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An analysis of Lur'e stability, including the derivation of the Lime-optimal regulator law and coordinate transformations in vector-matrix and vector-scalar forms. It covers topics such as eigenvectors, Lur'e coordinates, and the inverse of the system matrix.
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Z
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N66-
(ACCEB_IONc__NUMB R)
|NABA CR OR TMX OR AD NUMBER)
(C_DE) ICATEGORY)
NASA (R-
GPO PRICE $ CFSTI PRICE(S) $
Hard copy (HC) Microfiche (MF) ff 653 Jul'f 65
MINIMAX ATTITUDE CONTROL OF
AEROBALLISTIC LAUNCH VEHICLES
4#• !
For sale by the Clearinghouse for Federal Scientific and Technical Information Springfield, Virginia 22151 - Price $6.
INTRODUCTION
An engineer confronted with the problem of de signing an autopilot system for a rocket vehicle whose ratio of length to diameter is moder- ate to high is faced with a host of new problems. These problems arise principally from the fact that rocket structures are usually highly flex- ible because of the requirement for a low ratio of structural weight to total vehicle weight (usually only 10 percent of the vehicle weight is made up of structure that can resist elastic deformation of the vehicle under the various forces to which it is exposed in flight). For such sys- tems, the control engineer can no longer concern himself only with the pitch, yaw, roll, and translation of the vehicle to obtain the requirements that his control system must satisfy, but rather he must broaden the scope of his analysis to include problems associated with the elastic structure, both to discover the control system requirements and to assure the compatibility of the control system with the dynamic charac- teristics of the elastic structure. Specifically, if he does not consider the effects of the elastic structure on the control system requirements, he may well design a system that will be unstable when actually installed in the vehicle; if he does not consider the compatibility of the control system with the dynamic characteristics of the elastic structure, he may design a stable control system that will cause structural failure because of dynamic elastic deformations arising from this operation. The above problems are not academic, but very real, and occur almost universally in rocket vehicle design. The basic reason for this is that the highly elastic heavy structure will usually have an elastic response mode whose frequency falls well within the control frequency band and cannot be neglected, in contrast to conventional aircraft design where the control band can usually be chosen well below the important elastic response frequency bands. One mathematical formulation of a problem within the framework outlined above was stated in the original procurement request, and is
in essence repeated here, paraphrased to conform to the particular problem considered. The steady-state missile dynamics are represented by a homo- geneous system of differential equations.
x=Ax
where x is an n-vector and A an n×n matrix, together with an initial state x(0) = x °. The system expresses various forces and torques due to structural and aerodynamic effects. The time interval 0 -_ t -T for the problem is assumed to be sufficiently short that A is considered constant. A scalar disturbance y is introduced to represent wind effects in the missile where, analytically, y y(t) is restricted to a class of functions F. In order to maintain stability under a disturbance as des- cribed above, control elements are introduced into the system as a scalar _ in some class of functions _. The system is now rewritten to include y and d# as
= Ax + a@+by , x(O) = x °
where a and b are constant n-vectors. Let L. I be a given set of constants
such that Ixil < L i, i = 1,2,'''n assures that the system remains stable. The basic control problem may now" be posed as an optimum control one; that is, a control must be found that will ensure x i = I I for all functions y in the class where L_:"is the I least such bound for each i. Since the control law which gives L"J 1 _ may be different for each i, it maybe necessary to specify constraints on Z..S- For example, if ;,,. 1 L = cL where 0 < c < i, the control law that minimizes c could be I i ;,,_ determined. Or the control law that minimizes L. (^) 3 only while satisfying I xilp < L i, i = l'''n could be found. The particular problem formulation for the present investigation is to find _ that realizes
rainmax llxll
A recapitulation of the results of this investigation_along with suggestions for further study, is given in Section 7. In addition,several appendices are included for background material, for detailed deriva- tions, or because they were published as papers based on the material
generated under this contract.
