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=1
~w
0
4 DECEMBER 1979
BY
S. V. CHEREMNYKH
CFOUOy
_ES
i OF 3
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JPRS L/8794
4 December 1979
- Translation
Stabilization afSpace Vehicles
_ By
S.V. Gheremnykh
FBIS FOREIGN BROADCAST INFORMATION SERVICE
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JPRS L/8i94
4 December 1979
STABILIZATION OF SPACE VEHICLES
- MosCOw STABILIZIRUYEMOST' KOSMICHESKIK.H LETATEL'NYKH APPARATOV
in Russian 1978 signed to press 31 Oct 78 pp 1-208
Book by S. V. Cheremnykh, Mashinostroyeniye Publishing House,
1200 copies
CONTENTS
PAGE
Foreword
2
Chapter
l. Simplest IJonconservative Oscillatory Systems
6
1.1.
Examples from Mechanics. Autonomous Systems with Ttao
Degrees of Freedom
6
1.2.
Oscillations of Unstable Systems
lg
1.3.
Effect of Dissipative Forces on Stability
32
1.4.
Controllability of the Oscillatory System with Two
Degrees of Freedom
40
Chapter
2. Stabilizability of Oscillatory Systems with One Input
52
- 2.1.
Terminology: Stabilizability, Structural Properties,
Stability. Formulation of the Basic Problems
52
2.2.
Study of the Stabilizabality of Oscillatory Systems with
One Input
63
2.3.
Dynamic Instability as a Form of Nonstabilizability of
the Object of Control
81
- 2.4.
Investigation of the Dynamic Instability of an Object of
~
Control
83
Chapter
3. Stabilizability of Space Vehicles
102
3.1.
Mathematical Models of Space Vehicles (Movement in the
Active Segment
102
3.2.
Simplest Cases of Investigatian of the Structural
Stability of the Space Vehicle
113
' a ' [2 - USSR - A
FOUO]
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3.3.
Stabilizability for the Classical System (Two
124
3.4.
Fuel Compartments
Space Vehicle with Engine on an Elastic Suspension
136
- 3
5
Other Examples of Investigation of the Stabilizability
.
.
of a Space Vehicle
146
rhapter
4. Application o� the Theory of Stabilizability to the
Problems of Space Vehicle Design
155
4
1
Investigation of the Structural Properties of the
.
.
Designed Space Vehicles '
155
4.2.
Stabilizability Criterion as the Quality Criterion of
162
the Composxtional Layout of a Space Vehicle
4.3.
Stabilization of Dynamically Unstable Space Vehicles
(Calculation of the Damping Coefficients and the
-
Parameters of the Automatic Stabilization System)
170
4
4
Stabilization of Structurallq Unstable Space Vehicles
.
.
Using a Discrete Statilization A].gorithm
194
Chapter
5. Stabilizability and Stability of Sg3ce Vehicles in
216
the
!~ctive Segment of Flight
Approximate Method of Investigating the 5tability of
Space Vehicles. Amplitude and Phase Stabilization
216
Effect of the Fuel Mobility in the Compartments on the
21g
Stability of a Space Vehicle
'
Consideration of Elasticity of the Space Vehicle
223
Structure
Stability of the Space Vehicle with 9ngular Position
Control System (Nonlinear Case)
228
230
Bibliography
- b -
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PUBLICATION DATA
English title
: STABIZIZATION OF SPACE VEHICLES
Russian title
: STABII,IZIRUYEMOST' KOSMICHESKIKH
LETATEL'NYKH APPARATOV
AuLhor (s)
. S. V. Cheremnykh -
Ed:Ltor (s)
,
Publishing House
: Mashinostroyeniye
Place of Publication
: Moscow
~
Date of Publication
: 1978
Signed to press
. 31 Oct 78
Copies
: 1200
COPYRIGHT
. Izdatel'stvo "Mashinostroyeniye", -
1978
- c -
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UDC 629.78.017.2.001
.
