Scholarly article on topic 'Comparison of MDO Methods for an Earth Observation Satellite'

Comparison of MDO Methods for an Earth Observation Satellite Academic research paper on "Earth and related environmental sciences"

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Procedia Engineering
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{"Multidisciplinary Design Optimization (MDO)" / "Earth observation satellite" / "Optimization method"}

Abstract of research paper on Earth and related environmental sciences, author of scientific article — X.H. Wang, R.J. Li, R.W. Xia

Abstract It is shown in this paper the application of several multidisciplinary design optimization (MDO) methods for an earth observation satellite to optimize the performance of the earth observation. Based on the best earth observation criteria, a mathematical model of the satellite is proposed, which involves design variables, coupling variables and constraint functions. The analysis models of each subsystem are given, including the payload, attitude control, propulsion, power, and structure subsystem. The optimization is conducted by utilizing three MDO methods. Moreover the comparison among these methods is given to show the different features of these methods.

Academic research paper on topic "Comparison of MDO Methods for an Earth Observation Satellite"

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Procedía Engineering 67 (2013) 166-177

7th Asian-Pacific Conference on Aerospace Technology and Science, 7th APCATS 2013

Comparison of MDO Methods for an Earth Observation Satellite

X. H. Wanga*, R. J. Lia, R. W. Xiaa

aSchool of Astronautics, Beihang University, Beijing, 100191, China

Abstract

It is shown in this paper the application of several multidisciplinary design optimization (MDO) methods for an earth observation satellite to optimize the performance of the earth observation. Based on the best earth observation criteria, a mathematical model of the satellite is proposed, which involves design variables, coupling variables and constraint functions. The analysis models of each subsystem are given, including the payload, attitude control, propulsion, power, and structure subsystem. The optimization is conducted by utilizing three MDO methods. Moreover the comparison among these methods is given to show the different features of these methods.

© 2013 TheAuthors.PublishedbyElsevier Ltd.

Selectionandpeer-reviewunder responsibilityofthe NationalChiaoTungUniversity Keywords: Multidisciplinary Design Optimization (MDO); Earth observation satellite; Optimization method

Nomenclature

x design variable

X vector of design variables

y coupling variable

Y vector of coupling variables

g constraint function

G vector of constraint functions

P objective function Subscripts

i the i-th discipline

j the j-th discipline

ij from the i-th discipline to the j-th discipline.

Superscript

* Corresponding author. Tel.: +86-10-8233-8763. E-mail address: xhwang@buaa.edu.cn

1877-7058 © 2013 The Authors. Published by Elsevier Ltd.

Selection and peer-review under responsibility of the National Chiao Tung University doi:10.1016/j.proeng.2013.12.016

S_system-level_

1. Introduction

Satellites are such complex systems involving multiple disciplines with complex couplings [1-2]. It is impossible to obtain an optimal design for a multidisciplinary system by means of optimizing each discipline separately [3-5]. In order to optimally design and analyze this kind of complex engineering systems, MDO methods have emerged [6-7], such as Collaborative Optimization (CO), All-At-Once (AAO), and Multidisciplinary Feasible Method (MDF) [8].

Among these methods, CO is a multilevel MDO one, which has advantage of reducing the scale of the problems to be optimized. This kind of methods, however, requires a large amount of iterative computations, and hence is not possible to precisely prove its convergence. It is because that during the optimization process it has been "crafted" through the engineer's insight into the problem, as opposed to rigorous mathematical derivations

AAO and MDF are single-level optimization methods which solve the MDO problem as a whole, excluding the human factors mentioned above so that the convergence can be mathematically proved [10]. The drawback lies in that the single-level optimization model for MDO problem would have a large number of variables and constraints, which results a heavy burden computation.

In this paper, we apply the above MDO methods to optimize the observation performance of the earth observation satellite, where our interests are the engineering applicability, the accuracy of solutions, and the computational efficiency of those methods.

The rest of the paper is organized as follows. In Section 2, a problem formulation is given. In Section 3, each subsystem of the satellite is analyzed. Then several optimization methods are applied to such satellite to optimize its earth observation performance, and the comparison is given followed by conclusion.

2. Problem formulation

To begin with, we briefly introduce the satellite system under consideration, where the MDO problem for the satellite consists of five subsystems: the payload, attitude control, propulsion, power, and structure subsystem, as shown in Fig. 1, where ytj is the coupling variable from the i-th discipline to the j-th discipline.

Fig.1. The coupling relationship diagram of the satellite's subsystems.

The optimization design problem is complex due to a large number of functions and design variables. In specific, it includes 1 objective function, 10 design variables, 15 coupling variables, and 10 constraint functions. The formulation is given as follow:

Find : X, Y Maximize : P (X) = A > Subject to : G(X, Y)

GSD (X )

where the objective is to optimize the performance of the earth observation, which can be expressed by the objective function P(X), with coefficients A=0.6, B=0.4, ground resolution GSD(X), its reference value GSD0, the satellite observation covering the width of the central angle ¥(X) and its reference value W0 [11].

