TAANSACTIONS
IRE
AUTOMATIC
ON
COATTROL
8
Novena ber
Analysis of aNonlinearControl
System for
Stabilizing a Missile"
LEONARD ATRANT
Summary-An autopilot with attitude and rate feedback, representativesystem lags, and a two-way relay servo with inherent
hysteresis is considered for roll control of a missile with peripheral,
tangentially operating jets.
This type of control system is shown to produce a steady-state
oscillation. Missile dynamics in the presence of
hunting are developed and the relationships governing angular position and rates
are found to be functions of the oscillation frequency, control force
magnitude, and missile constants (geometry and weight).
The describing function techniqueis utilized to determinegraphically the relationship among frequency, hysteresis band, and system
time delays. A comparison is made between the root locus and amplitude-phase presentation. An analog computer study of system behavior is presented to illustrate the agreementbetween the analysis
and systemperformance.
INTRODUCTION
N E of the foremost considerations in the design
Fig. 1-3lissile coordinate system.
of a missile control system is that itbe a s simple
and inexpensive as possible. This can be achieved
b y using only those components which are absolutely
necessarytoperformthecontroltaskadequately.
L (summation of roll moments). (1)
Gyros, resolvers, amplifiers, filter networks relays, and
actuator hardware should be kept to a minimum.
Because of symmetry, we may assume
This paper shall restrict
itself to the discussion of
I,
I , and I,,
0.
roll stabilization by means of a relatively simple and
reliable autopilot system designed to provide attitude
Eq.
in simplified form becomes:
control,whereasthe
missile maneuvercommandsin
this case are applied to the pitch and yaw channels.
Although in aircraft operation almost any continuous
oscillation is intolerable, in an unmanned missile i t is where
lj Lj/Ir (control moment effectiveness),
possible t h a t a steady-state oscillationconfined
to
I , =L,/I, (misalignment torque factor),
reasonablelimitsisentirelysatisfactory.Thecontrol
fj
force produced by control jets,
system under discussion consists in partof a relay servo
f,=net unbalanced force due to possible thrust misdriving a jet nozzle valve in such a manner as to proalignment,
duce a continuous dither. The nozzles themselves are
L j control force moment arm,
located on the periphery of a plane normal to the main
L, =effective moment arm upon
which unbalanced
thrust line. Control of the jets results in a fixed torque
forces act.
in either of opposite directions.
W i t h no unbalanced torques acting on the system,
The autopilotsignal requiredt o roll stabilize is derived
from an attitude gyro; however, a derivative signal is fm = O ; therefore, upon integrating ( l ) , we arrive at the
necessary to augment the
negligible aerodynamic damp- following expression for roll rate:
ing of the missile.
P Ijjj[t
switching]
C$
(missile roll rate). (3)
0
EIISSILE
ROLLD n x m c s
During any "on-time"
Let us assume a general
missile configurationas shown
fj
k F (pounds) constant.
in Fig. 1.
n/w
T h e roll moment equation, taken about the body axis Theon-time foreachunidirectionalblastis
where
is
the
frequency
of
oscillation
in
rad/sec.
T
he
is
peak roll rate is, therefore:
Manuscript received by the PG.IC, September 9, 1956. Presented a t WESCON, Los Angeles, Calif., August, 1956.
t Air Arm Div., IVestinghouse Electric Corp., Baltimore, Md.
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90ZjF
@mal
w
(deg.:'sec).
1357
Atran:
Analysis
ofNonlinear
a
Control
System
for Stabilizing
Missile
a
Fig. 4-Relay servo controlling missile in roll (externally
applied torque of 49 pounds-feet).
(b)
Fig. 2-(a)
Waveforms of force and acceleration. (b) Waveforms
roll position and rate.
SERVOSYSTEM
Theinput-outputwaveformsoftherelay-actuator
combination are illustrated in Fig. 5 , next page.
The autopilot control system is shown in block diagram form in Fig. 6.
