Error Formulation Research Articles

Adaptive control of A class of nonlinear systems with guaranteed parameter estimation: A concurrent learning based approach

AbstractRecent advances in concurrent learning based adaptive controllers have relaxed the persistency of excitation condition required to achieve exponential tracking and parameter estimation error convergence. This was made possible via the use of additional concurrent learning stacks in the parameter estimation algorithm. However, the proposed concurrent learning components, that is, the history stacks, needed to be filled with “selected” values dependent on the actual system states. Therefore, the previously proposed concurrent learning adaptive controllers required the system to be stable initially for a finite time so that the corresponding history stacks can be filled (finite excitation condition). In this work, motivated to remove the finite excitation condition, a novel desired system state based concurrent learning adaptive controller is proposed. In order to remove the system state dependencies in the controller and estimation algorithms, a filtered version of the dynamics and a novel prediction error formulation have been designed. The overall exponential stability, parameter error convergence and boundedness of the system states during closed loop operations are ensured via Lyapunov based arguments. The main advantages of the proposed method are its dependence on the desired system states and the overall stability results that paved the way in removing the need for finite excitation condition. Numerical studies performed on a two link robotic device are also presented to illustrate the feasibility of the proposed method.

Ameliorated Snake Optimizer-Based Approximate Merging of Disk Wang-Ball Curves.

A method for the approximate merging of disk Wang-Ball (DWB) curves based on the modified snake optimizer (BEESO) is proposed in this paper to address the problem of difficulties in the merging of DWB curves. By extending the approximate merging problem for traditional curves to disk curves and viewing it as an optimization problem, an approximate merging model is established to minimize the merging error through an error formulation. Considering the complexity of the model built, a BEESO with better convergence accuracy and convergence speed is introduced, which combines the snake optimizer (SO) and three strategies including bi-directional search, evolutionary population dynamics, and elite opposition-based learning. The merging results and merging errors of numerical examples demonstrate that BEESO is effective in solving approximate merging models, and it provides a new method for the compression and transfer of product shape data in Computer-Aided Geometric Design.

Thermoelastic Stress Analysis in the Presence of Biaxial Stresses in Titanium: Effect of the Mean Stress on Errors in Stress Evaluation

BackgroundThermoelastic Stress Analysis (TSA) is a contactless technique capable of estimating superficial stresses on components subjected to dynamic loads. Surface stresses can be obtained by means of calibration methods that require the assessment of the thermoelastic constants. However, these methods can lead to errors in stress evaluation in those materials where the effect of the mean stress on the thermoelastic signal cannot be neglected (e.g., titanium).ObjectiveThe aim is the development of an analytic formulation of error made in first stress invariant amplitude evaluation for a biaxial stress state, when neglecting mean stress effect.MethodsBy considering the general theory of thermoelastic stress analysis accounting for the mean stress effect, the formulation of the thermoelastic effect in the presence of a biaxial stress state was obtained. The results were compared to those obtained by a numerical simulation and the proposed formulation has been validated for titanium by means of experimental tests.ResultsFirstly, the new formulation of thermoelastic temperature variations accounting for mean stress effect in presence of a biaxial stress state was provided. Secondly, an error analysis provided an analytical formulation for the error made in case mean stress effect is neglected for different case studies.ConclusionsThe error in stress evaluation can be considered as the error originating from the use of an incorrect calibration formula (traditional one)”. The new analytical formulation accounting for the general theory of thermoelastic stress analysis allows to account for the mean stress on titanium in the presence of a uniaxial and biaxial stress states and to evaluate the error made in neglecting such a second order effect when using TSA.

A recursive system identification inertia estimator for traditional and converter-interfaced generators

This paper proposes a method to estimate the inertia constant of synchronous generators (SGs) and the virtual inertia provided by converter-interfaced generators (CIGs). It can be applied when the system is under normal operating conditions or after power imbalance events. The algorithm is based on a recursive system identification approach and uses the active power and bus frequency measurements proceeding from phasor measurement units. The measurements are preprocessed to improve the estimation accuracy. Subsequently, a model-independent rotor speed estimator is used to compute rotor speed variations (or internal frequency variations, in the case of CIGs). Afterwards, a recursive least-square equation error formulation is implemented to compute the SG and CIG parameters in the z-domain, which are later converted into their respective counterparts in the s-domain. The method is applied to several generation devices in the IEEE 39-bus and IEEE 118-bus test systems, including traditional SGs and CIGs providing virtual inertia. Numerical results show the accuracy and the low computational burden of the proposed method.