STABILIZATION OF SPACE VEHICLES NEW PROBLEMS AND METHODS
Moscow STABILIZIROYEMOST' KOSMICHESKIKH LETATEL'NYKH ApPARATOV in Russian
1978 signed to press 31 Oct 78 pp 1-208
[Book by S. V. Cheremnykh, Mashinostroyeniye, 208 pages, 1200 copies]
[Text] A study is made of the problems of stabilizing spacecraft in the
- active segments of the flight from the point of view of some new methods of
motion control theory.
A new approach to investigating the dynamic characteristics of a space
vehicle as an object of control is discussed which is a development of the
controllability and observability theory of Kalman as applied to the given
class of obj ects.
ihe study is made of various problems in tre analysis of the spacecraft
dyanmics encountered in various planning and design stages.
The book is intended for engiueerii.g and techr.ical workers involved in designing rockets and other flight vehicles.
1 p
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- FOREWORD
In this paper a discussion is presented of the methods of investigating the
stabilizability of linear dynamic systems including an oscillating controlled
target and controls : of given structure as applied to the problems of
conCrolling a spacecraft with a liquid-propulsion rocket engine.
- It is known that liquid-propellant rockets are very difficult subjects for
stabilization because of the unfavorable dynamic characteristfcs arising
from the mobility of the fuel in the tanks, the elasticity of structure and
also the nonsteady-state nature of the characteristics of the vehicle and
the environment. Therefore in spite of using the latest methads of
_ synthesizing control systems for such objects, frequently the optimal quality
indexes which could be achieved are not achieved.
At the same time, in the design phase of the space vehicle as an object of
control in practice there are always unused possibilities for selecting the
structure and elements of the composite system basically determining its
dynamic characteristics in the process of controlled movement.
The problem of how these possibi].ities can be used is the starting point for
the studies, the results of which are discussed in this paper.
Of course, it would be desirable to solve the problem of optimizing the
~ dynamic characteristics of a;pace vehicle in the most general form, consider-
- ing the cl-osed target-controller system as a whole, also taking into account
ths ballistic, strength and other requireinents. For many reasons, primarily
as a result of the "crrs,~ of size," as R. Bellman puts it, this is in
- practice impossible an4 against oux wills it is necessary to limit ourselves
to more modest goals. Let us note that in the liquid-propulsion rocket design developments the
situation is typical where the structure of the controls is rigidly given
for one reason or another. For example, a spacecraft for a different pur-
pose designed on the basis of some basic version can have the same auto-
matic stabilization system with respect to structure with, perhaps, only
the values of the paraffieters altered.
2
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In such situati.ons, when the structure of the controls is defined, it is
natural to ask the following question: how are the structural parameter.s
of the object of control selected so that it will have the best character-
istics with respect to the controls of the given structure? To solve tliis
problem i*_ is desirable to have an optimalneas criterion which does not
_ depend on the specific parametere of the controls determined only by thc
str,.:cture of the controls and the characteristic parameters of the ohject.
Farmally, it is possible to construct such a criterion only after provirig
the theoretical possibility of separating the regions of stability and space
of the parameters of the object and the centrols (under assumptions that
are reasonable for the investigated class of systems).
So far as the author knows, this was done for the first time in the papers
by B. I. Rabinovich [57], I. M. Sidorov, I. P. Korotayeva [67] for controlled
oscillatory systems with a controller which slightly disturbs the natural
frequencies of the auxiliary oscillators.
There were other prerequisites for studying the problem of optimizing the
dynamic properties of the object of control from general points of view.
The fact is that in general control theory as a result of the work of
R. Kalman and other researchers, there is a tendency at the present time to
separate the investigation of the problems pertaining to the controlled
system itself as the object of control into a separate region (the problems
of controllability, observability [27], invariance [64], directivity [51],
and so on).
_ Although the structure of the control system itself is completely ignored
- here, the corresponding criteria provide valuable information about the
behavior of the object of control in the control process.
