A Bayesian Approach to Non-Deterministic Hypersonic Vehicle Design

Affordable, reliable endoand exoatmospheric transportation, for both the military and commercial sectors, grows in importance as the world grows smaller and space exploration and exploitation increasingly impact our daily lives. However, the impact of disciplinary, operational, and technological uncertainties inhibit the design of the requisite hypersonic vehicles, an inherently multidisciplinary and non-deterministic process. Without investigation, these components of design uncertainty undermine the designers’ decision-making confidence. In this paper, the authors propose a new probabilistic design method, using Bayesian Statistics techniques, which allows assessment of the impact of disciplinary uncertainty on the confidence in the design solution. The proposed development of a two-stage reusable launch vehicle configuration highlights the means to first quantify the fidelity of the disciplinary analysis tools utilized, then propagate such to the vehicle system level. BACKGROUND For the second time since its inception, the United States Air Force (USAF) Scientific Advisory Board (SAB) researched the future needs and present shortcomings in the USAF’s overall mission effectiveness. This assessment includes definitions of and requirements for future mission areas. The SAB compiled their findings in a series of volumes, collectively known as the New World Vistas, describing aerospace systems concepts with the potential to provide the greatest mission capabilities to the future Air Force [1]. For many cases, the United States needs to develop and field hypersonic systems to best equip its Air Force for its missions. In fact, the SAB considers hypersonic vehicles one of seven top-level system categories to answer the challenges facing tomorPh.D. Candidate, ASDL Director, ASDL; Professor and Boeing Chair in Advanced Aerospace Systems Analysis row’s Air Force. Specifically, its members envision hypersonics taking operational form as tactical and strategic missiles, global-reach spaceplanes, and reusable launch vehicles. These systems could fulfill the strike, bombing, mobility, reconaissance, and space access roles more effectively than other candidates, owing to their higher speeds and lethality. Market forecasts reveal strong growth in worldwide passenger travel. This growing demand, and the desire for shorter trip times, point to a hypersonic commercial transport to satisfy passenger’s needs. A correspondingly stronger world air cargo market, and the desire for faster route travel, indicate a possible freighter role for the hypersonic transport as well. Growing opportunities exist for commercial vehicles for access to space as well. Taking the form of reusable launch vehicles (RLVs), commercial hypersonic systems would fulfill the missions of rapid intercontinental transport, satellite delivery and on-orbit maintenance, and civil space missions including manned spaceflight and the International Space Station [2]. Cost drives the desire to shift the world’s space launch burden off of its fleet of expendable launchers. The expense of operating the Space Shuttle, and expendable launchers such as the Titan IV, constrains commercial and government efforts in space, especially with today’s declining budgets [3]. Reusable launch vehicles (RLV’s) potentially offer more reliable and affordable access to space. However, NASA cannot retire the veteran Shuttle until commercial RLVs not only become available, but have demonstrated safe and reliable operation [2]. MOTIVATION DISCIPLINARY UNCERTAINTY The lack of a credible multidisciplinary simulation capability for hypersonic vehicles stands between those vehicles in service today and those needed tomorrow. Any aircraft design process re1 quires a multidisciplinary approach to vehicle synthesis and sizing to best evaluate the compromises between the various disciplines. Hypersonic vehicle design takes this “necessary evil” to the extreme, for it exhibits an exceptionally tight coupling of disciplines [4], as shown in Figure 1. The more stringent requirements and constraints of the design force this close coupling, with greater sensitivity to changes. Figure 1: Coupling of disciplines in hypersonic vehicle design [4] A synthesis and sizing capability takes form as an automated computational environment of integrated disciplinary analysis tools [5]. The term “disciplines” implies both those in the classic engineering sense, such as aerodynamics and propulsion, and the product life-cycle sense, including manufacturing and operations. Analysis tools may run the gamut from simple regression equations, to physicsand process-based analyses (e.g., computational fluid dynamics), to experimental databases. The fidelity of the synthesis and sizing environment’s constituent analyses comes into question [6], as do their algorithmic accuracy. The use of lower-fidelity tools results from trading accuracy for computational speed, or from the utter lack of tools of higher analytical fidelity. Lowerfidelity tools implement first-order analyses in the form of regression of historical data or oversimplifications of the physics involved. As such, they utilize only a minimal input vehicle configuration, and in turn, require a minimum of time for problem setup and execution. These tools thus run quickly, yet provide dubious results. The inability to analytically predict the exact value of parameters defines disciplinary uncertainty, the uncertainty ultimately manifest in the use of lower-fidelity tools. Other than accounting for the impact of uncertainty, only the use of higher-fidelity tools mitigates this problem. However, their physics-based analyses require detailed input and long setup and execution times, rendering their use at each design iteration