In specific, the vector of design variables X is listed in Table 1, the vector of the coupling variables Y is given in Table 2, and the vector of the constraint functions G(X,Y) is given in Table 3.

Table 1. The design variables of the satellite system.

Design variables Physical symbols Unit Physical meaning Min Values Max

Xi h km Orbital altitude 600 800

X2 Hw N-m-s Angular momentum of flywheel 0 100

X3 Hsa m Height of a single solar panel 1 2

X4 Lsa m Length of a single solar panel 3 10

X5 DNT — Descending node time 8 11.5

X6 hm m Satellite body height 1.5 5

X7 b m Satellite body width 1.2 2.475

X8 Dc m Diameter of center bearing cylinder 0.24 0.825

X9 Fp N Thrust of orbit control engine 0 100

X10 Lt year Design life of the satellite 3 5

Table 2. The coupled variables of the satellite system.

Coupling variables Physical symbols Unit Physical meaning Min Values Max

yi2 Fd N Thrust of the attitude control engine 0 0.001

yi3 Padcs W Power of attitude control subsystem 15 40

yi4 mADCS kg Mass of attitude control subsystem 10 30

y23 Pp W Power of the propulsion subsystem 10 30

y24,1 mp kg Mass of propellant 50 400

y24,2 mp,net kg Net mass of propulsion subsystem 1 3

y31 A-.m m2 Area of the solar panels 8 15

y34,1 mpow kg Mass of the power subsystem 50 70

y34,2 mSa kg Mass of the solar panels 50 70

y41,1 Ld m Unloading arm of the attitude control engine 1.2 2.475

y41,2 Aw m2 Satellite windward area 8 14

y41,3 Iy kg-m4 Rotational inertia around y-axis 100 3000

y41,4 Iz kg-m4 Rotational inertia around z-axis 100 4000

y42 mtotal kg Total mass of the satellite 1050 2000

y43 m„t kg Net mass of the satellite 1000 1600

Table 3. The constraint functions of the satellite system.

Constraint functions Expression Unit Constraint functions Expression Unit

gi SNR — g6 b/5-Dc m

g2 Tc-Td N-m g7 Dc-b/3 m

g3 Psa-P W g8 Fa Hz

g4 2C--C Wh g9 Fl Hz

g5 Hsa~Lsa M gio aoc m/s2

In Table 3, SNR is the signal to noise ratio; TC and TD are control and disturbance torque; Psa is the output power of solar panels at the end of life; P is required power of the entire satellite; Cs is capacity of each battery; C is the total required capacity of batteries; Fa and Fl are the axial and lateral natural frequency of the satellite respectively; aoc is the linear acceleration of the satellite.

3. The analysis of the subsystems

Since the analysis models of the above mentioned subsystems are called in the duration of the optimization, we in what follows briefly describe the analysis models of them.

3.1. Payload subsystem

In payload subsystem, the SNR is a crucial performance index for the CCD camera [12], which is defined as

SNR = e

V^625 (2)

where Ne is the electronic flow of a signal with the following definition

N = 1727.17 x GSD(X)4 ¡(Re + h)3

0.0735 x h3 xRe \ ^ (3)

with the average radius of the earth's equator Re and the earth's gravitational constant ^.

3.2. Attitude control subsystem

The main function of the attitude control subsystem is to produce the control torque TC to overcome the disturbance torque TD [13]. And TC and TD are defined as

T _ 4Hw

0637T (h) (4)

Td - Tg (h, Iy, Iz) + Ts (Asa) + Tm (h) + Ta ( Aw , h)

where T(h) is the orbital period; Tg (h, Iy, Iz) is gravity gradient torque; Ts(Asa) is solar radiation pressure torque; Tm(h) is geomagnetic disturbance torque; Ta (Aw,h) is aerodynamic torque.

3.3. Propulsion subsystem

The linear acceleration of the satellite aoc and the mass of propellant mp are the main constraints in the propulsion subsystem. And aoc and mp are given as

aoc = Fp / mtotal (6)

m = m + m + m + m

Hip !Upo~ ,npa~ "lpm~ pr (7)

where mpo is required mass of propellant for orbit control; mpa is required mass of propellant for momentum wheel unloading; mpm is mass of margin propellant; mpr is mass of residual propellant.

3.4. Power subsystem

In the power subsystem, the output power of solar panels at the end of life Psa should be larger than the required power of the entire satellite P. And Psa and P are given as

Psa = 127.61X 0.9725Lt x Asa (8)

P = {PeTe/0.65 + PdTd/0.85)/Td (9)

where Pe and Pd are required power in shadow and in sunshine respectively, and Te and Td are time of shadow and sunshine in one cycle respectively with Te + Td =T.