4
In the analysis or synthesis of any physical system, one of the first and most important steps is the selection of an appropriate mathematical model. For initial synthesis and feasibility analysis, a rather gross approximation to the actual dynamics may suffice; for final analysis and simulation, a more faithful description is usually necessary. When the physical object is as complex as the non-rigid aeroballistic vehicle treated herein, the problem of choosing the model is especially difficult, not necessarily from the point of view" of the dynamic description, but rather from that of determining how" much fidelity is required to achieve a sufficiently accurate assessment of the behavior.
The model derived in this section is intended to be complete enough for an accurate determination of the dynamics of the vehicle, and yet of a low enough order to allow" computer simulation. Included in this model are:
i) Aerodynamic forces (considered to be located at the vehicle center of pressure). Z) Inertia reaction torques on vehicle motion due to nozzle dynamics ("tail wags dog" effect).
i) A distributed aerodynamic force on vehicle motion.
i-I
LOCAL VERTICAL IANGENT TO REFERENCE TRAJECTORY I X BOOSTER^ LOCKEDUNDEFLECTED^ CENTER^ LINE
CENTER (^) NOZZLE LINE OFDEFLECTED RIGID BOOSTER
C.G. OF TOTAL
BOOSTER CENTER LINI
GIMBAL POINT
NOTE: YAXIS FORM (^) AIS RIGHT OUT (^) OFHAND THE COORDINATE.X, Z PLANE TOSYSTEM
NoNz%L._IMBAL POINT_ /C,G. OF TOTAL BOOSTER
___ ,n I ./0 ___ pX..l _AERODYNAMIC CENTERoF PRESSURE
Figure t-1. Coordinate system for analysis of the flexible booster,
a( )
a
&w
qi 1 g N D T T (^) c V M I
M n In m. (^) i x. 1 _Si
_Bi COBi
Table l-l. Nomenclature.
i-
Normal Acceleration
The vehicle acceleration in the z direction is given by:
a (^) Z = -V_ - a (^) X _ + V_ + V_ "vV^ (1-4)
The remaining equations require some explanation of the repre- sentation used for the bending and sloshing dynamics. Both effects re- sult from the motion of continuous media, and the fundamental dynamical equations. Application of the technique of separation of variables to these Dartia] differential _q11_tin_s re_u!ts in an infinite sequence of sets of total differential equations for each. Each set in the sequence consists of differential equations in spatial coordinates and a differential equation in time; the solution to such a set is called a mode. For a detailed dis- cussion, see the references. The resulting time equations are:
Body Bending Equation The bending equations are found by applying Hamilton's principle. This yields for, the k th bending mode,
M _ik + Z_Bk WBk _ik + (_Bk) qk - M (ax + g cos 0) kkg v
TCkg g -^ (EI•^ F^ i^ kkx.^ I )^ (^ +^ _)^ +^ >2'j Zi^ F.^ I k^ kx. 1 k^ Jxi.^ qj
X m_ [Okx _ - (kkx f + kkg)(ax + g cos O) _]
(i -5)
= - (Mn n kg +I^ n k^ kg )'^ -^ Tc C kg
Fluid Sloshing Equations The sloshing equations are obtained by the same method as the bending equations. This method yields for, the k th sloshing mode,
2 _k + Z_SkC0Sk_k + (c0Sk) _k + (ax + gcos 0) w +XkW + ;$ - az
V@- xk ¢ + Ei (ax + g cos O) k.IXk qi - _ixk'qi : 0
The spatial equations may be solved to determine the positions and slopes necessary to solve (I-5) and (I-6); or, more likely, they may be found expe rimentally. Figure i-i plus the quantities entering into (i-5) and (i-6) lead to the following geometrical equations:
Center Line Deflection The deflection of the vehicle center line (displacement of c.g.) due to nozzle deflection and sloshing fluid is given by
Mv = M _ 13- Em. ( 1 _7) n n j J -O
