Support Vector Regression-Driven Multidisciplinary Design ...

AIAA 2009-6238

AIAA Modeling and Simulation Technologies Conference 10 - 13 August 2009, Chicago, Illinois

Support Vector Regression-driven Multidisciplinary Design Optimization of Multistage Ground Based Interceptor Qasim Zeeshan*, Dong Yunfeng†, Saqlain Ghumman‡, Amer Farhan Rafique§, Ali Kamran** Beijing University of Aeronautics and Astronautics (BUAA), Beijing 100191, China

In this paper we propose meta-model based design and optimization strategy for multistage ground based interceptor comprising of three stage solid propulsion system for an exo-atmospheric boost phase intercept. An efficient Least Square Support Vector Regression technique is used to approximate the current problem. The mission of Ground Based Interceptor is to deliver Kinetic Kill Vehicle to an optimal position in space to allow it to complete the intercept. The modules for propulsion characteristics, mass properties and flight dynamics have been integrated to produce a high fidelity model of the entire vehicle. For the present effort, the design objective is to minimize the Gross Lift off Mass of the ground based interceptor under the mission constraints of miss distance, intercept time, lateral divert, velocity at intercept, g-loads and stage configuration requirements as Solid Rocket Motor envelope constraints which comprise of length to diameter ratios, nozzle expansion ratios, propellant burn rates and grain geometry constraints like web fraction, volumetric loading efficiency etc. Though, the optimization results and performance are to be considered as preliminary (proof-of-concept) only, but they can be compared to existing systems and used for conceptual design of ground based interceptors. The proposed design and optimization methodology provides the designer with a time efficient and powerful approach for the design of interceptor systems.

Keywords: Boost Phase, Interceptor, Meta modeling, Multidisciplinary Design Optimization, Solid Rocket Motor, Trajectory

I.

Introduction

Simulation-based analysis tools are finding increased use during preliminary design to explore design alternatives at the system level. In spite of advances in computer capacity and speed, the enormous computational cost of complex, high fidelity scientific and engineering simulations makes it impractical to rely exclusively on simulation codes for the purpose of Multidisciplinary Design Optimization (MDO). A preferable strategy is to utilize approximation models which are often referred to as meta-models, replacing the expensive simulation model during the design and optimization process. Meta-modeling techniques have been widely used for design evaluation and optimization in many engineering applications; a comprehensive review of meta-modeling applications in mechanical and aerospace systems can be found in Ref.1. A review of meta-modeling applications in multidisciplinary design optimization is given in Ref. 2. The design of missile systems capable of intercepting fast moving targets is a complex problem that must balance competing objectives and constraints. It involves teams of specialists working separately on their specialized design domains (like propulsion, guidance etc), although coordinated through a system level set of design requirements such as physical size or weight. This type of segmented design process requires rigorous iterations to make the missile sub-systems compatible with each other while meeting the mission specifications. Therefore the *

Ph.D. Student, School of Astronautics, AIAA Student Member, [email protected]. Professor, School of Astronautics. ‡ Ph.D. School of Astronautics. § Ph.D. Student, School of Astronautics, AIAA Student Member. ** Ph.D. Student, School of Astronautics, AIAA Student Member. †

1 American Institute of Aeronautics and Astronautics Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

need arises for a Multidisciplinary Design Optimization approach that can control the design domains concurrently and configure an optimum design. Simulation-based analysis tools are finding increased use during preliminary design to explore design alternatives at the system level. In spite of advances in computer capacity and speed, the enormous computational cost of complex, high fidelity scientific and engineering simulations makes it impractical to rely exclusively on simulation codes for the purpose of multidisciplinary design optimization. A preferable strategy is to utilize approximation models which are often referred to as meta-models, replacing the expensive simulation model during the design and optimization process. References 3, 4 have implemented meta-modeling for the design and optimization of multistage space launch vehicle. Most authors have considered the design of ground and air launched configurations for interceptors 5, 6, 7, 8. but still the potential of using meta-modeling for the MDO of a multistage long range exo-atmospheric interceptor have not yet been gauged. In recent years, evolutionary techniques have been used to solve multitude of design optimization problems. Significant research has been performed in rocket based vehicle design optimization using various evolutionary techniques 9, 10, 11, 12. Almost every discipline in aerospace from Guidance, Navigation, Control, Propulsion and Structures has yielded itself to the power of computational intelligence 13. Population based, non-gradient, stochastic direct search optimization methods are attractive choices for such problem as they are easy to implement and effective for highly nonlinear problems. In this paper we propose a multidisciplinary design and optimization strategy for conceptual design of multistage Ground Based Interceptor (GBI) comprising of three stage solid propulsion system for an exoatmospheric boost phase intercept. Since multidisciplinary design optimization of multistage launch vehicles is a complex and computationally expensive, an efficient Least Square Support Vector Regression (LS-SVR) technique 14, 15 is used to approximate the current problem. Genetic Algorithm (GA) is used as optimizer for its global search efficiency. The optimized configurations based on the exact analysis and meta-model are compared.

