Therefore

e ( t) = z 1 ( t) - z2 ( t)

z 1 (t) = 0

69

(25)

e(t) = -z2 (t) - -0.01[(-.5- jl0)(-.625- jl.03)e-( .5+ jlO)t

(-.5+ jl0)(-.625+ jl.03)e-(.5- jlO)t]

_ . 121e-.5t[ej(lOt + 33.8°) + e-j(lOt + 33.8°~

- .242e-.5tcos(l0t + 33.8°)

B-2. Inverse Modal Matrix of Linearized System and Solution

of x(o)

0 0 20 40

100 0 0 0 -1 0.01 M -

j5 -j5 1 + j .05 -.5- j .025

-j5 j5 -1 + j. 05 . 5- j. 025

0 0

M- 1~(0) M-1 -.2 0

_y( 0) - - - 0.01 -.5 -.625- jl.03

.25 -.625 + jl.03

-.__ ~¥-' - ~ , --------- --.~ ·- ·-- --- ---.....---~·-- ------.._, --._,----.--- .. ~- -- -· -- ·-- --- .

..

B-3. Adjoint of AI - J for the Compensated System eq

~(A 3+21A+310A I +2000)

1710A

0

A(A+l)(A 2+20A+290)

0

-21. (A+l) (A+20)

Adj[Al-J ] =I -684A(A+l) 684A(A+l) (A+l)(A 3+21A 2+18A eq +400)

342A(A+l) -342A(A+l) (A+l) (116A+800)

17.H2(A+l) -17.H 2(A+l) A(A+l)(5.8A+40)

0

A(A+l)(A+20)

40(A+l)(2.9A+20)

(A+l)(A 3+21A 2+252A +1600)

-A(A+l)(2.9A+20)

0

100A(A+l)

-40A(A+l) 2

20A(A+l) 2

A2(A+l) 2

-'l 0

-1 M =

B-4. Inverse Modal Matrix for the Example Compensated System

0 0 • 2 .4 0

1 0 0 0 0

-.00526 .00526 .00056 .00028 .00277

.00263- j.00091 -.00263 + j.00091 .00086 + j.00015 -.00043- j.000076 -.00138- j.00177

.00263 + j.00091 -.00263 - j.00091 .00086 - j.00015 -.00043 + j.000076 -.00138 + j.00177

-..J j-J

B-5. Solution of e(t) for the Compensated System from

Equations ( 44) _, {_!:22_, and (51)

72

w(t) = Mv(t) - - (45)

Using M_, equation (44)_, and v(t)_, equation (51), it is

:found that

w1 (t) = 0

Therefore from equation (52)

Then

e(t) = - ~40.5- jll7.3)(-.00037- j.000736)e-(5.5+ jl3.03)t

+(L~0.5+ jll7.3)(-.00037+ j.000736)e-(5.5- jl3.03)t

+190(.000018)e- 10t]

_ -.l03 e-5.5t~j(l3.03t + 187.7°) + e-j(l3.03t + 187.7°)]

-. 003'-~2 e -lOt

-- -. 206e-5 .Stcos ( 13.03t + 187. 7°)- .00342e-lOt

- .206e-5 · 5tcos (13.03t+ 7.7°)- .00342e-lOt

The error should contain only the oscillatory mode since

eigenvector excitation was used. The term -.00342e-lOt

is negligible and has entered the analytic solution due to

round-off error in the calculations.

73

APPENDIX C

Develo£~ent o~ Pseudo-Random Signals

The pseudo-random signals used ~or r and b were ob­

tained by passing digital signals ~rom a Scienti~ic Control

Corporation 650 computer across D-A lines and then through

a series o~ low-pass ~ilters on the analogue computer. The

digital computer was used to generate pseudo-random numbers

by the ~allowing scheme. In a twelve bit register there arc

2 12 di:ff'erent binary numbers that can be represented. All

o:f these combinations except zero can be obtained in a ran-

dom manner by an exclusive-or of the last two bits, a right

shif't o:f the register, and then a trans:fer of' the result of

the exculsive-or into the first bit position. This is

shown schematically as

The maximtun output on the D-A lines of the SCC 650 is ±10

volts, therefore 2 12-1 different voltage levels between +10

and -10 can be transferred to analogue low-pass filters.

The program used to generate pseudo-random nwnbers

from the sec 650 by the scheme mentioned above is shoivn on

the :following page.

Memory Machine Location Instruction Language

Memory Machine I_JOC:_?.t io~ Inst rue t ion _Ig l1£:;Ua(Se

7000 r-(001 7002 7003 7004 7005 7006 7007 7010 'lOll '7012 7013 7014 7015 '7016 7017 7020 7021 7'022 1023 '(024 '(025

LDA +D CAX 8AR SRA LCR CAX STA LDL SEL LDA TPA MIN JMP MTN LDX LAS

3

+D 1

32 +D 32 +A

*-1 +B +B

SRA ·7 _, Z JMP -X--6 CIA STA MTN Jl11P

+B +C

-x--25

61l1-5 02LJ-O OOllJ. 0031 0310 o1I~o 3537 3201 0732 6134 0472 4133 2501 4132 1131 0050 0371 2506 0003 3524 1+124 2525

,-(026 'T027 7030 7031 '(032 7033 7031+ 7035 '(036 '(037 70lt0 '(041 70h2 7043 '['Ol+l+ 70h5 '(046 701~7 ~1050 ,--(051 'T0 52

LDA JMP CAX SAR SRA LCR CXA sr.rA LDL SEL LDA TFA LDA STA JTl[p PAR PAH PAR PAR PAH PAR

3

+E 2

32 +E 32 +P +C

·X--lj)J-

1)

A B c E F

6123 2101 0214-0 001/t 0031 0310 0111-0 3514 3202 o-(32 6111 3505 6110 3505 251Jh

D A B c

F

All nDmbers used j_n the SCC 650 program are in octal or

base eight ~arm. Initial values used were

A - 0000 B - 0000

C - 7772 E - 1000 D - 1000 F 7772

SVJr.rc H SE'Tr.r ING -- 0020

~ Jl · drawina represents the analogue L o .. _ oN J. ng o ]_O\'l-p8. SS

~ilters. The outputs o~ the filters_, shown as r and b, were

then used in the model-referenced system.

D- A __ --"l0 ___ _J Chan~!_

No. l

D-A ~ r1 c~ .... No. 2

rj/ LG)

VITA

The author was born September 29 _, 19}_~3 in Houston_,

Texas. He received hi.s secondary education in Tulsa_,

Oklahoma. His college education has consisted of' two

semesters at the United States Air Force Academy_, seven

semesters at the University o:f Tulsa_, and three semesters

aJc the Uni.versi.ty o:f Missouri at Rolla. He recelved the

Bachelor o:f Sci.ence Degree in Electrical Engineering in

January 1967 at the Unj_versity o.f Tulsa.

He has been enrolled in the Graduate School or the

University o:f Mi8souri. at Rolla since January 1967 and

has held a Graduate Assistantship in the Department of'

Electrical Engi.neering since that time. He is a member

o:f Eta Kappa Nu_, Kappa r.'Iu Epsilon_, and the Institute of'

Electrical and Electronics Engineers.