Center Line Rotation The rotation of the vehicle center line due to nozzle deflection and sloshing fluid is
Iw = - (In +M (^) n _ (^) n _ (^) g ) _ - Ej m (^) J x. (^) ] _j +Z (^) i Ej k. (^) jx.l F. 1 qj^ (i-8)
Engine Gimbal Point Deflection The deflection of the engine gimbal point from the undeflected center line of vehicle is given by
Ug = v - f g w + E. 1 ¢ig qi
Engine Gimbal Point Slope
The slope at the engine gimbal point is
(i-9)
qag w - E. 1 k ig qi^ (I -10)
i-
Bending Deflection Equation (kth mode):
_z° + + z(Y[li{i)+ {k +_zz_k + z (_i73+i " qi) Yz4 qk
½ j •
Sloshing (k th mode):
]"79- _+_48 _ + _^ • + ri (Y30i^ qi ) +.-^ Ei (_31i^ qi ) + _'j (_J3 Z [j)
+_ I_{3%1+ [k+ _k+ _ : _ _ +'
Sensor Equations
In addition to the equations of motion, it is necessary to introduce equations that describe the output of any sensors used. Sensors mounted on the vehicle sense the rigid body motion, the motion of the vehicle arising from the bending modes, the sloshing modes and the engine dynamics. The following three equations show the total inputs to the various sensors. It should be noted that these equations apply only to "perfect" sensors. No sensor dynamics have been included. The addition of sensor dynamics requires the cascading of the sensor dynamics with the output of the 'Iperfect" sensor defined by Equations 1-17, 1-18, and 1-19.
-Angular Displacement
The angular displacement sensed by an instrument located at sta- tion P along the missile's longitudinal axis is given by
_sensor = _ + 2(Y39qi)^ i^ + >2,(y O_j){ + 41 (I-17)
Rate
The input to an angular rate sensing instrument located at station P along the missile's longitudinal axis is given by
i 1 o L _sensor (^) = + z %9qi + + y4,P i 3
Normal Acceleration The input to an accelerometer which senses acceleration normal to the longitudinal axis of missile located at station A is given by
n A = 4Z + "{43 _ - _43 _ + /44 +
i Z (Y45 _ii) i (1-19)
The equations governing the sensor dynamics depend upon the mechanization of the sensor; the following are typical sensor dynamics:
Angular Displacement If it is assumed that the angular orientation is measured by a position gyro, the sensor dyna:nics can be taken as a pure gain. The attitude reference is a "free" gyro (really a three-axes gimballed gyro), and the orientation of the body relative to the gyro is read out by means of synchro pickoffs. Any dynamics associated with the motion of the gyro would arise as a result of manufacturing inperfections, e.g., bearing torques and mass unbalance. The synchro is essentially a vari- able transformer; the dynamics associated with the synchro signal arise in the signal processing circuitry and are quite high in frequency. Synchro pickoffs, or analogous linear devices can be used to measure engine gimbal angles. I-I
_I
(In + _ (^) n _ Mn) n Mn
-N (^) Ol Vll M
i = I^ n +_^ n _^ g Mn "i12 IMn _n (Ek Fkkix k) - _ig
I k.
v._ = c /E F, k. \ + (T c^ % ir D)^ k. ±v± ig
7J15 - (T c (^) MI N) m.j x.j
M (^) n n M
B (^) n YI7 M (^) n n
K (^) n +K & M (^) n n
(T c - No_) M (In^ +^ _^ n _^ g M n)
K (^) a YI9 M (^) n n
_ l Y20 M[_ (F_ kkx_)]
Table I-2. Coefficient definitions. (continued)
1-
I_ (F, k kx,) l
"f22 = 2 _Bk°_Bk
Y23^ i^ _^ T%kgM k^ ig +^ M!.^ (F£k
"24 = (°Bk)
mx[i
____ - m. MI _ F_ kkx _ - --_J _kx. J
m. (^) J x.J
Y27 =^ --_n^ k^ kg nM^ n^ qbkg
( -In ÷ M n (^) n _ (^) g) MI
T ( c+T M, "{28 _ M c _kg + M nM n k^ kg In+M _ _ 1 T n n g _kg
x k _29 - V
i 1 k Y30 - V Xk IV F_ ix_
Table i-2. Coefficient definitions. (continued)