II. Design Requirements for Ground Launched Boost Phase Intercept The boost phase is that part of flight when the rocket motors of ballistic missile are ignited and propel the entire missile system towards space. Boost Phase Intercept (BPI) is alluring because rocket boosters are easy to detect and track, they are relatively vulnerable due to the large axial loads on a missile under powered flight, the entire payload (single or multiple warheads) may be destroyed in a single shot, and countermeasures to defeat boost phase defense are more difficult to devise than for midcourse ballistic missile defenses. Moreover, if intercepted several seconds before booster burnout, the debris will land well short of the target area. On the other hand, boost-phase ballistic missile defense is technically challenging because the intercept timelines are very short i.e. 1-3 minutes for shortrange ballistic missiles and 3-5 minutes for intercontinental ballistic missiles (ICBM) 16, 17.The burn time of an ICBM's booster, coupled with the distance that interceptor must travel to reach its target (which results from the geography of a particular scenario), determines the response time and interceptor speed necessary for a BPI system. More technologically advanced ICBMs would require higher-performance interceptor systems because those ICBMs will usually have shorter burn times and faster acceleration. To intercept a target in boost phase the interceptor must be solid-fueled for responsiveness, have high thrust, high acceleration, and a short startup time (fast ignition), be easy to start, require little preparation time, and be easy to maintain and reliable 17. The considerations involved in a GBI design differ for other surface-based systems and space-based Minimize Maximize systems. GBI must be able to survive the high mechanical and thermal stresses associated with flying Size, GLOM and Payload Speed and acceleration, burn through the atmosphere at supersonic speeds. Space- (Kill vehicle) mass out velocity based Interceptors (SBI), by contrast, have little or no Thrust, specific impulse, interaction with the atmosphere because intercepts Intercept Time combustion speed usually occur at very high altitudes. From the standpoint of effectiveness, a primary trade-off in designing an Preparation and Start up time Propellant burning rate and Burn time interceptor is between speed and acceleration on the one hand and size on the other. A number of characteristics G-Loads Maneuverability affect the details of the tradeoff between interceptor speed and size. They include the propulsion efficiency of the booster, the mass of its structures, the number of Table 1. Design Requirements and Tradeoffs. booster stages, and the mass of the payload (KKV). The 2 American Institute of Aeronautics and Astronautics

design requirements and tradeoffs are summarized in Table.1. The Interceptor’s Gross Lift off Mass (GLOM) varies as a function of structural mass, payload mass, and speed and acceleration (booster burn time). The system characteristics that provide the desired operational performance should be optimized. The selection of burn time is a compromise between the desire for high acceleration (to increase interceptor reach) and the penalty of high acceleration (larger and heavier boosters to provide greater thrust and withstand greater thermal and mechanical stresses). Interceptors with shorter burn times typically require greater maneuverability on the part of the KKV because the ability to make trajectory corrections with steering commands to the booster ends earlier in the interceptor's flight. Trajectory corrections after booster burnout must be made by the KKV. Greater KKV maneuverability in turn results in greater KKV weight. The need for higher acceleration increases as the time available for interceptor flyout decreases. For a given interceptor speed, decreasing payload mass can provide substantial reductions in total interceptor mass. On the contrary, for a fixed KKV size and mass, the total interceptor configuration must be optimized to meet the required velocity and intercept time. The impact that those characteristics have on interceptor’s speed and mass has to be quantified, analyzed and the performance of the interceptor has to be optimized.

A. Design Objective In aerospace vehicle design minimum take-off mass concepts have traditionally been sought. Since vehicle development costs tend to vary as a function of GLOM, it is considered a minimum development cost concept 12. For the present effort, the design objective is to minimize the GLOM (kg) of the interceptor under the mission constraints and SRM envelope constraints. In doing so, we try to configure an optimum propulsion system for interceptor missile to achieve our goal i.e. effective intercept of the target in boost phase. The mission of the interceptor is to deliver a 200kg payload (KKV) to the vicinity of the target to complete the effective intercept. The Baseline Design here is that all three stages are made of sequentially stacked Solid Rocket Motors. The payload (KKV) is enclosed in a fairing whose shape is known beforehand. The number of stages is fixed as three. The discipline wise design variables are shown in Table 2.

B. Design Constraints Interceptor design is constrained by physical and/or performance requirements. The constraints can be categorized as  Mission Constraints  SRM envelope constraints Mission Constraints comprise of miss distance (m), intercept time (min), lateral divert (m/s), velocity at intercept (km/s), G-Loads etc. SRM envelope constraints include stage configuration requirements which comprise of Length to Diameter Ratios, Nozzle expansion ratios, propellant burn rates and Grain geometry constraints like web fraction, volumetric loading efficiency etc. Weight to thrust ratios (ν0) and propellant mass ratio (µp) was constrained to be within allowable ranges. Nozzle exit diameters are constrained to be less than stage diameters. Dynamic penalty function is used to handle in flight and terminal constraints. A symbolic problem statement can be expressed as follows:

Design Variable

Symbol Units

Relative Mass Coefficient of Grain

μki

ratio

Body Diameter

Di

m

Chamber Pressure

pci

bar

Exit Pressure

pei

bar

Coefficient of Grain Shape

Ksi

Grain Burning Rate

ui

Discipline Structure Propulsion Structure Propulsion Aerodynamics Structure Propulsion Structure Propulsion Structure Propulsion

mm /s

Propulsion

Table 2. Design Variables Discipline wise

m

min f ( x)  f ( x)  h(k ) max0, g i ( x)

(1)

i 1

Where f(x) is the objective function, h(k) is a dynamically modified penalty value; k is the current iteration number of the algorithm. The function gi(x) is a relative violated function of the constraints 18.