Deterministic bridge regression for compressive classification

Pattern classification using a compact representation is a crucial component of machine intelligence. Specifically, it is essential to learn a model with well-regulated parameters to achieve good generalization. Bridge regression provides a mechanism for regulating parameters through a penalized ℓp-norm. However, due to the nonlinear nature of the formulation, an iterative numerical search is typically used to solve the optimization problem. In this work, we propose an analytic solution for bridge regression based on solving a penalized error formulation using an approximated ℓp-norm. The solution is presented in primal form for over-determined systems and in dual form for under-determined systems. The primal form is suitable for low-dimensional problems with a large number of data samples, while the dual form is suitable for high-dimensional problems with a small number of data samples. We also extend the solution to problems with multiple classification outputs. Numerical studies using simulated and real-world data demonstrate the effectiveness of our proposed solution.

Probability Formulation of Soft Error in Memory Circuit

Downscaling threatens the designers invested in integrity and error mitigation against soft errors. This study formulated the probability of soft error changing the logic state of a Differential Logic with an Inverter Latch (DIL). Using Cadence Virtuoso, current pulses were injected into various nodes in stages until a logic flip was instigated. The voltage and temperature parameters were increased to observe the current level changes over time. The critical charge from each stage was obtained, and a method to formulate the probability of each instance was developed. The voltage produced a higher effect of the change to the critical charge of any instance as compared to temperature. The findings revealed that the N-channel metal-oxide semiconductor (NMOS) drain is more vulnerable to temperature and voltage variation than P-channel metal-oxide semiconductor (PMOS).

Using the (Iterative) Ensemble Kalman Smoother to Estimate the Time Correlation in Model Error

Numerical weather prediction systems contain model errors related to missing and simplified physical processes, and limited model resolution. While it has been widely recognized that these model errors need to be included in the data assimilation formulation, providing prior estimates of their spatio-temporal characteristics is a hard problem. We follow a systematic path to estimate parameters in the model error formulation, specifically related to time-correlated model errors. This problem is more difficult than the standard parameter estimation problem because the model error parameters are only visible through the random model error realisations. By concentrating on linear and nonlinear low-dimensional systems, we are able to highlight the many aspects of this problem, using state augmentation in an ensemble Kalman smoother (EnKS) and its iterative variant (IEnKS). It is not possible to estimate the model error parameters in one assimilation window because enough information has to be gathered to see the parameters through the random errors, even when every time step is observed. If only one parameter is estimated in a linear one-dimensional system the EnKS works well, but when we try to estimate two parameters the method fails. An IEnKS is able to find the correct parameter values for the linear system. For the highly nonlinear logistic map the IEnKS can get stuck in local minima, but with careful tuning of the step length in the iterations and careful transformation of the solution space the correct parameter values can be found. The main conclusion is that estimating model error parameters –even in low-dimensional systems– is a difficult problem, but via careful reformulation of the problem practical solutions can be found.

Generalized n-Dimensional Rigid Registration: Theory and Applications.

The generalized rigid registration problem in high-dimensional Euclidean spaces is studied. The loss function is minimized with an equivalent error formulation by the Cayley formula. The closed-form linear least-square solution to such a problem is derived which generates the registration covariances, i.e., uncertainty information of rotation and translation, providing quite accurate probabilistic descriptions. Simulation results indicate the correctness of the proposed method and also present its efficiency on computation-time consumption, compared with previous algorithms using singular value decomposition (SVD) and linear matrix inequality (LMI). The proposed scheme is then applied to an interpolation problem on the special Euclidean group SE(n) with covariance-preserving functionality. Finally, experiments on covariance-aided Lidar mapping show practical superiority in robotic navigation.

Application of stochastic time dependent parameters to improve the characterization of uncertainty in conceptual hydrological models