- The methods discussed in this paper occupy an intermediate position between
the corresponding methods of controllability (observability) theory and the
classical theory of stability. The structure of the control system here is
significant in contrast to the Kalman theory; at the same time the specific
' values of the parameters of the control system do not enter into various
criterial relations, and the results of the investigations are formulated
in terms of the regions in space of the parameters of the object of control
itself.
In order to emphasize this fact, and also considering that the concept of
stability is too overworked, in this book, following the lead of reference
[56], we shall call the criteria characterizing the object of control the -
stabilizability criteria.
The idea of the proposed approach consists in the following. Some formal
analog a quadratic form with coefficients which depend on the parameters
of the object of control is placed in correspondence to this object. The
positive (or negative) definability of this form is identified with the
concept of perfection of the object (for example, the space vehicle) in the
dynamic sense.
3
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Testing the Sylvester conditiona far the mentioned quadratic form and
the construction of the corresponding regions in the space of the parameters
of the object of control make up the content of the methods of investigating
the stabilizability of controlled oscillatory systems.
If the terminology adopted in space vehicle dynamics is used, the discussed
= theory is the theory of "phase" stabilization of oscillatory systems. In
pracrlice when "phase" stabilization of the space vehicle is impossible,
usually the problems of "amplitude" stabilization are investigated which
are essentially the classical problems of the analysis of the stability of
moving objects.
The methods of investigating stabiZizability are to some degree analogous
to the above-mentioned methods of investigating controllability, obaerva-
bility and so on in general control theory: both permit the general analysis
of the properties of the object of control as the first step in solving the
classical problems of stability of motion or various problems of optimal
control.
Chapter 2 of this book contains a discussion of stabilizabilit; theory.
Chapter 1 is an auxiliary chapter. The simplest model including two
connected oscillators is used to investigate some of the problems
characteristic of the modern theory of linear controlled systems (the prob-
- lems of dynamic instability, controllability, observability, modal control
in various situations, and so on).
Chapters 3 and 4 are of an applied nature. In these chapters a study is
made of the problems of the stabilizability of various models of space
vehicles and also adjacent problems pertaining to the design of optimal
(in the dynamic sense) obj ects.
The mathematical models of :space vehicles are used to the degree of complete-
ness which corresponds to the level of the initial design phases of objects
of this type: as a rule the equat'-ons are assumed to be linear, the
coefficients are considered constant ("frozen" for some characteristic point
in time T of the activ,. segment).
The oscillatory natura c` the object of control in the given case comes
from the presence of moving fuel components used to operate the sustainer
- engines and also elasticity of the hull and other structural elements.
Automatic stabilization systems (in the transverse oscillation mode) are
used as the control systems here, and in the case of longitudinal oscilla-
- tions, the engine is used directly. The perfoxmed studies of specific
composite systems of space vehicles and the standard conditions of space
' vehicle movement provide a basis for considering that the discussed methods
are a quite effective tool for investigating the dynamic properties of
flight vehicles with liquid-propulsion rocket engines under the conditions
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of incompleteness of a priori information about the stabilization system.
It is appropriate to emphasize here that the effectiveness of the methods
of inveatigating the stabilizability is especially noticeable when they are
used in combination with the classical methods of analysis, preceding tttem
in the general process of investigating the stability of the investigntc:d
class of objects. Therefore, a discussion is presented below (see Chapter 5)
of a number of the traditional methods of analyzing the stability of the
_ closed system made up of the space vehicle and its control system, and t11e
problems of amplitude stabilization are investigated. The author gives
special attention here to the interpretation of the regions of stabilizabil-
ity of space vehicles when investigating the stability of the control pro-
cesses in the active segment.
On the whole, in this book the author would like to attract the attention
of the readers to the new possibilities which are offered by successive.
(physical) analysis of such characteristics as controllability, observabil-
ity, stabilizability, and so on as applied to dynam,ic systems of a large
number of oscillatory degrees of freedom and with limited possiUilities of
modal control.
