High-dimensional uncertainty quantification for Mars atmospheric entry using adaptive generalized polynomial chaos

Mixed uncertainty quantification for terminal flight state of Mars atmospheric entry

Mars entry trajectory planning using robust optimization and uncertainty quantification

<p>A robust self-tuning controller for a chemical process is developed based on a generalized Polynomial Chaos (gPC) model that accounts for probabilistic time-invariant uncertainty. Using this model, it is possible to calculate analytical expressions of the one-step ahead predicted mean and variances of controlled and manipulated variables. The key idea is to consider these predicted values for performing online robust tuning of the controller through a quadratic optimization procedure. The gPC model is also used to identify overlap between consecutive probability density functions (PDFs) of manipulated variables and to find trade-offs between the aggressiveness of the self-tuning controller and robustness to uncertainty based on this overlap. The proposed methodology is illustrated by a continuous stirred tank reactor (CSTR) system with stochastic variations in the inlet concentration. The efficiency of the proposed algorithm is quantified in terms of control performance and robustness.</p> <p> </p>

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<p>A robust adaptive controller is developed for a chemical process using a generalized Polynomial Chaos (gPC) expansion-based Markov decision model, which can account for time-invariant probabilistic uncertainty and overcome computational challenge for building Markov models. To calculate the transition probability, a gPC model is used to iteratively predict probability density functions (PDFs) of system’s states including controlled and manipulated variables. For controller tuning, these PDFs and controller parameters are discretized to a finite number of discrete states for building a Markov model. The key idea is to predict the transition probability of controlled and manipulated variables over a finite future control horizon, which can be further used to calculate an optimal sequence of control actions. This approach can be used to optimally tune a controller for set point tracking within a finite future control horizon. The proposed method is illustrated by a continuous stirred tank reactor (CSTR) system with stochastic perturbations in the inlet concentration. The efficiency of the proposed algorithm is quantified in terms of control performance and transient decay.</p> <p> </p>

Generalized Polynomial Chaos (gPC) is known to fail for problems involving strong nonlinear dependencies on stochastic inputs, which especially arise in the context of long term integration or stochastic discontinuities. There are various attempts in the literature which address these difficulties, such as the time-dependent generalized Polynomial Chaos (TD-gPC) and the multi-element generalized Polynomial Chaos (ME-gPC) both leading to higher accuracies but higher numerical costs in comparison to the classical gPC approach. A combination of these methods is introduced, which leads to a powerful solution method since high accuracies can be maintained and computational cost can be distributed by utilizing parallel computation. However, to be able to apply the hybrid method to all types of ordinary differential equations subject to random inputs, new modifications with respect to TD-gPC are carried out by creating an orthogonal tensor basis consisting of the random input variable as well as the solution itself. Such modifications allow TD-gPC to capture the dynamics of the solution by increasing the approximation quality of its time derivatives.

A multiresolution approach is presented to study the uncertainty propagation problem in dynamic systems. The proposed approach is an extension of the generalized polynomial chaos and involves a separation of random variables from deterministic ones in the solution algorithm for a stochastic dierential equation. The random variables are expanded in various local polynomial expansions which are further blended together to obtain a global approximation for the random state variables. These polynomials are associated with the assumed probability density functions for the input variables. Galerkin projection is used to generate a system of deterministic dierential equations for the expansion coecients. We rst present this method to linear systems with parametric uncertainties, and subsequently we generalize it to nonlinear systems with parametric and initial condition uncertainties. The eectiveness of the proposed method is demonstrated by considering dierent numerical examples.

Uncertainties in dynamic response of buildings with non-linear base-isolators

Linear quadratic regulation of systems with stochastic parameter uncertainties

Efficient Spectral-Chaos Methods for Uncertainty Quantification in Long-Time Response of Stochastic Dynamical Systems

<p>A robust self-tuning controller for a chemical process is developed based on a generalized Polynomial Chaos (gPC) model that accounts for probabilistic time-invariant uncertainty. Using this model, it is possible to calculate analytical expressions of the one-step ahead predicted mean and variances of controlled and manipulated variables. The key idea is to consider these predicted values for performing online robust tuning of the controller through a quadratic optimization procedure. The gPC model is also used to identify overlap between consecutive probability density functions (PDFs) of manipulated variables and to find trade-offs between the aggressiveness of the self-tuning controller and robustness to uncertainty based on this overlap. The proposed methodology is illustrated by a continuous stirred tank reactor (CSTR) system with stochastic variations in the inlet concentration. The efficiency of the proposed algorithm is quantified in terms of control performance and robustness.</p> <p> </p>

In this paper we present a Multi-Element generalized Polynomial Chaos (ME-gPC) method to deal with stochastic inputs with arbitrary probability measures. Based on the decomposition of the random space of the stochastic inputs, we construct numerically a set of orthogonal polynomials with respect to a conditional probability density function (PDF) in each element and subsequently implement generalized Polynomial Chaos (gPC) locally. Numerical examples show that ME-gPC exhibits both p- and h-convergence for arbitrary probability measures

In last decades, there has been an increasing interest in the use of micromachining technology. Intelligent materials such as shape memory alloy are considered for the design of micro-actuators. It is worth mentioning that the numerical study of such actuator without taking into account the uncertainties of different parameters has shown a great attention. Nevertheless, material parameters can suffer from a lack of accuracy due to the experiment measurement of these parameters. As a result, this paper proposes a methodology for considering uncertainty in a shape memory alloy materiel for a micro-pump actuator. The proposed approach is used in order to determine the response of this micro device with uncertainty related to material parameters. The simulation results are obtained by the generalized Polynomial Chaos (gPC) method for the response of such actuator under uncertainty. The proposed technique is an efficient probabilistic tool for uncertainty propagation. It is well shown that the gPC technique is an attractive alternative to the parametric studies. For more accuracy, the generalized Polynomial Chaos results are compared with Monte Carlo simulation.

Left ventricular assist devices (LVADs) have been used for end‐stage heart failure patients as a therapeutic option. The aortic valve plays a critical role in heart failure and its treatment with a LVAD. The cardiovascular‐LVAD model is often used to investigate the physiological demands required by patients and predict the hemodynamic of the native heart supported with a LVAD. As it is a “bridge-to-recovery” treatment, it is important to maintain appropriate and active dynamics of the aortic valve and the cardiac output of the native heart, which requires that the LVAD pump be adjusted so that a proper balance between the blood contributed through the aortic valve and the pump is maintained. In this paper, we investigate how the pump power of the LVAD pump can affect the dynamic behaviors of the aortic valve for different levels of activity and different severities of heart failure. Our objective is to identify a critical value of the pump power (i.e., breakpoint) to ensure that the LVAD pump does not take over the pumping function in the cardiovascular‐pump system and share the ejected blood with the left ventricle to help the heart to recover. In addition, the hemodynamic often involves variability due to patients’ heterogeneity and the stochastic nature of the cardiovascular system. The variability poses significant challenges to understanding dynamic behaviors of the aortic valve and cardiac output. A generalized polynomial chaos (gPC) expansion is used in this work to develop a stochastic cardiovascular‐pump model for efficient uncertainty propagation, from which it is possible to rapidly calculate the variance in the aortic valve opening duration and the cardiac output in the presence of variability. The simulation results show that the gPC‐based cardiovascular‐pump model is a reliable platform that can provide useful information to understand the effect of the LVAD pump on the hemodynamic of the heart.

Purpose – The purpose of this paper is to propose a novel approach to improve prediction accuracy of grey power models including GM(1, 1) and grey Verhulst model. Design/methodology/approach – The modified new models are proposed by optimizing the initial condition and model parameters. The new initial condition consists of the first item and the last item of a sequence generated by applying the first-order accumulative generation operator on the sequence of raw data. Findings – It is shown that the newly modified grey power model is an extension of the previous optimized GM(1, 1) and grey Verhulst model. And the optimized initial condition reflected the principle of new information priority. Practical implications – The result of a numerical example indicates that the modified grey model presented in this paper with better prediction performance. Originality/value – The new initial condition are derived by weighted combination of the first item and the last item. The coefficients of weight obtained by the least square method.