3 American Institute of Aeronautics and Astronautics

III.

Optimization Approach

The goal of design optimization is to find the optimum (from the Latin word optimus, meaning best) solution to the design problem. The optimization problem as stated above is solved by using evolutionary Genetic Algorithm. First the problem is optimized using exact analysis and second by incorporating Support Vector Machine as a metamodel tool. Both optimization algorithms are shown in Fig. 1. In the first case a set of design variables with upper and lower bounds is passed to Genetic Algorithm which create initial random population and perform its further operations. These candidate design vectors are then passed to calculate exact analysis for weight and sizing, propulsion and trajectory modules. While in second case a set of design variables with upper and lower bounds is passed to Latin Hypercube Sampling to generate a set of design vectors based on design of experiments. These candidate design vectors are then passed to calculate exact analysis for weight and sizing, propulsion and trajectory modules. The corresponding output data is then passed for vector regression. The GA then instead of performing exact analysis will get approximate information from meta-modeling. The constraints are calculated and handled by external penalty function. The algorithm runs in a closed loop via optimizer until optimal solution is obtained. The Genetic Algorithm evolutions for the two optimization algorithms are presented in Fig. 7.

OPTIMIZATION MODULE

Design Variables (X)

______________________

______________________

Genetic Algorithm Find:

Optimum Design Variables (X*)

Satisfy: Constraints Minimize: GLOM, Mg

OPTIMIZATION MODULE

Multidisciplinary Design Analysis Module Vehicle Configuration

DESIGN OF EXPERIMENTS ______________________ Latin Hypercube Sampling

Latin Hypercube Sampling

Trajectory Analysis

Exact Analysis based Optimal Design (X*EXACT)

Find:

Optimum Design Variables (X*)

Satisfy:

Weight Analysis

Propulsion Analysis

Genetic Algorithm

META-MODELING MODULE

Constraints Minimize: GLOM, Mg

______________________ Support Vector Regression

Support Vector Machine

Meta-Model based Optimal Design (X*SVM)

Fig.1 Overall Design and Optimization Strategy

A. Genetic Algorithm Calculus-based optimization (CBO) e.g. gradient descent methods, schemes use sensitivity derivatives in the immediate vicinity of the current solution and can therefore easily fall into local optima from which they cannot recover. To avoid these local optima and to increase the odds of obtaining an acceptable solution these CBO methods require a reasonable starting scheme. GA requires neither sensitivity derivatives nor a reasonable starting solution 19. Before specifying Genetic Algorithm parameters, an extensive parametric study was conducted for the problem in hand by varying one design parameter at a time. Selecting an optimal combination of GA parameters is very difficult because each of the GA parameters is varied individually and the number of combinations of the GA parameters is infinite. Although the values used here are not necessarily optimal but based partly on practices. A population size of 100 with two point crossover function and uniform mutation with rate of 0.025 is used. A tournament selection was used to allow for minimization and to avoid potential scaling concerns Both the population size and the string length influence the choice of the mutation probability, so this also varied with each problem. The Genetic Algorithm stops when the stall generation limit of 50 exceeds, or after 200 generations, whichever occurs first. This allows for a comparison of computational cost for the different strategies.


IV.

Multidisciplinary Design Analysis

The MDO process requires that analysis from separate disciplines be integrated into design optimization process, i.e. modules for propulsion characteristics, aerodynamics, mass properties and flight dynamics have been integrated to produce a high fidelity system model of the entire vehicle. The fundamental requirements for computer programs used in conceptual design are fast turnaround time and ease of use. Fast turnaround is necessary to search a broad solution space with sufficient number of iterations for design convergence. A good design code connects the vehicle physical parameters directly to trajectory code that calculates flight performance. Baseline vehicle data should be imbedded in the code, to facilitate startup. More detailed computational methods are used later in design, when the number of alternative geometric, subsystem, and flight parameters has been reduced to a smaller set of alternatives 20 . A Multidisciplinary Design and Optimization Strategy is envisaged for multi-stage Interceptor analysis which includes: (a) Weight Analysis (b) Propulsion Analysis (c) Intercept Trajectory analysis and (d) Optimization Techniques are included in which the configurations are "optimized" to achieve the maximum performance with minimum GLOM. 21

A. Weight Analysis The heart of any design optimization is the ability to synthesize the weights of the various solid-rocket components through "scaling" equations. In addition, length and volume relationships are included in the analysis. Using a combination of physics-based methods and empirical relations, the weight of the SRM components and propulsion analysis for solid stages is determined 21. The total mass of a multistage interceptor includes the masses of propellants and their tanks, related structures and payload mass. The mass equation for a multi-stage interceptor can be written as: m0i  m pi  mki  m0(i 1)