If we are talking about the general proUlem of stability, which in no way
replaces the classical methods, this approach helps us to find the primary
causes of instability and either to eliminate them or determine the direction
of further research.
- In conclusion, the author expresses his deep appreciation to doctor of
_ technical sciences, Prof B. I. Rabinovich for valuable suggestions made
when reviewing the manuscript of the book and also engineer Yu. V. Shchetinin
for his assistance in preparing the manuscript for publication.
It is requested that all critical comments and suggestions be sent to the
- following address: Moscow, GSP-6, 1-y Basmannyy per., d.3, izd-vo
"Mashinostroyeniye."
5
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CHAPTER 1. SIMPLEST NONCONSERVATIVE OSCILLATORY SYSTEMS
1.1. Examples from Mechanics. Autonomous Systems with Two Degrees of
- Freedom
Aircraft Wing Flutter
Flutter is the classical example of a phenomenon which cannot be explained
by investigating the system with only one degree of freedom. This form of
instability is characterized by intensive interaction of at Ieast two
oscillatory elements under the effect of external nonpotential forces.
The names given to the var.ious forms of flutter illustrate this fact:
Bending-aileron wing flutter (bpnding vibrations of the wing combined with
aileron vibrations);
Bending-rudder flutter of the horcizontal empennage (bending vibrmtions of
the fuselage in the vertical plane jointly with vibrations of the elevator
around the axis of suspension);
Torsional-rudder flutter of the horizontal empennage (torsional vibrations
of the fuselage combined with vibrations of the elevator and tail aesembly),
and 30 on.
This is how this phenomEnon uppears to observers from the outside [30]:
"While testing an expe-imental aircraft, a twin-engine monoplane, the wings
_ began to vibrate unexpt,tedly. This occurred while the aircraft was flying
a measured base line At -aaximum speed near the ground. The wing vibrations
began abruptly and were of an antisymmetric nature, that is, if the right
half of the wing went up, the left half went down at the same time. Power-
- ful vibrations of the ailerons occurred at the same time, so that the
controls were jerked out of the pilot's hands. In the given case the
pilot made the right decision: he throttled down and in spite of the fact
tbat the controls had been jerked out o� his hands, he suceeded in signif-
- icantly taking up the elevator. As a result, the spee3 b.egan to drop
sharply, and when it had decreased by about 20%, the vibration4 stopped
almost as abruptly as they had begun. Five to eight seconds passed between
the beginning and end of the uibrations. After the vibrations stopped,
- the aircraft behaved normally, and af ter 5 minutes of flight the pilot
- made a good landing at ':he airport.
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c,(yt Zslt
ZBl 6 ~
y
al I---~ ( I
I J cz
. m~i B~2 B~Z -
_ Figure 1.1. Force diagram on an inclined plate
"Examination of the aircraft revealed that at soffie points in the structure
of the center wing section rupture of the skin and cowlings had started
- (cracks had formed), and some of the rivets holding the skin to the longerons,
- stringers and ribs had sheared off. Residual 3eformations in the form of
waves in the skin had appeared on the surface of the center wing section
between the engines and the fuselage. Clay was detected in the suspension
of the ailerons and the aileron servotabs.
"However, flutter does not always end so favoYably. Cases are known where
vibrations that began in flight have led to complete diaintegration of the
_ aircraft in one or two seconds or less. It appears to observers of such
an accident from the ground gs if the part of the aircraft where the
_ vibrations started has exploded."
Thus, observations indicate that under def ined flight conditions vibrations
o� the fuselage and control surfaces which are extraordinarily intense can
occur under def ined flight conditions.
The complete theory of flutter is highly complex..[9, 25, 76]. Here, only =
a suitable mechanical model [50] which can be used to explain the primary
- aspects of the nature of this phenomenon which once was a threatening
obstacle on the path of increasing the speed of aircraft, will be
investigated. _
- Let the plate depicted in Fig 1.1 have two degrees of freedom. We shall
characterize its position by two coordinates the angle of rotation ~
and vertical displacement y of the center of the plate.
