Title: Model reference adaptive control of a nuclear rocket engine
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 Material Information
Title: Model reference adaptive control of a nuclear rocket engine
Physical Description: xiv, 119 leaves : illus. ; 28 cm.
Language: English
Creator: Humphries, Jack Thomas, 1929-
Publication Date: 1966
Copyright Date: 1966
 Subjects
Subject: Rocket engines   ( lcsh )
Nuclear Engineering Sciences thesis Ph. D   ( lcsh )
Dissertations, Academic -- Nuclear Engineering Sciences -- UF   ( lcsh )
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis - University of Florida.
Bibliography: Bibliography: leaves 115-117.
Additional Physical Form: Also available on World Wide Web
General Note: Manuscript copy.
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Bibliographic ID: UF00097859
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000549750
oclc - 13291336
notis - ACX4048

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MODEL REFERENCE ADAPTIVE CONTROL OF

A NUCLEAR ROCKET ENGINE






















By
JACK THOMAS HUMPHRIES











A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY









UNIVERSITY OF FLORIDA


April, 1966





















DEDICATION


To my wife, Mildred,

my son, David,

my daughter, Jacqueline,

and my mother,

for their unfailing

understanding and encouragement











ACKNOWLEDGMENTS


The author wishes to express his gratitude to

the co-chairmen of his supervisory committee,

Dr. Robert E. Uhrig and Dr. Andrew P. Sage, who sug-

gested this work and offered continued help and encour-

agement throughout its development. The demonstrated

concern and interest of the remaining members of the

committee, Professor Glen J. Schoessow, Dr. William

H. Ellis, Dr. Walter Mauderli, and Dr. Robert G. Blake,

had a most beneficial effect upon the work's progress

and is sincerely appreciated.

Special thanks go to Mr. Kenneth Fawcett for his

inspired maintenance of the analog computing equipment,

to Mr. Reese Waters for his help with the MIMIC code,

and to Mr. Lawrence Fitzgerald for his assistance with

some of the digital computation. Discussions with my

good friend Mr. Jagdish Saluja provided some of the most

rewarding moments of the investigation.

It is a pleasure to extend thanks to Mrs. Wilma

Smith for her capable assistance in typing the manu-

script and to Mr. Richard Schoessow for his execution


iii








of the drawings. Finally, the author expresses his

appreciation to the United States Air Force for pro-

viding this opportunity for graduate study at the

University of Florida.











TABLE OF CONTENTS


Page

ii

iii


DEDICATION . .

ACKNOWLEDGMENTS

LIST OF TABLES

LIST OF FIGURES

LIST OF SYMBOLS

ABSTRACT . .

CHAPTER


. . vii

. . viii

* . xi
. . xiii


I. INTRODUCTION . . . . .. 1

II. THE NUCLEAR ROCKET ENGINE . .. 10

III. ADAPTIVE CONTROL FEASIBILITY
ANALYSIS . .. . . . 19

IV. SIMULATION OF THE NUCLEAR ROCKET
ENGINE . . ... . . 33

The Constant-parameter System . 34
The Reference Model .. .. . 38
Adaptive Loop Design . . . 42
Adaptive Loop Performance . . 44

V. PERFORMANCE OF MODEL REFERENCE
ADAPTIVE CONTROL SYSTEM . . 47

Adaptive Control for a Varying
Pressure Time Constant . . 47
Stability Considerations . .. 64

VI. SUMMARY AND CONCLUSIONS . . 71

Summary of Results . . .. .71
Implications of Results . .. 72
Suggestions for Further Study . 73


. .


: : : : : : : :









TABLE OF CONTENTS (Continued)


Page

APPENDICES . . . . . . . 75

APPENDIX

A. PARAMETER ADJUSTMENT ADAPTIVE
CONTROL . . . . . . 76

B. EQUATION NORMALIZATION . . .. 85

Reactor Kinetics Equations . .. 85
Temperature and Pressure Equations 87
Reactivity and Controller
Equations . . . . . 88

C. ANALOG SIMULATION ...... ... 89

Equipment . . . . . . 89
Analog Circuit Symbolism . . .. 89
Problem Scaling . . . . 92
Constant-parameter System Simulation 95
Reference Model Simulation ... 102
Adaptive Loop Simulation . .. 102

D. DIGITAL CHECK . . . . .. 111

LIST OF REFERENCES . . . . ... 115

BIOGRAPHICAL SKETCH . . . . ... .118












LIST OF TABLES


TABLE Page

1. ROCKET ENGINE CONSTANTS ...... 14

2. INITIAL CONDITIONS . . . ... 96

3. POTENTIOMETER SETTINGS FOR CONSTANT-
PARAMETER ENGINE SIMULATION . .. 100

4. POTENTIOMETER SETTINGS FOR MODEL
SIMULATIONS . . . . ... 104

5. POTENTIOMETER SETTINGS FOR ADAPTIVE
LOOP SIMULATION . . . . .. 107


vii











LIST OF FIGURES


Figure Page

1. Parameter Adjustment Model Reference
Adaptive Control System . . . 7

2. Schematic Diagram of a Nuclear Rocket
Engine . . . . . . . 11

3. Block Diagram of Linearized Nuclear
Rocket Engine . . . . . 23

4. Block Diagram of Simplified Linearized
Nuclear Rocket Engine . . . .. 26

5. Effect of G2 on Performance Index with
Tmod 5 Seconds . . . . 30
6. Effect of G on Performance Index with
Tmod = 1i Seconds . . . .. 31
7. Response of Constant-parameter Nuclear
Rocket Engine . . . . . . 37

8. Responses of Models and Constant-
parameter System . . . .... 40

9. Block Diagram of Adaptive Control System 43

10. Nuclear Rocket Engine Response for
T7 (normal) . . . . . . 45
11. Control Device Response for Tp (normal) 46

12. Nuclear Rocket Engine Response for 1.2
rp (normal) . . . . . . 49

13. Control Device Response for 1.2 7p
(normal) . . . . . . . 50

14. Nuclear Rocket Engine Response for
1.4 p (normal) . . . . . 51

15. Control Device Response for 1.4 rp
(normal) . . . . . . . 52


viii









LIST OF FIGURES (Continued)


Figure Page

16. Nuclear Rocket Engine Response for
1.6 Tp (normal) . . . . . 53

17. Control Device Response for 1.6 7T
(normal) . . . . . .p. . 54

18. Nuclear Rocket Engine Response for
1.8 Tp (normal) . . . . . 55
19. Control Device Response for 1.8 7p
(normal) . . . . . . . 56

20. Nuclear Rocket Engine Response for
2 7p (normal) . . . . . . 57

21. Control Device Response for 2 Tp
(normal) . . . . . . . 58

22. Nuclear Rocket Engine Response for
0.8 7 p (normal . . . . . 60

23. Control Device Response for 0.8 7p
(normal) . . . . . . . 61

24. Constant-parameter and Adaptive Control
Responses with 7p Variation . .. 62

25. Adaptation of Control Poison Proportional
Control Gain . . . . .... .65

26. System Instability Due to Negative G 67

27. Stability Boundary for Constant-
parameter System . . . ... .69

28. Generation of Response Error in Model
Reference Adaptive Control System 77

29. Block Diagram of Second-order System with
Adjustable Parameter in Feedforward
Path . . . . . . . . 77









LIST OF FIGURES (Continued)


Figure Page

30. Block Diagram of Second-order System
with Adaptive Control Loop . . .. 83

31. Adaptive Response of Second-order System 84

32. Symbols Used in Analog Computer Diagrams 90

33. Analog Computer Diagram for Constant-
parameter Nuclear Rocket Engine .. 97

34. Analog Computer Diagram for Constant-
parameter Control System . . . 98

35. Analog Computer Diagrams for Reference
Models . . . . . .. ... . 103

36. Analog Computer Diagram for Adaptive
Loop . . . . . . . . 105

37. MIMIC Instructions . . . . ... .113











LIST OF SYMBOLS

C' Normalized delayed neutron precursor density

G1 Temperature integral control gain (sec- )

G2 Temperature proportional control gain

G3 Pressure integral control gain (sec-1)