Where m0i = gross mass of ith stage rocket mpi = mass of propellant of ith stage rocket mki = mass of structure of ith stage rocket m0(i+1) = payload of the ith stage rocket

(2) Forward and aft skirts (thrust skirt) Forward and aft closures

Chamber

Cylindrical section (shell) Attachment joint

The gross lift-off mass m01 of multistage solid fuelled interceptor is calculated as:

Thermal insulation/liner

n

m01  mPAY   (mgni  msti  msvi  masi  m fei  m fsi )

Conic shell

i 1

(3)

Nozzle

Attachment flange Thermal insulation/liner

mo1 

mPAY

Thrust vector control system

n

[1  Ni  K gniuki (1   sti )]

Igniter (4)

i 1

th

Grain

Where m01 is gross mass of the i stage vehicle; mgni Grain is mass of the ith stage SRM grain; msti is mass of the ith Coating stage SRM structure; msvi is mass of control system, safety self-destruction system, servo, and cables inside Figure 2. Mass Model of SRM. the ith stage aft skirt; masi is mass of the ith stage aft skirt including shell structure, equipment rack, heating protect structure, and directly subordinate parts for integration; mfei is mass of equipment and cables inside the ith stage forward skirt; mfsi is mass of the ith stage forward skirt including shell structure, equipment rack, and directly subordinate parts for integration. The mass of the payload mPAY is known from the design mission. Skirt mass ratio 5 American Institute of Aeronautics and Astronautics

Ni, and propellant reserve coefficient Kgni have small dispersions which can be selected from statistical data as described in Ref. 21. Relative mass coefficient μki of effective grain is given below in Eq. (5). It is a design parameter which should be optimized. u ki 

e m gni

moi

(5)

Structure mass fraction (αsti) is the main design parameter. It is dependent upon structural material, grain shape, as well as the parameters of internal ballistics of SRM. This structure mass fraction is the ratio of the sum of the chamber case mass (mcc), cementing layer mass (mcl), nozzle mass (mn) and insulation liner mass (min) to the grain mass (mgni), as shown in Eq. (6):  sti 

mcc 

mcl 

mn 

mcc  mcl  mn  min mgni

(6)

fpc cc      1 D3  i  2 gni  i

(7)

 2

cl gni 1    Di3

(8)

ks. gi ui  gni nav RcTc  n  Ae  3   1 gni Di o pc sin  n A  t 

mincc  Kin  2  gni  in Di3

(9) (10)

Where, f is the factor of safety, ρ is the density, σ is the strength, ε is ratio of cementing layer to SRM diameter, (T) is the combustion temperature, (αn) is ratio of nozzle wall thickness to stage diameter, KIN is ratio of insulation layer thickness to stage diameter (Di) and (ψi) is grain volumetric efficiency. To begin with, we have not restricted to a particular shape of grain at preliminary design level, rather a variable ksi is used to represent the burning surface area (Sri) of grain as a function of grain length (Li) and diameter (Di). The chamber Pressure (pc) is an important design variable which has an affect on the motor specific impulse. Increasing the pc will reduce the losses at the nozzle exit and increase the specific impulse. The pc, however, also has effects on the burning rate of the propellant, combustion stability, size of expansion nozzle and the thickness of the casing materials to with-stand the pressure stresses. Burning surface area of the propellant grain mainly dictates the performance of propulsion system in SRM. mgni 

 4

 gni i gni Di3

(11)

Di   4K gni ki moi  gn i gni 

13

(12)

The grain mass consumed rate is 

m gni   gniui S ri   gniui K si  gni Di

2

(13)

B. Propulsion Analysis Propulsion analysis describes important parameters like thrust, burn time, mass flow rate and nozzle parameter 22 . In order to calculate average specific impulse Isp, the process of a real SRM should be simplified and abstracted to be an ideal SRM. Some assumptions are needed: 6 American Institute of Aeronautics and Astronautics

 The grain is burned perfectly in combustion chamber and its gas is ideal gas, the “specific heat ratio” of the gas keeps constant during expansion.  Average expansion, between the gas phase and condensed phase there are no velocity-lag and temperature-lag.  One dimension flow, the exhaust is parallel to the axis of SRM.  Without viscosity, between gas flow and internal wall, no friction loss and heat dispersion loss. Under the same conditions, the specific impulse of and ideal SRM, which is named “theoretic impulse”, is easy to set up the relationship of thrust (F), specific impulse w.r.t. design parameters. It is recommended to use mass flow rate and specific impulse for calculation of thrust using following relations. 