In an effort to determine the behavior of the system
in the presence of a highly nonlinear servo, the system
transfer functions are combined and the characteristic
equation rearranged to thefollowing expression for neutral stability:’
-1
N(x)
Fig. 3-Plot
of maximum roll angle and roll rate vs frequency of
acceleration.
1)
IjB(71S
syns
(73s
(6)
where
A/B.
Thedescribingfunction
N ( x ) has beenderivedin
the Appendix; in addition, both sides of (6) are plotted
inFig. 7. T h e crossover point of the describing functioncurveandsystemtransferfunctiondetermines
boththefrequencyandamplitude
of theresultant
steady-state oscillation.
AUTOPILOT
PARAMETERS
Further integration leads to the following expression
for the peak roll angle:
I th a s beenshown thatforthenonlinearsystem
under discussion the magnitude of t h e roll angle excursions isa function of the control force and the frequency
of oscillation. In turn, the frequencyis a function of the
hysteresis band, system time lags, and autopilot gain.
The interrelation will now be explored.
A steady-statetorqueunbalance,dueinpart
to a
misalignment of the thrust axis, will compel t h e missile
to assume a steady-state angular error. However, the
system will continue to oscillate in accordance with t h e
Thewaveforms of thesystemundersteady-state
conditions are illustrated in Fig.
2 (a) and 2 (b)
Plots of
and dmaxvs modefrequencyareshown
in Fig. 3 for values of l j -0.58 and F = 100 pounds.
Superposition of an unbalanced torque requires the
development of a counter moment of the same average
value, thereby necessitating that the symmetry of the
E. C. Johnson, “Sinusoidal analysis of feedback control systems
control force vs time curve be destroyed. This is illuscontaining nonlinear elements,” Trans. A I E E , vol. 71, part 2, pp.
trated in the REAC recordings of Fig. 4.
169-181 July, 1952.
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TRANSACTIONS
IRE
Fig. 5-Control
November
ON A U T O M A T I C CONTROL
system waveforms.
Ii
Fig. 6-Control
system block diagram.
requirements imposed by( 6 ) , and thebias from its original zero steady-stateroll angle may be determined in the
following manner.
Assume for the moment that roll
the oscillation caused
by system nonlinearity is
sufficientlyremoved in frequency from the actual
roll controlmodecornerfrequency for complete decoupling to exist. That
is, the
response to a roll command input is not influenced by
the higher frequency dither. Under these conditions we
may write that
$=O
Fig. i--.lmplitude
and phase of servo describing function and
system transfer function.
F=-
Lmj7n
(pounds).
Lj
T h e roll mode,regarded as linear and
Aljs -Blj 0, has a damping factor
of the form
and average
fjse
Substituting the above into (2) we have:
L,
LiB
(rad).
where
wn
d--Blj.
Thus for a specified
the value of A that is necesThus if the maximum allowable roll angle is specified sary is
together with the magnitude of the unbalanced torques
likely tobeencountered,therequiredautopilotgain
A4=-.2i-nwn
becomes
lj
B E - Lmfm (pounds/rad).
L&AV
T o counterbalance the moment Lmfmrequires a force
capability of
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DISCUSSION
A missile controlsystemwasselectedforanalysis
and the following values were assigned to the various
parameters.
of a Nonlinear Control System for Stabilizing
Atran:
Analysis
I957
Missile
5
0
J
0
I
Fig. 9-Relay
-5
Fig. 8-Root locus plot of representativecontrolsystem
showing
effect of nonlinearity. -4=linear system, B =nonlinear system.
72=0.05 sec, F=lOlt.
lj
0.58
1
pound
sec2
F
A
42 pounds/rad/sec,
B
420 pounds/rad,
r2
0, 0.05, 0.1 second,
73
0.01 second for r 2
h
100 pounds,
0.05,
0.1F, 0.5F.
Time lags 7 2 and 7 3 were considered as representative
delays for this type system.
Autopilotand missile characteristics, as illustrated
by Figs. 7-9, result in thefollowing observations.
T o limit the amplitudeof the roll angle oscillation t o
3 . 5 O requires the missile t o oscillate at approximately
5.5 cps. Thus an autopilot lag
of 0.05 second and a
hysteresis/force ratio of 0.1 will satisfy this condition.
