Multidisciplinary modeling and surrogate assisted optimization for satellite constellation systems

Abstract

Satellite constellation system design is a challenging and complicated multidisciplinary design optimization (MDO) problem involving a number of computation-intensive multidisciplinary analysis models. In this paper, the MDO problem of a constellation system consisting of small observation satellites is investigated to simultaneously achieve the preliminary design of constellation configuration and the satellite subsystems. The constellation is established based on Walker-δ configuration considering the coverage performance. Coupled with the constellation configuration, several disciplines including payload, power, thermal control, and structure are taken into account for satellite subsystems design subject to various constraints (i.e., ground resolution, power usage, natural frequencies, etc.). Considering the mixed-integer and time-consuming behavior of satellite constellation system MDO problem, a novel sequential radial basis function (RBF) method using the support vector machine (SVM) for discrete-continuous mixed variables notated as SRBF-SVM-DC is proposed. In this method, a discrete-continuous variable sampling method is utilized to handle the discrete variables, i.e., the number of orbit planes and number of satellites, in the satellite constellation system MDO problem. RBF surrogates are constructed and gradually refined to represent the time-consuming simulations during optimization, which can efficiently lead the search to the optimum. Finally, the proposed SRBF-SVM-DC utilized to solve the satellite constellation system MDO problem is compared with a conventional integer coding based genetic algorithm (ICGA). The results show that SRBF-SVM-DC significantly decreases the system mass by about 28.63% subject to all the constraints, which greatly reduces the cost of the satellite constellation system. Moreover, the computational budget of SRBF-SVM-DC is saved by over 85% compared with ICGA, which demonstrates the effectiveness and practicality of the proposed surrogate assisted optimization approach for satellite constellation system design.

Appendix

1.1 Thermal network model parameters (Wu and Huang 2012)

Table 10 Parameters of thermal network model

1.2 Structural FEA model parameters

Table 11 Material parameters for FEA model

1.3 Radial basis function surrogate

Radial basis function (RBF) is an interpolation method on the acquired information at discrete sample points x_k (Gutmann 2001). A RBF surrogate can be formulated as

$$ {\displaystyle \begin{array}{c}\widehat{y}={\boldsymbol{\beta}}^T\boldsymbol{\phi} \left(\boldsymbol{x}\right)\\ {}\boldsymbol{\phi} \left(\boldsymbol{x}\right)={\left[\phi \left(\left\Vert \boldsymbol{x}-{\boldsymbol{x}}_1\right\Vert \right)\kern0.5em \phi \left(\left\Vert \boldsymbol{x}-{\boldsymbol{x}}_2\right\Vert \right)\kern0.5em \dots \kern0.5em \phi \left(\left\Vert \boldsymbol{x}-{\boldsymbol{x}}_{n_s}\right\Vert \right)\right]}^T\\ {}\boldsymbol{\beta} ={\left[\begin{array}{cccc}{\beta}_1& {\beta}_2& \dots & {\beta}_{n_s}\end{array}\right]}^T\end{array}} $$

(A1)

where n_s is the number of sample points, ϕ(‖x − x₁‖), i = 1, 2. . n_s is radial basis function and β is the coefficient vector of RBF.

Since RBF should satisfy the interpolation condition at sample points, (A1) can be written as

$$ \boldsymbol{A}\boldsymbol{\beta } =\boldsymbol{y} $$

(A2)

where the matrix A is the radial basis function matrix shown as below

$$ \mathbf{A}={\left[\begin{array}{ccc}\phi \left(\left\Vert {\boldsymbol{x}}_1-{\boldsymbol{x}}_1\right\Vert \right.& \dots & \phi \left(\left\Vert {\boldsymbol{x}}_1-{\boldsymbol{x}}_{n_s}\right\Vert \right.\\ {}\vdots & \ddots & \vdots \\ {}\phi \left(\left\Vert {\boldsymbol{x}}_{n_s}-{\boldsymbol{x}}_1\right\Vert \right.& \cdots & \phi \left(\left\Vert {\boldsymbol{x}}_{n_s}-{\boldsymbol{x}}_{n_s}\right\Vert \right.\end{array}\right]}_{n_s\times {n}_s} $$

(A3)

The vector y consisting of the actual response values at sample points is formulated as

$$ y={\left[{y}_1\kern0.5em {y}_2\kern0.5em \dots \kern0.5em {y}_{n_s}\right]}^T $$

(A4)

Coefficient vector β can be calculated as

$$ \boldsymbol{\beta} ={\boldsymbol{A}}^{-1}\boldsymbol{y} $$

(A5)

Commonly-used radial basis functions can be found in (Wu et al. 2013b).