G4 Pressure proportional control gain

1 Mean effective neutron lifetime (sec)

N' Normalized neutron density

P Core inlet pressure (psi)

Pd Design core inlet pressure (psi)
P' Normalized core inlet pressure

Px Desired normalized core inlet pressure

s Laplace transform variable (sec-1)

T Maximum core surface temperature (OR)

Td Design maximum core surface temperature (OR)

Tr Mean core temperature (OR)

T' Normalized maximum core surface temperature

Tt Desired maximum core surface temperature

V Valve setting

V' Normalized valve setting

V Desired normalized valve setting







LIST OF SYMBOLS (Continued)


(Ot Temperature coefficient of reactivity (OR-l)
C t Normalized temperature coefficient of reactivity
(dollars)

X h Propellant density coefficient of reactivity (oR/psi)
C'h Normalized propellant density coefficient of
reactivity (dollars)
f Delayed neutron fraction
Decay constant (sec-1)
Damping coefficient
p Total reactivity
'P Normalized total reactivity (dollars)
Pc Control poison reactivity
Pc Normalized control poison reactivity (dollars)

P ~ Desired normalized control poison reactivity
cx (dollars)
Time constant (sec)

T Pressure time constant (sec)
t Thermal time constant (sec)
T Control poison time constant (sec)
Tv Valve time constant (sec)
Wn Natural frequency (sec-1)


xii








Abstract of Thesis Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy

MODEL REFERENCE ADAPTIVE CONTROL OF A
NUCLEAR ROCKET ENGINE

By

Jack Thomas Humphries

April, 1966


Co-chairmen: Dr. Robert E. Uhrig and Dr. Andrew P. Sage

Major Department: Nuclear Engineering Sciences


Parameter adjustment model reference adaptive

control of a nuclear rocket engine is investigated. The

engine considered is similar to the solid-core, 25,000

megawatt, million pound thrust model developed by Smith

and Stenning. Proportional control gain for the control

poison, i.e., rods or drums, is the parameter adaptively

adjusted, and maximum core surface temperature is the

adaptively controlled variable. Acceptable engine per-

formance is achieved by using a second-order model with

one finite zero.

Engine response to step commands is particularly

sensitive to change in the system's pressure time

constant. With constant-parameter control, doubling

this time constant results in doubling the system's

transition time from an idling condition to full power.


xiii







Adaptive control for the same condition achieves a

transition time within 20 per cent of the desired,

or "normal," response. Estimates of propellant savings

with adaptive control are given, and dynamic stability

of the system is discussed.


xiv










CHAPTER I


INTRODUCTION


The "case" for nuclear rocket propulsion has been

firmly established (1, 2, 2).* Arguments in favor of

nuclear rocketry convincingly point up the dramatic

increase in specific impulse and burnout velocity obtain-

able with nuclear-hydrogen rockets as compared with

chemical rockets. There is general agreement that nuclear

systems offer great promise for planetary missions

requiring large payloads and high burnout velocities.

With the assumption that development of nuclear rocket

systems is essential to man's exploration of space, this

study is directed to one of the problem areas associated

with nuclear rocketry, namely, control of the rocket

engine.

Mohler and Perry (4) observe that a nuclear rocket

engine "requires a fairly sophisticated control system."

They point out that rapid start-up (less than 1 minute)

is necessary to conserve propellant and to reduce the


*Underlined numbers in parentheses refer to entries
in the List of References.











complexity of attitude control problems. One imme-

diately recognizes this to be a much more difficult

control problem than that encountered in operation of

stationary power reactors. Investigators are using

conventional feedback control techniques as well as

modern automatic control theory in their attack upon

the problem.

The control problem is complicated by a lack of

agreement among investigators concerning certain key

areas. For example, different opinions have been

expressed as to the nature of temperature reactivity

effects, a factor which significantly affects the design

of any reactor control system. Stenning and Smitn (5)

contend that temperature reactivity is directly propor-

tional to the square root of the core exit stagnation

temperature, while Mohler and Perry (4) hold that tempera-

ture reactivity is directly proportional to this tempera-

ture. Weaver (6) observes, "both contentions could give

fairly accurate results simply by choosing appropriate

reactivity coefficients"; however, this is not a satis-

fying answer to the problem, and exhaustive analysis of

the data available from rocket reactor tests is indicated.

Perry (7) feels that temperature formulations in addition

to those mentioned above will be considered before the

matter is clarified. This all "adds up" to the fact that











nuclear rocket engines possess certain characteristics

which are not at all well understood, even as design of

the control system progresses.

Other considerations may also tend to complicate

the design of an effective rocket engine control system.

Are there effects of environment which tend to alter

characteristics of the nuclear rocket engine? During a

typical mission, for example, the engine will operate

in low- or zero-gravity fields as well as in high-gravity

fields. Effects of such environmental extremes upon

flow and heat transfer have only recently received

attention in depth (8). It may develop that gravita-

tional effects can be expected to alter certain thermal

characteristics of a nuclear heat-exchanger rocket engine.

It appears then that a control system must be

designed for a vehicle with dynamics that are not

thoroughly understood and with characteristics that may

be dependent upon environmental conditions. Such a situa-

tion would seem to indicate the need for what is termed

an "adaptive" control system.

The term "adaptive" first appeared in the control

literature around eleven years ago, however, Draper (2)

points out that adaptive features were employed in anti-

aircraft fire-control systems as early as 1939. A

concise definition of adaptivity has apparently been









difficult to formulate, and Truxal (10) suggests the

following rather general idea:

A system is an adaptive control
system if a means is provided to moni-
tor, in addition to the state variable,
its performance and/or process (internal
and/or external) characteristics in
order to accordingly modify the control
action in an attempt to be an acceptably
performing system.

Or, according to Whitaker (12):

An adaptive control system is defined
to be a system which either by measurable
adjustments of controllable parameters
or by measurable and well-defined changes
in dynamic characteristics can adapt to
a changed environment, a changed character
of input signals, or a changed system
or component characteristic in such a
manner that the specified desired per-
formance is maintained.

Most of the adaptive control systems which have

been studied can be included in the following three

categories (11): 1) high gain systems, 2) model

reference systems, and 3) optimum adaptive systems.

The complexity of these systems increases with the order

of the above listing. Donalson and Kishi (11) present

rather comprehensive discussions of all three types of

adaptive control, and it is not appropriate to include

such a treatment here. Uhrig and Sage (13) have observed

that the high gain systems do not appear particularly

applicable to nuclear rocket control because the system

would have to continually limit cycle at some frequency.










They further point out that the complexity of the

optimum-adaptive schemes coupled with the fact that they

are basically guidance techniques, and not control

techniques, renders them also unattractive for this

application. Such immediately obvious disadvantages do

not exist in the case of model reference control systems.

As the name implies, the model reference control

technique employs a model which embodies the design

specifications for the system to be controlled. A

command input is applied simultaneously to the reference

model and to the basic control loop. The system output

is compared with the reference model output, and a signal

representing the error between actual and desired per-

formance is made available. Requirement for parameter

adjustment is determined by generating functions of the

error, and parameter-variation commands are sent to

the parameter adjustment devices in the control system.

Thus the controller "adapts" in such a manner that the

system output closely matches that of the model.

Model reference techniques have been classified in

three categories (11): 1) parameter adjustment techniques,

2) parameter perturbation techniques, and 3) control

signal synthesis techniques. The most popular technique,

and the one used in this research, is the parameter









adjustment technique. Figure 1 depicts a model reference

adaptive control system with provision for parameter

adjustment. The basic control loop is enclosed in a

dashed box. A response error signal is operated upon to

bring about "optimum" control parameter variation, and

as mentioned above, system output is closely matched

with reference model output. Note that extent of

parameter adjustment is determined solely by changes in

characteristics of the controlled system since those of

the model remain fixed.

Del Toro and Parker (14) have noted the following

advantages, in addition to that of self-adaptability,

of such a control system:

1. The adaptive loop functions independently

of the main control loop. If the adaptive loop

fails, operation is still possible.

2. The response error signal is immediately and

continuously available, making disturbing test

signals unnecessary.