F1  I spo1 mgi   po  ph  Ae1 F2.. N  I

vac sp

(14)



m gi  ph Aei

p  I sp01  I spv1   e1 pc1  

 1 

p I spvac  I spa   e   pc 

(15)

RcTc po

 1 

RcTc

go2 I spa pe1

(16)

go2 I spa

(17)

I spa  I spred .std  19.4  0.76 pc  0.003 pc2  70 pe  25 pe2

I spred .std  I spstd 1   4.3  0.17 Al  0.009 Al 2 102 

(18) (19)

The working time (tki), grain mass consumption rate, nozzle throat area (At) and expansion ratio (), of ith stage are also calculated in propulsion analysis module: tki   i Di 4ugi ks.gi

(20)



m gni   gn ugi ks.gi gni D

2 i

(21)

At  RcTc  gnugi ks.gi gni D pc 2 i

1



Ae  o At



o   2

 pe    p  c  



 1

2

 

 pe

  1 1   

(22)  pc 

 1 

   

(23)

 1 2 1

(24)

C. Trajectory Analysis Trajectory is the yardstick for evaluating the relative merits of various alternative designs. Since detailed data is not available at beginning of conceptual design, it is inappropriate to use 6DOF trajectory simulation during the conceptual design for the convenient evaluation of guided flight. The development of the required autopilot for 6 DOF guided flight is time consuming, diverting emphasis from other more appropriate considerations 20. Therefore, a 3D model is developed for both interceptor and target with boost phase acceleration profile that depends on total mass, propellant mass and specific impulse in the gravity field. The BPI architecture is modeled as shown in Fig.3.


Space Based Infrared [SBIR] System detects the Target (ICBM) and provides information to the Ground Station

SBIR 2

SBIR 1

Ground Based Radar [GBR] establishes a more accurate initial position, and hands this information to the Ground Station

Ground Station updates the track, determines the Target/ICBM flight path, and sends position and velocity vectors of the target to Interceptor

Interceptor Target

At the appropriate time, the Ground Station commands the launch of Interceptor

GBR continues to track the targets, and provides target track data to the Ground Station to update the Target track, and send updates to the Interceptor GBR boosts to the proximity of the Target and delivers the KKV

GBR2

KKV uses its on-board radar sensor and the target data i.e. position and velocity provided by the Ground Station to identify, home and hit the Target

Ground Station

GBR1

Figure 3. Intercept Scheme.

D. Guidance Algorithm The guidance algorithm used is standard proportional navigation. The system commands accelerations normal to Line of Sight (LOS) between the interceptor and the target, proportional to the closing velocity (Vc) and the line of site rate. Mathematically, the guidance law can be stated as

Lateral Acceleration Achieved nL Flight Control System

nC  N 'VC 

Lateral Acceleration Commanded nc

(25)

where N' is the effective navigation ratio or gain. It is assumed that there is a perfect seeker and a perfect radar system so that the target position and velocity are known exactly. For preliminary design studies these two assumptions are appropriate. Typical ranges for N' are 3 to 5 (unitless) according to Ref. 23 for tactical weapon systems.

Interceptor Dynamics

-

Navigation Coefficient N’ LOS Rate ’ Closing Velocity Vc

Figure 4. Guidance Algorithm.


+

True Target Position

V.

Meta-modeling

Much of today’s engineering analysis requires running complex and computationally expensive analysis and simulation codes. Despite continuing increases in computer processor speeds and capabilities, the huge time and computational costs of running complex engineering codes maintains pace. A way to overcome this problem is to generate an approximation of the complex analysis code that describes the process accurately enough, but at a much lower cost. Such approximations are often called “meta-models” in that they provide a “model of the model” 24. Mathematically, if the inputs to the actual computer analysis are supplied in vector x, and the outputs from the analysis in vector y, then the true computer analysis code evaluates: y  f (x)

(26)

Where f(x) is a complex engineering analysis function. The computationally efficient meta-model approximation is:

such that:

ŷ g (x)

(27)

y  ŷ+ ε

(28)

Where εincludes both approximation and random errors. There currently exist a number of meta-modeling techniques to approximate f(x) with g(x), such as polynomial response surface models, multivariate adaptive regression splines, radial basis functions, kriging, and neural networks, and a recent comparison of the first four of these metamodeling techniques can be found in Ref. 25. All of these techniques are capable of function approximation. In particular, although neural networks are able to approximate very complex models well, they have the two disadvantages of (i) being a “black box” approach and (ii) having a computationally expensive training process. “Black box” means that little can be seen and understood about the model, because an exact function is not generated, only a trained “box” that accepts inputs and returns outputs. Support Vector Machines 26 is a powerful methodology for solving problems in nonlinear classification, function estimation and density estimation which has also led to many other recent developments in kernel based methods in general. Originally, it has been introduced within the context of statistical learning theory and structural risk minimization. In the methods one solves convex optimization problems, typically quadratic programs. The computationally efficient theory behind SVR is presented, and SVR approximations are compared against the aforementioned four meta-modeling techniques using a test bed of 22 engineering analysis functions in Ref. 14 and it was concluded that SVR achieves more accurate and more robust function approximations than above mentioned four meta-modeling techniques and shows great promise for future meta-modeling applications. Least Squares Support Vector Machines (LS-SVM) are reformulations to the standard SVMs which lead to solving linear Karush-Kuhn-Tucker (KKT) systems. LS-SVMs are closely related to regularization networks and Gaussian processes but additionally emphasize and exploit primal-dual interpretations. Links between kernel versions of classical pattern recognition algorithms such as kernel Fisher discriminant analysis and extensions to unsupervised learning. Robustness, sparseness and weightings can be imposed to LS-SVMs where needed 15. To cut down the computational cost, surrogate models (LS-SVR), are constructed from and then used in lieu of the actual simulation models. B. Least Square Support Vector Machine Some typical choices of positive definite kernels are linear, polynomial, Radial Basis Function (RBF) and Multi Layer Perceptrons (MLP) kernel. In classification problems one frequently employs the linear, polynomial and RBF kernel. In nonlinear function estimation and nonlinear modeling problems one often uses RBF kernel 15. In order to make a LS-SVM model with RBF kernel, we need two important parameters: γ (gam) is the regularization parameter, determining the trade-off between the fitting error minimization and smoothness. In the case of RBF kernel, σ2 (sig2) is the bandwidth. The tuning of these two parameters is utmost important to get good results.