Horizontal displacements will be considered imposaible.
Zet us write the equations def ining � and y as a�unction of time:
_ �=0(t); Y=Y (t)� _
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The tensions of the two springs cl and c2 will be assumed to be different,
r and the mass of the plate, uniformly distributed over its entire surface.
Let us denote by m the mass corresponding to a unit area of tlie central
plane of the plate.
I'uring its movement, the plate is acted on hy the list
Y = da Q bltP,
running at a distance a from the left edge of the plate and a7.so the
force of reaction of the elastic supports proportional to the displacements
of the long sides of the plate:
2 T )C11; Rs = - (y - 2- CP) c11.
Reducing these reactions to the center of gravity of the platc:, we obtain
the force
R=Rl-f-R2=-(CI-4-C2)ly- 2 (c1-cs) t'f _
and the moment _
M = (y + 2 cpl c,/. l (y - i ~pl c2l ? ,
~ J
- tlow let us select the equations of motion of the plate. One of them
describes the center of the gravity of the plate: . Y-{-R=mbl dd 2
where mbl is the mass of the erizlre plate and the other, the rotations of
the plate around the horizontal axis z passing through the center of mass:
b _al+. M=mb''1 d2~ , (1.2)
2 f 2 dt2
Substituting the expre6:.ions for Y, R and M in equations (1.1) and (1.2),
we obtain the following system of differential equations:
d~ ~aijy+aiz'?=0; (1.3)
d2T . 72 ,ra2ly--a22'?=ol
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where
11 C2 3(cl-f- c2) dc4 v'' b- 2a
~ti = mb ~ a21- mb ^ da e 2 M2 '
~ CC, _dcy va 1 6 (ct-c�t)
t2 - 21n d~t ~ 2 m'~21= rnb2 ' (1.4~
When solving the problems of aeroelasticity, to which the discussed topic
belongs, first the conditions are defined to which the static form of the
stability loas, the so-called divergence, is poasibl.e.
For this purpose it is propoaed that y and 0 are constants. Then the
second derivatives vanish and the equations assume the form
aiiy-i-a12?=0, a21Jd-a22?=0. (1.5)
The condition of nonzero solutions of system (1.5) has the form
ajjaza - ~j,_UZ1= 0.
Substituting the expressions (1.4) for the coefficients, the following
formula is obtained, which defines the critical divergence rate:
~ Ci
v~=2 Q dcy 1- cl .
da l cZ )
Key: 1. critical
~ y 9
~
t -~`'t t
(1.6)
-y- ~
i-~C_======7C_~ _-s-
-s ~
--Jiiii T -s-. i 1J771
d ''b .F
Figure 1.2. Types of movements of the plate after in3.tia1
deflection:
the axis of rigidity is denoted by the x, the center of mass
of the plate is in the middle of the span.
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FOR OFFICIAL USE ONLY Divergence as a static form of loss of stability is possible obviously when
- cl 0; . . . . (2.39)
Substituting the values of u=0, u=-v12;...; u=-o 2 in the function gl(u),
successively, we find that the inequalities [2A), and tagether with them,
the conditions of stability of the given system (2.34) are satisf ied if:
a) . Xo >O; xi >0 (2.40)
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_ (the requirements on the parameters of the control system);
b) bspl; >O; 1=1, 2,..., rn (2.41)
~ (the requirements on the parameters of the object of control).
As is obvious, the situation is such that the stability of the given system
(2.28)-(2.29) can be insured as a result of the successive performance of
two operations:
Ad3ustment of the cGntrol system parameters [in order that the conditions
(2.40) be satisfied];
Selection of the parameters of the ob3ect of contirol in accordance with
the inequalities (2.41) (having the senae of conditions of structural
stability of the object).