An increase in either time delay or hysteresis results
in a lowering of the natural frequency and a consequent
increase in the signal amplitude level,
a factor of importance for a system operating close to saturation.
The root locusplot,2 while providing the requisite
W. R. Evans, “Graphical analysis of control systems,” Trans.
AIEE, vol. 67, p. 85; January, 1948.
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servo controlling missile in roll.
stability information, proves to be somewhat more
difficulttoconstructas
a result of thisparticularnonlinearity. The actual procedure consisted
of obtaining
a family of loci for a series of describing function phase
angles,thenplotting
oneach“phase”locusthegain
corresponding tothe describingfunctionamplitude.
T o analyzethe effects of additionallagsdemands
a
complete new family of loci, whereastheamplitudephase presentation requires but a modified plot of the
system transfer function.
A computer(REAC)studyresultedinreasonable
agreementbetweentheanalysisandactualsystem
performance,thefrequencyandamplitudechecking
t o within 10 per cent thereby justifying the assumption
regarding the filteringof higher harmonics.
For example,inFigs.
7 and 8, a resonantpoint
exists a t
36 rad/sec when
0.05 sec and F = 1Oh
for the system under discussion.Fig. 9 illustrates the
results of a REAC study with agreement as to frequency
of oscillation; the theoretical amplitudes may be found
in Fig. 3.
CONCLUSIOS
1) Unbalancedtorquesshouldbeminimized
duce the control force.
t o re-
2) The frequencyof the roll oscillation must be maintained at a reasonably high value so as to restrict
the magnitude of the roll oscillation and to reduce
coupling with the roll control mode.
3 ) Hysteresisandtimelagsdueto
nonidealcomponents must be minimized within the autopilot
in order for the system
t o achieve the necessary
resonant frequency.
4) T h e describing function technique may be readily
employed as an analytic toolin the study of a
missile autopilot system having a relay servo.
Nouember
IRE TRANSACTIORTS ON A U T O M A T I C C O N T R O L
APPENDIX
DERIVATION
OF
THE
and
DESCRIBISG
FUNCTION
-1
Considering the waveform t o be a periodic odd function, with the period divided equally, the time function
in terms of a Fourier series is
f(t)
an sin n ( w t
x
4-412
B12
!a
p
a)
Cpon substituting for A1 and B1, we arrive at the
required relationship t h a t
where
2F
(1
cos nn)
From (9) we have that one term of f(t) is:
A,
B, sin m t
nut
where
An=
B,
a, sin m ~ .
ncy.
a,,
Consider initially the effects of harmonics that areof
a higher order than the fundamental as being
negligible;
i e . , n 1 only.
Substitutinginto (10) wefind t h a t
4F
a1
T
and
4F
A1
a
sin
4F
B1
a
However, from Fig.
5,
sin
k
GLOSSARY
Roman Symbols
A =roll rate gain (pounds/rad/sec).
AI, A , Fourier coefficients.
al, a, Fourier coefficients.
B =roll attitude gain (pounds/rad).
F , f=control force (pounds).
h =hysteresis (pounds).
I,, Iu,I,, I,, =moments and product of inertia(slug
feet2).
=jet control subscript, alsod-1.
L =roll moment (pound -feet).
N ( x ) servodescribing function.
n =integer of harmonic.
P roll rate (rad/sec).
Q pitch rate (rad/sec).
R =yaw rate (rad/sec)
=Laplace operator.
=real time (sec).
X=amplitude of servoinput signal, rollaxis
of missile.
Y = missile pitch axis.
missile yaw axis.
Greek Symbols
and
phase lag.
phase angle (deg).
T =system time delay (sec).
roll angle (rad).
=radian frequency (rad/sec)
=damping factor (nondimensional).
a
A1
4Fh
TX
B1
4F49
The describing function is
h2
ACKXOWLEDGMENT
The author wishes to acknowledge theencouragement and advice provided byA. M. Fuchs now of CDC
Control Services, Inc.
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