Bilinear Approximation Research Articles

Data-driven predictor of control-affine nonlinear dynamics: Finite discrete-time bilinear approximation of koopman operator

This paper introduces a novel and efficient data-driven approach for approximating a finite discrete-time bilinear model of control affine nonlinear dynamical systems with a tunable parameter that balances model dimension and prediction accuracy. An approximation of the Koopman operator based on the evolutions of the nonlinear system measurements used to lift a control-affine nonlinear system to a higher dimensional model. However, higher dimensional spaces can result in a long learning time and the curse of dimensionality in control analysis. The proposed approach addresses these challenges by introducing a convex optimization which identifies informative observable functions. This technique allows for the adjustment of a parameter to strike a balance between model dimension and accuracy in prediction. The main contribution of this study is to introduce a reduced dimensional bilinear model for a nonlinear complex system. This achievement is made possible by implementing convex sparse optimization, enabling the exploration of informative estimated Koopman eigenfunctions while minimizing the number of system measurements required. The optimization problem is solved using the alternating direction method of multipliers. The effectiveness of the proposed method is evaluated on three different nonlinear systems: a numerical nonlinear system, a Van der Pol oscillator, and a Duffing oscillator. In the last simulation, an estimation of the Koopman linear model is considered as a special case, and the policy iteration algorithm is employed to evaluate optimal control designed for different reduced-dimensional models.

Calibration of Thermal Viscoelastic Material Models for the Dynamic Responses of PVB and SG Interlayer Materials.

Laminated glass interlayer materials polyvinyl butyral (PVB) and SentryGlas® (SG, kuraray, Houstan, TX, USA) exhibit thermal viscoelastic behavior under dynamic tensile loading. Significant temperature and strain rate effects on the behavior of these interlayer materials pose a challenge for accurately modeling the dynamic response of laminated glass. Many researchers have simplified their approaches by modeling the response of the interlayer material using a bilinear approximation or established hyperelastic models. However, temperature and strain rate effects can be captured using the three-network viscoplastic (TNV) model. Therefore, the objective of this study is to calibrate material models for the thermal viscoelastic dynamic responses of PVB and SG interlayer materials. Uniaxial tensile tests were performed at strain rates of 2, 20, and 45 s-1 and temperatures of 0, 23, and 60 °C, and material models were calibrated using the experimental data. Finite element analysis using the calibrated material models successfully predicted the dynamic responses of PVB and SG under the experimental test conditions within a 10% error margin. This suggests that the calibrated models using the TNV model represent significant improvements over existing approaches to modeling the dynamic response of laminated glass. Similar procedures can be applied to other thermoplastics, laying the groundwork for establishing a standard calibration guide.

Sampling discretization of the uniform norm and applications

Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. Previous known results show that for any N-dimensional subspace of the space of continuous functions it is sufficient to use eCN sample points for an accurate upper bound for the uniform norm by the discrete norm and that one cannot improve on the exponential growth of the number of sampling points for a good discretization theorem in the uniform norm. In this paper we focus on two types of results, which allow us to obtain good discretization of the uniform norm with polynomial in N number of points. In the first way we weaken the discretization inequality by allowing a bound of the uniform norm by the discrete norm multiplied by an extra factor, which may depend on N. In the second way we impose restrictions on the finite dimensional subspace under consideration. In particular, we prove a general result, which connects the upper bound on the number of sampling points in the discretization theorem for the uniform norm with the best m-term bilinear approximation of the Dirichlet kernel associated with the given subspace.

Performance analysis of digitally controlled nonlinear systems considering time delay issues

In this paper, a comprehensive investigation into discretization, effective sample time selection considering delays in the system, and time and frequency domain analysis of a DC-DC buck converter, which plays a vital role in photovoltaic (PV) systems, is conducted to enhance the understanding of their dynamic behavior, optimize control algorithms, improve system efficiency, and ensure reliable power conversion in photovoltaic applications. To effectively address the non-linear behavior and enhance digital control of a buck converter by selecting the best sample time, several approaches can be employed. These include accurate modeling and identification of non-linear elements, development of advanced control algorithms that account for non-linearities, implementation of adaptive control techniques, and utilization of feedback mechanisms to compensate for deviations from linearity. By considering and mitigating the non-linear behavior, digital control systems can achieve improved accuracy, stability, and transient behavior in regulating the buck converter's output waveforms (voltage or current). The results of the study demonstrated that the trapezoidal integration method which is also known as bilinear approximation, or Tustin's approach outperformed other commonly used discretization methods, such as first-order hold (FOH), zero-order hold (ZOH), impulse response matching (impulse invariant), and matched pole-zero (MPZ) technique, in dual-domain (both time and frequency) analysis. The key finding highlighting the superiority of the bilinear approximation was its ability to achieve the closest match in the frequency domain bridging the continuous-time and discrete systems. This finding emphasizes the significance of the bilinear approach in preserving the frequency characteristics of the original continuous-time system during discretization. By employing this method, the discrete system closely approximated the behavior of its continuous-time counterpart, ensuring accurate frequency-domain representation.

