Set Of Unknown Parameters Research Articles

Optimal model parameter estimation and performance analysis of PEM electrolyzer using modified honey badger algorithm

The electrochemical model of a proton exchange membrane (PEM) electrolyzer consists of a set of different unknown parameters, and identification of these parameters is needed. However, obtaining the optimum values of these parameters is critical, and this involves a complex, multi-variate, non-linear, and multimodal optimisation problem. In this article, a novel model parameter estimation problem for a PEM electrolyzer with eight unknown model parameters is formulated and solved using modified honey badger algorithm (MHBA). A mean squared error-based objective function is formulated and considered as the mean squared error (MSE) between the experimental voltage and the estimated voltage using MHBA algorithm. The accuracy of the proposed model is validated using polarization curves (J-V curve). It is found that the MHBA outperforms with a good mapping between experimental and estimated voltage. Also, the reliability and robustness for parameter estimation algorithm is validated considering two case studies with four different operating conditions. Additionally, the results obtained using MHBA are compared to those obtained with other competing algorithms. A statistical study including different statistical indices is also carried out to prove the robustness for the proposed approach. The results show that the minimum obtained values of MSE for different operating conditions are 8.73E-06, 8.75E-06, 6.17E-05 and 6.44E-05. Box plot study and convergence curves have also been drawn to check the effectiveness of the algorithm. Furthermore, to prove the superiority of the proposed approach, the PEM electrolyzer's performance is explored at different operating conditions, such as operating temperatures and output hydrogen pressures.

Adaptive Tracking of Large-Angle Maneuvers for Spacecraft Equipped with an Active Pointing Ultraquiet Platform

An adaptive controller for the tracking of large-angle maneuvers is proposed for spacecraft equipped with an active pointing ultraquiet platform (AQP). The inertial parameters of the spacecraft and the payload (supported by the AQP) are estimated online, whereas the parameters of the AQP are assumed to be precisely known. To enable the derivation of the proposed controller, a novel dynamic model of AQP spacecraft was derived, which employs relative motion variables for the payload and is linear in the unknown inertial parameters. A Lyapunov-like argument was used to prove the local asymptotical stability of the closed-loop system, and the original controller was simplified further to a physically more intuitive proportional-differential (PD) plus adaptive feed-forward controller. The superior tracking performance of the proposed controller compared with that of a standard PD plus nonadaptive feed-forward controller was shown via numerical simulation. It also was shown that the estimates of a selected set of unknown inertial parameters will converge to their true values if a certain kind of maneuver trajectory is tracked.

A Generalized Multiple Criteria Data-Fitting Model With Sparsity and Entropy With Application to Growth Forecasting

In this article, we present an extended data-fitting model which involves different and conflicting criteria, and we propose an algorithm based on a scalarization technique to solve it. Our model integrates in a unique framework three different criteria, namely, a data-fitting term, and the entropy and the sparsity of the set of unknown parameters. This model can be analyzed by means of multiple criteria decision-making techniques. We then validate the proposed modified algorithm using two computational experiments: We analyze the problem of handwritten digit recognition using a logistic regression model and a deep neural network model, respectively. In the final part of the article, we employ this methodology to forecasting instead. Given the importance of forecasting techniques to predict the future, which in turn can lead to positive impacts on firm performance, we propose two numerical experiments focusing on the forecast of the US GDP. In the first one, we proceed by means of a modified iterated function system with grayscale maps-type fractal operator, and, in the second one, we implement a modified neural network-based model.

State and Parameter Estimation of Non-Uniform Transmission Line Using Kronecker Product Based Modeling

The distributed parameters of a non-uniform transmission line (NTL) are assumed to depend on a finite set of unknown parameters, for example, their spatial Fourier series coefficients. The line equations are set up taking random voltage and current line loading into account with the random loading being assumed to be white Gaussian noise in the time and spatial variables. Bypassing over to the spatial Fourier series domain, these line stochastic differential equations (SDEs) are expressed as a truncated sequence of coupled stochastic differential equations for the spatial Fourier series components of the line voltage and current. The coefficients that appear in these stochastic differential equations are the spatial Fourier series coefficients of the distributed line parameters and these coefficients in turn are linear functions of the unknown parameters. By separating these complex stochastic differential equations into their real and imaginary part, the line equations become a finite set of SDEs for the extended state defined to be the real and imaginary components of the line voltage and current as well as the parameters on which the distributed parameters depend. The measurement model is assumed to be white Gaussian noise corrupted versions of the line voltage and current sampled at a discrete set of spatial points on the line and this measurement model can be expressed as a linear transformation of the state variables, i.e, of the spatial Fourier series components of the line voltage and current plus a white measurement noise vector. This work uses sparse matrix factorization using Kronecker product to implement the Fourier unitary transform. This work implements perturbation theory for non-uniform transmission line. The extended Kalman filter (EKF) equations are derived for this state and measurement model and our real timeline voltage and current and distributed parameter estimates show excellent agreement with the true values based on MATLAB simulations. The use of sparse matrix factorization using Kronecker product ease the mathematical complexity and provides compact representation.

