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  • 1
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    Uncertainty Quantification for Polynomial Systems via Bernstein Expansions (2012)
    In:  CASI
    Details
    Publication Date: 2019-07-13
    Description: This paper presents a unifying framework to uncertainty quantification for systems having polynomial response metrics that depend on both aleatory and epistemic uncertainties. The approach proposed, which is based on the Bernstein expansions of polynomials, enables bounding the range of moments and failure probabilities of response metrics as well as finding supersets of the extreme epistemic realizations where the limits of such ranges occur. These bounds and supersets, whose analytical structure renders them free of approximation error, can be made arbitrarily tight with additional computational effort. Furthermore, this framework enables determining the importance of particular uncertain parameters according to the extent to which they affect the first two moments of response metrics and failure probabilities. This analysis enables determining the parameters that should be considered uncertain as well as those that can be assumed to be constants without incurring significant error. The analytical nature of the approach eliminates the numerical error that characterizes the sampling-based techniques commonly used to propagate aleatory uncertainties as well as the possibility of under predicting the range of the statistic of interest that may result from searching for the best- and worstcase epistemic values via nonlinear optimization or sampling.
    Keywords: Numerical Analysis
    Type: AIAA Paper 2012-1851 , NF1676L-13334 , 13th AIAA Gossamer Systems Forum; Apr 03, 2012 - Apr 26, 2012; Honolulu, HI; United States|53rd Structures, Structural Dynamics, and Materials Conference (SDM); Apr 23, 2012 - Apr 26, 2012; Honolulu, HI; United States|20th AIAA/ASME/AHS Adaptive Structures Conference; Apr 23, 2012 - Apr 26, 2012; Honolulu, HI; United States|14th AIAA Non-Deterministic Approaches Conference; Apr 23, 2012 - Apr 26, 2012; Honolulu, HI; United States
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  • 2
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    Stochastic Optimal Control via Bellman's Principle (2003)
    In:  CASI
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    Publication Date: 2019-07-10
    Description: This paper presents a method for finding optimal controls of nonlinear systems subject to random excitations. The method is capable to generate global control solutions when state and control constraints are present. The solution is global in the sense that controls for all initial conditions in a region of the state space are obtained. The approach is based on Bellman's Principle of optimality, the Gaussian closure and the Short-time Gaussian approximation. Examples include a system with a state-dependent diffusion term, a system in which the infinite hierarchy of moment equations cannot be analytically closed, and an impact system with a elastic boundary. The uncontrolled and controlled dynamics are studied by creating a Markov chain with a control dependent transition probability matrix via the Generalized Cell Mapping method. In this fashion, both the transient and stationary controlled responses are evaluated. The results show excellent control performances.
    Keywords: Numerical Analysis
    Type: NASA/CR-2003-212419 , NIA Report No. 2003-04 , NAS 1.26:212419
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  • 3
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    A First and Second Order Moment Approach to Probabilistic Control Synthesis (2005)
    In:  CASI
    Details
    Publication Date: 2019-07-13
    Description: This paper presents a robust control design methodology based on the estimation of the first two order moments of the random variables and processes that describe the controlled response. Synthesis is performed by solving an multi-objective optimization problem where stability and performance requirements in time- and frequency domains are integrated. The use of the first two order moments allows for the efficient estimation of the cost function thus for a faster synthesis algorithm. While reliability requirements are taken into account by using bounds to failure probabilities, requirements related to undesirable variability are implemented by quantifying the concentration of the random outcome about a deterministic target. The Hammersley Sequence Sampling and the First- and Second-Moment- Second-Order approximations are used to estimate the moments, whose accuracy and associated computational complexity are compared numerically. Examples using output-feedback and full-state feedback with state estimation are used to demonstrate the ideas proposed.