• We test stochastic processes to improve modeling of uncertainty in a real case study. • Bayesian inference and cross-validation are used to calibrate and assess performances. • Results highlight clear benefits for appropriate modeling choices. • We make recommendations to uncover/avoid issues due to overparameterization. • We test a novel parallel framework that simplifies access to HPC resources. The traditional description of a hydrological system with a deterministic, conceptual model and a lumped output error term does not explicitly consider the main mechanisms of uncertainty generation due to approximate process representation, unobserved variability in processes and influence factors, and input uncertainty. In this study, we test the description of such intrinsic uncertainty in conceptual models by making process rates stochastic through stochastic, time-dependent rate parameters. We analyze the advantages and challenges of this approach by using Bayesian inference to jointly estimate model parameters, parameters of the stochastic processes, and the time series of the stochastic parameters. Numerically, we use a particle filter to infer the stochastic time series and to approximate the marginal likelihood for Markov chain sampling of the constant parameters. Compared to the lumped error formulation, we achieve a more realistic description of uncertainty, as we obtain larger errors in intrinsic states, larger uncertainty during prediction than calibration periods (a feature missing for simple lumped error models), and autocorrelated model outputs. However, the additional degrees of freedom introduced with stochastic parameters can lead to an unintentional compensation for model deficits or input errors. This problem is symptomatic of model structure inadequacy or poor selection of the parameters that are made stochastic, and can be diagnosed through cross-validation and careful posterior analysis. The proposed approach is computationally more demanding and its implementation more challenging than the traditional description of hydrological systems using a deterministic model with a lumped error term. However, its advantages both in providing a more realistic representation of uncertainties and in diagnosing model deficiencies suggest its adoption and further development in future studies.

Minimal error partially resolving simulation methods for turbulent flows: A dynamic machine learning approach

A significant extension of previously introduced continuous eddy simulation methods is presented by introducing minimal error partially and fully resolving simulation methods for turbulent flows. This approach represents a machine learning strategy for the hybridization of modeling-focused and resolution-focused simulation methods. It can be applied to well-known equation structures (Spalart–Allmaras type equations, usually applied two-equation models), and it can be used for different hybridization types and in different computational versions. Physically, minimal error methods implement a mode interplay, which ensures that the resolution imposed by a model equals the actual flow resolution. Differently formulated simulation methods reveal two typical errors, and they cannot be expected to provide reliable predictions under conditions where validation data are unavailable. These problems can be avoided by minimal error formulations of model structures considered.

Modeling and numerical investigation of a conformable co‐infection model for describing Hantavirus of the European moles

The aim of this work is to investigate theoretically and numerically a conformable‐order epidemiological model of the Susceptible‐Infected (SI) type. The qualitative properties of the solutions are studied. Besides, the dynamics of the problem is discussed in detail. The conformable version of the implicit and explicit Adams multistep methods is proposed with the formulation of the truncation error. An inverse optimization problem using the least square method is introduced to estimate the model parameters. Moreover, to show the effectiveness of the conformable order in describing the studied phenomena and to calibrate the model, some created data of Nova and Bruges virus in Belgium were used.

Reinforcement learning-based finite-time tracking control of an unknown unmanned surface vehicle with input constraints

In this paper, subject to completely unknown system dynamics and input constraints, a reinforcement learning-based finite-time trajectory tracking control (RLFTC) scheme is innovatively created for an unmanned surface vehicle (USV) by combining actor-critic reinforcement learning (RL) mechanism with finite-time control technique. Unlike previous RL-based tracking which requires infinite-time convergence thereby rather sensitive to complex unknowns, an actor-critic finite-time control structure is created by employing adaptive neural network identifiers to recursively update actor and critic, such that learning-based robustness can be sufficiently enhanced. Moreover, deduced from the Bellman error formulation, the proposed RLFTC is directly optimized in a finite-time manner. Theoretical analysis eventually shows that the proposed RLFTC scheme can ensure semi-global practical finite-time stability (SGPFS) for a closed-loop USV system and tracking errors converge to an arbitrarily small neighborhood of the origin in a finite time, subject to optimal cost. Both mathematical simulation and virtual-reality experiments demonstrate remarkable effectiveness and superiority of the proposed RLFTC scheme.

Block coordinate descent algorithm improves variable selection and estimation in error-in-variables regression.

Medical research increasingly includes high‐dimensional regression modeling with a need for error‐in‐variables methods. The Convex Conditioned Lasso (CoCoLasso) utilizes a reformulated Lasso objective function and an error‐corrected cross‐validation to enable error‐in‐variables regression, but requires heavy computations. Here, we develop a Block coordinate Descent Convex Conditioned Lasso (BDCoCoLasso) algorithm for modeling high‐dimensional data that are only partially corrupted by measurement error. This algorithm separately optimizes the estimation of the uncorrupted and corrupted features in an iterative manner to reduce computational cost, with a specially calibrated formulation of cross‐validation error. Through simulations, we show that the BDCoCoLasso algorithm successfully copes with much larger feature sets than CoCoLasso, and as expected, outperforms the naïve Lasso with enhanced estimation accuracy and consistency, as the intensity and complexity of measurement errors increase. Also, a new smoothly clipped absolute deviation penalization option is added that may be appropriate for some data sets. We apply the BDCoCoLasso algorithm to data selected from the UK Biobank. We develop and showcase the utility of covariate‐adjusted genetic risk scores for body mass index, bone mineral density, and lifespan. We demonstrate that by leveraging more information than the naïve Lasso in partially corrupted data, the BDCoCoLasso may achieve higher prediction accuracy. These innovations, together with an R package, BDCoCoLasso, make error‐in‐variables adjustments more accessible for high‐dimensional data sets. We posit the BDCoCoLasso algorithm has the potential to be widely applied in various fields, including genomics‐facilitated personalized medicine research.

