Robust Optimal Onboard Reentry Guidance of a Space Shuttle: Dynamic Game Approach and Guidance Synthesis via Neural Networks

Robust optimal control problems for dynamic systems must be solved ifmodeling inaccuracies cannot be avoided and/or unpredictable andunmeasurable influences are present. Here, the return of a future Europeanspace shuttle to Earth is considered. Four path constraints have to beobeyed to limit heating, dynamic pressure, load factor, and flight pathangle at high velocities. For the air density associated with theaerodynamic forces and the constraints, only an altitude-dependent rangecan be predicted. The worst-case air density is analyzed via an antagonisticnoncooperative two-person dynamic game. A closed-form solution of the gameprovides a robust optimal guidance scheme against all possible air densityfluctuations. The value function solves the Isaacs nonlinear first-orderpartial differential equation with suitable interior and boundaryconditions. The equation is solved with the method of characteristics in therelevant parts of the state space. A bundle of neighboring characteristictrajectories yields a large input/output data set and enables a guidancescheme synthesis with three-layer perceptrons. The difficult andcomputationally expensive perceptron training is done efficiently with thenew SQP-training method FAUN. Simulations show the real-time capability androbustness of the reentry guidance scheme finally chosen.     

Robust optimization for routing problems on trees

The prize-collecting travelling salesman problem (pc-tsp) is a known variant of the classical travelling salesman problem. In the pc-tsp, we are given a weighted graph \(G=(V,E)\) with edge weights \(\ell :E\rightarrow {\mathbb {R}}_+\), a special vertex \(r\in V\), penalties \(\pi :V\rightarrow {\mathbb {R}}_+\) and the goal is to find a closed tour K such that \(r\in V(K)\) and such that the cost \(\ell (K)+\pi (V{\setminus } V(K))\), which is the sum of the weights of the edges in the tour and the cost of the vertices not spanned by K, is minimized. In this paper, we study the pc-tsp from a viewpoint of robust optimization. In our setting, an operator must find a closed tour in a (known) edge-weighted tree \(T=(V,E)\) starting and ending in the depot r while some of the edges may be blocked by “avalanches” defining the scenario \(\xi \). The cost \(f(K,\xi )\) of a tour K in scenario \(\xi \) is the cost resulting from “shortcutting” K, i.e. from restricting K to the nodes which are reachable from r in scenario \(\xi \), i.e. in the graph \(T {\setminus } \xi =(V,E{\setminus }\xi )\). In the absolute robust version of the problem one searches for a tour which minimizes the worst-case cost over all scenarios, while in the deviation robust counterpart, the regret, that is, the deviation from an optimum solution for a particular scenario, is sought to be minimized. We show that both versions of the problem can be solved in polynomial time on trees.     

区间动力系统的鲁棒稳定性分析

本文研究区间系统的鲁棒稳定性问题 ,把连续区间系统的鲁棒Hurwtiz稳定和离散区间系统的鲁棒Schur稳定的问题等价转换于一参数扰动矩阵集的鲁棒非奇异问题 ,然后给出鲁棒Hurwtiz稳定和鲁棒Schur稳定的基于 μ 分析的充分必要条件 .     

Extended and Unscented Kalman filtering based feedforward neural networks for time series prediction

With the ability to deal with high non-linearity, artificial neural networks (ANNs) and support vector machines (SVMs) have been widely studied and successfully applied to time series prediction. However, good fitting results of ANNs and SVMs to nonlinear models do not guarantee an equally good prediction performance. One main reason is that their dynamics and properties are changing with time, and another key problem is the inherent noise of the fitting data. Nonlinear filtering methods have some advantages such as handling additive noises and following the movement of a system when the underlying model is evolving through time. The present paper investigates time series prediction algorithms by using a combination of nonlinear filtering approaches and the feedforward neural network (FNN). The nonlinear filtering model is established by using the FNN’s weights to present state equation and the FNN’s output to present the observation equation, and the input vector to the FNN is composed of the predicted signal with given length, then the extended Kalman filtering (EKF) and Unscented Kalman filtering (UKF) are used to online train the FNN. Time series prediction results are presented by the predicted observation value of nonlinear filtering approaches. To evaluate the proposed methods, the developed techniques are applied to the predictions of one simulated Mackey-Glass chaotic time series and one real monthly mean water levels time series. Generally, the prediction accuracy of the UKF-based FNN is better than the EKF-based FNN when the model is highly nonlinear. However, comparing from prediction accuracy and computational effort based on the prediction model proposed in our study, we draw the conclusion that the EKF-based FNN is superior to the UKF-based FNN for the theoretical Mackey-Glass time series prediction and the real monthly mean water levels time series prediction.     