Moment estimation for parameters in high-order uncertain differential equations

As a type of differential equations with high-order derivatives of uncertain processes, high-order uncertain differential equations are widely applied to modelling dynamic systems in uncertain environment, which usually involve unknown parameters to be estimated. Since observations are always discrete in practice, based on these discrete observations of solution processes, we propose moment estimations for unknown parameters by Euler method approximation of high-order uncertain differential equations. Matching sample moments with corresponding population moments, a system of equations whose solution is the moment estimation of the set of unknown parameters is derived. Finally, some examples illustrate our method in detail.

Gradient-based reconstruction of molecular Hamiltonians and density matrices from time-dependent quantum observables

We consider a quantum system with a time-independent Hamiltonian parametrized by a set of unknown parameters $\alpha$. The system is prepared in a general quantum state by an evolution operator that depends on a set of unknown parameters $P$. After the preparation, the system evolves in time, and it is characterized by a time-dependent observable ${\cal O}(t)$. We show that it is possible to obtain closed-form expressions for the gradients of the distance between ${\cal O}(t)$ and a calculated observable with respect to $\alpha$, $P$ and all elements of the system density matrix, whether for pure or mixed states. These gradients can be used in projected gradient descent to infer $\alpha$, $P$ and the relevant density matrix from dynamical observables. We combine this approach with random phase wave function approximation to obtain closed-form expressions for gradients that can be used to infer population distributions from averaged time-dependent observables in problems with a large number of quantum states participating in dynamics. The approach is illustrated by determining the temperature of molecular gas (initially, in thermal equilibrium at room temperature) from the laser-induced time-dependent molecular alignment.

Consideration of specific key points improves outcome of decompression treatment in patients with endocrine orbitopathy: pre-/post-OP comparison and biomechanical simulation

Endocrine orbitopathy is typically treated by resecting orbital walls. This procedure reduces intraorbital pressure by releasing intraorbital tissue, effectively alleviating the symptoms. However, selection of an appropriate surgical plan for treatment of endocrine orbitopathy requires careful consideration because predicting the effects of one-, two-, or three-wall resections on the release of orbital tissues is difficult. Here, based on our experience, we describe two specific orbital sites (’key points’) that may significantly improve decompression results. Methodological framework of this work is mainly based on comparative analysis pre- and post-surgery tomographic images as well as image- and physics-based simulation of soft tissue outcome using the finite element modelling of mechanical soft tissue behaviour. Thereby, the optimal set of unknown modelling parameters was obtained iteratively from the minimum difference between model predictions and post-surgery ground truth data. This report presents a pre-/post-surgery study indicating a crucial role of these particular key points in improving the post-surgery outcome of decompression treatment of endocrine orbitopathy which was also supported by 3D biomechanical simulation of alternative two-wall resection plans. In particular, our experimental results show a nearly linear relationship between the resection area and amount of tissue released in the extraorbital space. However, a disproportionately higher volume of orbital outflow could be achieved under consideration of the two special key points. Our study demonstrates the importance of considering natural biomechanical obstacles to improved outcomes in two-wall resection treatment of endocrine orbitopathy. Further investigations of alternative surgery scenarios and post-surgery data are required to generalize the insights of this feasibility study.

Robotic Tool Tracking Under Partially Visible Kinematic Chain: A Unified Approach

Anytime a robot manipulator is controlled via visual feedback, the transformation between the robot and camera frame must be known. However, in the case where cameras can only capture a portion of the robot manipulator in order to better perceive the environment being interacted with, there is greater sensitivity to errors in calibration of the base-to-camera transform. A secondary source of uncertainty during robotic control are inaccuracies in joint angle measurements which can be caused by biases in positioning and complex transmission effects such as backlash and cable stretch. In this work, we bring together these two sets of unknown parameters into a unified problem formulation when the kinematic chain is partially visible in the camera view. We prove that these parameters are nonidentifiable implying that explicit estimation of them is infeasible. To overcome this, we derive a smaller set of parameters we call lumped error since it lumps together the errors of calibration and joint angle measurements. A particle filter method is presented and tested in simulation and on two real world robots to estimate the lumped error and show the efficiency of this parameter reduction.