Penerapan Metode Fuzzy Tsukamoto untuk Prediksi Jumlah Produksi Tanaman Cabai

Inaccuracy in predicting the amount of chili production is one of the problems found in chili production. Based on the data obtained from observations and interviews conducted with the supervisor of chili plantations, there are 3 variables that can be used to predict the amount of chili production, namely land area, number of seeds, and maintenance costs. The application of the Fuzzy Tsukamoto method to this problem is one of the most appropriate and effective ways. To find out the accuracy of the application of the Tsukamoto fuzzy method for predicting the amount of chili production, it was carried out by comparing the prediction of the amount of chili production obtained from the application of the Fuzzy Tsukamoto method with the actual data on the amount of chili production using the Mean Absoulute Percentage Error (MAPE) formulation, the results obtained were 14%. Can be categorized as a good forecast. The existence of a system for predicting the amount of chili production will be more effective and efficient in its use, the system testing was carried out on 30 users using the Post-study System Usability Quistionnaire (PSSUQ)instrument, it was obtained a value of 80,15% was obtained which can be categorized as feasible to be implemented.

Consistent Right-Invariant Fixed-Lag Smoother with Application to Visual Inertial SLAM

State estimation problems without absolute position measurements routinely arise in navigation of unmanned aerial vehicles, autonomous ground vehicles, etc., whose proper operation relies on accurate state estimates and reliable covariances. Unaware of absolute positions, these problems have immanent unobservable directions. Traditional causal estimators, however, usually gain spurious information on the unobservable directions, leading to over-confident covariance inconsistent with actual estimator errors. The consistency problem of fixed-lag smoothers (FLSs) has only been attacked by the first estimate Jacobian (FEJ) technique because of the complexity to analyze their observability property. But the FEJ has several drawbacks hampering its wide adoption. To ensure the consistency of a FLS, this paper introduces the right invariant error formulation into the FLS framework. To our knowledge, we are the first to analyze the observability of a FLS with the right invariant error. Our main contributions are twofold. As the first novelty, to bypass the complexity of analysis with the classic observability matrix, we show that observability analysis of FLSs can be done equivalently on the linearized system. Second, we prove that the inconsistency issue in the traditional FLS can be elegantly solved by the right invariant error formulation without artificially correcting Jacobians. By applying the proposed FLS to the monocular visual inertial simultaneous localization and mapping (SLAM) problem, we confirm that the method consistently estimates covariance similarly to a batch smoother in simulation and that our method achieved comparable accuracy as traditional FLSs on real data.

Inverse differential quadrature method: mathematical formulation and error analysis.

Engineering systems are typically governed by systems of high-order differential equations which require efficient numerical methods to provide reliable solutions, subject to imposed constraints. The conventional approach by direct approximation of system variables can potentially incur considerable error due to high sensitivity of high-order numerical differentiation to noise, thus necessitating improved techniques which can better satisfy the requirements of numerical accuracy desirable in solution of high-order systems. To this end, a novel inverse differential quadrature method (iDQM) is proposed for approximation of engineering systems. A detailed formulation of iDQM based on integration and DQM inversion is developed separately for approximation of arbitrary low-order functions from higher derivatives. Error formulation is further developed to evaluate the performance of the proposed method, whereas the accuracy through convergence, robustness and numerical stability is presented through articulation of two unique concepts of the iDQM scheme, known as Mixed iDQM and Full iDQM. By benchmarking iDQM solutions of high-order differential equations of linear and nonlinear systems drawn from heat transfer and mechanics problems against exact and DQM solutions, it is demonstrated that iDQM approximation is robust to furnish accurate solutions without losing computational efficiency, and offer superior numerical stability over DQM solutions.

Data-Driven Performance-Prescribed Reinforcement Learning Control of an Unmanned Surface Vehicle.

