Uncertain Initial Conditions Research Articles

Probability analysis of ex-vessel steam explosion induced pressure load based on analysis code SEINA

In nuclear reactor severe accidents, the interaction of molten core melt and coolant may lead to steam explosion, which seriously threatens the safety of the reactor. The four initial parameters that affect the steam explosion were randomly sampled and calculated multiple times by steam explosion analysis code SEINA, and the probability distributions of steam explosion pressure peak and pressure impulse under uncertain initial conditions are obtained. The results show that the peak pressure of ex-vessel steam explosion is roughly normally distributed, and the pressure peak of 90% of the cases under submerged and non-submerged conditions within the range of 10–60 MPa and 20-80Mpa. respectively. By calculating the four key correlation coefficients between input parameters and output parameters, it is concluded that the influence of melt temperature and breach size on output parameters is greater than that of system pressure and coolant temperature.

Study of a Fractional System of Predator-Prey with Uncertain Initial Conditions

In this manuscript, we study a nonlinear fractional-order predator-prey system while considering uncertainty in initial values. We derive the feasibility region and the boundness of the solution. The suggested model’s equilibrium points and the basic reproduction number are calculated. The stability of equilibrium points is presented. We use the metric fixed point theory to study the existence and uniqueness results concerning the solution of the model. We use the notion of UH-stability to show that the model is Ulam–Hyres type stable. To attain the approximate solution of the proposed model, we construct a method that uses the fuzzy Laplace transform in collaboration with the ADM (Adomian decomposition method). Finally, we simulate our theoretical results using MATLAB to show the dynamics of the considered model.

Large-scale three-dimensional hydraulic tomography analyses of long-term municipal wellfield operations

Hydraulic tomography (HT) has been shown to be an effective approach for subsurface heterogeneity characterization. However, only a few HT studies have been performed for large-scale field problems due to the difficulty in conducting dedicated HT surveys at large-scale sites, as well as the uncertainty regarding estimated initial and boundary conditions for inverse modeling. To overcome these issues, this study advocates the utilization of existing long-term municipal well records as alternative datasets for large-scale heterogeneity characterization, along with novel data processing and analyses strategies that are proposed to minimize the effect of uncertain initial and boundary conditions on inverse modeling. Specifically, geology-based zonation and geostatistical models are adopted for site heterogeneity characterization through HT analyses of municipal well records, and the estimated hydraulic parameters from both models are validated through the prediction of head response data that have not been used for calibration efforts. Our results reveal that existing field municipal well records could be utilized for large-scale subsurface heterogeneity characterization using the approach of HT when uncertainties regarding initial and boundary conditions are well addressed for inverse modeling. In comparison to geology-based zonation model, the geostatistical model reveals greater details of intralayer heterogeneity where head response data are available, yielding significantly improved calibration and validation results. Overall, this study provides a general framework of using existing hydrographs for large-scale heterogeneity characterization through HT, and advocates the utilization of geostatistical inverse modeling as the second step over traditional zonation modeling approach to reveal intralayer heterogeneity details of hydraulic parameters.

Combining set propagation with finite element methods for time integration in transient solid mechanics problems

The Finite Element Method (FEM) is the gold standard for spatial discretization in numerical simulations for a wide spectrum of real-world engineering problems. Prototypical areas of interest include linear heat transfer and linear structural dynamics problems modeled with partial differential equations (PDEs). While different algorithms for direct integration of the equations of motion exist, exploring all feasible behaviors for varying loads, initial states and fluxes in models with large numbers of degrees of freedom remains a challenging task. In this article we propose a novel approach, based in set propagation methods and motivated by recent advances in the field of Reachability Analysis. Assuming a set of initial states and inputs, the proposed method consists in the construction of a union of sets (flowpipe) that enclose the infinite number of solutions of the spatially discretized PDE. We present the numerical results obtained in five examples to illustrate the capabilities of our approach, and compare its performance against reference numerical integration methods. We conclude that, for problems with single known initial conditions, the proposed method is accurate. For problems with uncertain initial conditions included in sets, the proposed method can compute all the solutions of the system more efficiently than numerical integration methods.

An adaptive wavelet optimized finite difference B-spline polynomial chaos method for random partial differential equations

In this paper, wavelet optimized finite difference B-spline polynomial chaos method is proposed for solving stochastic partial differential equations. The generalized polynomial chaos is applied by considering linear B-spline wavelet basis. Then, stochastic Galerkin projection is executed for evaluating the deterministic coefficients of the generalized polynomial chaos. In the next step, the system of equations is discretized by using Crank-–Nicolson scheme for time integration and for approximating the differential operators, central finite difference matrices are considered. An adaptive grid is generated using the linear B-spline generalized polynomial chaos for optimizing the numerical solution. The method is then tested on three problems namely heat equation with uncertain initial conditions, Burger’s equation with random initial conditions and Burger’s equation with random viscosity as well as uncertain initial conditions. For the three test problems, grid modifications are displayed by taking periodic boundary conditions into consideration. Mean and standard deviation are plotted for each test problem. Moreover, for the third test problem, computational time comparison is performed by computing CPU time taken by the proposed method and the CPU time taken by the finite difference method on a uniform grid.