3.5. Structure subsystem

The structure subsystem should guarantee that the axial natural frequency fa and lateral natural frequency f of the satellite are big enough to avoid resonance. And fa and f are given as

f = 0.25

fl = 056

a (Dc )■ E

m total ■ hm

1 (Dc )' E

mtotai ■ hm

where a(Dc) is cross-section of main structure; E is elasticity modulus of material; I(Dc) is moment of inertia of cross-section.

We have shown the analysis models of all subsystems under consideration, and in the next section, we will show the application of the optimization methods to the problem.

4. Application of optimization methods to an earth observation satellite

In this section, we apply CO, AAO, and MDF method to optimize the problem of the earth observation satellite. 4.1. Collaborative Optimization (CO) (1) Optimization model

The optimization model of the CO consists of two levels. The first one is for the system-level optimization model, that is,

Find :

X™ = K > y?2 > y'a > y?4 > > y^4,2> K > > ySl.1> y4i,2> y4i,3> y4i,4 > y«}

Maximize : P (X) = 0.6 x

GSD (xi)

+ 0.4 x

Kx) i2

Subject to : JAc < 10"4, Jpmer < 10"4, J„ < 10" J ,. < 10"4, J , , < 10"4

propulsion * payload

And the second one is for the subsystem-level optimization model, i.e.

Find :

Minimize : Jd = (1 - x;. / x* )2 + (1 - / ys )2 Subject to : g( Given as Target: x*, yS

where Jd is objective function of every subsystem; x; is design variable; yij is coupling state variable; gi is constraint

function.

(2) Optimization results

Conduct the optimization in iSIGHT™, and terminate the iterations according to the internal convergence criteria. The iteration processes of objective function P(X) are shown in Fig. 2. The number of iteration steps is small because it only counts the iterations of the system-level.

4.2. All-At-Once (AAO) (1) Optimization model

Fig. 2. CO objective function iterative processes.

X sys ~

Maximize : P (X) = 0.6;

GSD (x1

+ 0.4)

Subject to : & > 50, g2 > 0, ft ^ 0, ^4 > 0,3 < < 5, ^6 < 0,< 0, ft > 25,> 24,ft0 > 0.05,

S11 =1 y23 / y23 - ^ 0, gl2 =1 y31 / y3l - 0, &13 =| y41,1 / y4l,1 - 0,gl4 =| y41,2 / y4l,2 - 0 gi5 =| y41,3 / y4:,3 -0, g16 =| y41,4 / y4:,4 - 0,

g\7 =^42/ y42 - ^ 0

(2) Optimization results

Conduct the optimization in iSIGHT™, and the iteration processes of objective function P(X) are shown in Fig. 3. The jumps in the figure are the processes to adjust the step length.

Fig. 3. AAO objective function iterative processes.

4.3. Multidisciplinary Feasible Method (MDF) (1) Optimization model

Find : X = {x^...,x10}

Maximize : P(X) = 0.6 x

GSD (x1 )Y J^(x1)

V 30 y

+ 0.4 x

V 12 y

Subject to : g, > 50,g2 > 0,g3 > 0,g4 > 0,3 < g5 < 5, g6 < 0, g7 < 0, g8 > 25, g9 > 24, g10 > 0.05

(2) Optimization results

Conduct the optimization in iSIGHTTM, and the iteration processes of objective function P(X) are shown in Fig.

Fig. 4. MDF objective function iterative processes.

4.4. Comparison results

In this section the comparison results of the optimization of the above three methods are shown in Table 4.

Table 4. Comparison results of three optimization methods.

Parameters Initial values Optimized values

CO AAO MDF

P(X) 1 1.026 1.016 1.010

h(km) 700.00 656.61 675.61 688.48

H„,(N-m-s) 5 1 89.21 4.97

HSa(m) 1.5 1.28 1.45 1.40

Lsa(m) 3 3.85 4.46 4.30

DNT 8 8 8.81 8

hm(m) 3.5 2.06 2.11 3.49

b(m) 1.4 1.29 1.72 1.40

Dc(m) 0.4 0.30 0.41 0.40

Fp (N) 85 69.95 13.44 86.03

Lt(year) 3 3.46 3.48 3

Optimization time(s) — 566 94 210

It can be seen that, the ground resolution is improved and the observation coverage area is reduced by decreasing the orbital altitude. In the sense of orbital altitude and objective function, CO method is the best as the orbital altitude is reduced from 700 to 656.61 km with the largest decrease of 6.2%, and objective function is reduced from 1.000 to 1.026 with the largest increase of 2.6%.

The same results of the three MDO methods can be observed from Table 4 that, the height of solar panels decreases while the length increases and the body height of the satellite decreases.

Meanwhile there are some differences among the methods. First of all, compared with the results of MDF and CO, the angular momentum of the flywheel optimized by AAO is much larger but the thrust of orbit control engine is much smaller, and the descending node time increases. Second, the diameter of the center bearing cylinder and the body width of the satellite are almost unchanged by means of MDF, however that of AAO increases and that of CO decreases. In addition, there is a 15% increase of the life of the satellite by AAO and CO.

For the computation time, AAO is the fastest, and CO requires the most. Although the iteration number of MDF is much less than that of AAO, its optimization time is longer than that of AAO, because MDF needs disciplinary analysis in every iteration step.

From the optimization compilation process, with respect to single-stage optimization, CO needs the discipline-level analysis and additional consistency constraints, making compilation process complex. AAO method only optimizes at the system level, but does not effectively use the analysis process to reduce the computation scale. MDF does not require calculation in the discipline level, and the design variables even do not involve the coupling variables. It only requires an analytical process without hard decoupled, which results the simplest compilation process.

5. Conclusion

In this paper, we have compared three MDO methods such as CO, AAO, and MDF based on the formulation and the analysis models of the subsystems of the earth observation satellite. We have shown that the observation performance of the satellite is improved by the optimization processes, which validates the effectiveness of the methods. Among the three MDO methods, CO is the most effective in the sense of the objective function, it however needs complex compilation process and largest calculation time, and there is no guarantee of convergence and equivalence between the original and the decomposed problem. In contrast, single-level methods is less effective than multi-level methods, while they can obtain convergent and more accurate results independent of the influence of the subjective effect. In single-level methods, MDF has simpler compilation process and smaller scale of variables, more suitable to be used in practical applications. In the future work, the improvement of the calculation efficiency of the MDF will be conducted.

Acknowledgements

This work is supported by the Fundamental Research Funds for the Central Universities of China under grant No.YWF-13-ZY-02.

References

[1] Wu, W.R., Huang, H., Chen, S.Y., Wu, B.B., 2013. Satellite Multidisciplinary Design Optimization with a High-Fidelity Model, Journal of Spacecraft and Rockets, Vol. 50, No. 2, pp. 463-466.

[2] Jafarsalehi, A., Zadeh, P. M., Mirshams, M., 2012. Collaborative Optimization of Remote Sensing Small Satellite Mission Using Genetic Algorithms, Iranian Journal of Science and Technology-Transactions of Mechanical Engineering, Vol. 36, No. M2, pp. 117-128.

[3] Ravanbakhsh, A., Mortazavi, M., 2008. Multidisciplinary design optimization approach to conceptual design of an LEO earth observation microsatellite, Space Ops 2008 Conference. Heidelberg, Germany.

[4] Wang, X.H., Xia, R.W., 2006. A novel MDO method and its testing results, In Proceedings of Asian-Pacific Conference Aerospace Technology and Science. Guilin, China.

[5] Wang, X.H., Lin, Z.W., Xia, R.W., 2013. SIMP based Topology Optimization of a Folding Wing with Mixed Design Variables, In Proceedings of the 17th IEEE International Conference on Computer Supported Cooperative Work in Design, Whistler, BC, Canada.

[6] Balling, R., Sobieski, S., 1996. Optimization of coupled systems: a critical overview of approaches, AIAA Journal, Vol. 34, No. 1, pp. 6-17.

[7] Lambe, A., Martins, J., 2012. Extensions to the design structure matrix for the description of multidisciplinary design, analysis, and optimization processes, Structural and Multidisciplinary Optimization, Vol. 46, pp. 273-284.

[8] Tosserams, S., Hofkamp, A., Etman, L., Rooda, J., 2010. A specification language for problem partitioning in decomposition-based design

optimization, Structural and Multidisciplinary Optimization, Vol. 42, No. 5, pp. 707-723.

[9] Balesdent, M., Berend, N., Depince, P., Chriette, A., 2012. A survey of multidisciplinary design optimization methods in launch vehicle design, Structural and Multidisciplinary Optimization, Vol. 45, No. 5, pp. 619-642.

[10] Chen, Q.F., 2003. Distributed coevolutionary multidisciplinary design optimization methods for flying vehicles, Ph.D. dissertation, National University of Defence Technology, Changsha, China. (in Chinese)

[11] Wertz, J., Larson, W., 1999. Space mission analysis and design, 3rd Editon, Microcosm Press.

[12] Richards, A., Richardson, M., 2013. EMCCD and sCMOS imaging systems confront color night vision, Laser Focus World, Vol. 49, No. 2, pp. 38-42.

[13] Wood, M., Chen, W.H., 2013. Attitude control of magnetically actuated satellites with an uneven inertia distribution, Aerospace Science and Technology, Vol. 25, No. 1, pp. 29-39.