3. Adaptive loops can be added to existing systems

without appreciably altering the basic control

loop.

4. Where several adaptive loops are required,

several different models can be used.














3


0
O


----I

I ii I


IL


cp
1CE
4- E -I
0-i C








,--

o C


Q)
H-0
.'
0U 0
Cr
0 0*S










5. Since noise in the input command is treated

essentially in the same manner in the model as

in the system, little, if any, deterioration

in the optimizing process should occur.

This control technique is inherently nonlinear,

time-varying, and dependent upon the time variation of

the system parameters. The problem is usually too diffi-

cult for conventional analysis and so computer simula-

tion is essential for design and analysis. The prolif-

eration of model reference adaptive control schemes

has not yet resulted in a generally accepted theory

or procedure for design and analysis of the adaptive

system. However, Osburn (15) has presented a design

procedure which is quite attractive for many applica-

tions, and his approach was used in this investigation.

A description of the derivation of his method along

with a design example is included as Appendix A.

The objective of this research was to investi-

gate the use of model reference parameter adjustment

techniques in the control of a nuclear rocket engine.

Chapters II and III contain a description of the nuclear

rocket engine employed in the study, and an analysis

of the feasibility of adaptive control for this appli-

cation. Simulation of the engine is described in

Chapter IV, and performance of the adaptive control






9



system is examined in Chapter V. Chapter VI includes a

discussion of the results and suggestions for further

study.












CHAPTER II


THE NUCLEAR ROCKET ENGINE


The nuclear rocket engine model employed in this

study is based upon the work of Smith and Stenning

(16, 17, 18, 19, 20). Their system has been extensively

treated in the open literature, and exhibits the princi-

pal features of the solid-core reactor technology being

developed in the United States for nuclear rockets (21).

Basic components of the engine are depicted in

Figure 2. Liquid hydrogen, stored in low pressure

tankage, is pumped to high pressure, and is used to

regeneratively cool the nozzle and reflector. The

propellant gas is further heated as it passes through

the reactor core and is exhausted through the nozzle.

A portion of the hot gas is bled off from the main

stream and is used to power the turbopump, a feature

giving rise to the name, "bleed turbine system." The

bleed flow, and consequently the system pressure are

controlled by the valve in front of the turbine.












































3:
0

LL.


C
(U


0)







0.


0-4







0 C:
I.. M
H,











The following equations describe the Smith-Stenning
nuclear rocket engine in terms of time dependent normalized
quantities. An account of the normalization procedure
is contained in Appendix B.


I dN'

1 dC'
_ dt N +


tdT = N' p'(T')


r dPt P'V'(T')1/2
'P dt '


(2.1)


(2.2)

(2.3)


p (p')2
(T )1/2


(2.4)


Sc dt
1





t2
vI G3 (Px -


dV'. v
dt x


1/2


- (T'12 dt +Pc (0)


P)dt + G4(Px P') + V (0)


Equations 2.1 and 2.2 are the reactor kinetics equations.


(2.5)


(2.6)


(2.7)


(2.8)


(2.9)


+ Ch IT











Equation 2.3 gives the time rate of change of maximum

core surface temperature, and equation 2.4 gives the

time rate of change of the core inlet stagnation

pressure. In the discussions which follow, "temperature"

always refers to the maximum core surface temperature,

and "pressure" always refers to the core inlet stagna-

tion pressure. Reactivity is given by equation 2.5, and

the engine's control system is represented by equations

2.6 through 2.9.

Table 1 contains important constants for the

nuclear rocket engine. Note that the delayed neutron

fraction is smaller than the value typically associated

with a uranium-235 reactor. This reduced value was used

by Smith and Stenning to account for diffusion of some

precursors into coolant channels and their subsequent

expulsion from the system before emission of delayed

neutrons.

The above dynamic equations describe a nuclear

rocket engine with thrust of 1,500,000 pounds, specific

impulse of 640 seconds, and total power of 25,000 mega-

watts. These design specifications significantly

exceed those of nuclear engines presently under develop-

ment and will probably be first achieved through

clustering of smaller engines. However, as indicated









TABLE 1

ROCKET ENGINE CONSTANTS


Design core surface temperature

Design core inlet stagnation pressure

Thermal time constant

Pressure time constant

Control poison time constant

Valve time constant

Decay constant (one group)

Temperature reactivity coefficient

Hydrogen reactivity coefficient

Neutron lifetime

Delayed neutron fraction


0


5000 OR

1500 psi

0.68 sec

2.50 sec

2.00 sec

0.20 sec
-1
0.10 sec-1

-10-6 OR-1

0.3 OR/psi

10-4 sec

0.005











above, the great amount of information in the open

literature concerning the Smith-Stenning engine makes

it attractive as the test vehicle for a control system

study.

The design maximum core surface temperature at

full power, or simply the design temperature, is 50000R.

Precise temperature control with small overshoot, i.e.,

less than 5 per cent, is required since the system

operates near its limiting temperature in order to attain

maximum specific impulse. Design core inlet stagnation

pressure is 1500 psi. The terms "design temperature"

and "design pressure" are used in the following discus-

sions to denote conditions at full power.

It has been shown (17) that specific impulse is

proportional to exit temperature and is independent of

entrance pressure, while thrust is proportional to

pressure, but independent of temperature. The feedback

control system proposed by Smith and Stenning does not

provide for independent control of specific impulse and

thrust because coupling exists between propellant flow

and reactivity. Their system is controlled by commanding

step changes in both temperature and pressure, after

which the engine proceeds to full power under its

conventional feedback control system. The square root










of temperature used for temperature commands is obtained

from measurements of propellant flow and pressure.

Designers of the original engine considered this tech-

nique preferable to the use of slower thermocouple

measurements.

The Smith-Stenning nuclear rocket engine model

per se was not used in this research, but rather a

modified version was deemed more suitable for this

control system study. These modifications are now

described.

Equation 2.5 indicates that temperature reac-

tivity is directly proportional to the square root of

temperature. As mentioned in the previous chapter,

differences of opinion exist concerning this dependence.

If one examines equation 2.5 and determines control

poison reactivity at an idling condition where

T' P' 0.2 and then compares this with control

poison reactivity at full power where T' = P' = 1.0, it

is noted that control poison movement is simply that

of removal. If the (T')1/2 term of equation 2.5 is

replaced by T', as in the Mohler-Perry (4) formulation,

and if a conservative (low) value of the temperature

reactivity is used, i.e., -10- R control poison

movement is dramatically affected. During the transition










to full power, control poison is removed, but is finally

returned to very nearly its original position. This

means that the T' formulation, with its temperature

coefficient, lessens the control effect of temperature

and places more importance upon the control poison. Since

this study was concerned with the practicability of a

control technique, and the T' formulation appeared more

rigorous, this modification was incorporated in the nuclear

rocket engine model. Note that the term "control poison"

is being used to avoid the limiting of control devices

to conventional control rods. It is generally agreed that

control drums in the reflector offer excellent control

possibilities, and Bussard (3) has calculated that satis-

factory response frequencies can be obtained with reason-

able power requirements.

Another modification of the nuclear rocket engine

model concerned proportional control of control poison.

Smith and Stenning did not provide for such control in

their model, however, in this study, proportional control

of control poison was deemed practical.

The above modifications required that equations 2.5

and 2.6 be rewritten as follows:


p' = p + C T' +oiCh (2.10)










t2 ( 1/2 1/2
c G J (Tx) (T') dt
tl


+ G2 (T)1/2 (T')/2 + ) (0) (2.11)

One remaining modification had to do with turbine
size. Smith and Stenning have presented results (19) of
engine simulations requiring turbines capable of signifi-

cantly greater flows than those encountered at design
point operation, e.g., Vax = 1.33 and 3.0. They point
out that although this increases speed of transition to
full power, the heavier turbines lead to reduced burnout
velocities. In this investigation, V' was limited to
max
a value of 1.10. Chapter III is an adaptive control
feasibility analysis for the modified nuclear rocket
engine described above.