C. Latin Hypercube Sampling (LHS) Design of Experiments strategies are often used to sample the design space to generate sample data to fit an approximate model to each of the output variables (responses) of interest. Thus, sample points should be chosen to fill the design space for computer experiments. Sampling is a statistical procedure which involves the selection of a finite number of individuals to represent and infer some knowledge about a population of concern. Space-filling designs are used when there is little or no information about the underlying effects of factors on responses, as in the case under study. The aim is to spread the points as evenly as possible around the operating space. These designs literally fill out the n-dimensional space with points that are in some way regularly spaced. Random Sampling generated from the marginal distributions, is also referred to as pseudo random, as the random numbers are machine generated with deterministic process. Statistically, random sampling has advantages, as it produces unbiased estimates of the mean and the variance of the output variables. Latin hypercube sampling (LHS) is a stratified random procedure that provides an efficient way of sampling variables from their multivariate distributions. LHS is better than random sampling for estimating the mean and the population distribution function. LHS is asymptotically better than random sampling in that it provides an estimator (of the expectation of the output function) with lower variance. In particular, the closer the output function is to being additive in its input variables, the more reduction in variance. LHS yields biased estimates of the variance of the output variables. It was initially developed for the purpose of Monte-Carlo simulation, efficiently selecting input variables for computer models 27, 28 and has been used 29, 30. LHS follows the idea of a Latin square where there is only one sample in each row and each column. Latin hypercube generalizes this concept to an arbitrary number of dimensions. In LHS of a multivariate distribution, a sample size n from multiple variables is drawn such that for each variable the sample is marginally maximally stratified. A sample is maximally stratified when the number of strata equals the sample size n and when the probability of falling in each of the strata is n-1. Given k variables X1; . . . ;Xk the range of each variable X is divided into n equally probable intervals (strata), then for each variable a random sample is taken at each interval (stratum). The n values obtained for each of the variables are then paired with each other either in a random way or based on some rules. Finally we have n samples, where the samples cover the n intervals for all variables. This sampling scheme does not require more samples for more dimensions (variables) and ensures that each of the variable in X is represented in a fully stratified manner. The LHS algorithm is as follows: divide the distribution of each variable X into n equiprobable intervals; for the ith interval, the sampled cumulative probability is: Probi = ( 1/n ) ru + ( i - 1) / n

(29)

where ru is a uniform random number ranging from 0 to 1; transform the probability into the sampled value X using the inverse of the distribution function F-1: X = F-1 (Prob)

(30)

the n values obtained for each variable X are paired randomly or in some prescribed order with the n values of the other variables. LHS offers flexible sample sizes while ensuring stratified sampling, i.e., each of the input variables is sampled at n levels. These designs can have relatively small variance when measuring output variance. LHS “space filling” design strategy is used to treat all regions of the design space equally.

D. Training of Least Square Support Vector Machine Least Square Support Vector Machine is proposed with multiple output regression in this study. Training data of space-filling design points is generated using Latin Hypercube Sampling (LHS) technique. The LS-SVM are trained from a set of design points obtained from LHS. The performance of trained LS-SVM is evaluated by comparing its simulated output with predicted values from test data of Latin Hypercube Sampled design points. This training and evaluation procedure, as shown in Fig.5, is repeated for different values of σ2 (sig2) and γ (gamma) combinations to find out the best combination with least Mean Absolute Error (MAE). After training, the support value and bias term for each LS-SVM is saved for LS-SVM cascade being used in optimization routine. 10 American Institute of Aeronautics and Astronautics

Initialize Design Space _____________________

SELECT EXPERIMENTAL DESIGN _____________________________________  Central Composite Design  Orthogonal Array  Latin Hypercube Sampling

Lower bound ≤ xi ≤ Upper bound

Pruning _____________________ Removal of unreasonable designs

Sample Design Space

CONSTRUCT SURROGATE MODEL

__________________________ Training Data

_____________________________________  Response Surface  Kriging  Least Square – Support Vector Machine

α k, b Validate Approximations _____________________ Using validation data

Figure 5. Meta-Model Developing Sequence

The performance of trained LS-SVM is evaluated by comparing its simulated output with target values from additional test data of Latin Hypercube Sampled design points. 