A cross-validation statistical framework for asymmetric data integration.

The proliferation of biobanks and large public clinical data sets enables their integration with a smaller amount of locally gathered data for the purposes of parameter estimation and model prediction. However, public data sets may be subject to context-dependent confounders and the protocols behind their generation are often opaque; naively integrating all external data sets equally can bias estimates and lead to spurious conclusions. Weighted data integration is a potential solution, but current methods still require subjective specifications of weights and can become computationally intractable. Under the assumption that local data are generated from the set of unknown true parameters, we propose a novel weighted integration method based upon using the external data to minimize the local data leave-one-out cross validation (LOOCV) error. We demonstrate how the optimization of LOOCV errors for linear and Cox proportional hazards models can be rewritten as functions of external data set integration weights. Significant reductions in estimation error and prediction error are shown using simulation studies mimicking the heterogeneity of clinical data as well as a real-world example using kidney transplant patients from the Scientific Registry of TransplantRecipients.

Parameter Estimation for Dynamical Systems Using a Deep Neural Network

The deep neural network (DNN) was applied for estimating a set of unknown parameters of a dynamical system whose measured data are given for a set of discrete time points. We developed a new vectorized algorithm that takes the number of unknowns (state variables) and number of parameters into consideration. The algorithm, first, trains the network to determine weights and biases. Next, the algorithm solves the systems of algebraic equations to estimate the parameters of the system. If the right hand side function of the system is smooth and the system have equal numbers of unknowns and parameters, the algorithm solves the algebraic equation at the discrete point where absolute error between the neural network solutions and the measured data is minimum. This improves the accuracy and reduces computational time. Several tests were carried out in linear and non-linear dynamical systems. Last, we showed that the DNN approach is more successful in terms of computational time as the number of hidden layers increases.

Virtual Evaluation of Deep Learning Techniques for Vision-Based Trajectory Tracking

<div class="section abstract"><div class="htmlview paragraph">Artificial intelligence (AI) enhanced control system deployments are emerging as a viable substitute to more traditional control system. In particular, deep learning techniques offer an alternate approach to tune the ever increasing sets of control system parameters to extract performance. However, the systematic verification and validation (to establish the reliability and robustness) of deep learning based controllers in actual deployments remains a challenge. This is exacerbated by the need to evaluate and optimize control systems embedded within an operational environment (with its own sets of additional unknown or uncertain parameters). Existing literature comparisons of deep learning against traditional controllers, where they may exist, do not offer structured approaches to comparative performance evaluation and improvement. It is also crucial to develop a standardized controlled test environment within which various controllers are evaluated against a common metric. Hence, in this paper, we evaluate deep learning based controllers by a structured selection of pantheon of evaluation metrics and employ the insights to propose further modifications to the deep learning algorithms. We first evaluate a high resolution proportional-integral-derivative (PID) controller to serve as a common benchmark and develop the suite of evaluation metrics for a mobile robot to create a point-to-point trajectory within in the simulation environment. Against this backdrop, we then evaluate alternate variants of imitation learning and deep reinforcement learning controllers which use camera image stream as an input to develop or tune trajectory tracking output parameters. Subsequent deployments of control strategies on actual hardware (in a Sim2Real transition) is anticipated, where the developed evaluation metrics can be used as a validation tool when deploying the control strategy on actual hardware.</div></div>

Delay Identification in Thermoacoustics

Abstract The stability of thermoacoustic systems is often regulated by the time delay between acoustic perturbations and corresponding heat release fluctuations. An accurate estimate of this value is of great importance in applications since even small modifications can introduce significant changes in the system behavior. Different studies show that the nonlinear delayed dynamics typical of these systems can be well captured with low-order models. In this work, a method is introduced to estimate the most likely value of the time delay of a single thermoacoustic mode from a measured acoustic pressure signal. The mode of interest is modeled by an oscillator equation, with a nonlinear delayed forcing term modeling the deterministic flame contribution and an additive white Gaussian noise to embed the stochastic combustion noise. Additionally, other thermoacoustic relevant parameters are estimated. The model accounts for a flame gain, for a flame saturation coefficient, for linear acoustic damping, and for the background combustion noise intensity. The pressure data time series is statistically analyzed and the set of unknown parameters is identified. Validation is performed with respect to synthetically generated time series and low order model simulations, for which the underlying delay is known a priori. A discussion follows about the accuracy of the method, in particular, a comparison with existing methods is drawn.