LPV control and analysis under uncertain initial conditions

This paper is focused on the control of discrete-time linear parameter-varying (LPV) systems with uncertain initial conditions. The synthesis objective is to design a feedback LPV controller that ensures that the closed-loop system is asymptotically stable and some performance inequality is satisfied for all permissible parameter trajectories. The performance measure used is similar to the ℓ2-induced norm except that the initial state and the disturbance input are considered to reside in two separate norm balls; that is, the supremum of the ℓ2-norm of the performance output is minimized subject to the initial state taking values in a unit Euclidean ball and the disturbance input taking values in a unit ℓ2-norm ball. The paper provides convex analysis and synthesis conditions for solving this problem and demonstrates that using nonstationary LPV controllers instead of standard LPV ones could result in a significantly improved performance. The paper also shows how to determine bounding envelopes for the performance output using the analysis result. An illustrative example concludes the paper.

Design framework for optimizing waypoints of vehicle trajectory considering terminal velocity and impact angle constraints

Ballistic missiles often require the terminal velocity and impact angle to be confined to a certain region around a desired value considering a wide variety of uncertain initial conditions and operational ranges. This study presents a design framework to determine optimum waypoints that satisfy constraints for given launch conditions and mission profiles. As a systematic approach to this kinodynamic motion planning problem, the proposed framework deterministically samples the waypoints. The trajectory generated by the determined waypoints in consideration of various conditions satisfies the terminal constraint. Numerical simulations are performed to demonstrate the effectiveness of the proposed method.

A Data-Driven Aero-Engine Degradation Prognostic Strategy.

Degradation prognostics of aero-engine are a well-recognized challenging issue. Data-driven prognostic techniques have been receiving attention because they rely on neither expert knowledge nor mathematic model of the system. But they are highly dependent on the quantity and quality of degradation data. To solve the problems caused by unlabeled, unbalanced condition monitoring (CM) data and uncertainties of the prognostics process, a novel data-driven aero-engine degradation prognostic strategy is proposed in this article. First, two indicators are defined to remove redundant degradation features. Then, the number of discrete states of health is determined by a fuzzy c -means algorithm, and the health state labels can be automatically assigned for health state estimation, where the uncertain initial condition and the uncertainty of health state's transition are fully considered. Finally, a multivariate health estimation model and a multivariate multistep-ahead long-term degradation prediction model are proposed for remaining useful life estimation for aero-engines. Verification results using the aero-engine data from NASA can show that the proposed data-driven degradation prognostic strategy is effective and feasible.

An Interannual Probabilistic Assessment of Subsurface Water Storage Over Europe Using a Fully Coupled Terrestrial Model

AbstractThe years 2018 and 2019 were two of the hottest and driest in Mid‐Europe, highlighting the need for a comprehensive assessment of available water resources. In this study, we propose a probabilistic, terrestrial water assessment method, which utilizes a terrestrial forward model that closes the coupled water and energy cycles, from groundwater to the top of the atmosphere. In this methodology, the model is initialized with the current state of the water year and forced with a climatologic ensemble of atmospheric forcing to account for atmospheric uncertainty and natural variability. The simulations result in an ensemble of ensuing water years that are analyzed for subsurface water storage anomalies. The methodology was applied to the water years 2011/2012 and 2018/2019 and showed an increased probability of a significant water deficit in regions that had a water deficit in the previous year. This was also observed in an evaluation simulation. The results were compared to simulations with perfect forcing and uncertain initial conditions, and showed predictability at the interannual timescale and beyond, depending on the strength of the anomaly. The methodology was then applied to 2019/2020 to provide an outlook of the evolution of the current anomalies. The results emphasize the importance of accounting for groundwater dynamics in applied terrestrial models to account for long‐term memory effects in the terrestrial water cycle in forward simulations, over large spatial scales. This method of probabilistic subsurface water storage assessment may provide crucial information to public and industrial sectors for long‐term water resource planning.

Robustness analysis of uncertain time‐varying systems using integral quadratic constraints with time‐varying multipliers

SummaryThis article presents a dissipativity approach for robustness analysis using the framework of integral quadratic constraints (IQCs). The derived results apply for linear time‐varying nominal systems with uncertain initial conditions. IQC multipliers are used to describe the sets of allowable uncertainty operators, and signal IQC multipliers are used to describe the sets of allowable disturbance signals. The novel concepts of dichotomic nodes and their corresponding factorizations are introduced, which allow for the aforementioned multipliers to be general time‐varying operators. The results are illustrated via the robustness analysis of a flight controller for an unmanned aircraft system tasked to perform a Split‐S maneuver.