Average Absolute Error(MAE)

Root Mean Square Error(RMSE)

max | yi  y i | , i = 1,…,nerror



nerror

i 1

(31)



( yi  y i ) 2

(32)

nerror

The difference between each predicted value (ŷi) and the actual function value (yi) calculated as the error for that test point is shown in Fig.6. This training and evaluation procedure is repeated by different values of sig2 and gam combinations. The error measures calculated to assess accuracy by using MAE are listed in Table 3.

S.No 1 2 3 4

RBF Kernel Least Square Support Vector Machine (Performance Parameters) Miss Distance, m Range, km Intercept Time, s GLOM, Mg

σ2 (sig2)

γ (gam)

Mean Absolute Error (MAE)

15 15 15 15

10 10 10 5

0.0187 0.0113 0.0001 0.0173

Table 3. Least Square Support Vector Machine Architecture and Regression Performance 11 American Institute of Aeronautics and Astronautics

180

55

65

160

60

50

120 100 80

MISS DISTANCE (m)

55 MISS DISTANCE (m)

MISS DISTANCE (m)

140

50 45 40

60

35

40

30

45

40

35

30

0

10

20

30

40

50

60

70

80

90

25

100

0

10

20

30

40

50

60

70

80

90

25

100

0

418

416

416

414

414

414

412

412

412

410 408

GBI Range (km)

418

416

GBI Range (km)

418

410 408

406

404

404

404

402

402

10

20

30

40

50

60

70

80

90

100

0

10

20

30

40

50

60

70

80

90

402

100

2.52

2.52

2.5

2.5

2.5

2.48

2.48

2.48

2.44 2.42

2.46 2.44 2.42

2.4

2.38

2.38

30

40

50

60

70

80

90

2.36

100

0

10

20

30

40

50

60

70

80

90

2.36

100

36

36

36

34

34

34

32

32

32

30

30

28

28

26 24

26 24

22

22

20

20

18

18

16

0

10

20

30

40

50

60

70

80

90

100

16

70

80

90

100

0

10

20

30

40

50

60

70

80

90

100

0

10

20

30

40

50

60

70

80

90

100

30 GLOW (Mg)

GLOW (Mg)

GLOW (Mg)

20

60

2.42

2.38

10

50

2.44

2.4

0

40

2.46

2.4

2.36

30

2.52

Intercept Time (sec)

2.46

20

408

406

0

10

410

406



GBI Range (km)

20

28 26 24 22 20

0

10

20

30

40

50

60

70

80

90

100

18

0

10

20

30

40

50

60

70

80

90

100

Figure 6. Performance Validation of LS_SVM Metamodel

E. Results and Discussion Each optimization algorithm architecture is executed five times with exact function and meta-model to find performance index for conceptual design problem of interceptor. Multiple number of runs are used in order to reduce some of the stochastic error that can normally be present in each individual run. The best feasible fitness value is the lowest fitness i.e., minimum GLOM ever encountered that does not violate the constraints. The total number of function counts required to reach the stopping criterion was also computed. Both of these values were used to compare the performance of the computational algorithm. Most significant contribution is the drastic reduction in computational time required for convergence by using meta-model based on Support Vector Machine. The number of function counts needed to converge reflected the performance of these two architectures shown below. Meta-model based optimization architecture was observed computationally more attractive choice than exact 12 American Institute of Aeronautics and Astronautics

analysis architecture. The proposed scheme is especially beneficial for complex, nonlinear and expensive engineering analysis required for optimization. The proposed scheme is especially beneficial for resource hungry expensive engineering analysis required for optimization. 40

40 Exact Analysis Based

Meta-Model Based 35 GLOM [Mg]

GLOM [Mg]

35 30

30

25

25

20

20

0

50 100 150 Number of generations

200

0

50 100 150 Number of generations

200

Figure 7. Convergence Comparison between Exact Analysis and Meta-Model based optimization A comparison of the Exact analysis and Meta-model based optimized configuration using is shown in Fig. 8 and Table 4. The stage-wise weight distribution is shown Fig.8 and performance is depicted in Fig. 9. Exact Analysis Optimized

Meta-Model (SVM) Optimized

GLOM (Mg)_________________________________ 23.139 Kill Vehicle_____________________________________ Mass (kg): 200

GLOM (Mg)_________________________________ 24.644

Stage III_________________________________________ Mass kg 940 Propellant Mass kg 793 Thrust kN 46.78

Stage III_________________________________________ Mass kg 1017.3 Propellant Mass kg 863.2 Thrust kN 51.6

Stage II_________________________________________ Mass kg 4542 Propellant Mass kg 3951 Thrust kN 208.06

Stage II_________________________________________ Mass kg 4818.5 Propellant Mass kg 4203.3 Thrust kN 220.4

Stage I__________________________________________ Mass kg 17457 Propellant Mass kg 15547 Thrust kN 596.15

Stage I__________________________________________ Mass kg 18610.8 Propellant Mass kg 16578.9 Thrust kN 649.1

Kill Vehicle _____________________________________ Mass (kg): 200

Figure 8. Comparison between Exact Analysis and Meta-Model based optimized configurations