A new semi-analytical technique for nonlinear systems based on response sensitivity analysis

In this paper, a new semi-analytical method, namely the time-domain minimum residual method, is proposed for the nonlinear problems. Unlike the existing approximate analytical method, this method does not depend on the small parameter and can converge to the exact analytical solutions quickly. The method is mainly threefold. Firstly, the approximate analytical solution of the nonlinear system $${\varvec{F\left( \ddot{x},{\dot{x}},x\right) }}={\varvec{0}}$$ is expanded as the appropriate basis function and a set of unknown parameters, i.e., $${\varvec{x(t)}}\approx \sum _{i=0}^{N}{\varvec{a_i\chi _i(t)}}$$ . Then, the problem of solving analytical solutions is transformed into finding a set of parameters so that the residual $${\varvec{R}}={\varvec{F}}\left( \sum _{i=0}^{N}a_i\ddot{\chi }_i,\sum _{i=0}^{N}a_i{\dot{\chi }}_i,\sum _{i=0}^{N}a_i\chi _i\right) $$ is minimum over a period, i.e., $$\underset{{\varvec{a}}\in {\mathscr {A}}}{\min }\int _{0}^T {\varvec{R}}({\varvec{a}},t)^{T} {\varvec{R}}({\varvec{a}},t) \mathrm {d} t$$ . The nonlinear equation $${\varvec{F\left( \ddot{x},{\dot{x}},x\right) }}={\varvec{0}}$$ is regarded as the objective function to optimize, and the process of solving the analytic solution is transformed into a nonlinear optimization process. Finally, the optimization process is iteratively solved by the enhanced response sensitivity approach. Four numerical examples are employed to verify the feasibility and effectiveness of the proposed method.

Robust parameter estimation of a PEMFC via optimization based on probabilistic model building

In this work, we approximated a set of unknown physical parameters for a semi-empirical mathematical model of a PEMFC. We used an Estimation of Distribution Algorithm (EDA) known as UMDAG to find the tuple that best reproduces the experimental polarization curve. We tackled non-derivable objective functions to perform robust parameter estimation. We compared the sum of the squared error with published results, and the sum and the median of the absolute error values were used to diminish or remove the effect of possible noise or outliers. Since the UMDAG requires a single user-given parameter (the population size) and presents a natural reduction of the variance, it was possible to introduce a variance-based stopping criterion. The obtained results were compared with the most up-to-date evolutionary algorithms, demonstrating that this proposal is competitive. We used four previously reported experimental datasets to get the parameters or validate them. Two of them were used to test the method and to compare it with reported results of recent bio-inspired metaheuristics. Then, we used the identified parameters to simulate the cases of the remaining data sets validating the correct estimation. Finally, we introduced a posterior statistical analysis (hypothesis test), which provided further information about dependencies and the impact of each parameter on the cell performance.

Solving inverse problems for steady-state equations using a multiple criteria model with collage distance, entropy, and sparsity

In this paper, we extend the previous method for solving inverse problems for steady-state equations using the Generalized Collage Theorem by searching for an approximation that not only minimizes the collage error but also maximizes the entropy and minimizes the sparsity. In this extended formulation, the parameter estimation minimization problem can be understood as a multiple criteria problem, with three different and conflicting criteria: The generalized collage error, the entropy associated with the unknown parameters, and the sparsity of the set of unknown parameters. We implement a scalarization technique to reduce the multiple criteria program to a single criterion one, by combining all objective functions with different trade-off weights. Numerical examples confirm that the collage method produces good, but sub-optimal, results. A relatively low-weighted entropy term allows for better approximations while the sparsity term decreases the complexity of the solution in terms of the number of elements in the basis.

Identification of a Class of Hybrid Dynamical Systems

This paper presents a novel identification procedure for a class of hybrid dynamical systems. In particular, we consider hybrid dynamical systems which are single flowed and single jumped and whose flow and jump maps linearly depend on two sets of unknown parameters. A systematic way to determine whether the system is flowing or jumping is introduced and used to identify the unknown parameters by employing a linear recursive estimator. Simulations have been performed to prove the validity of the proposed methodology. Results proved the efficiency and accuracy of the developed identification procedure.

