The benefits are demonstrated of a multiobjective optimization based control structure integrated design methodology. An application of the proposed CSI methodology to the integrated design of the Spacecraft COntrol Lab Experiment (SCOLE) configuration is presented. Integrated design resulted in reducing both the control performance measure and the mass. Thus, better overall performance is achieved through integrated design optimization. The mutliobjective optimization approach used provides Pareto optimal solutions by unconstrained minimization of a differentiable KS function. Furthermore, adjusting the parameters gives insight into the trade-offs involved between different objectives

Gupta, Sandeep

Joshi, Suresh M.

English

NASA Technical Reports Server

N91 - 22322
An Integrated Control/Structure Design
Method using Multi-objective Optimization
Sandeep Gupta
ViGYAN, Inc.
30 Research Drive,
Hampton, VA 23666
and
Suresh M. Joshi
NASA Langley Research Center
Hampton, VA 23665
Presented at the
4th NASA Workshop on Computational Control
of Flexible Aerospace Systmes.
Kingsmill Resort, Williamsburg, Virginia.
July 11-13, 1990.
https://ntrs.nasa.gov/search.jsp?R=19910013009 2020-03-19T17:52:54+00:00Z
Structural design variables are the diameters of tubular beam elements.
Hence, there are 10 structural design variables. Controller design variables deter-
mine the attitude and attitude rate gain matrices for the constant gain dissipative
feedback control law. Since Gp and Gr have so be positive definite for this con-
troller, we use Cholesky factors of the gain matrices as follows: Gp = LpL T, and
G,. = L,.L T, where Lp and Lr are lower triangular matrices. The controller de-
sign variables are the elements of Lp and L_ matrices. Subsequently, there are 24
controller design variables in this study.
Design Variables
Structural:
10 structural design variables -- diameters of tubular beam elements
Controller:
Controller design variables -- the attitude and rate gain matrices
Since Gp, G r , are positive definite, we use Cholesky factors for design
T LrLTrGp= LpLp and Gr=
Hence, controller design variables are elements of Lp and Lr
24 controller variables for this case.
Four control/structure objectives were selected for this study. The first
structural objective is to minimize mass of the flexible mast and the actuators,
MT; obtained by subtracting the fixed masses of the shuttle and the reflector from
the total mass of the system. Another structural objective is to maximize the first
open-loop frequency, wl, so that the structure can be made as stiff as possible within
allowable values of the mass and the control objectives. The first control objective
is to minimize a measure of transient response decay time, r. This measure is the
sum of the reciprocal of absolute real parts of the closed-loop eigenvalues, as shown
below. The last CSI objective is to minimize a noise attenuation measure, a, which
is the steady-state root mean square attitude error due to a white noise input at
the sensors.
CSI Performance Objectives
Structural:
1) Mass of flexible mast and actuator mass, MT
2) First open-loop structural frequency, co1
Controller:
3) A measure of transient response decay time, 't
n 1
":=T_,
i=1 [ Re _-i]
4) A noise attenuation measure -- root mean square attitude
error due to a white noise input at the sensors, <_
24.5
The main issue in using approaches which employ incremental scaling par
rameters a i and bj is to select values for these parameters. Although arbitrary
values of these parameters would still lead to a Pareto optimal solution with our
approach, some preliminary optimizations are performed to establish trends in the
behavior of various objectives, which assist in choosing reasonable values for the
scaling parameters. Minimizing mass, MT, alone takes the structural variables to
their lower bounds (for least possible structural mass), and the controller variables
close to zero (for least actuator mass). However, this makes the structure very
flexible. Maximizing first open-loop frequency, _al, sends the structural variables to
their upper bounds and the controller variables close to zero (near zero point masses
at the tips). This leads to a very massive and stiff structure. These opthnizations
show the tradeoff between the first two objectives. Optimizing both, i.e. minimiz-
ing mass and maximizing the first open-loop frequency, simultaneously, results in
reasonable values for both objectives.
Trends from Optimization
* Minimizing mass MT"
structural variables--> lower bounds
controller variables --> close to zero
Mass MT~IO 4
-3 g
very flexible structure, 0)1-10 ; _ ~10
-5
; c =1.65x 10
* Maximizing first open loop frequency ml •
structural variables --> upper bounds
controller variables --> close to zero
First open loop frequency m1 = 12.24 -s
very massive structure, MT~ 15300, "¢ = 134.26, a = 2.3 x 10
Structural optimization -- minimizing mass, maximizing frequency
structural variables -- thick close to shuttle end, thin out at the
reflector end
controller variables -- close to zero
MT = 2827.3, (01 = 0.173,
-5
= 99.9, c = 2.15x10
The previous optimizations were primarily structural optimizations. In
controls, first a rigid body controller is designed (with no optimization) as follows:
Gp = w2j and Gr = 2pwJ, where J is the moments of inertia matrix for the
structure and p, w are closed-loop damping ratio and frequency. With p = 0.707
and w = 0.05 for the nominal SCOLE configuration, we get r = 121.34 and
cr = 2.16 x 10 -5. Next, an optimization is performed with respect to control vari-
ables only, while using the nominal structural configuration for SCOLE, resulting
in reduction of v to 104.37. Finally, performing simultaneous optimization with
respect to structural and controller variables reduces r further to 65.86, but this
results in a massive and stiff structure with _'V_T = 5537.6, since there was no re-
striction on mass. Thus, there is another tradeoff involved between mass, _'V/T, and
the transient response decay measure, v.
Minimizing transient response decay time measure, "= :