    Keywords: Numerical Analysis
    Type: AIAA Paper 2005-6133 , AIAA Guidance, Navigation, and Control Conference and Exhibit; Aug 15, 2005 - Aug 18, 2005; San Francisco, CA; United States
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  • 4
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    Application of Interval Predictor Models to Space Radiation Shielding (2016)
    In:  CASI
    Details
    Publication Date: 2019-07-13
    Description: This paper develops techniques for predicting the uncertainty range of an output variable given input-output data. These models are called Interval Predictor Models (IPM) because they yield an interval valued function of the input. This paper develops IPMs having a radial basis structure. This structure enables the formal description of (i) the uncertainty in the models parameters, (ii) the predicted output interval, and (iii) the probability that a future observation would fall in such an interval. In contrast to other metamodeling techniques, this probabilistic certi cate of correctness does not require making any assumptions on the structure of the mechanism from which data are drawn. Optimization-based strategies for calculating IPMs having minimal spread while containing all the data are developed. Constraints for bounding the minimum interval spread over the continuum of inputs, regulating the IPMs variation/oscillation, and centering its spread about a target point, are used to prevent data over tting. Furthermore, we develop an approach for using expert opinion during extrapolation. This metamodeling technique is illustrated using a radiation shielding application for space exploration. In this application, we use IPMs to describe the error incurred in predicting the ux of particles resulting from the interaction between a high-energy incident beam and a target.
    Keywords: Statistics and Probability; Space Radiation
    Type: NF1676L-21631 , AIAA SciTech; Jan 04, 2016 - Jan 08, 2016; San Diego, CA; United States
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  • 5
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    Robust Design Optimization via Failure Domain Bounding (2007)
    In:  CASI
    Details
    Publication Date: 2019-07-13
    Description: This paper extends and applies the strategies recently developed by the authors for handling constraints under uncertainty to robust design optimization. For the scope of this paper, robust optimization is a methodology aimed at problems for which some parameters are uncertain and are only known to belong to some uncertainty set. This set can be described by either a deterministic or a probabilistic model. In the methodology developed herein, optimization-based strategies are used to bound the constraint violation region using hyper-spheres and hyper-rectangles. By comparing the resulting bounding sets with any given uncertainty model, it can be determined whether the constraints are satisfied for all members of the uncertainty model (i.e., constraints are feasible) or not (i.e., constraints are infeasible). If constraints are infeasible and a probabilistic uncertainty model is available, upper bounds to the probability of constraint violation can be efficiently calculated. The tools developed enable approximating not only the set of designs that make the constraints feasible but also, when required, the set of designs for which the probability of constraint violation is below a prescribed admissible value. When constraint feasibility is possible, several design criteria can be used to shape the uncertainty model of performance metrics of interest. Worst-case, least-second-moment, and reliability-based design criteria are considered herein. Since the problem formulation is generic and the tools derived only require standard optimization algorithms for their implementation, these strategies are easily applicable to a broad range of engineering problems.
    Keywords: Numerical Analysis
    Type: 9th AIAA Non-Deterministic Approaches Conference; Apr 23, 2007 - Apr 26, 2007; Waikiki, HI; United States
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  • 6
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    Stochastic Control Synthesis of Systems with Structured Uncertainty (2003)
    In:  CASI
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    Publication Date: 2019-07-10
    Description: This paper presents a study on the design of robust controllers by using random variables to model structured uncertainty for both SISO and MIMO feedback systems. Once the parameter uncertainty is prescribed with probability density functions, its effects are propagated through the analysis leading to stochastic metrics for the system's output. Control designs that aim for satisfactory performances while guaranteeing robust closed loop stability are attained by solving constrained non-linear optimization problems in the frequency domain. This approach permits not only to quantify the probability of having unstable and unfavorable responses for a particular control design but also to search for controls while favoring the values of the parameters with higher chance of occurrence. In this manner, robust optimality is achieved while the characteristic conservatism of conventional robust control methods is eliminated. Examples that admit closed form expressions for the probabilistic metrics of the output are used to elucidate the nature of the problem at hand and validate the proposed formulations.