CHAPTER III


ADAPTIVE CONTROL FEASIBILITY ANALYSIS


According to Osburn (15),

Optimum performance of a model
reference adaptive system is defined
to be that which results when the system
parameters are adjusted to produce a
minimum of the performance index for
the input or class of inputs for which
performance is specified.

If this control technique is to be applied to a nuclear

rocket system, then the designer must ensure the exis-

tence of a minimum of some selected performance index

with respect to parameter variations. A preliminary

analysis at this point will serve to indicate the feasi-

bility for this particular application of model reference

adaptive control.

The performance index (PI) used in this analysis

is the integral squared response error, i.e.,

t2
PI e(t)2 dt (3.1)
1
where the response error e(t) is formed by comparing

system output with that of the reference model.










As is typically the case, development of needed
transfer functions requires "linearization" of the
rocket engine dynamic equations of Chapter II. Each
time dependent quantity in these equations is assumed
to be the sum of a steady-state value and a small devia-
tion from that steady-state value. For example, N' is
replaced by N6 + N'. These substitutions are now
performed to obtain the linearizedd" nuclear rocket
engine model. Equation 2.1 becomes

Tn(N + 6N') (p + 6P')(N + 6N')

(NO + 6 N') + (CO + 6C') (3.2)

where the dot indicates differentiation with respect
to time, and Tn = 1/l Remembering that

N )N N + C (3.3)
O 00 0 0 C0

and neglecting second-order terms, equation 3.2 becomes

n 6'N' P6N + N06P' 6N' + 6c' (3.4)

In a similar fashion, equation 2.2 becomes

Tc6C' = 6 N 6C' (3.5)

where Tc = 1/1. Linearizing equation 2.3 gives

t(TO + 6T') = (NO + 6N)
S+ 6 1/2
(PO + 6P')(T + 6T') (3.6)








and noting that
(T 6T)1/2 (T1/2 6T' (3.7)
(T + 6T') (To) (3.7)
2(T;)

equation 3.6 becomes
1/2 PO
t 6T' 6N'- (T)12 6P' 1' (3.8)
2(TO)
Similarly, by using
-1/2 ,-1/2 T1'
(TO + 6T r (TO) (1 ) (3.9)
2(T;)

and by dropping steady-state terms, equation 2.4 becomes

T('p6 2(T)P T' + V(T)1/26 p
S(T,)1/2 V + ( 2 2PO P'
0 6v + 0.6T ` T
2(T) (TO)
(3.10)
The reactivity equation, i.e., equation 2.10,
with the temperature and hydrogen coefficients in
expanded form, is written as
-.' + EctTd T hdp
= c tTd + hPd T (3.11)
'a )0 Td T
Linearizing equation 3.11 results in

6P' = c + At6T' + Ah6P' (3.12)








where

At = XtTd hPd PO (3.13)
p Td (TO)2
and
Ah ,= h Pd 1 (3.14)
STd TO

The controller equations are treated similarly,
and equations 2.7, 2.8, 2.9, and 2.11 become, respec-
tively,
t2
56P' G 6Tx T dt
Pcx 2 (TO)1/2 (T0)1/2


G2 56 Tx tiT'
+ 6T-- 6V, (3.15)
2 (Tx0)1/2 (T)' /2

Tr6p' :6Pcx -Pi (3.16)
t, P''

6V = G3 (6Px 6P' )dt + G4(P 6P'
1 (3.17)
7T = 6v 6v' (3.18)

A block diagram of the linearized nuclear rocket
engine, in terms of the Laplace transform variable s,
is contained in Figure 3. Note that the engine is
essentially a reactor-temperature loop coupled with a
turbomachinery-pressure loop.







23


F.
LO






ol In
+ F)


U)U
00
00)
CC) z

~+ w
$CD


+ L

+ 4-)





c u ac

Q4.4
0
LO 10 -0- r







N
ca
> zz







+ 0




CY a+
aU
HL
Z0











LOO









The set of linearized equations does not easily
lend itself to analysis, and the following simplifying
assumptions are employed. The prompt neutron lifetime
is set equal to zero, an assumption which is valid for
reactivities less than 0.6/(22). Further, effects of
delayed neutrons are neglected, a conservative assump-
tion, and no time lag is associated with the control
poison. The reasonableness of these assumptions was
supported by analog computer simulation. Finally, the
valve opening is held constant, and temperature is the
controlled variable.
With these assumptions and restrictions, the
linearized model equations for the feasibility study
become
NO 6P' 6N (3.19)

rt6T 6N' (TO)1/26 -P /6T' (3.20)
2(T0)

rp6P = P(V; + (p32
2(TO)1/2 2(T)3/2

+ V(TO)1/2 2P6 )6 (3.21)

6P + At6T + Ah( p (3.22)/
6)o' 6Pc + At6T' + Ah P (3.22)








t
G 2l 6T 6T
6Pcx =6Pc = f ( T ) T dt
2 (T)1/2


G2 6T~ 6T- (3.23)
2 (T )1/2

Figure 4 is a block diagram of the simplified linearized
rocket engine model.
Determination of the performance index, i.e., the
integral squared response error, requires that a
reference model be chosen. For this analysis, a first-
order model with transfer function (TFmod) equal to
1/(Ts + 1) was selected to represent the desired
temperature response, and a unit step was used as the
command input. This means that the transformed response
error is given by

e(s) = 1(TF TF ) (3.24)
s sys mod
The block diagram of Figure 4 can be used to determine
the transfer function of the simplified linearized
rocket engine system (TFss), where

TF )6T'(s) (3.25)
sys 6 T(s)


The system transfer function is given by








26



U,





0 c
+

Q)
04~
C


w
(N,







-441-
-O~O 0



)cr

















M
U*) -4






->o-4
alhi a

(I,








ccc



+ oo



C4
-4
Q












(n f(










x0
0u




0)

-4
LL,








TF G2s2 + (C1 + G2R2)s + R2G1
Tsys = F (3.26)
s3 4 Xs2 + Ys + FG1R2

where
F NO
S 1/2 (3.27)
27-t(TO)

X = R1 + R2 + FG2 + AtO- (3.28)
't
Y = R1R2 Ah N6E + F(G1 + R2G2)
7t
N'R2 (T;)1/2
+ AtNO2 + E (3.29)
Tt Tt

R 1= (3.30)
1 2 7 (TO)1/2
t 0

R2 0T03/ (3.31)

E (Po0)2
(T)3/2 (3.32)
,T

Substitution of equation 3.26 in equation 3.24
gives

e(s) = (FG2 1)s2 + (F [GrT + G2R2 + G2] X)s

+ F(G1 + G2R2 + GIR22) Y) /

Ts4 + (TX + i)s3 + (X + TY)s2

+ (Y + FR2Gl* )s + F (3.33)









The performance index, as given by equation 3.1,

can now be determined as a function of the proportional

control gain G2 by using Parseval'stheorem (23). This

represents a convenient way of expressing the integral

squared value of a time function in terms of its

transform, i.e., if
00
I = dtX2(t) (3.34)
-00
then
j06
I = 1 dsX(-s)X(s) (3.35)
27(j _
-joo

Equation 3.35 can be written as

joo
1 ds c(s)c(-s)
n 2j J d(s)d(-s)
-jco
where

n-1
c(s) = c sn- + .- + cO (3.36)

and

d(s) = dnsn + ... + dO (3.37)

Fortunately, values of this integral have been tabulated,

and for the simplified rocket engine, where n = 4, the

integral squared response error is given by










3(d0d3 + dd1d2) + (C2 2clc3)d0dd4
2dod4 (-dOdj djd4 + dld2d3)

(c1 2cOc2)dod3d4 + c2(-dld + d2d3d4)
2 2
2dod4 (-d0d3 d d4 + dld2d3)

Values of the performance index were calculated as a

function of the proportional gain G2 and are presented

in Figures 5 and 6. The number associated with each

performance index curve identifies the steady-state

condition of the engine when it was perturbed, i.e.,

T;= 0.1, TO PO = 0.2, etc.
Figure 5 represents the response error for a

reference model with time constant equal to 5 seconds,

while the model for Figure 6 has a 10-second time constant.