Let us introduce the space of the parametera of the object of control af
dimensionality 2m( 2m (b,l, bem, ni', �..,'~m J+ , in which the conditions of
uncontrollabi.lity or observability of the syatem
bQJn;=O (j=1, 2,..., m) (2.42).
are isolated by certain boundaries.
The conditions of structural stability (2.41) of the.object of control
provide the decoding of the regions on both sides of the boundary (2.42),
which permits investigation of them 4s a generalization of the Kalman
conditions as applied to the investigated apecial problem.
The establishment of the conditiona of the type of (2.41) in the general
case of oscillatory systema including n oscillatora and conatituting the
object of comtrol with one input is the basic problem of further analysis.
Here the central event is the theoretical possibility of the separation _
of investigation of the object of control from analysis of the closed
system as a whule within the framework of reasonable assumptiona, which in _
the given case leads to the necessity for introducing the concept of
controllability and observability of the system (problem 1), structural
stability (problem 2), and then the concept of stabilizability of the
object of control.
_ Formalization of the Object of Control and the Control System. Statement
of the Problem `
Let us consider the system of differential equations
x =eBx-{-Ax+bu, (2.43)
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where x is the vector of the generalized coordinates of the system:
t is the control vector; u(t) is the control input; A is the matrix of
dimensionality nXn, the elements of which depend, possibly, oii r parameters
v1,v2,...9vr; e is a small parameter. -
Let uH make the following assumptione:
I. The system (2.43) is a set of connected oscillators characterized by
- the frequency vi (i=1, 2,..., n), and it can be (for u(t)=0) both stable
and unstable as a result of the effect of positional nonconservative
forces.
- H. The elements of the matrix B of generalized dissipative .4orces are -
small, which is characterized by the introduction of the small parameter-E.
III. The variability of the coefficients of the system (2.43) is small in
the characteristic time interval T-2ff/a. -
_ Then let us propose that the measuring device of the control system receives
the signal , v(E)=(g, x), (2.44)
~
which is a physically observable value; the vector g=(gl,...,gn) is the
observation vector for the investigated system.
The equation of the control system will be assumed in the following form:
L (P) u= Li (P) V, (L). -
where L(p)=L1(p)/L0(p) is the transfer function of the control system
given by its frequency characteristic _
L(tw)=A(w)(COSy(w)-}-isinp(u,)],
With respect to the control ~ystam we shall assume the following:
_ 1. The eigenvalues of the operator Lp(p) belongs to the region of
stability QZ which dceG not intersect with the region of eigenvalues QA
in the matrix A for ail the variations of the parameters vl, v2,...,vr in
the given region.
_ 2. The disturbances Apj (j=1,2,...,n) o� the eigenvalues (pi 2) of the
- matrix A caused by the effect of the control system are small in the
- sense that (Apjl�Q, where Q is the characteristic frequency.
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Figure 2.1. Roots of the characteristic equation of an
open system: �
o-- object of control; x-- control system
3. The condition sign [Im L(iwj]=const is satisfied, where wj=lm pj;
pj are the eigenvalues of the matrix A.
The condition 1 means that the control system as an element of the cloaed
automatic control circuit is asymptotically stable, and it retains this
property under all conditiona of movement of the investigated system.
_ Fig 2.1 shows the roots of the c.haracteristic equation of this system (2.43)
including three connected oscillators characterized by the frequencies -
al, 02, Qg (the symbols Q) (u(t)=0, e=0). On the same figure the multipli-
cation symbol denotes the eigenvaluea of the operator Lp(p) satisfying
the condition l.
The condition "2" obviously imposes restrictions on the amplification
- coefficient of the control systems and means tha.t the eigenvalues of the
- closed system made up of the object of control and the control system are
closed to their rated values (Fig 2.1) calculated in the open state of the
system (u (t ) =0) .