1) No optimization. Using rigid body controller as
G p = co2J and Gr = 2po) J
where o) = 0.05 and p = 0.707
"_ = 121.34 and (_ =2.16x10 "s
2) Optimization with controller variables only.
structural variables at nominal value
"_ =104.37 and _ =2.12x105
3) Optimization with both structural and controller variables
"_ = 65.86 and (_ = 2.16 x 10"s
but since there is no restriction on mass, MT = 5537.67
With some idea of the tradeoffs among different objectives and some insight
into the numerical values of the objectives involved for selecting the parameters aj
and bj, multi-objective optimizations are performed, next. The control-optimized
design is used as the initial design. In order to reduce both mass and response decay
time, lower values of desired mass and desired response decay time are used. The
desired values for the first open-loop frequency and the noise attenuation measure,
are used more as constraint values than performance objectives. The parameters bj
were chosen to make the incremental variations from the desired values commensu-
rable. The optimization results in lower values for both mass, MT = 2847.7, and
response decay time measure, r = 94.622. However, the first open-loop frequency is
lower than its desired value. To emphasize this objective more, we reduce the value
of parameter b2. Now, the optimal first open-loop frequency is much closer to its
desired value; but the mass and the response decay time measure, are not reduced
quite as much.
Multi-objective Optimization Results
INITIAL
1
a i
b1i
1
2
ai
2
bj
2
f_
MT
2881.0
2800.0
10.0
2_7.7
2800.0
10.0
2880.6
0.2055
0.2
0.01
0.153
0.2
0.001
0.197
"C
104.37
90.0
1.0
94.622
90.0
1.0
i
97.92
G
-5
2.1x 10
2.2 x 10"s
10 "=
2.14x 10.5
2.2 x 10s
10 -4
2.21 x 10s
Previous results demonstrated that mass was the constraining factor in im-
proving the response decay time. In fact, the optimizer was reducing the structural
mass making the mast more flexible and adding the mass to the actuators to im-
prove r. Therefore, in the next series of optimizations, the desired mass is increased
to allow reduction in response decay time. Optimization resutls indeed show the
trend of the mass going up while r is reduced. In a similar manner, by varying the
values of a i and hi, the designer can place different emphasis on various objectives,
and perform parametric tradeoiT studies with Pareto optimal designs.
Multi-objective Optimization Results
INITIAL
3
a.
I
b.3
I
3
4
ai
MT
2881.0
3000.0
4
fo=
10.0
2996.11
3500.0
0.2055
0.2
0.01
0.292
0.2
104.37
90.0
1.0
89.74
80.0
_3
-5
2.1x 10
2.2 x 10s
..410
2.15 x 10.5
2.2 x 10s
4 -4
b j 10.0 0.01 1.0 10
3575.2 0.485 87.53 2.15 x 10 s
This paper demonstrates the benefits of a multi-objective optimization-
based control/structure integrated design methodology. An application of the pro-
posed CSI methodology to the integrated design of the SCOLE configuration is
presented here. Integrated design resulted in reducing both the control perfor-
mance measure, r, and the mass, MT. Thus, better overall performance is achieved
through integrated design optimization.
The multi-objective optimization approach used here provides Pareto opti-
mal solutions by unconstrained minimization of a differentiable KS function. Fur-
thermore, adjusting the parameters aj and bI gives insight into the trade-offs in-
volved between different objectives.
Concluding Remarks
" Control/Structure Integrated Design:
--> Example application of CSI methodology to SCOLE
--> Integrated design optimization gives better overall
performance
Multi-objective Optimization Approach:
--> Pareto optimal solutions
--> Unconstrained optimizations
(constraints can be included as desired values with large
weights)
--> Adjusting a i and b j gives insight into tradeoffs
involved between different objectives
Benefits of the CSI design were observed in a structurally simple SCOLE
configuration. Greater opportunity for such benefits exists in the CSI design of more
complex space structures. We will be applying this methodology to: 1) the EOS
(Earth Observing System) structure, which is the ADMT/CSI focus configuration,
and 2) the phase-zero evolutionary model at NASA Langley Research Center. This
methodology will also be used with more sophisticated control laws such as dynamic
dissipative controllers, as well as, LQG and Hoo optimal controllers. ALso, open-
loop plant dynamics could be refined by including sensor/actuator dynamics (which
would include filtering of input and output signals).
Future Work
Apply this technique to more complex structures
1) EOS structure --> ADMT/CSI focus configuration
2) Phase zero evolutionary model
Use with more sophisticed control laws
1) Dynamic dmsipative controllers
2) LQG and H._ optimal controllers
Optimization including sensor/actuator dynamics
(which would include filtering of input and output signals)
References
1. Schy, A. A. and Giesy, D. P.: Multicriteria Optimization Methods for De-
sign of Aircraft Control Systems, pp 225-262, Multicriteria Optimization in
Engineering and in the Sciences, Plenum Press, New York, 1988.
2. Gembicki, F. W. and Haimes, Y. Y.: Approach to Performance and Sensi-
tivity Multiobjective Optimization: The Goal Attainment Method, pp 769-
771, IEEE Transactions on Automatic Control, December 1975.
3. Kreisselmeier, G. and Steinhauser, R.: Systematic Control Design by Opti-
mizing a Vector Performance Index, pp 113-117, Computer Aided Design
of Control Systems, Zurich, Switzerland, August 1979.
4. Sobieszczanski-Sobieski, J., Dovi, A. R. and Wrenn, G. A.: A New Algo-
rithm for General Multiobjective Optimization, NASA Technical Memoran-
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5. Joshi, S. M. and Maghami, P. G.: Dissipative Compensators for Flexible
Spacecraft Control, Proceeding of 1990 American Control Conference, San
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6. Joshi, S. M.: Control of Large Flexible Space Structures, Vol. 131, Lecture
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An integrated control/structure design method using multi-objective optimization

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