In both cases, well defined minima were obtained for

nearly all conditions, indicating that adaptive control

of temperature with a variable gain in the proportional

controller for control poison is feasible. These curves

point up the fact that the engine's response is dependent

upon power level. This is, in effect, another argument

for adaptive control of nuclear rocket engines since this

can be likened to a system characteristic that changes

with environment.









1.4 0.1


1.3


1.2


1.1


1.0 1.0
1.0

0.9 0.9

S0.80.8

x 0.7
S0.7 0

0. 0.6

S 0.6 -
0.40.
0
0.5 -


0.4 -
0.4

0.3


0.2 0.3


0.1 -.2

0.2

10 20 30 40 50 60 70 80 90 100 110
Proportional Gain G2

Figure 5. Effect of G2 on Performance Index with T'mod = 5 Seconds








0.9


0.2


0.1


10 20 30 40 50 60 70 80 90
Proportional Gain G2


Figure 6.


Effect of G2 on Performance Index with
Tmod = 10 Seconds


1.4


1.3


0.7





32



The results of this preliminary analysis indi-

cated that adaptive control of the simplified nuclear

rocket engine was feasible, and simulation of the

complete system followed. This simulation is described

in Chapter IV.













CHAPTER IV


SIMULATION OF THE NUCLEAR ROCKET ENGINE


Simulation of the nuclear rocket engine and its

control system was performed with an analog computer.

A description of the analog computer equipment used in

this work is presented in Appendix C. Block diagrams

as well as detailed computer wiring diagrams are presented

as the simulation is described. The investigation pro-

ceeded in the following manner. A constant parameter

nuclear rocket engine was simulated for the purpose

of determining a desired, or "normal," performance with

a conventional feedback control system. A reference

model closely approximating this desired behavior was

then chosen, and an adaptive loop including the parameter

adjustment mechanism was designed and simulated. Finally,

performance of the model reference adaptively controlled

nuclear rocket engine was studied as a system parameter

was varied.











The Constant-parameter System


The system equations of Chapter II, including the
constants of Table 1, are repeated here for convenience.


N '= 50(p'N' N' + C')

C' 0.1(N' C')

T' 1.471 N' P'(T')1/2)


P' = 0.4 P'V'(T')1/2


p' p' T' 18
P C+T'


P x G1
0


(T')1/2


(4.1)

(4.2)

(4.3)


(4.4)


(4.5)


(p )2
(T')1/2


- (T')1/2 dt


+ G2 ((T)1/2


Sc 0.5(Px Pc)


II
Vx 2 G3
0


(Px P')dt + G4(P; P') + Vx(O)


V' = 5(V V')


(4.6)

(4.7)

(4.8)

(4.9)


Simulation of a system on an analog computer
requires amplitude scaling of the system equations. This
scaling is necessary because an analog computer estab-
lishes mathematical relations between machine voltages


- (T')1/2 +,c (O)








which represent problem variables. These voltages must
be kept within specified limits (+100 volts for the
equipment used in this work) in order to prevent over-
loading of machine components. Such overloading, or
saturation, leads to incorrect computation. Details of
the amplitude scaling for this problem are included in
Appendix C.
Amplitude scaling of equations 4.1-9 results in the
following set, where the bars indicate that quantities
are now expressed in volts, i.e., the equations are
written in terms of machine variables.


N = 100Q 50N + 50C

C = 0.l(N )

T 14.71N 4.16R

T 1.7S 11.322

p = 1Opc T + 180Y
t
G- i (- 1/2
Pcx 21/2 ( TX)
0
G2 1/2
2 ()72


P ca 0.5(P)cx


(4.10)

(4.11)

(4.12)

(4.13)
(4.14)


1/2
- (T) dt


(4.15)


(4.16)


-1/2 1
(T) I + Xx (0)


PC)











x = 1.33G3 f (P, P)dt + 1.33G4(P P) + Vx(0)
0
(4.17)
o
V z 5(x V) (4.18)

where

N
S '1o (4.19)
=P T100
1/2
S(T) (4.20)
10


Y 10 (4.21)
T


S V (4.22)
100

Z R- (4.23)
100

The rationale underlying the grouping of quantities as

expressed in equations 4.19-23 is discussed in

Appendix C.

The above equations were used for simulation of

the constant-parameter nuclear rocket engine. Figure 7

depicts response of the system to a step command calling

for transition from an idling condition (T 1000 OR,

P = 275 psi) to the full power condition (T = 5000 0R,

P = 1500 psi). Controller gain settings are listed in























































0 5 10 15 20 25 30 35 40 45

Time after Step Command (sec)


Figure 7.


Response of Constant-parameter Nuclear Rocket
Engine


1.0


0.8

a) M
N
* 0.6

E .
64E
o 0 0.4
z2


0.2


-e
N O

Eo
o
z








EO<
0o4j

z0
q)-
*Z


1.0
0.8
0.6
0.4
0.2
0

1.0

0.8

0.6
0.4
0.2
0









the figure. Figure 7 does not indicate the existence of

intolerable temperature transients, and the overshoot is

less than an acceptable 5 per cent. This response was

chosen as the "best" attainable with this design and

is hereafter referred to as the system's "normal"

response.


The Reference Model


The normal performance depicted in Figure 7 is the

performance that the model reference adaptive control

system is to maintain despite changes in characteristics

of the nuclear rocket engine. This means that a reference

model must be developed with a response closely resembling

this normal response.

An important criterion for the reference model is

that it be as simple as possible, i.e., that it require

a minimum of control equipment above and beyond the

equipment requirements of the conventional feedback

control system. The analysis of Chapter III, together

with the system's simplified transfer function, suggests

that a reasonable first choice for the reference model

would be one with the following transfer function:


TFmod = 1 (4.24)
m s + 1










The response of such a first-order model with a time

constant of 10 seconds is contained in Figure 8 along

with the normal system response and the responses of

two other models which are discussed below. The

responses are for a step command corresponding to the

command calling for transition from an idling condition

to the full power condition.

It is obvious that the "tailing off" of the first-

order model's response as it approaches the full power

condition would result in excessive transition times if

used as the reference model in an adaptive control

system. If attempts are made to reduce this "rise time,"

severe temperature transients and their attendant thermal

shock problems would undoubtedly arise. These observa-

tions cause one to consider a more complicated, yet

simple, reference model. This is a second-order model

with transfer function as follows:

t2
TFmod 2 n2--T (4.25)
s2 + 2 2~Jns + 2

Figure 8 contains the response of this model for

0.7, L/n 0.179 sec-1. This model is an improve-

ment over the first-order model in approximating the

normal rocket engine response. Overshoot and time for

the idle to full power transition are acceptable. How-

ever, note that at the instant of command, i.e., t 0,













Constant-parameter System
-- First-order Model
-. -- "Pure" Second-order Model
S- Second-order Model with Finite Zero


0


- -


0.8 I


0.6


0.4


0.2 h


0 5 10 15 20 25 30 35


Figure 8.


Time after Step Command (sec)

Responses of Models and Constant-
parameter System











the slope of the model's transient response is zero.

This means that at the instant of command, and for a

short time thereafter, such a reference model would

demand a degradation of the nuclear rocket engine's

normal response. The model would later counteract this

action; however, this situation does not appear attrac-

tive.

It appears that a second-order reference model

might suffice for adaptive control of the rocket engine,

provided that its initial response can be made to better

approximate the normal system response. This is accom-

plished by adding a zero to the transfer function of the

second-order model, i.e.,

(Cj2/a) (s + a)
TFmod 2 2 (4.26)
s + 2 ns +

Figure 8 contains the response of such a model with

= 0.74, )n = 0.16 sec-1, and L 2/a 0.049. It is

observed that this second-order model with a zero in its

transfer function very closely approximates the rocket

engine's normal response during the early portion of the

idle to full power transition. This is desirable and

means that the system's normal response will not be

affected in this important phase of the transition by

the adaptive controller unless, of course, changed












characteristics have altered the nuclear rocket engine's

transient response. With the realization that a perfect

system-model match can be attained only when the two

transfer functions are identical, the second-order model

with a zero was chosen as the reference model for the

nuclear rocket engine. Detailed analog computer diagrams

for the three reference models discussed above are con-

tained in Appendix C.