Hybrid Modelling for Robust Solving

We study a balanced academic curriculum problem and an industrial steel mill slab design problem. These problems can be modelled in different ways, using both Integer Linear Programming (ILP) and Constraint Programming (CP) techniques. We consider the utility of each model. We also propose integrating the models to create hybrids that benefit from the complementary strengths of each model. Experimental results show that hybridization significantly increases the domain pruning and decreases the run-time on many instances. Furthermore, a CP/ILP hybrid model gives a more robust performance in the face of varying instance data.     

Robust linear algorithms for cutsets

Shamir's algorithm for finding a minimum cutset in a (quasi)reducible graph runs in linear time and aborts if it discovers nonreducibility. A slight modification of the algorithm is more useful in contexts where valid input graphs may fail to be (quasi)reducible. The modified algorithm never aborts, always finds a cutset in linear time, and comes close to finding a minimum cutset when the input graph is close to being an acceptable input for the original algorithm. We study the size of the cutset as a function of the choices made in the search order. Examples suggest that the modified algorithm will do well in practice, especially if it is followed by another linear algorithm that scans a given cutset and removes any obviously unnecessary nodes. Detecting and breaking deadlocks in multiprogramming is one of the applications. In particular, the periodic centralized detect-and-break strategy can be implemented with linear time and space bounds and without any risk of starvation. Previous proposals for this strategy have exponential bounds.     

Robust analysis on promotion duration for two competitive brands

There are few studies that provide a useful tool or model to determine the promotion duration during the transition state of customers' switching between different brands. This implies that marketing managers usually decide the promotion duration based on their past experiences. The study integrates the Markov chain, entropy, and diffusion theory to model the problem and find a solution. Furthermore, the Taguchi method is also used to capture the uncertain parameters of the model to solve the problem. A numerical example is used to illustrate how the model determines optimal promotion duration.     

Robust Bounds for Krylov Methods

In this paper we give bounds for polynomials of operators. These bounds are robust in low rank perturbations. This problem is encountered in the study of the convergence of Krylov methods. The central idea here is to view the resolvent as a meromorphic function.     

Robust classification for skewed data

In this paper we propose a robust classification rule for skewed unimodal distributions. For low dimensional data, the classifier is based on minimizing the adjusted outlyingness to each group. In the case of high dimensional data, the robustified SIMCA method is adjusted for skewness. The robustness of the methods is investigated through different simulations and by applying it to some datasets.     

Guidance Trajectories for Spacecraft Rendezvous

In a previous paper of Miele et al. (J. Optim. Theory Appl. 132(1), 2007), we employed the single-subarc sequential gradient-restoration algorithm to optimize the three-dimensional rendezvous between a target spacecraft in a planar circular orbit and a chaser spacecraft with an initial separation distance and separation velocity. The achieved continuous solutions are characterized by two, three, or four subarcs depending on the performance index (time, fuel) and the constraints. In this paper, based on the solutions in Miele et al. (J. Optim. Theory Appl. 132(1), 2007), we employ the multiple-subarc sequential gradient-restoration algorithm to produce pieced guidance trajectories implementable in real time via constant control components. In other words, in this investigation, we force the controls to behave as parameters in each subarc. With the above understanding, we investigate four problems: (P1) minimum time, fuel free; (P2) minimum fuel, time free; (P3) minimum time, fuel given; (P4) minimum fuel, time given. Problem P1 results in a two-subarc solution, each subarc with constant controls: a max-thrust accelerating subarc followed by a max-thrust braking subarc. Problem P2 results in a four-subarc solution, each subarc with constant controls: an initial coasting subarc, followed by a max-thrust braking subarc, followed by another coasting subarc, followed by another max-thrust braking subarc. Problems P3 and P4 result in two, three, or four-subarc solutions depending on the performance index and the constraints, albeit with constant controls in each subarc. For each of the problems studied, the performance index of the multiple-subarc pieced guidance trajectory approximates well the performance index of the single-subarc continuous optimal trajectory of Miele et al. (J. Optim. Theory Appl. 132(1), 2007) as well as the performance index of the multiple-subarc pieced optimal trajectory: the pairwise relative differences in performance index are less than 1/100. This research was supported by NSF under Grant CMS-0218878.     