Optimizing autocatalysis with uncertainty by derivative‐free estimators

SummaryA derivative‐free estimator was introduced to alleviate the drawbacks of the conventional Kalman filter when performing nonlinear analyses under different circumstances. In this work, the scaled Unscented Kalman Filter, Divided Difference Kalman filter, and Cubature Kalman filter (CKF) were selected to investigate the effectiveness of these filters in predicting the states of a complex semi‐batch reaction between propionic anhydride and 2‐butanol. The estimator's performance was evaluated under four different case studies, that is, under normal condition, under poor estimator initialization, under disturbances, and under parameter uncertainty. Results from the study show that CKF was the best option for an online dynamic optimization because of its highest degree of accuracy and stability under the normal and noisy conditions. Under normal condition, CKF yielded the lowest root mean square error of 0.61 × 10−2. Under uncertain initial condition, disturbance and parameter uncertainty, the lowest error of 1.83 × 10−2, 1.04 × 10−2, and 0.81 × 10−2 were obtained, respectively.

Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems

In this paper, we establish that for a wide class of controlled stochastic differential equations (SDEs) with stiff coefficients, the value functions of corresponding zero-sum games can be represented by a deep artificial neural network (DNN), whose complexity grows at most polynomially in both the dimension of the state equation and the reciprocal of the required accuracy. Such nonlinear stiff systems may arise, for example, from Galerkin approximations of controlled stochastic partial differential equations (SPDEs), or controlled PDEs with uncertain initial conditions and source terms. This implies that DNNs can break the curse of dimensionality in numerical approximations and optimal control of PDEs and SPDEs. The main ingredient of our proof is to construct a suitable discrete-time system to effectively approximate the evolution of the underlying stochastic dynamics. Similar ideas can also be applied to obtain expression rates of DNNs for value functions induced by stiff systems with regime switching coefficients and driven by general Lévy noise.

Data-driven surrogates for high dimensional models using Gaussian process regression on the Grassmann manifold

This paper introduces a surrogate modeling scheme based on Grassmannian manifold learning to be used for cost-efficient predictions of high-dimensional stochastic systems. The method exploits subspace-structured features of each solution by projecting it onto a Grassmann manifold. This point-wise linear dimensionality reduction harnesses the structural information to assess the similarity between solutions at different points in the input parameter space. The method utilizes a solution clustering approach in order to identify regions of the parameter space over which solutions are sufficiently similarly such that they can be interpolated on the Grassmannian. In this clustering, the reduced-order solutions are partitioned into disjoint clusters on the Grassmann manifold using the eigen-structure of properly defined Grassmannian kernels and, the Karcher mean of each cluster is estimated. Then, the points in each cluster are projected onto the tangent space with origin at the corresponding Karcher mean using the exponential mapping. For each cluster, a Gaussian process regression model is trained that maps the input parameters of the system to the reduced solution points of the corresponding cluster projected onto the tangent space. Using this Gaussian process model, the full-field solution can be efficiently predicted at any new point in the parameter space. In certain cases, the solution clusters will span disjoint regions of the parameter space. In such cases, for each of the solution clusters we utilize a second, density-based spatial clustering to group their corresponding input parameter points in the Euclidean space. The proposed method is applied to two numerical examples. The first is a nonlinear stochastic ordinary differential equation with uncertain initial conditions where the surrogate is used to predict the time history solution. The second involves modeling of plastic deformation in a model amorphous solid using the Shear Transformation Zone theory of plasticity, where the proposed surrogate is used to predict the full strain field of a material specimen under large shear strains.

Control of a Whole-Body Task with Uncertain Initial Conditions: Application to the Upstart On Bars

The aim was to determine whether operating a feedforward schema for generating movement pattern parameters was more successful than an open loop strategy for coping with uncertain initial conditions. A computer simulation model was used to determine the optimal solutions that maximised the likelihood of performing a successful upstart. Feedforward schema were established between movement pattern parameters and initial angular velocity. The success of modifying a pre-planned movement pattern based on the parameter relationships (feedforward) was compared with optimal solutions unable to adapt (open loop) to initial angular velocity. The open loop solution was successful 28% and 20% of the time for a full strength (elite) and weaker gymnast. The feedforward strategy had success rates of 99% and 96% respectively.