Adaptive Loop Design


With a reference model selected, it is now possible

to design the adaptive loop, which includes the reference

model, weighting function filter, and parameter adjust-

ment mechanism. These components are discussed in detail

both in Chapter III and in Appendix A. The example con-

tained in Appendix A should be of help in following the

design of the adaptive loop for the nuclear rocket engine

control system. A block diagram of the adaptive loop is

contained in Figure 9. Design of the loop closely follows

Osburn's procedure. The weighting function filter has the

reference model transfer function. Signal input to the

adjustable parameter is passed through this filter and is

multiplied by the error signal resulting from subtraction

of the model output from that of the system. This product












43
























6
k
a)








C:
0
44




0
E
a)









-o
k







0)
m
CD









.1-
U











0





al

1-4
Lz









is integrated and serves to vary the adjustable gain

automatically. The analog computer diagram for the

adaptive loop is contained in Appendix C.


Adaptive Loop Performance


It is now possible to study the effect on normal

system performance of addition of the adaptive control

loop. Figures 10 and 11 depict response of the nuclear

rocket engine going from idle to full power with and

without the adaptive controller. Note that there is

slight upgrading of the temperature response, but, in

general, the normal response of the system is relatively

unchanged by addition of the adaptive control loop. With

the adaptive controller operative, it is now appropriate

to evaluate the control system's effectiveness in main-

taining normal performance of the nuclear rocket engine

as an important engine characteristic is varied. This

evaluation is described in the following chapter.


























































Figure 10.


Time after Step Command (sec)

Nuclear Rocket Engine Response for
7p (normal)


1.0


N 4-

E .
0 a
oa0.


0.2


"0
N (







4
E a
k H
o












z
N <



Z3


0.8
0.6
0.4
0.2
0

1.0
0.8

0.6
0.4
0.2







46







1.0


or











-2-
-0 4-
-o













*. -84
N4-,
,-I a)
M 0.5














0 -1
Z













0
S-2-
--- Non-adaptive Control





SAdaptive Control-
-6











0
4-








0
*^






0 -10 -
t-






S-182
c I I l












0 5 10 15 20 25 30 35 40

Time after Step Command (sec)

Figure 11. Control Device Response for Tp (normal)
Q< / p












CHAPTER V


PERFORMANCE OF MODEL REFERENCE ADAPTIVE CONTROL SYSTEM


Study of the applicability of model reference tech-

niques to control of the nuclear rocket engine has

developed in the following manner. An engine including

a conventional feedback control system was simulated on

the analog computer. A "normal," or desirable, per-

formance was established, and this performance is to be

achieved in the face of changed rocket engine charac-

teristics by means of a model reference adaptive control

system. A reference model matching this normal per-

formance was chosen and utilized in the design and

simulation of an adaptive control loop. It now remains

to be seen whether or not the adaptive control system is

effective in achieving the specified performance.


Adaptive Control for a Varying Pressure Time Constant


The nuclear rocket engine model used in this study

is particularly sensitive to changes in the pressure

time constant, and it is this parameter which was varied











to evaluate the effectiveness of the adaptive control

system. The parameter was varied from its expected,

i.e., normal, value, to two times this value.

Figures 12 through 21 depict response of the

nuclear rocket engine for various values of the pressure

time constant. Response of the constant-parameter

system, as well as the reference model response, is

presented along with the adaptively controlled response

for each case. In addition, pressure, neutron density,

valve position, and control poison reactivity are shown

for each type of control.

Note that the constant-parameter, i.e., non-adaptive,

system's response deteriorates and becomes more sluggish

as the time constant increases. On the other hand, the

adaptively controlled system "sees" this deterioration

and acts to achieve the desired performance. Although

temperature is the controlled variable, the coupled

system variables reflect this same behavior. It is

interesting to observe the control poison reactivity as

a function of time after the step command to proceed

from the idling condition to full power. In each case,

more positive reactivity is introduced to compensate for

the sluggish turbomachinery.










49





1.0-


0.8


a)30.6
N +)
- k --- Non-adaptive Control
S0.4- ----Adaptive Control

z- ------Model Response
0.2


0-
S1.0-
Te f 0.8- e
m" 0.6-
0 0.4-
0.2-
0




e 0.6-
E o 0.4-
o
O-
z 2 0.2-
S1.0-



0 5 10 15 20 25 30 35 40 45

Time after Step Command (sec)

Figure 12. Nuclear Rocket Engine Response for 1.2
op (normal)















1.0







0) -
P-2
a ()









-I C
.-4
Sa 0.5




~0
>










-4 Non-adaptive Control




0







-o -8
0 10"









4-
U I












I -18 -


0 5 10 15 20 25 30 35 40

Time after Step Command (sec)

Figure 13.. Control Device Response for 1.2
Tp (normal)
C:P

























































0 5 10 15 20 25 30 35 40 45

Time after Step Command (sec)


Figure 14.


Nuclear Rocket Engine Response for
1.4 7p (normal)
P


1.0


0.8
0)
V-k
N4 0.6

E a
4E 0.4
0 a)
ZH

0.2


0


.0

N H
--I (fl
0 Q)

2





N0)
'rv
ro c
E0

2


1.0
0.8
0.6

0.4

0.2
0

1.0
0.8
0.6
0.4

0.2
0






























Non-adaptive Control

Adaptive Control


0 5 10 15


Time after Step Command (sec)

Figure 15. Control Device Response for
1.4 7p (normal)


1.0






o
N 4-

' 0.5
(A >
z M

0


-2


-4


-6


-8


-10


-12


-14


-16


-18













1.0 .


0.8 -

0 -k
N4 0.6

'D .. NNon-adaptive Control
S.0.4 Adaptive Control
z -.-.- Model Response
0.2


0


:3 0.8

E 0.6
o0. 0.4 -
0.2
0

-^1.0 -- --
o0.8 8
(D n L
N (D^ 0.6
(m c 0.4
Eo
0 +0.2
w 0

0 5 10 15 20 25 30 35 40

Time after Step Command (sec)

Figure 16. Nuclear Rocket Engine Response for
1.6 p (normal)
































- Non-adaptive Control

----Adaptive Control


-
/


0 5 10 15 20 25 30 35 40

Time after Step Command (sec)


Figure 17.


Control Device Response for
1.6 p (normal)


C
. ,-I
N +-.



Zo


-t
















1.0


0.8



0.0
rI ro0.6

E0 0C
S0.6
o 0.4


0.2


0

1.0
wU cU 0.8



0.4
on
2 0.2
0


a) 0.8
N )
- 0.6

4o0.4
04-
Z 0.2
2 0


0 5 10 15 20 25 30 35 40 45

Time after Step Command (sec)


Figure 18. Nuclear
1.8 T


Rocket Engine Response for
(normal)



























































0 5 10 15 20 25 30 35 40


Time after Step Command (sec)


Figure 19. Control
1.8 rp
1p


Device Response for
(normal)


1.0


c
-a *H
N 4J
ro
Ea)
>


0.5


-6


-8


-10


-12


-14


-16


-18














1.0


0.8-
0k
^N 0.6 /,

O Non-adaptive Control
S0. 4 /

z ---Adaptive Control

0.2- --- Model Response


0 -

S 1.0 .
N* 0.8
0.8- -

o W- 0.6
Ok 0.4-
z 0.2:
0)

.4 1 0 .0- _.-
3c 0.8-
N Q)
* a 0.6

Eo 0.4-

2o3 0.2
z 0


0 5 10 15 20 25 30 35 40 45

Time after Step Command (sec)

Figure 20. Nuclear Rocket Engine Response for
2 r- (normal)
P




















Nc
1O -1
1.0












-12 O
S 0.5


0 0




-2
--2 Non-adaptive Control


-4 - Adaptive Control



V) -6


o -18

c-4-

















0 5 10 15 20 25 30 35 40 45
-14











Time after Step Command (sec)

Figure 21. Control Device Response for

2 Tp (normal)
04ue2.CnrlDeieRsos o
-1 -p - -a










Figures 12 through 21 represent action by the

adaptive control system to upgrade, or speed up,

response of the system. Figures 22 and 23 illustrate

the performance of the two control systems for the case

where the pressure time constant is only 80 per cent of

its normal value. This means that the constant-parameter

nuclear rocket engine is more responsive to the full

power command, and the adaptive controller acts to

degrade, or slow down, its response to match that of

the reference model. The reactivity contribution of

the control poison follows a markedly different pattern

in this case.