1.4 Support vector machine

Support vector machine (SVM) developed by Vapnik has been widely used for pattern classification problems (Burges 1998). Consider a group of training samples belonging to two different classes

$$ \left({\boldsymbol{x}}_1,{y}_1\right),\dots, \left({\boldsymbol{x}}_l,{y}_l\right),{\boldsymbol{x}}_i\in {\boldsymbol{R}}^n,{y}_i\in \left\{+1,-1\right\} $$

(A6)

where x_i is the i-th sample point, y_i is the discrete classification value at sample point x_i, and l is the number of training samples.

First, assume that the training samples could be separated linearly by a hyperplane as shown in (A7)

$$ \left(\boldsymbol{w}\cdot \boldsymbol{x}\right)+b=0,w\in {\boldsymbol{R}}^n,b\in \boldsymbol{R} $$

(A7)

where w = [w₁, w₂, …w_n] is the coefficient vector of the hyperplane. The optimal hyperplane can be obtained by solving the constrained convex quadratic optimization problem in (A8), and the equation can be rewritten in (A9).

$$ {\displaystyle \begin{array}{c}\min \kern0.36em \frac{1}{2}{\left\Vert \boldsymbol{w}\right\Vert}^2\\ {}s.t\kern0.48em {g}_i={y}_i\left(\left(\boldsymbol{w}\cdot {\boldsymbol{x}}_i\right)+b\right)\ge 1,i=1,\dots, l\end{array}} $$

(A8)

$$ {\displaystyle \begin{array}{l}\min \kern0.36em \sum \limits_{i=1}^l{\alpha}_i-\frac{1}{2}\sum \limits_{i,j=1}^l{\alpha}_i{\alpha}_j{y}_i{y}_j\left({\boldsymbol{x}}_i\cdot {\boldsymbol{x}}_j\right)\\ {}s.t\kern0.84em \begin{array}{c}{\alpha}_i\ge 0,\kern0.36em i=1,\dots, k\\ {}\sum \limits_{i=1}^l{\alpha}_i{y}_i=0\end{array}\end{array}} $$

(A9)

In (A9), α_i is the Lagrange multiplier of constraint g_i, which is obtained by solving the dual optimization problem. Then it is easy to obtain w and b (Burges 1998). The classification function of linear SVM classifier is shown in (A10)

$$ f\left(\boldsymbol{x}\right)=\operatorname{sgn}\left(\sum \limits_{i=1}^l{y}_i{\alpha}_i\left(\boldsymbol{x}\cdot {\boldsymbol{x}}_i\right)+b\right) $$

(A10)

where sgn(x) ∈ {−1, +1}, x is an arbitrary design point to be classified.

When the training samples cannot be linearly separated by a hyperplane in Euclidean space Rⁿ, a nonlinear map** φ(x) : Rⁿ → χ is utilized to achieve the linear classification of training samples in another space, denoted as feature space χ. Thus, the optimal hyperplane in the feature space is expressed as

$$ {\displaystyle \begin{array}{c}\min \kern0.36em \frac{1}{2}{\left\Vert \boldsymbol{w}\right\Vert}^2\\ {}s.t\kern0.48em {g}_i={y}_i\left(\left(\boldsymbol{w}\cdot \varphi \left({\boldsymbol{x}}_i\right)\right)+b\right)\ge 1,i=1,\dots, l\end{array}} $$

(A11)

The procedure of determining SVM classification function in χ is the same as that in Euclidean space Rⁿ as shown in (A12).

$$ f\left(\boldsymbol{x}\right)=\operatorname{sgn}\left(\sum \limits_{i=1}^l{y}_i{\alpha}_i\left(\varphi \left(\boldsymbol{x}\right)\cdot \varphi \left({\boldsymbol{x}}_i\right)\right)+b\right) $$

(A12)

According to Mercer’s conditions, the inner product of nonlinear map** could be substituted by a certain kind of kernel function K(x,x_i). Based on Mercer’s conditions, (A12) is rewritten as below. Commonly-used kernel functions can be found in (Burges 1998).

$$ f\left(\boldsymbol{x}\right)=\operatorname{sgn}\left(\sum \limits_{i=1}^l{y}_i{\alpha}_iK\left(\boldsymbol{x},{\boldsymbol{x}}_i\right)+b\right) $$

(A13)

1.5 Algorithm of interesting sampling region

ISR is a relatively small hypercube sub-region where the global optimum probably is located. ISR is determined by the distance between the current pseudo optimum and the cluster center of superior cheap points. The algorithm to identify ISR is exhibited in Appendix Table 12 (Shi et al. 2016).

Table 12 Algorithm of ISR