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 cochairmen 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 Constantparameter 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
Constantparameter 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 Constantparameter 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. Constantparameter 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 Secondorder System with
Adjustable Parameter in Feedforward
Path . . . . . . . . 77
LIST OF FIGURES (Continued)
Figure Page
30. Block Diagram of Secondorder System
with Adaptive Control Loop . . .. 83
31. Adaptive Response of Secondorder 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 (sec1)
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 (sec1)
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 (ORl)
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 (sec1)
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 (sec1)
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
Cochairmen: 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 solidcore, 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 secondorder model with
one finite zero.
Engine response to step commands is particularly
sensitive to change in the system's pressure time
constant. With constantparameter 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 nuclearhydrogen 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 startup (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 zerogravity fields as well as in highgravity
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 heatexchanger 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 firecontrol 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 welldefined 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
optimumadaptive 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 parametervariation 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 selfadaptability,
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
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IL
cp
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4 E I
0i C
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.'
0U 0
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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,
timevarying, 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 solidcore 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
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0)
0.
04
0 C:
I.. M
H,
The following equations describe the SmithStenning
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 uranium235 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 sec1
106 OR1
0.3 OR/psi
104 sec
0.005
above, the great amount of information in the open
literature concerning the SmithStenning 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 SmithStenning 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 MohlerPerry (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 steadystate value and a small devia
tion from that steadystate 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 secondorder 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 steadystate 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 reactortemperature loop coupled with a
turbomachinerypressure loop.
23
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HL
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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
>o4
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)
27t(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
n1
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 steadystate
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 10second 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 Constantparameter 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 4.19 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 4.1923 is discussed in
Appendix C.
The above equations were used for simulation of
the constantparameter 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 Constantparameter 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 firstorder 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 secondorder model
with transfer function as follows:
t2
TFmod 2 n2T (4.25)
s2 + 2 2~Jns + 2
Figure 8 contains the response of this model for
0.7, L/n 0.179 sec1. This model is an improve
ment over the firstorder 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,
Constantparameter System
 Firstorder Model
.  "Pure" Secondorder Model
S Secondorder 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 secondorder 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
secondorder 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 sec1, and L 2/a 0.049. It is
observed that this secondorder 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
systemmodel match can be attained only when the two
transfer functions are identical, the secondorder 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
14
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
S2
 Nonadaptive Control
SAdaptive Control
6
0
4
0
*^
0 10 
t
S182
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 constantparameter
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 constantparameter, i.e., nonadaptive,
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  Nonadaptive 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) 
P2
a ()
I C
.4
Sa 0.5
~0
>
4 Nonadaptive 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)
Vk
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
Nonadaptive 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 .. NNonadaptive 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)
 Nonadaptive 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 Nonadaptive 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 Nonadaptive Control
4  Adaptive Control
V) 6
o 18
c4
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 constantparameter
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., constantparameter
versus model reference control. The figure indicates
that the reference model requires 14 seconds to make the
transition. The constantparameter 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
Nonadaptive 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
(N,
U)
0 Wk
0)
k
co
0
~ 0 0
CC~
kC
a) 00)0
E. 4) 
4
00 k m mO
U) 
(N c C
U) +) a
0 (E C.
r_ P
4 H 04
ok~
ac,
L)
'~~O .
0 
I r(N
"4
C(N (D 00 \0 C1 4 0
0 4 N (N 04 C(N I r 
(Oas) 6*0 =il 04 Z*O = &I W0iJ OWTJ uoTTITsueIJ
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
constantparameter 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
1 4 
E E E
C C C
o 0 0
"0. 0. "
E E
0 0
C Co C
C>.
0 0 0 0 0 0 0 0
Zo uTeq
0 0

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 RouthHurwitz
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 constantparameter
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 constantparameter 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 steadystate 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 stableunstable boundary. It appears that the
constantparameter system has a limited region of
stability for this configuration. A stability map with
the system initially at a steadystate 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 secondorder 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 constantparameter 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 gaincontrol 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 Secondorder 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 selfadaptive
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 secondorder 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 notsowellknown 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