The results contained in Figures 12 through 23 are

summarized in Figure 24. Here, the time required by

the nuclear rocket engine to proceed from 20 per cent

to 90 per cent of its design temperature is compared

for the two modes of control, i.e., constant-parameter

versus model reference control. The figure indicates

that the reference model requires 14 seconds to make the

transition. The constant-parameter system's rise time

varies from 14.4 seconds to 29 seconds, or is nearly

doubled as the pressure time constant is doubled. The

adaptively controlled system's rise time varies from

13.6 seconds to 16.2 seconds for the same variation in























































0 5 10 15 20 25 30 35 40 45


Time after Step Command (sec)


Figure 22,


Nuclear Rocket Engine Response for
0.8 7p (normal)
p


Q)
k
+ O.
cv
a)
a0.
a)


1.0
a a 0.8
N 1
H3 ~ 0.6

E 0.4
^a 0.2

0

a 1.0
N Q 0.8
mc 0.6
eo
0
-40.4
S0.2
2































Non-adaptive Control


--Adaptive Control


0 5 10 15 20 25 30 35 40


Time after Step Command


(sec)


Figure 23.


Control Device Response
0.8 p (normal)


1,0



CP
o-. 0.5






-2
-2


-4


-6


-8


-10


-12


-14


-16


-18


for


~~~






62












(n
a)
(n
CA c
0
04 CL
0
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time constant. This amounts to a 19 per cent increase

in rise time compared with the normal response. Figure 24

also indicates the degrading action of the adaptive

control system in the case of the time constant's being

80 per cent of its expected value.

Figure 24 can also be used as a basis for esti-

mating propellant savings due to action by the adaptive

controller. Smith and Stenning determine that the pro-

pellant mass rate of flow in the rocket engine is propor-

tional to P'/(T')1/2, and that the flow rate at full

power is 2300 pounds per second. This means that, at

the idling condition, a mass flow rate of approximately

1000 pounds per second is to be expected. If the flow

rate increases linearly with power, and this is conserva-

tive, one can speak of an average mass flow rate of 1600

pounds per second during the transition from idle to

full power. The rocket engine in its "normal" state

then uses slightly more than 22,000 pounds of propellant

during the transition. In its degraded, or sluggish

condition, with the pressure time constant doubled,

about 44,000 pounds are expended. Now with the adaptive

control system, transition to full power in the sluggish

condition is accomplished with about 26,000 pounds of

propellant, representing a savings per transition of











18,000 pounds, when compared with performance of the

constant-parameter control system. Or, to look at it

another way, the savings represents 8 seconds of full

power operation.

It is important to observe the variation of the

adjustable gain G2 for these different conditions.

Figure 25 shows the adjustment action brought about by

the adaptive loop. The gain is expressed as G which
1/2
is equal to G2/(2) as indicated in equation 4.15.

As expected, greatest adjustment is required and exe-

cuted for the most sluggish condition, i.e., for the

doubled pressure time constant.

The variation of G2 is particularly interesting

in the case of the 80 per cent time constant. Here, it

is observed that, for a time, the gain becomes negative.

Recall that, in this configuration, the control system is

degrading the performance exhibited by the constant-

parameter system. This brings up the question of system

stability, which is discussed below. It should be noted

that the maximum excess reactivity involved in these

transitions to full power amounts to 0.6 dollar.


Stability Considerations


The transition for the 80 per cent time constant

results in G* being driven negative by the adaptive













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o 0 0

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control system. An indication of system instability
1/2 *
for negative G2, i.e., (2) G2, comes from the simpli-
fied transfer function of Chapter III. The Routh-Hurwitz

stability criterion applied to the transfer function

given by equation 3.26 establishes that, for stability

of the nuclear rocket engine, X>0, which implies

G2 > -30.6, and Y=0, which implies G2> -5.78. The

more restrictive of these two necessary conditions is

that G2='-5.78. It is not advisable to rely too heavily

upon a linearized approximation for stability analysis

of a nonlinear system, and so the following analog

simulation was undertaken for both the constant-parameter

system and the adaptively controlled system.

As the nuclear rocket engine makes its transition

from idle to full power, a dollar step of positive reac-

tivity is introduced 5 seconds after the transition

command. This is accomplished by passing 5 volts to

the input of amplifier #20 of Figure 33. Figure 26

depicts the nuclear rocket engine's response.

The constant-parameter control system allows a

dangerous temperature overshoot, but succeeds in

limiting the temperature surge. Model reference adap-

tive control appears to handle the added reactivity






67




I
I
/
/


1.0 ----


0.3-





0.4
N 0.4 Adaptive Control

0.2 --.- Model Response
O
o






20


10




C
'~ -10


-20




0 5 10 15 20 25 30 35
Time after Step Command (sec)

Figure 26. System Instability Due to
Negative G2












satisfactorily until the gain G2 becomes negative.

The system becomes unstable at a G2 value of approx-

imately -6.0.

This simulated configuration is one in which the

system temperature is greater than called for by the

temperature command. The adaptive control system acts

to match the system's response with that of the reference

model and drives the gain G2 negative. This adaptive

action results in an unstable situation.

The stability boundary is presented as Figure 27

where values of G2 are plotted versus the difference

between command and system temperatures. The constant-

parameter system was simulated at steady-state design

temperature and then disturbed with step signals repre-

senting the indicated temperature differences. The gain

G2 was varied through a small negative range and values
leading to unlimited temperature response are used for

the stable-unstable boundary. It appears that the

constant-parameter system has a limited region of

stability for this configuration. A stability map with

the system initially at a steady-state temperature of

0.2Tdesign has essentially the same appearance.
Although Figure 27 indicates a stable region for

negative G2, this does not mean that acceptable nuclear


































*c'J


to
N


-8


-10


-12


-400


Figure 27.


-300 -200 -100 0
Tx T (oR)


Stability Boundary for Constant-
parameter System






70


rocket engine operation can be attained with these

values when the system approaches design temperature.

This is due to the fact that intolerable temperature

overshoot is encountered for nearly every negative

value of G. This means that, for the nuclear rocket

engine under study, negative values for G2 are to be

avoided, and limiting devices appear to be necessary.












CHAPTER VI


SUMMARY AND CONCLUSIONS


Summary of Results


Model reference parameter adjustment techniques

were found effective in controlling a nuclear rocket

engine. A relatively simple second-order model (2 poles

and 1 finite zero) was suitable as a reference model for

the step command inputs utilized in the investigation.

Acceptable operation of the nuclear rocket engine was

possible with a limited valve opening due to the addition

of proportional control of control poison.

The system's pressure time constant was varied to

test performance of the adaptive control loop. Acceptable

system performance was achieved with adaptive control as

the value of this time constant was increased to twice

its normal value. Savings in time required for transition

from idle to full power and the attendant savings in

propellant were found significant when compared with

results of constant-parameter control.









Dynamics of the nuclear rocket engine permitted

the adaptive system to be more effective in upgrading

engine response than in degrading its response. In the

latter case, with the gain-control parameter selected

for adaptation, it was observed that the adaptive con-

troller could cause unstable operation. This instability

could be avoided through use of limiting circuits in

the adaptive loop.

Implications of Results


The results of this investigation are encouraging;

however, this control concept must be evaluated within

the framework of the total body of knowledge pertaining

to the development of nuclear rocketry. For example,

this study indicates that propellant savings of up to

20,000 pounds per transition from idle to full power are

possible with adaptive control. But, is there a mission,

and will there be a vehicle, where such savings warrant

the addition of a possibly more complicated control system?

For the same case, it has been shown that adaptive

control reduces the transition time from idle to full

power by as much as 14 seconds. Does this shorter rise

time substantially reduce the complexity of attitude

and guidance control problems so that requirements for

elaborate guidance control programming are reduced