Optimal Guidance for Quasi-planar Lunar Ascent

The minimum-time controls (thrust pitch angle and thrust yaw angle) for the three-dimensional transfer of a constant-thrust rocket from one state to another over a flat moon are used to develop guidance laws for operation over a spherical moon. The objective is to evaluate the effect of making approximations on the size of the thrust pitch angle on the suitability of the resulting control law as a guidance law. After assuming small out-of-plane motion (small yaw angle), three pitch angle control laws (exact, first-order, and zeroth-order) are developed. The three laws are employed in the sample and hold guidance of a lunar ascent vehicle. All three laws satisfy the final conditions and give essentially the same pitch and yaw control histories. Since the zeroth-order law can be obtained completely analytically (no iteration processes), it merits consideration for ascent guidance.     

Robust clustering for functional data based on trimming and constraints

Advances in Data Analysis and Classification - Many clustering algorithms when the data are curves or functions have been recently proposed. However, the presence of contamination in the sample of...     

Robust competence assessment for job assignment

Allocating the right person to a task or job is a key issue for improving quality and performance of achievements, usually addressed using the concept of “competences”. Nevertheless, providing an accurate assessment of the competences of an individual may be in practice a difficult task. We suggest in this paper to model the uncertainty on the competences possessed by a person using a possibility distribution, and the imprecision on the competences required for a task using a fuzzy constraint, taking into account the possible interactions between competences using a Choquet integral. As a difference with comparable approaches, we then suggest to perform the allocation of persons to jobs using a robust optimisation approach, allowing to minimise the risk taken by the decision maker. We first apply this framework to the problem of selecting a candidate within n for a job, then extend the method to the problem of selecting c candidates for j jobs (c ? j) using the leximin criterion.     

不确定时滞Lurie系统的鲁棒绝对稳定性分析

研究了一类不确定Lurie时滞系统的鲁棒绝对稳定性问题，不确定项范数有界。通过构造特殊的李亚普诺夫函数，分别考虑了时滞相关和时滞无关两情况，得到了系统鲁棒绝对稳定的充分条件。     

Diagnostic Plots for Robust Multivariate Methods

Robust techniques for multivariate statistical methods—such as principal component analysis, canonical correlation analysis, and factor analysis—have been recently constructed. In contrast to the classical approach, these robust techniques are able to resist the effect of outliers. However, there does not yet exist a graphical tool to identify in a comprehensive way the data points that do not obey the model assumptions. Our goal is to construct such graphics based on empirical influence functions. These graphics not only detect the influential points but also classify the observations according to their robust distances. In this way the observations are divided into four different classes which are regular points, nonoutlying influential points, influential outliers, and noninfluential outliers. We thus gain additional insight in the data by detecting different types of deviating observations. Some real data examples will be given to show how these plots can be used in practice.     

Robust estimation for nonparametric generalized regression

This paper focuses on nonparametric regression estimation for the parameters of a discrete or continuous distribution, such as the Poisson or Gamma distributions, when anomalous data are present. The proposal is a natural extension of robust methods developed in the setting of parametric generalized linear models. Robust estimators bounding either large values of the deviance or of the Pearson residuals are introduced and their asymptotic behaviour is derived. Through a Monte Carlo study, for the Poisson and Gamma distributions, the finite properties of the proposed procedures are investigated and their performance is compared with that of the classical ones. A resistant cross-validation method to choose the smoothing parameter is also considered.     

Robust Estimation for Generalized Additive Models

This article studies M-type estimators for fitting robust generalized additive models in the presence of anomalous data. A new theoretical construct is developed to connect the costly M-type estimation with least-squares type calculations. Its asymptotic properties are studied and used to motivate a computational algorithm. The main idea is to decompose the overall M-type estimation problem into a sequence of well-studied conventional additive model fittings. The resulting algorithm is fast and stable, can be paired with different nonparametric smoothers, and can also be applied to cases with multiple covariates. As another contribution of this article, automatic methods for smoothing parameter selection are proposed. These methods are designed to be resistant to outliers. The empirical performance of the proposed methodology is illustrated via both simulation experiments and real data analysis. Supplementary materials are available online.