A closed form analytical expression for the minimum attainable attitude error (as well as the error rate) in a flexible beam by feedback control using co-located rate sensors is announced. For simplicity, researchers consider a beam clamped at one end with an offset mass (antenna) at the other end where the controls and sensors are located. Both control moment generators and force actuators are provided. The results apply to any beam-like lattice-type truss, and provide the kind of performance criteria needed under CSI - Controls-Stuctures-Integrated optimization

Balakrishnan, A. V.

English

NASA Technical Reports Server

N90-23038
MINIMUM ATTAINABLE RMS ATTITUDE ERROR
USING CO-LOCATED RATE SENSORS
A. V. Balakrishnant
Abstract
In this paper we announce a closed form analytical expression for the minimum attain-
able attitude error (as well as the error rate) in a flexible beam by feedback control using
co-located rate sensors. For simplicity, we consider a beam clamped at one end with an
offset mass (antenna) at the other end where the controls and sensors are located. Both
control moment generators and force actuators are provided. The results apply to any beam-
like lattice-type truss, and provide the kind of performance criteria needed under CSI --
Controls-Structures-Integrated optimization.
t Research Supported in part under NAS1-18585 Task Assignment 49.
449
https://ntrs.nasa.gov/search.jsp?R=19900013722 2020-03-19T22:04:37+00:00Z
1. Introduction
One of the challenges in the Design Challenge For Flexible Flight Structure Control
System Design formulated in the inaugural paper on SCOLE [1] was to hold the antenna
pointing error within =L,0.02 degrees after slewing by appropriate feedback control. In this
paper we derive a closed form expression for the minimal achievable mean square pointing
error using co-located rate sensors. A slightly simplified form of the SCOLE article (which
eliminates rigid-body modes) is used: a cantilevered beam with an offset mass where the
controls m both c.m.g.'s and force actuators m and the rate sensors are located. Our results
are in terms of continuum model parameters m the uniform Bernoulli version is used. The
beam dynamics are given in Section 2. The main results are in Section 3. We note that a
technique for deriving equivalent Bernoulli beam parameters for various types of trusses is
described by Noor and Anderson in [4]. Recently Noor and Russell [5] presented equivalent
anisotropic Timoshenko beam models for beam-like lattice tresses with an arbitrary degree of
modal coupling, which appear to yield excellent agreement with modal frequencies derived
from finite element models. Our theory is able to handle these Timoshenko models, and
moreover we can also use it for rigid-body modes, although they are not included here. Thus
our results can be used for any beam-like lattice truss structure.
450
2. TheModel
Weconsidera uniformBernoullibeamclampedat oneendwithanoffsetmass(antenna)
at the other end which also houses the sensors and actuators. See Figure 1. We allow for both
force actuators and moment actuators. The sensors are rate gyros. Because of the clamping at
one end, no rigid-body modes are involved and hence no attitude sensors are needed.
We allow bending in two mutually perpendicular planes containing the beam axis, as
well as torsion in the plane perpendicular to the beam axis, all uncoupled. The continuum
model (uniform Beroulli beam) dynamics can then be described by the following partial differ-
ential equations (similar to those in [2, 3]). Let the beam extend along the z-axis, 0 < s < L,
and let u¢(s, t), uo(s, t), denote the bending displacements and uv(s, t) the torsion angle
about the beam axis. Let in the usual notation (cf. [I]), EI¢, EI o denote the flexural stiffness
and GI w the torsional rigidity. Let p denote the mass per unit area and A the cross-sectional
area. Then we have:
_F%(s, t) _u¢(s, t)
pA at2 + EI¢ (_s4 - O, O<s<L; O<t
;Fuo(s, 0 a4uo(s, 0
pA at 2 + E! 0 as 4 = o, 0<s<L; 0<t
/Fuv (s, 0
ply at2 - GIvu_,(s,t) = O, O < s < L ; O < t
with the clamped boundary conditions at s = O:
u¢(O,O = uo(O,O = _(0,0 = 0
e #
u,(O,O "- u0(O,t) - O.
The antenna center of gravity is located at
(rx, ry, L) .
The distance from the beam tip to antenna center of gravity is denoted by
451
Figure1: Shuttle/AntennaConfiguration
452
The force balance equation at s = L yields
m 1 0 rx0 1 ry
_,(L,t)
ao (L,O
iiv (L, O
+ If_(t)f2(t)
where m is the antenna mass and fl('), f2(') are the applied control forces. The torque
balance equations yield
et, ui'(L, t)
El o u_'(L, t)
GI v u_ (L, t)
+ la_) + M(t) +r®
f_(t)
ii, (L, t) + rxiiv (L, t)+ r ® ii0 (L, t) + ry iiv (L, 0
where the superdots indicate time derivatives and the primes the derivatives with respect to
the spatial variable s; ® denotes the vector cross-product and co the angular rate vector
"_' (L, 0
co = _'(L,O ,
_,CL, 0
la denotes the moment of inertia of the antenna about the beam tip (s - L) and finally, M(t)