Fault-tolerant tracking control with guaranteed performance for nonlinearly parameterized systems under uncertain initial conditions

In this work, by integrating a virtual parameter separation technique and the error-shifting transformation, we develop a smooth fault-tolerant control (FTC) strategy for nonlinearly parameterized systems subject to actuation failures and unknown initial conditions. The essential step in control design lies in the introduction of a novel rate function and the utilization of which to build the error-shifting transformation, enabling the developed design method with the ability to steer the tracking error into a prescribed performance boundary within the pregiven settling time under arbitrary bounded (yet possibly uncertain/unknown) initial tracking conditions. Different from existing related methods, the key features of our adaptive tracking control are its independence of the initial condition and its minimum number of tuning parameters, namely, no matter how many unknown parameters are involved, only a single parameter estimation appears in each step of the backstepping design.

Exploiting nonlinear invariants and path constraints to achieve tighter reachable set enclosures using differential inequalities

This article presents a new method for computing sharp bounds on the solutions of nonlinear dynamic systems subject to uncertain initial conditions, parameters, and time-varying inputs. Such bounds are widely used in algorithms for uncertainty propagation, robust state estimation, system verification, global dynamic optimization, and more. Recently, it has been shown that bounds computed via differential inequalities can often be made much less conservative by exploiting state constraints that are known to hold for all trajectories of interest (e.g., path constraints that describe feasible trajectories in the context of dynamic optimization, or constraints that explicitly describe invariant sets containing all system trajectories). However, effective bounding algorithms of this type are currently only available for problems with linear constraints. Moreover, the theoretical results underlying these algorithms do not apply to constraints that depend on time-varying inputs and rely on assumptions that prove to be very restrictive for nonlinear constraints. This article contributes a new differential inequalities theorem that permits the use of a very general class of nonlinear state constraints. Moreover, a new algorithm is presented for efficiently exploiting nonlinear constraints to achieve tighter bounds. The proposed approach is shown to produce very sharp bounds for two challenging case studies.

Reach-Set Estimation for DAE Systems under Uncertainty and Disturbances Using Trajectory Sensitivity and Logarithmic Norm

Trajectory sensitivity analysis is useful for analyzing the dynamic behaviour of differential-algebraic equation (DAE) systems under uncertain initial conditions and/or parameters. However, the approximate trajectories obtained using trajectory sensitivities are not accompanied by explicit error bounds. In this paper, we provide an efficient method to obtain a numerical error bound for the first-order trajectory approximation. This approach uses second-order trajectory sensitivities. A theoretical result quantifying the excursion of trajectories induced by uncertain initial conditions and external disturbances is derived based on the logarithmic norm, and is extended to DAE systems. Although this result itself provides a guaranteed over-approximation of the reach-set of nonlinear DAE systems, by combining this result with the efficient bound obtained from trajectory sensitivities, we are able to provide a much less conservative reach-set estimate for systems under uncertain initial conditions and/or parameters, and external disturbances.

Parameter Identification in Uncertain Scalar Conservation Laws Discretized with the Discontinuous Stochastic Galerkin Scheme

We study an identification problem which estimates the parameters of the underlying random distribution for uncertain scalar conservation laws. The hyperbolic equations are discretized with the so-called discontinuous stochastic Galerkin method, i.e., using a spatial discontinuous Galerkin scheme and a Multielement stochastic Galerkin ansatz in the random space. We assume an uncertain flux or uncertain initial conditions and that a data set of an observed solution is given. The uncertainty is assumed to be uniformly distributed on an unknown interval and we focus on identifying the correct endpoints of this interval. The first-order optimality conditions from the discontinuous stochastic Galerkin discretization are computed on the time-continuous level. Then, we solve the resulting semi-discrete forward and backward schemes with the Runge Kutta method. To illustrate the feasibility of the approach, we apply the method to a stochastic advection and a stochastic equation of Burgers' type. The results show that the method is able to identify the distribution parameters of the random variable in the uncertain differential equation even if discontinuities are present.

Direct multiple shooting transcription with polynomial algebra for optimal control problems under uncertainty

This paper proposes a novel approach to the solution of optimal control problems under uncertainty (OCPUUs). OCPUUs are first cast in a general formulation that allows the treatment of uncertainties of different nature, and then solved with a new direct transcription method that combines multiple shooting with generalised polynomial algebra to model and propagate extended sets. The continuity conditions on extended sets at the boundary of two adjacent segments are directly satisfied by a bounding approach. The Intrusive Polynomial Algebra aNd Multiple shooting Approach (IPANeMA) developed in this work can handle optimal control problems under a wide range of uncertainty models, including nonparametric, epistemic, and imprecise probability ones. In this paper, the approach is applied to the design of a robust low-thrust trajectory to a Near-Earth Object with uncertain initial conditions. It is shown that the new method provides more robust and reliable trajectories than the solution of an analogous deterministic optimal control problem.

An estimation accuracy of state observers under uncertain initial conditions

An estimation accuracy of state observers under uncertain initial conditions