and the addition of adaptive control is justified?












Finally, would the addition of adaptive control repre-

sent a significant increase in system weight and com-

plexity, or can this "cost" be reduced through utiliza-

tion of computing equipment already on board? These

questions cannot be answered here, but as mentioned

above, the results of this study are sufficiently

encouraging to prompt further investigation of this

application of model reference adaptive control.

Of course, it can be reasonably argued that

adaptive control should not be considered for applica-

tion in systems which can be suitably controlled with

conventional techniques. Adaptive control does bring

with it problems in analysis and requirements for addi-

tional equipment. However, such arguments become less

significant if the conventional systems require elaborate

programming of controller gain changes or depend upon

humans for this adjustment when they might possibly be

under severe stress, or even incapacitated for periods

of time. A nuclearly powered rocket with its crew

presents a provocative topic for such discussion.


Suggestions for Further Study


A logical sequel to this work would involve adap-

tive adjustment of additional parameters. System










response with control emphasis shifted from control

poisons to the hydrogen propellant should be studied.

Adaptive control should be studied for application in

all phases of the rocket engine's operating regime, i.e.,

startup and shutdown control.

The nuclear rocket engine model should be up-

graded within the bounds of available computing equip-

ment. The stability considerations encountered in this

study point up the need for some knowledge of the useable

range of values for the parameters to be varied in any

new engine model. To all of this should be applied

recently developed techniques which purport to allow the

control system designer to determine compensation for

improving adaptation and to predict responses of model

reference adaptively controlled systems (24). Finally,

consideration should be given to the use of model

reference techniques for determination of nuclear reactor

system parameters.































APPENDICES











APPENDIX A


PARAMETER ADJUSTMENT ADAPTIVE CONTROL


Osburn's derivation (15) of a model reference

adaptive control system design procedure is summarized

here for completeness. The problem is "to define opti-

mum performance and provide the means for automatic

enforcement of this performance."

System performance is considered optimum when a

selected performance index is minimized. The greatly

simplified schematic diagram of Figure 28 depicts the

quantity which is used to generate the performance

index (PI), i.e., the response error e(s) between

system and model outputs. The selected PI is given by
t2
PI = F(e)dt (A.1)

tl
where F(e) is an even function of e.

The definition of optimum performance for a system

with an adjustable parameter K requires that


PI) -2 F(e)dt = 0 (A.2)
afl














q(s)
sys


Figure 28. Generation of Response Error in Model
Reference Adaptive Control System


Figure 29. Block Diagram of Second-order System with
Adjustable Parameter in Feedforward Path










This leads to the following definition of error quantity

(EQ).
t2
EQ / F(e)dt (A.3)
6K t
tl

Parameter adjustment is then performed according to
/K -Sa(EQ) (A.4)

where Sa is the adaptive loop gain. It should be noted

that Osburn's derivation provides for adjustment of
n parameters; however, this summary describes the deri-
vation for adjustment of only a single parameter.
Equation A.3 is rewritten as

EQ ft2 OF(e) dt (A.5)
tl 3K

with the assumptions that tl and t2 are independent
of the adjustable parameter K and that 8F(e)/3K are
continuous functions of both K and t. Now

aF(e) F,'(e)e_ (A.6)
3K 3K

F'(e) dF(e) (A.7)
de

Methods for obtaining signal indications of these
various quantities are now described. As indicated in
Figure 28, the response error is defined as
e(s) q(s)ys q(s)mod (A.8)










where q(s)sys is the system output, and q(s)mod is the

model output. Since q(s)mod is not affected by any

parameter of the system,
ae(s) C q(s)sys (.9)
K dK

this partial derivative of the system output with

respect to the variable parameter K must be obtained.

Osburn shows that generation of dq(s)sys/)K

requires the use of the system characteristic operator

and various derivatives of the system input and output

quantities. He points out that
it is exactly the ignorance of
detailed information about these
coefficients and system characteristics
that leads to the necessity for pro-
viding the system with a self-adaptive
capability.

This apparent impasse is circumvented by noting that

the control system can meet its design specifications

whenever the variable parameter K is suitably adjusted.

Since the model is designed to represent these specifi-

cations, Osburn reasons that the model dynamics can

provide a good approximation to the system dynamics

and model characteristics can be employed to generate

an approximation to the needed dq(s)sys/c K "in some

neighborhood of the system optimum response state."

His work has shown that this approach makes possible











a practical and effective design technique. The approx-

imate, or indicated, partial derivative, is called the

error weighting function (WE).

All quantities required for parameter adjustment

have been treated, and as is shown below, are available.

Equation A.5 is rewritten as

t2
EQ = F'(e)WE(t)dt (A.10)
tl

and for F(e) E e2, as is the case in Osburn's work

and in the present study, equation A.10 becomes

EQ = 2/ e(t)WE(t)dt (A.11)
tl

Equation A.4, together with equation A.11, indicates

that the parameter K is adjusted according to

d = -Sae(t)WE(t) (A.12)

The following example is used to demonstrate

Osburn's design technique, and it also points up the

availability of signals required for parameter adjust-

ment as called for in equation A.12. Figure 29 is the

block diagram of a second-order system with an adjustable

parameter in the feedforward path. It is desired to

adjust K by using a reference model with the open loop

transfer function 10/s(0.ls + 1). The design procedure









calls for generation of aq(s) ys/K. For the system
of Figure 29,

q(s) K q(s) (A.13)
sys s(O.ls + 1) + K in

The required partial derivative is found to be
_q(s)S s s(O.ls + 1) (A.
)K-[ ( q(s)in (A.14)
SK s5(0.is + 1) + K 2
or equivalently,

$q(s)
K sys ) (TFsys 1 S0.s + 1) q(s)in
K K s(O.is + 1) + K in
(A.15)
where TFsy is the closed loop transfer function of the
controlled system. In terms of the input signal to K,
i.e., q(s)K, equation A.15 can be written as
q(s)sys (TFsys)(--)q(s)K (A.16)
aK
At this point, Osburn's approximation is introduced,
i.e., the typically not-so-well-known transfer function
of the controlled system is replaced by the transfer
function of the reference model. The error weighting
function is now written as

WE (TFmod)( ) q(s)K (A.17)

Osburn found it advantageous to replace the 1/K term of
equation A.17 by unity. This improves the adaptive










response, and division by a varying quantity is not

necessary. This modification is employed here and

equation A.17 becomes

WE = (TFmod)q(s)K (A.18)

Parameter adjustment calls for integration of

the product of the response error and the error weight-

ing function, according to equation A.12. Osburn's

design procedure was derived with the assumption that

parameter variation during adaptive response by the

system is zero. The nonlinear nature of the system

requires this assumption, which is obviously not true

in an actual control system. Willis (25) investigated

this matter and found that the parameter variation

"seen" by the adaptive system is an "apparent" parameter

variation which is "very small compared to the actual

parameter variation," thus establishing the validity

of the assumption.

Figure 30 is a block diagram depicting all compo-

nents of the example system with its adaptive control

loop. The system was simulated on an analog computer

and Figure 31 illustrates the adaptive response of the

system for two offset values of the variable gain K.

No further analysis is presented, since this material

is provided to describe the basis for the adaptive

loop design contained in Chapter IV.










83





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APPENDIX B


EQUATION NORMALIZATION

Reactor Kinetics Equations

The kinetics equations used by Smith and Stenning

to describe the time behavior of the neutron and pre-

cursor populations for a point reactor model are written

in the following form:

dN N + i (B.1)
dt 1

dCi N C (B.2)


Since the magnitudes of quantities in the above equa-

tions differ considerably and vary over a wide range,

some type of normalization is advantageous for analog

computer simulation. In the problem under discussion,

all quantities are normalized to design point conditions,

i.e., maximum core surface temperature of 5000 OR and

inlet stagnation pressure of 1500 psi. This specifica-

tion of design point conditions solely in terms of

temperature and pressure points up the fact that all

other variables are dependent and are determined by the




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