impractical. Also, higherfidelity codes don’t completely mitigate the problem of design uncertainty. For example, such codes include calibration factors to finely adjust the values of the outputs; these factors exist to circumvent the uncertainty in calculations of the given values. Also, particularly in the hypersonic flight regime, highfidelity disciplinary analysis tools elude use by vehicle designers. The hypersonic flight literature documents a number of efforts to develop such analyses, including those for aerodynamics (References [7, 8]) and propulsion (References [9, 10]). Meanwhile, several more efforts seek experimental solutions, as described in References [11–14]. Yet Blankson et al. discuss the “inability of ground facilities to generate hypersonic test data at real flight conditions for validating design tools” [15]. The tendency of flight test data “to raise more questions than answers” compounds the problem [16]. Until recent efforts come to fruition, disciplinary uncertainty will hamper conceptual and preliminary hypersonic vehicle design. The inadequacies of test facilities contribute to the lack of knowledge of the hypersonic aerothermodynamic environment, resulting in extreme limitations of current multidisciplinary simulation capability [15]. ACCOUNTING FOR UNCERTAINTY NASA Administrator Daniel Goldin states that [17] . . . the hypersonic and space environments are filled with uncertainty, so traditional numerical approaches will not work. . . In order to account for the uncertainty and to quantify the risk level, we need to move from the traditional deterministic methods to non-deterministic methods. . . Deterministically derived, single-value solutions fail to capture the effects of uncertainty, a random phenomenon by definition. Thus, probability distributions must serve to represent uncertain quantities, for “probabilities are the language of uncertainty; probability laws are the grammar of that language” [18]. Past research by the Aerospace Systems Design Laboratory asserts and demonstrates this point in modeling disciplinary, operational, and technological uncertainty [19–21]. Figure 2, based on Reference [22] and professional experience, depicts a representative design space exploration process for probabilistic system design. From a given mission, and a “baseline” vehicle sized to that mission, the process starts with the definition of a design space. The design variables of interest, and ranges of values for those variables, define the design space. Inclusion of operational uncertainty requires further definition by way of relevant noise factors, and ranges of possible values for them. Probability distributions further define each variable. Control factors receive uniform distributions, for they lie within the designer’s control and thus may take any value in the range with equal likelihood. Noise factors receive such probability distributions as Normal or Beta to represent their randomness. Commencement of the pro2 Figure 2: Generic probabilistic design environment ! " # $&% #!' ( ) * + , # * -.+ ' ' $ % # * , # / 0 1 2 3 4 576 8 9 :<;>= ? @ 6 A B C DFE G HJI>K LNM O P Q P R O&S T M U R!V V W X Y Z [ Z \ Y ] ^ X _ \ ` ` a b b c dfe g hji c k l g m n k o a e g o p.a m m c l g o e g h q r s t u v7w x y z<{>| } ~ w  €  ‚Fƒ „ …J†>‡ ˆ ‰>ŠJ‹!Œ  Š>‹ Ž    ‘“’ ‰>ŠJ‹ ŽJŒ  >‹ ” ” ‰J” •“–“—J˜ ™›š —Jœ>>žJ– Ÿ ¡ ¢ £ £ £ ¤“¥“¦J§ ̈›© ¦Ja>«>¬J¥ ­ ® ̄ ° ± ± ± Figure 3: Modification for design uncertainty quantification cess includes metric identification, by defining objectives and constraints for the system. The vehicle simulation capability comprises a series of disciplinary analysis codes, linked to a synthesis and sizing tool (e.g., a trajectory analysis code), and further linked to an economics (with manufacturing) code for lifecycle cost analysis. The linking of codes in this manner creates an automated multidisciplinary simulation environment, an integral part of any probabilistic design effort [5, 6]. The disciplinary and economic analysis tools in Figure 2 need not be actual computer programs or experimental databases. These may take the form of analysis and cost modules internal to a monolithic design code, integrated with synthesis and sizing routines. Or, they may be metamodels, like Response Surface Equations (RSE’s), linked with a synthesis and sizing tool, a procedure well-documented in the ASDL literature (e.g. References [23–25]). Notice the interim step between the disciplinary analyses and the synthesis and sizing tool. The probability distributions following each analysis in Figure 2 represent the effects of design uncertainty. Instead of “point” values for disciplinary outputs, one set per set of input values, there exist distributions borne of fidelity and model fit error accounting. These distributions then proceed to the synthesis and sizing code. This detailed process of quantifying and propagating disciplinary uncertainty provide the foundation of the proposed work. A probabilistic tool (e.g., the Southwest Research Institute’s Fast Probability Integrator, or FPI) integrates disciplinary uncertainty information, obtained a priori, into the disciplinary outputs. Likewise, the probabilistic tool inputs this information to the synthesis and sizing, and economics codes, ultimately providing the distributions on the system-level responses, i.e., the previously defined objectives and constraints. “Technical feasibility” assessment entails comparison of performance objectives with imposed constraints [26]. For a launch vehicle, one such objective might be gross lift-off weight (GLOW), subject to the constraint of a maximum of 5 million pounds. “Economic viability” assessment likewise compares life-cycle cost objectives to constraints. Again for the launch vehicle, an example is cost per pound of payload to