denotes the applied control moment.
It is convenient to denote by b(t) the boundary vector:
b(t)=
u, (L, 0
u 0 (L, 0
ui (t., t)
u_ (L, 0
u_ (L, t)
The boundary rate vector would thus be/_(t). Hence our sensor model is:
v(t) = b(t) + No(t )
where we assume that No(t) is white Gaussian noise with spectral density matrix dol, where
1 is the identity (5)<5) matrix. Similarly we assume that the control actuators are also
453
characterizedby additive white Gaussian noise. Denoting the applied control vector by u(t)"
ul(L, t)
u2(L , t)
u(O - u3(L, t)
u4(L,t)
us(L,0
we have
f i(0
f2(t) - u(O + N,(O
M(t)
where Ns(t)is white Gaussian with spectraldensityds. We shallalsouse M b to denote the
actuator mass/inertia matrix
Mb B
m 0
0 m
0 0
0 0
_x mry
0 0 mr,,
0 0 mry
/a
where
la " Ia +
,._ -r:,'y 0
-r:y r_ 0
o o :,+:,
where 1a is the antenna moment of inertia about its center of gravity. For any control input
u(') (which must perforce be a "feedback" control, based on the sensor data v(.)) the mean
square pointing error is then expressed by:
lira _ f ut(L,t)2dt + f uo(L,t)2dr + }rl2 f u_(L,t)2tit
T-_'* 0 0 0
and the mean square pointing rate is given by
454
l im _ f i%(L,O 2dt + f i_o(L,O 2dt + Irl2f i_(L,O 2dr .
T-_ 0 0 0
From the results in [6] it follows that the minimal attainable mean square pointing error is
given by
aMbla *
whe re
a = row vector (1, 1, 0, 0, Irl)
a* - transpose of a
3. Main Results
We need some notation first. The mean square attitude response, whatever the feed-
back control used is defined by
1 im _ f uO(t, £)2 dt
T_** 0
+ f u o(t,£) 2dt + Irl2f u w(t,£) 2dr .
0 0
(3.1)
This is recognized as the mean square displacement of the center of gravity of the antenna
which is then also proportional to the mean square "pointing" error -- see [1] for the
relationships.
Next let u denote any (vector) of control inputs -- a constant "step" input:
Ul
u2
u = u3 (3.1)
U4
U5
Solve the exluatiom
El, u;'"(s) - 0 ]
Elou_'"(s) = 0
fr
GIw u w (s) - 0
0 < s < L, (3.2)
455
subject to the end conditions
EI, u_,"(L) - ul
Elou_"(L) = u2
Ei, u_'(L) + u3 = 0
Elou_'(L) + ,4 = 0
61,N,(L) + .s - o
(3.3)
Note that the solution can be recognized as the steady-state response of the system to the
step-input u, assuming that there is somedamping. We only need to calculate the response to
three specialized inputs:
Calculate the response to u, (L) to the special case, Case 1, where:
Ul - I
ui - O, 2<i<5.
Calculate the response us(L) to the special case, Case 2, where:
Ul - 0
u4 -us =0.
u2 - 1
Calculate the response uv (L) to the special case, Case 3, where
ul - u2 - u3 - u4 - 0
u6 - I.
Then the minimal achievable mean-square response whatever the choice of the feedback and
whatever the mean-square control effort, is given by
(.,(L) 2 + uo(L) 2 + r2u_(L) 2) . (3.4)
This is our main result. Unfortunately the derivation is beyond the scope of this report and
456
will be published elsewhere. To proceed further with (3.4) we calculam the solution of (3.2),
(3.3) explicitly. Thus for any u, the solution is of the form
whe re
Thus for Case 1 we have
and for Case 2 we have
and for Case 3:
u,(s) - a3: + az:,
uo(s) - h: + v2:,
%(s) - qs,
b3 . _ u._._._
6/il o
1[.__ ,it"Ia_ = _ El, + EI,j
1 __uL
b2- _[Elo "'/'1+ Eioj
L $
uo(L) 2" 3Eio
L
%(L)2" GTv
O<$<L
O<s<L
O<s<L
Hence the mean-square attitude error
" _ 3-gr;+ y_. + c_, j. o.5)
Note the appearance of the noise parameters in {3.5) in product form.
The technique for calculating the minimal mean square atttiude ©rror in morn complex
models than that illustrated is the same: calculate tim mean r,quare step mspons¢ (assuming
45"/
some damping) to unit step inputs.
In conclusion we suggest this result (3.5) can be the basis for combined structures-
controls optimization -- CSI, since the required structural parameters can be calculated for a
lattice truss from the material gage and physical dimensions as in [4, 5]. We omit the details
of these calculations.
458
References
1. L. W. Taylor and A. V. Balakrishnan. "A Mathematical Problem and a Spacecraft Control
Laboratory Experiment (SCOLE): NASAflEEE Design Challenge." Proceedings of the
NASA SCOLE Workshop, Langley Research Center, December 1984.
2. A. V. Balakrishnan. "A Mathematical Formulation of the SCOLE Control Problem,
Part I." NASA CR 172581. May 1985.
3. A. V. Balakrishnan. "Control of Flexible Flight Structures." In: Analyse mathematique et
applications. Paris: Gauthier-Villars, 1988.
4. A. K. Noor and C. M. Anderson. "Analysis of Beam-like Trusses." Computer Methods in
Applied Mechanics and Engineering, Vol. 20 (1979).
5. A. K. Noor and W. C. Russell. "Anisotropic Continuum Models for Beam-like Lattice
Structures." Computer Methods in Applied Mechanics and Engineering, Vol. 57 (1986).
6. A. V. Balakrishnan. "A Mathematical Formulation of the SCOLE Control Problem,
Part II: Optimal Compensator Design." NASA CR 181720. December 1988.
459


Minimum attainable RMS attitude error using co-located rate sensors

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server