    Keywords: Numerical Analysis
    Type: NASA/CR-2003-212167 , NIA-2003-01
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  • 7
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    Differential Flatness and Cooperative Tracking in the Lorenz System (2002)
    In:  CASI
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    Publication Date: 2019-07-10
    Description: In this paper the control of the Lorenz system for both stabilization and tracking problems is studied via feedback linearization and differential flatness. By using the Rayleigh number as the control, only variable physically tunable, a barrier in the controllability of the system is incidentally imposed. This is reflected in the appearance of a singularity in the state transformation. Composite controllers that overcome this difficulty are designed and evaluated. The transition through the manifold defined by such a singularity is achieved by inducing a chaotic response within a boundary layer that contains it. Outside this region, a conventional feedback nonlinear control is applied. In this fashion, the authority of the control is enlarged to the whole. state space and the need for high control efforts is mitigated. In addition, the differential parametrization of the problem is used to track nonlinear functions of one state variable (single tracking) as well as several state variables (cooperative tracking). Control tasks that lead to integrable and non-integrable differential equations for the nominal flat output in steady-state are considered. In particular, a novel numerical strategy to deal with the non-integrable case is proposed. Numerical results validate very well the control design.
    Keywords: Numerical Analysis
    Type: NASA/CR-2002-211920 , NAS 1.26:211920 , ICASE-2002-32
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  • 8
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    Optimization of Systems with Uncertainty: Initial Developments for Performance, Robustness and Reliability Based Designs (2002)
    In:  CASI
    Details
    Publication Date: 2019-07-10
    Description: This paper presents a study on the optimization of systems with structured uncertainties, whose inputs and outputs can be exhaustively described in the probabilistic sense. By propagating the uncertainty from the input to the output in the space of the probability density functions and the moments, optimization problems that pursue performance, robustness and reliability based designs are studied. Be specifying the desired outputs in terms of desired probability density functions and then in terms of meaningful probabilistic indices, we settle a computationally viable framework for solving practical optimization problems. Applications to static optimization and stability control are used to illustrate the relevance of incorporating uncertainty in the early stages of the design. Several examples that admit a full probabilistic description of the output in terms of the design variables and the uncertain inputs are used to elucidate the main features of the generic problem and its solution. Extensions to problems that do not admit closed form solutions are also evaluated. Concrete evidence of the importance of using a consistent probabilistic formulation of the optimization problem and a meaningful probabilistic description of its solution is provided in the examples. In the stability control problem the analysis shows that standard deterministic approaches lead to designs with high probability of running into instability. The implementation of such designs can indeed have catastrophic consequences.
    Keywords: Numerical Analysis
    Type: NASA/CR-2002-211952 , NAS 1.26:211952 , ICASE-2002-40
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  • 9
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    Random Variables with Moment-Matching Staircase Density Functions (2018)
    In:  CASI
    Details
    Publication Date: 2019-11-19
    Description: This paper proposes techniques for constructing non-parametric computational models describing the distribution of a continuous output variable given input-output data. These models are called Random Predictor Models (RPMs) because the predicted output corresponding to any given input is a random variable. One common example of an RPM is a Gaussian process (GP) model. In contrast to GP models however, we focus on RPMs having a bounded support set and prescribed values for the mean, and the second-, third-, and fourth-order central moments. The proposed RPMs are designed to match moment functions extracted from the data over a range of minimal spread. This paper presents the feasibility conditions that any random variable must meet in order to satisfy the desired constraints. Furthermore, a particular family of such variables, called staircase because their probability density is a piecewise constant function, is proposed. The ability of these variables to describe a wide range of probability density shapes, and their low computational cost enable the efficient generation of possibly skewed and multimodal RPMs over an input-dependent interval.
    Keywords: Numerical Analysis
    Type: NF1676L-26266 , Applied Mathematical Modelling (ISSN 0307-904X); 64; 196-213
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