low Earth orbit, constrained to under $1000/lb. The CDF’s 3 generated for each assessment indicate the probability of “success,” or the percentage of the design space defined in the design variable ranges that, subject to uncertainty, satisfies the imposed constraints on the objectives considered. From another perspective, the designer may view the system-level CDF’s as a measure of design confidence; for the point on the CDF curve intersected by the constraint, the corresponding probability value equals the level of confidence in the design achieving that value. In the $/lb example, if the constraint, say $1000/lb, meets the curve at the 30% probability level, then the designer may state he/she believes “the design can meet the target of $1000/lb to LEO with 30% confidence.” PROBLEM STATEMENT “Zoom in” on any one of the disciplinary analysis codes (and its probabilistic output) from Figure 2, and Figure 3 results. The upper half depicts the treatment of disciplinary analyses in existing design methods. The probabilistic tool employed accepts the uniformly distributed control factors of the design space, executes the code (or evaluates the metamodel) accordingly, and outputs the resulting distribution for the subsystem metric. Existing methods, without accounting for the uncertainty in the analysis, treat the output canonically; the point values defining the probability density function (PDF) curve are truly discrete points. The lower half of Figure 3 shows the effect of design uncertainty accounting. Application of the knowledge of the nature of the design uncertainty in question, obtained beforehand, alters the output PDF of the analysis. Each point on the PDF, accounting for uncertainty, becomes a distribution of its own. The output PDF, now a collection of distributions instead of discrete points, takes on a new form. Three questions result from the above consideration; it is these questions that ultimately motivate this work. 1. How does a designer systematically quantify design uncertainty for a disciplinary analysis? 2. By what process does one measure the uncertainty, and then recompute the analysis results in light of the uncertainty? 3. Upon quantifying design uncertainty, how does the designer propagate the results to the vehicle/system level? Through this paper, the authors present the future framework for systematically answering these questions. THEORY: BAYESIAN STATISTICS OVERVIEW Bernardo and Smith state that “Bayesian Statistics offers a rationalist theory of personalistic beliefs in contexts of uncertainty. . . The goal, in effect, is to establish rules and procedures for disciplined uncertainty accounting” [27]. From this scholarly assurance, the authors confidently assert Bayesian Statistics’ ability to answer the research questions posed. Over its long history, the statistics community derived several well-known probability distribution function (PDF) types. This work concerns itself only with the Uniform, Normal, and Beta distributions. Past work in aerospace systems design indicates that the Normal and Beta distributions adequately model uncertainty parameters, whereas the Uniform distribution represents control factors [24–26]. Also, Phillips reveals that in general practice, the Uniform, Normal, and Beta PDF’s suffice to solve any problem in Bayesian Statistics [18]. Equation 1 states Bayes’ Theorem in terms of discrete events, e.g., rolling dice or drawing cards. 2 ́3 and 2¶μ represent n discrete events partitioning a given sample space. These events are partitions because they are assumed to be mutually exclusive, i.e., they have no outcomes in common. B represents a further event, and ·1 ̧ o1⁄4» 2 ́3!1⁄2 equals the probability that event B occurs when it is known that event 2 3 has occurred, also known as the conditional probability of B, conditional on the occurrence of 2 3 . ·1 ̧ 23⁄43 » o 1⁄2À¿ ·1 ̧ 23⁄43 1⁄2ÂÁ ·1 ̧ o1⁄4» 2 ́3!1⁄2 ÃÅÄ μÇÆÉÈ ·1 ̧ 2 μ 1⁄2ÊÁ ·1 ̧ o1⁄4» 2 μ 1⁄2 (1) The above formulation of Bayes’ Theorem represents but one specific perspective on the theory. In general, “opinions are expressed in probabilities, data are collected, and these data change the prior probabilities, through the operation of Bayes’ Theorem, to yield posterior probabilities” [18]. In these more general terms, ·1 ̧ 2 ́3!1⁄2 , the prior probability, represents one’s hypothesis about the outcome of event 23⁄43 . ·1 ̧ o1⁄4» 2 ́3!1⁄2 represents the data collected or observations made when testing the hypothesis. Finally, ·1 ̧ 2 3 » o 1⁄2 , the posterior probability, equals the new hypothesis, or the original hypothesis now “corrected” in light of the new information provided by ·1 ̧ oË» 2 3 1⁄2 . QUESTIONS 1 & 2 QUANTIFYING UNCERTAINTY Past work [24, 25] indicate that consideration of disciplinary uncertainty typically rests with so-called expert opinion of that uncertainty. Expert opinion regarding a code’s fidelity typically stems from observation of a few cases, and thus omits the effects of dissimilar vehicle types and geometries, and flight conditions. In other words, the fidelity uncertainty varies with the specifics of the problem, so “blanket statements” about the fidelity become suspect. Therefore, the belief regarding the nature of the uncertainty requires revision based on data, collected for the purpose of testing the belief. This task is the raison d’être of Bayesian Statistics. In addition, proper accounting for fidelity uncertainty requires observations on multiple data sources which, together with expert opinion, will derive a more representative probability distribution to model disciplinary uncertainty. 4 Theoretical Background Lee presents Bayes’ Theorem reformulated in a form more suited to engineering and design, i.e., in terms of probability density functions (PDF’s) rather than discrete events [28]: