Fenfen Xiong

Optimizing Latin hypercube design for sequential sampling of computer experiments

Space-filling and projective properties are desired features in the design of computer experiments to create global metamodels to replace expensive computer simulations in engineering design. The goal in this article is to develop an efficient and effective sequential Quasi-LHD (Latin Hypercube design) sampling method to maintain and balance the two aforementioned properties. The sequential sampling is formulated as an optimization problem, with the objective being the Maximin Distance, a space-filling criterion, and the constraints based on a set of pre-specified minimum one-dimensional distances to achieve the approximate one-dimensional projective property. Through comparative studies on sampling property and metamodel accuracy, the new approach is shown to outperform other sequential sampling methods for global metamodelling and is comparable to the one-stage sampling method while providing more flexibility in a sequential metamodelling procedure.

Stability Limits of Spinning Missiles with Attitude Autopilot

Cr = control moment derivative coefficient, 1=rad C = static moment derivative coefficient, 1=rad C _ = damping moment derivative coefficient, s=rad IP, IQ = roll and lateral moment of inertia, kg m kr = dynamic gain of servo system ks = gain of servo system k! = gain of rate feedback kz = gain of attitude feedback L = reference length, m ncy, ncz = control command along Oy4 and Oz4 Ox5y5z5 = nonspinning velocity coordinate q = dynamic pressure, N=m S = reference area, m Ts = reciprocal of natural frequency of servo system, s = roll angle, rad c = coupling angle of the servo system, rad d = total delay angle of the system, rad l = lead angle of the command, rad m = roll angle measured by feedback element, rad = nutation angle, rad 2, 1 = nutation angle on plane Ox5y5 and plane Ox5z5, rad # = pitch angle, rad s = damping ratio of servo system cz, cy = command of servo in nonspinning body coordinate, rad 1, 2 = control surface angle in nonspinning body coordinate, rad = time delay of control system, s = yaw angle, rad

Enhanced probabilistic analytical target cascading with application to multi-scale design

Probabilistic analytical target cascading (PATC) is an approach for multi-level multi-disciplinary design optimization under uncertainty. In the original PATC approach, only the mean and variance of each interrelated response and linking variable are matched in a multi-level hierarchy. The ignorance of response correlation introduces difficulties in finding optimal solutions especially when the covariance of interrelated responses has a significant impact. In this article, an enhanced PATC (EPATC) approach is proposed. In addition to matching the first two statistical moments, the covariance between the interrelated responses is also considered by applying a modified updating strategy for estimating the statistical performance of an upper-level subsystem. A mathematical example and a multi-scale design problem are used to demonstrate the effectiveness and efficiency of the proposed EPATC approach. This study shows that the EPATC approach outperforms the original PATC by providing more accurate optimal solutions.

Parallel particle swarm optimization on a graphics processing unit with application to trajectory optimization

In order to reduce the computational time, a fully parallel implementation of the particle swarm optimization (PSO) algorithm on a graphics processing unit (GPU) is presented. Instead of being executed on the central processing unit (CPU) sequentially, PSO is executed in parallel via the GPU on the compute unified device architecture (CUDA) platform. The processes of fitness evaluation, updating of velocity and position of all particles are all parallelized and introduced in detail. Comparative studies on the optimization of four benchmark functions and a trajectory optimization problem are conducted by running PSO on the GPU (GPU-PSO) and CPU (CPU-PSO). The impact of design dimension, number of particles and size of the thread-block in the GPU and their interactions on the computational time is investigated. The results show that the computational time of the developed GPU-PSO is much shorter than that of CPU-PSO, with comparable accuracy, which demonstrates the remarkable speed-up capability of GPU-PSO.

Trajectory Optimization under Uncertainty based on Polynomial Chaos Expansion

A general procedure of trajectory optimization under uncertainty, which considers probabilistic uncertainties from both initial state and system parameter under both path and boundary constraints, is presented in this paper. With the proposed method, based on the robust design theory, the original stochastic trajectory optimization problem is transformed into an equivalent deterministic one in the expanded higher-dimensional state space by the polynomial chaos expansion method. Quantification of the stochastic cost, boundary and path constraints in terms of polynomial chaos expansion is described in detail in a straightforward way. Through the application of the proposed procedure to two examples of optimal trajectory generation, it is observed that the obtained optimal solutions are evidently less sensitive to uncertainties and more reliable compared to that of the deterministic optimization, which demonstrates the effectiveness of the proposed method.