Design Variables

X

Symbol

1

Relative Mass Coefficient of Grain 1 Stage

2

Ratio of Relative Mass Coefficient of Grain 2nd to1st Stage rd

nd

3

Ratio of Relative Mass Coefficient of Grain 3 to 2 Stage

4

Body Diameter 1st Stage st

LB

UB

X*EXACT

X*SVM

μk1

0.6

0.7

0.6719

0.6727

1

1.04

1.0350

1.0353

μk3 /μk2 D1

nd

Units

μk2 /μk1

st

m

1

1.08

1.0010

1.0183

1.2

1.8

1.5301

1.5301

5

Ratio of Body Diameter 2 to 1 Stage

D2 /D1

0.7

0.95

0.7029

0.7034

6

Ratio of Body Diameter 3rd Stage to 2nd Stage

D3 /D2

0.7

0.95

0.9400

0.9394

7

st

Pc1

bar

50

70

68.738

69.81

nd

Chamber Pressure 1 Stage

8

Chamber Pressure 2 Stage

Pc2

bar

40

60

59.145

59.07

9

Chamber Pressure 3rd Stage

Pc3

bar

30

50

30.103

30.97

10 Exit Pressure 1 Stage

Pe1

bar

0.5

0.9

0.7509

0.7605

11 Exit Pressure 2nd Stage

Pe2

bar

0.15 0.325

0.1589

0.1600

st

rd

12 Exit Pressure 3 Stage

Pe3

bar

0.1

0.25

0.2044

0.2095

13 Grain Burning Rate 1st Stage

u1

mm/sec

5

11

10.128

10.097

u2

mm/sec

5

10

9.105

8.961

15 Grain Burning Rate 3 Stage

u3

mm/sec

5

9

7.258

7.296

16 Coefficient of Grain Shape 1st Stage

ks1

1.5

2.3

1.6921

1.7297

17 Coefficient of Grain Shape 2nd Stage

ks2

1.5

2.3

1.5343

1.5539

18 Coefficient of Grain Shape 3rd Stage

ks3

1.5

2.3

2.0005

2.0169

nd

14 Grain Burning Rate 2 Stage rd

Table 4. Comparison between Exact Analysis and Meta-Model based optimized configurations 8

25 Configuration (EXACT ANLAYSIS) Configuration (METAMODEL BASED)

7

Configuration (EXACT ANLAYSIS) Configuration (METAMODEL BASED) 20

5

Mass (Mg)

Velocity (km/s)

6

4 3

15

10

2 5 1 0

0

50

100

0

150

0

Intercept Time (s)

50

100

150

Intercept Time (s)

600

120

500

100

400

80

Altitude (km)

Distance from Target (km)

Configuration (EXACT ANLAYSIS) Configuration (METAMODEL BASED)

300

200

100

0

50

100

Intercept Time (s)

40

20

Configuration (EXACT ANLAYSIS) Configuration (METAMODEL BASED) 0

60

150

0

0

50

100

150

Intercept Time (s)

Figure 9. Performance Comparison between Exact Analysis and Meta-Model based optimized configurations 14 American Institute of Aeronautics and Astronautics

All the optimized design variables (X*) lie between their upper (UB) and lower bounds (LB). Required velocity (≥ 6km/s) and intercept time (≤150s) are perfectly achieved. The meta-model based design and optimization strategy proved able to provide a conceptual design considering the performance objectives and design constraints. Although the training of the metamodel is time consuming and requires extensive runs of the model but once the training is achieved there is a tremendous reduction in optimization time, as Figure 10 shows that there is a significant reduction in computational cost when optimization is carried using SVM based meta-modeling without any penalty on the performance objectives.

16 14 12 10 8 6 4 2 0

Average Computational Time (hrs)

Exact Analysis Figure 10.

Meta-Model

Comparison of Computational Cost.

F. Concluding Remarks An integrated methodology for multidisciplinary optimization of multi-stage ground launched Boost Phase Interceptor was successfully implemented using Least Square Support Vector Regression. It proved able to provide conceptual design considering propulsion and mass features for each stage and trajectory/performance objectives and constraints. The most important accomplishment of this research is the optimization with meta-modeling at conceptual design phase for a solid fueled multistage interceptor. The most significant achievement gained through this analysis is tremendous reduction in computational cost by using meta-model during conceptual design level. Although there is a small percentage of error in the results obtained from the meta-model approach, this is tolerable because the methodology solves an approximation function, and the results are accurate enough in the conceptual design phase. Such a design strategy will allow vehicle designers to rapidly consider a number of fully converged design alternatives in a very short time without sacrificing design detail, thus improving the quality of conceptual design process. Due to ease of implementation and their potential to solve aerospace problems, non gradient methods like GA are more attractive choice than traditional design and analysis tools when complicated non-linear phenomena dominate the optimization space. Though, the optimization results and performance are to be considered as preliminary (proof-of-concept) only, but they can be compared to existing systems and used for conceptual design of ground based interceptors. The proposed design and optimization methodology provides the designer with a time efficient and powerful approach for the design of interceptor systems.

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