Condition "3" is the condition of "uniformity" of the phase shifts under
the effect of the operator L for all of the eigenvalues pi of the object
of control. �
Fig 2.2 shows the amplitude an3 phase characteristics of the control -
system satisfying condition 3. In the given case, as is obvious:
sign (Im L (iwj)] - -~-1 ( j =1, 2, 3). '
Let us investigate the problem of the so-called phase [56] stabilization of ~
_ the object of control (D), and accordingly let us consider the following
problems.
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Problem 2.1. Let the dynamic system
x Ax+ bu, v=(g, x), (D)
Lo(P)u = Li(P)v, (L)
be given, where the object of control (L) is def ined by the properties
I-III, the control system, by the properties 1-3.
It is required that the regions in which the closed system (D)-(L) will
be stable for any control system satisfying the conditions 1-3 be
isolated in the space of the parameters of the object (D).
Problem 2.2. Let the dynamic system be given:
x-Ax+bu' ----v_(g'.x) (D)
Lo(P)u = Li(P)v, (L)
where the object of control is defined by the properties I-III, the control
system, by the properties_1-2.
~ r-----~ -
~
2
0
x'
1
Figure 2.2. Standard phase-amplitude characteristics of the
control Gystem
What should the reqtiicements be on the regulator as alternative (3) in
order to insure stabil.Lty of the system (D)-(L) in the given region of
variation of the par4^_!eters of the object (D)?
Let
iDo(P2)=aoPs"`+ al.p2(m-1)+...-}-u2(M-1)P# -}-u�tm (2.45)
be the characteristic equation of the open system [for _
uy) = 0, V(t) = pJ; pj(I � 1, 2,..., n) , be its roots.
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_ Then the case
PJ= -wi (I =1, 2,. . n) (2,46)
-
corresponda to the dynamically stable object (D), the case
pi,s=a � lcu; P3,4= (2.47)
corresponda to its dynamic instability.
Then let us denote
l(P2E-A)Lo(P)-(bg)Li(P)I=0o(P2)Lo(A)+`Dk(P2)Li(P)�0 (2.48)
as the characteristic equation of the closed system (D)-(L);
wt
,p)=V/U the transmission function of the control object;
~
(D); v(p), u(p) the Laplacian transform of the general;ized coordinates
- v(t), u(t),
uk(k=1, 2,..., n-1) zeroes of the transmission function W(p);
Xk(k=1. 2,..., n) the ; onest of the transmission function W(p) ;
,
K=($, Ab*-, An'1b) the controllability matrix of the system (D); -
G=(J', A'l (A')1111) the obaervability matrix of the system (D).
Simultaneously with the system (D)-(L) we shall also consider the system
Y=AY-{-pu, v=h'7', u=L(P)v, . (2.49) Q3
(where A=diag {al, X2,..�, describing the movement of the dynamic
system in the base made up opthe eigenvectors of the matrix A.
Let us introduce the followfng definition.
Definition 2.1. The object of control (D) will be called stabilized, if:
a) The ones ak of the transmission function W(p2) of the ob3ect are _
prime,, real and negative;
b) The zeroa Ak and the ones uk of.the txansmissian function W(p2)
~ are permutated in the fdllowing ordert
~l< N < Xz 1 or c>0, �0, i=1,2) or lag (sin �i I~t ttl
Also assuming that the frequencies wi are numbered tn increasing order, -
let us calculate the values of the pt5lynomial Ok(w2) 'at the points w32 _
(J=1,2,...,m): m
(Dp(i"i) =aaja1j/(1,~ n (-u>i--o,2),
1tl
For structuraZ stabiliry of the {-nvestigated system it is necessary and
sufticient that the signs of k(w'Z) alternate on going from number j to
number j+l:
- (Dk (W1) > di 'Ok (0)2) < 0; ok (.)3) > 0; .
pT ok (Wl) < Q+ e,~k (0)2~ > 0; ODk (W3) < 0.
4) (~,j)-aQ u~; (1=1, 2,..., m),
Let us denote k jm
2 2 2
where 4I=W1 n (-�~I+01~~,
/..i
Since Di>0; A20; A4 0 (1=1~ 2,..., N).