A New Weighted Stochastic Response Surface Method for Uncertainty Propagation

Conventional stochastic response surface method (SRSM) based on polynomial chaos expansion (PCE) for uncertainty propagation treats every sample points equally during the regression process and may produce inaccurate coefficient estimations in PCE. A new weighted stochastic response surface method (WSRSM) to overcome such limitation by considering the sample probabilistic weights in regression is studied in this work. Techniques that associate sample probabilistic weights to different sampling approaches such as Gaussian Quadrature point (GQ), Monomial Cubature Rule (MCR) and Latin Hypercube Design (LHD) are developed. The proposed method is demonstrated by several mathematical and engineering examples. Results show that for various sampling techniques, WSRSM can consistently improve the accuracy of uncertainty propagation compared to the conventional SRSM without adding extra computational cost. Insights into the relative accuracy and efficiency of using various sampling techniques in implementation are provided.

A new probabilistic distribution matching PATC formulation using polynomial chaos expansion

In the existing probabilistic hierarchical optimization approaches, such as probabilistic analytical target cascading (PATC), all the stochastic interrelated responses are characterized only by the first two statistical moments. However, due to the high nonlinear relation between the inputs and outputs, the interrelated responses are not necessarily normally distributed. The existing approaches, therefore, may not accurately quantify the probabilistic characteristics of the interrelated responses, and would further prevent achieving the real optimal solution. To overcome this deficiency, a novel PATC approach, named PATC-PCE is developed. By employing the polynomial chaos expansion (PCE) technique, the entire distribution of interrelated response can be characterized by a PCE coefficients vector, and then matched and propagated in the hierarchy. Comparative studies show that PATC-PCE outperforms PATC in terms of yielding more accurate optimal solutions and fewer design cycles when the interrelated response are random non-normal quantities, while at a sacrifice of extra function evaluations.

Uncertainty propagation techniques in probabilistic design of multilevel systems

In hierarchical multilevel systems, information (interrelated responses) is passed among levels following a bottom-up sequence. One of the primary challenges for multilevel system design optimization under uncertainty is associated with the quantification of uncertainty propagated across multiple levels. Two newly developed uncertainty propagation techniques, the full numerical factorial integration method and the univariate dimension reduction method, are compared through their employment in probabilistic design of multilevel system. The Probabilistic Analytical Target Cascading (PATC) approach is used for solving the probabilistic multilevel hierarchical problems as well as demonstrating the two uncertainty propagation techniques. Covariance among the interrelated responses between neighboring levels is considered to improve the accuracy of the statistics estimation of upper-level outputs. Linear transformation of the correlated interrelated responses is adopted to facilitate the application of the uncertainty propagation techniques in PATC. The Monte Carlo method is used as the benchmark to verify the accuracy of these techniques.

An enhanced data-driven polynomial chaos method for uncertainty propagation

ABSTRACT As a novel type of polynomial chaos expansion (PCE), the data-driven PCE (DD-PCE) approach has been developed to have a wide range of potential applications for uncertainty propagation. While the research on DD-PCE is still ongoing, its merits compared with the existing PCE approaches have yet to be understood and explored, and its limitations also need to be addressed. In this article, the Galerkin projection technique in conjunction with the moment-matching equations is employed in DD-PCE for higher-dimensional uncertainty propagation. The enhanced DD-PCE method is then compared with current PCE methods to fully investigate its relative merits through four numerical examples considering different cases of information for random inputs. It is found that the proposed method could improve the accuracy, or in some cases leads to comparable results, demonstrating its effectiveness and advantages. Its application in dealing with a Mars entry trajectory optimization problem further verifies its effectiveness.

Notice of Retraction Gliding trajectory optimization based on hp-adaptive pseudospectral method

Trajectory design plays an important role in the overall design optimization of an aero-craft. As a classic method to solve trajectory optimization design problem, the indirect method is complex and very sensitive to the initial guesses of castrates. As opposed to indirect method, the direct method is convenient and can obtain the optimal solution with higher probability. Therefore, in the present paper, the recently developed direct method, hp-adaptive pseudospectral approach, is applied to a gliding trajectory optimization problem to generate an optimal gliding trajectory yielding maximum range. However, it has been frequently noticed that the direct method may end up with suboptimal or even poor solution since it is only an approximation of the original optimal control problem, which cannot be adopted in engineering and needs to be verified beforehand. To address this issue, the indirect method is employed to verify the optimality of the solution from the hp-adaptive pseudospectral method. Simulation results show that the verified optimal solution meets the dynamic constraints well and evidently improves the range compared to the original trajectory.