Controlled Stratification Based on Kriging Surrogate Model: An Algorithm for Determining Extreme Quantiles in Electromagnetic Compatibility Risk Analysis

An electromagnetic compatibility failure is a consequence of an applied interference level being in excess of the susceptibility level of the electronic equipment under investigation. Both interference and susceptibility levels depend on various configurations of coupling paths described by sets of unknown or uncertain parameters. It is therefore convenient to describe the applied interference and the susceptibility levels as random variables. As extreme values may have a strong impact on the risk of failure, we focus in this article on the estimation of extreme values of interference level (relevant applied fields, currents or voltages) by means of a restricted set of numerical simulations. The controlled stratification method aims at reducing the variance of estimation of extreme quantile, based on a correlated simple model. We recently highlighted that a kriging surrogate model was a good candidate to provide this simple model. Combined with controlled stratification, we obtained better estimation performances than using a standalone kriging model with the same output sample size. In practice, this sample size is limited due to the excessive simulation time of electromagnetic solvers. In this paper, we propose an original algorithm, which aims at checking whether the sample size is adequate to perform an acceptable estimation or not. We first validate the algorithm using analytical models. Finally, we apply this method to estimate the 99% quantile of the total radiated power of a source located inside an open cavity with 16 uncertain inputs. In that case, the algorithm reduces the number of calls to the initial model to approximately 40% of the budget that is required using a standard Monte Carlo approach. Moreover, it provides almost 4 times more extreme outputs. More remarkably, our proposed algorithm provides guidance for assessing the performance of quantile estimation according to the initially sample size of the design of experiment.

Adaptive Boundary Observer Design for coupled ODEs-Hyperbolic PDEs systems

We consider the state estimation of ηξ hyperbolic PDEs coupled with ηX ordinary differential equations at the boundary. The hyperbolic system is linear and propagates in the positive x-axis direction. The ODE system is linear time varying (LTV) and includes a set of ηθ unknown constant parameters, which are to be estimated simultaneously with the PDE and the ODE states using boundary sensing. We design a Luenberger state observer, and our method is mainly based on the decoupling of the PDE estimation error states from that of the ODEs via swapping design. We then derive the observer gains through the Lyapunov analysis of the decoupled system. Furthermore, we give sufficient conditions of the exponential convergence of the adaptive observer through differential Lyapunov inequalities (DLIs) and we illustrate the theoretical results by numerical simulations.

Parameter estimation and sensitivity analysis for dynamic modelling and simulation of beer fermentation

Beer fermentation efficiency improvements have the strongest potential to boost profitability, as its long batch time renders this particular unit operation the throughput bottleneck of this complex, multistage biochemical process which mankind has employed for several millennia. Accurate fermentation models are critical for reliable dynamic simulation and process optimisation: empirical trial-and-error approaches are not viable, and incrementally altering proven recipes implies prohibitively expensive campaigns, in terms of equipment use, off-spec production and personnel time for sampling and analysis. This paper considers parameter estimation for a published beer fermentation model, demonstrating that estimating the complete unknown parameter set is an ill-posed problem, which can lead to inconsistent solutions. Systematic sensitivity analysis is pursued, elucidating the relative significance of parametric discrepancy on the validity of key species concentration trajectories. Parameters have been identified and ranked by decreasing importance, and a high-fidelity estimation is performed for a published dataset.

Bayesian coherent and incoherent matched-field localization and detection in the ocean.

Matched-field processing is applied to source localization and detection of sound sources in the ocean. The source spectrum is included in the set of unknown parameters and is estimated in the localization/detection process. Bayesian broadband (multi-tonal) incoherent and coherent processors are developed, integrating the source spectrum estimation using a Gibbs sampler and are first evaluated in source localization via point estimates and probability density functions obtained from synthetic signals. The coherent performance is superior to the incoherent one both in terms of source location estimates and density spread. The two processors are also applied to real data from the Hudson Canyon experiment. Subsequently, using Receiver Operating Characteristic (ROC) curves, the two processors are evaluated and compared in the task of joint detection and localization. The coherent detector/localization processor is superior to the incoherent one, especially as the number of frequencies increases. Joint detection and localization performance is evaluated with Localization-ROC curves.