The characteristic equation of the system (3.44) will be represented in the
form (3.32). Setting p=iw, we find:
(Do (wZ)=
-w2
-a9s,w2
-a9s2(j)2
-aasNCU2
-(L,,Aw2
--w2-{-wi
0
0
-QS,gtu2
n -
- w2-~-wn
0.
.
. . . .
-(L1Npro2
. . . .
0
. . . .
0
. . . . .
...-W2--roN
-ae, -aesico2 -aes,(o2 , . . -llgsNtu2
' 0 -w2-}-wi 0 0
Ok0) z)= 0 0 -w2 -}-w~... ~
. . . . . . . . . . . . . . . .
0 0 0 -w2-~'~"~r
i
On the basis of the definition, the system (3.44) is structurally stable if -
the zeros and ones of the transmission.function W(pz) alternate.
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Obviously, this condition will be satisfied if the signs of the function
(po(w2) at the points w2=Wi2 (i=1,2,...,N) alternate.
We have
= - = - 2) (3.46)
(D0(1�?~ =a6s,as,,iio~~u~i-u11~...(to)~._I-~'~i~~O1r,1.1-ui~~...~u~, - y-~~~i .
On the basis of conservativeness of the system (3.44) aesiasi8>0'
Assuming that w1262); 0=001=v2); 0=1(Q10 (a body ot relatively large elongation with equal partial
oscillation frequenciea of *_he liquidint~oximated e b) the ellipseseg of _
the regions (3.68) are quit~ pre..ise1y aPP y the
- (k-~- ;h j~ky)7i-}-(l �,k)ZiZ2-}-(1-C1' 0)Z2+
~ /~.1..
-}-(21/k--CI, k),Z,-{-(--l l , C)Zs+(1+k~k-
z+
(k-Gk6~kY)Zi-~r,Yyk(-l--h)Z,Z:~~ (1--CYT) Z0; (3.69)
+(-21/-k-I-Ch)Zj-I'(2z2 -}'(1+ k)=0 (3.70) -
- with centers at the points
~ r 1 !
k - -7 ; r.; y 2 �
' - -f-
G; y-
-
The axes of the ellipses are rotared with respect to Z1 axis by the angle
- 1 ;Y(t -k') , iil, 2.
, 61= 2 arctg :V(~-t-k):F }"k(1-k)
Let us investigate the problem of the mutual arrangement of the regions of
~ structural and natural dynamic instability of the space vehicle. Let us
consider the simplest case of c-0; R=0 when the boundaries of the regions
of structural and natural instability are defined by the equations
~ .(Zi kZsi IZ,-- kZ2-- C (1-- k) 11 - Y (1-' k)J j _O+ -
[Zi-- kZz - f-( l +k)+C (Zl kZz))Z - 4k (ZZ - Zl?X
x [ 1+ GY (Zi + kZz)] =0.
i-
d e
Figure 3.16. Mutual arrangement of the regions of'dynamic
double croashatching) and structural instability
oi the space vehicle with two fuel tanks:
a, b, c t r_ameter c>0; d, e, f-- parameter c=0
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From the general theory discussed in Chapter 2 it follows that the regions
of natural instability are located inside the regions of structural
instability.
ZZ
~
'
-
~

Z
~

~
17.
i `r ~
_
.

f
/3-0 ~
o> !

~ ~
I
75
j I
I I
'
-Z J
�I ~
Q b
Figure 3.17. Nature of the effect of various oscillation frequencies
of the fuel in thz tank on the curvilinear boundaries
of the regions of structural instability of a
space vehicle (0=-1+Q22/a12);
a parameter SfO; b-- parameter 0;0
Zz
-3
_ 1 p
7~�lo
1
O, ,C J
~ 2 Z~
aQ~'
~
-2
Figure 3.18. MutLal arrangement of the regions of natural and
structural stab:ility of a space vehicle for c