An unmanned surface vehicle (USV) under complicated marine environments can hardly be modeled well such that model-based optimal control approaches become infeasible. In this article, a self-learning-based model-free solution only using input-output signals of the USV is innovatively provided. To this end, a data-driven performance-prescribed reinforcement learning control (DPRLC) scheme is created to pursue control optimality and prescribed tracking accuracy simultaneously. By devising state transformation with prescribed performance, constrained tracking errors are substantially converted into constraint-free stabilization of tracking errors with unknown dynamics. Reinforcement learning paradigm using neural network-based actor-critic learning framework is further deployed to directly optimize controller synthesis deduced from the Bellman error formulation such that transformed tracking errors evolve a data-driven optimal controller. Theoretical analysis eventually ensures that the entire DPRLC scheme can guarantee prescribed tracking accuracy, subject to optimal cost. Both simulations and virtual-reality experiments demonstrate the remarkable effectiveness and superiority of the proposed DPRLC scheme.

Competitive Algorithms for Large Scale Positive Definite Linear Systems of Equations Using an Error in Variables Optimization Model

We seek positive definite solutions of large scale overdetermined linear systems of equations with multiple right hand sides, where the coefficient and the right hand side matrices are respectively called data and target matrices. We recently proposed a new approach for solving such problems in small scales considering error in both the measured data and the target matrices and using a novel error formulation. We proposed four algorithms for solving small scale positive definite linear systems of equations with both full rank and rank deficient data matrices. Here, we extend our algorithms to solve large scale problems. The algorithms are implemented in Matlab and Fortran (using the ScaLAPACK library) respectively for small and large scale problems . ScaLAPACK is a parallel library written in Fortran to solve linear algebra problems such as computing matrix decompositions for large scale matrices. Block cyclic data distribution is used in ScaLAPACK to provide proper scalability, reliability, portability, flexibility and ease of use. Comparative numerical experiments with two available methods, the interior point method and a method based on quadratic programming, show that our approach produce a smaller standard deviation of the error entries and a smaller effective rank, as being desirable for control problems. Using the Dolan-More performance profiles we show that our proposed algorithms are more efficient and more accurate in computing proper solutions.

Beyond phase error compensation: pixel mapping-based error correction for high-accuracy 3D surface measurement

In optical-based phase-shifting 3D surface profilometry, phase errors are inevitable and lead to final measurement errors. Error compensation, based on pre-estimated phase errors, is an effective technique to alleviate the phase errors. However, the static and inaccurate nature of the pre-estimated errors lead to an implicit limit to the compensation accuracy. Furthermore, the procedures for error pre-estimation require additional projections, which reduces convenience in practice. In this paper, a novel pixel mapping-based error correction method is proposed. Different from compensation, with the proposed pixel-mapping errors inside each camera pixel can be directly corrected by online replacing with its corresponding ground-truth values input in the projector. This replacement technique is the key idea of our method, which establishes a shortcut to error-free ground-truth values, skipping the traditional laborious and imprecise compensation procedures. To achieve pixel-mapping, an optimization method is proposed and no additional projections are required. A phase error formulation is also introduced in the optimization method to eliminate the phase error distrubance when computing the pixel mapping. Extensive experimental results are shown to demonstrate that the phase accuracy can be obviously improved after utilizing the proposed method.

Fast Stokes Flow Simulations for Geophysical‐Geodynamic Inverse Problems and Sensitivity Analyses Based On Reduced Order Modeling

AbstractMarkov chain Monte Carlo (MCMC) methods have become standard in Bayesian inference and multi‐observable inversions in almost every discipline of the Earth sciences. In the case of geodynamic and/or coupled geophysical‐geodynamic inverse problems, however, the computational cost associated with the solution of large‐scale 3‐D Stokes forward problems has rendered probabilistic formulations impractical. Here we present a novel and extremely efficient method to produce ultrafast solutions of the 3‐D Stokes problem for MCMC simulations. Our approach combines the individual benefits of Reduced Basis techniques, goal‐oriented error formulations, and MCMC algorithms to produce an accurate and computationally efficient surrogate for the forward problem. Importantly, the surrogate adapts itself during the MCMC simulation according to the history of the chain and the goals of the inversion. This maximizes the efficiency of the forward problem and removes the need for preinversion off‐line computations to build a surrogate. We demonstrate the benefits and limitations of the method with several numerical examples and show that in all cases the computational cost is of the order of <1% compared to a traditional MCMC approach. The method is general enough to be applied to a range of problems, including uncertainty quantification/propagation, adjoint‐based geodynamic inversions, sensitivity analyses in mantle convection problems, and in the creating surrogate models for complex forward problems (e.g., heat transfer, seismic tomography, and magnetotellurics).