Sampling Discretization of the Uniform Norm

Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. We pay special attention to the case of trigonometric polynomials with frequencies from an arbitrary finite set with fixed cardinality. We give two different proofs of the fact that for any N-dimensional subspace of the space of continuous functions it is sufficient to use $$e^{CN}$$ sample points for an accurate upper bound for the uniform norm. Previous known results show that one cannot improve on the exponential growth of the number of sampling points for a good discretization theorem in the uniform norm. Also, we prove a general result, which connects the upper bound on the number of sampling points in the discretization theorem for the uniform norm with the best m-term bilinear approximation of the Dirichlet kernel associated with the given subspace. We illustrate the application of our technique on the example of trigonometric polynomials.

Two-stage distributionally robust strategic offering in pool-based coupled electricity and gas market

The intensified interactions between power and gas systems and the wide utilization of renewable energy introduce additional challenges in energy marketing and pricing mechanisms. Existing studies either neglect energy uncertainties in deterministic market-clearing models, use predetermined strategic offering prices in single-level models, or approximate gas dynamics in nonrealistic models. This paper proposes a bi-level two-stage distributionally robust electricity–gas market clearing (EG-MC) model considering energy uncertainties and strategic offering prices from energy producers. Strategic energy producers submit their offering prices in the upper-level problem to the EG-MC operator, who maximizes market profits under the realizations of renewable energy outputs while balancing the robustness and conservativeness of the day-ahead market decisions. The presence of gas dynamics in the two stages of the decision-making framework generates an intractable EG-MC problem. A novel triple-loop procedure, namely inner and outer columns & constraints generation and bilinear approximation algorithms, is proposed to sufficiently solve the formulated model. Finally, numerical analyses on an EG-MC model demonstrate the effectiveness of the distributionally robust strategic offers and the performances of the solution methodology.

Experimental verification of indirect tests for stress-crack opening curve of concrete in tension from a round robin test: application to railway sleeper elements

This paper presents experimental verification of indirect tests for the determination of the stress-crack opening curve of concrete, based on the method described in the RILEM TC 187-SOC report and submitted to ACI Committee 446 for Fracture Toughness Testing of Concrete, which relies on the standard cohesive model. In such method, a bilinear approximation of the softening curve is determined from three-point bending tests of notched specimens under crack-mouth-opening-displacement control and from tensile splitting tests, which produces good predictions of the complete crack evolution. The current work focusses on the brittleness length, which is directly related to the parameters of the linear curve defined by the first branch of the bilinear curve, and is one of the main fracture parameters of concrete, since it controls the behaviour of thin cracks. This length can be obtained from a simple version of experiments conducted up to peak load, thus, favouring its application in the industry. In particular, this work presents the results of a round-robin test carried out in collaboration with the Spanish Association of Railway Sleeper Element Manufacturers to determine the brittleness length of concrete. A specific implementation of the method has been proposed in order to perform the tests by using the facilities available in all the factories. The results of the round-robin test have been satisfactory, thus, supporting the viability of the RILEM and ACI proposals. Moreover, a method has been developed which allows the brittleness length of concrete to be determined easily in railway sleeper element factories.

Optimization for the minimum fuel consumption problem of a hybrid electric vehicle using mixed-integer linear programming

In this study, a mixed-integer linear programming (MILP) formulation for obtaining the operating points of hybrid electric vehicle (HEV) powertrains is proposed. This study focuses on linearizing the nonlinear terms, such as an engine fuel consumption map, or bilinear terms represented by the energy of the motor. To address this problem, a conventional piecewise linear (PWL) method and a multi-layer perceptron (MLP) regression approach are adopted. Although the optimal solution cannot be determined using a PWL approximation for the fuel consumption map, it can be obtained using an MLP regression. Furthermore, the PWL method achieves better results than the MLP approach in terms of the accuracy of its bilinear approximation. Obtaining the optimal solution using MILP helps in acquiring a Lagrange multiplication of the design variables by solving the dual problem, which allows an efficient design revision strategy to be obtained.

A Wavelet-Based Adaptive Finite Element Method for the Stokes Problems

In this work, we present the mathematical formulation of the new adaptive multiresolution method for the Stokes problems of highly viscous materials arising in computational geodynamics. The method is based on particle-in-cell approach—the Stokes system is solved on a static Eulerian finite element grid and material properties are carried in space by Lagrangian material points. The Eulerian grid is adapted using the wavelet-based adaptation algorithm. Both bilinear (Q1P0, Q1Q1) and biquadratic (Q2P-1) mixed approximations for the Stokes system are supported. The proposed method is illustrated for a number of linear and nonlinear two-dimensional benchmark problems of geophysical relevance. The results of the adaptive numerical simulations using the proposed method are in an excellent agreement with those obtained on non-adaptive grids and with analytical solutions, while computational requirements are few orders of magnitude less compared to the non-adaptive simulations in terms of both time and memory usage.

A new approach for stabilization of control-affine systems via integral inequalities

Abstract In this work, we use a bilinear approximation to examine the stability problem of a class of control-affine systems. We show that a continuous feedback may stabilize the system with given constraints on the term of perturbation. We provide a stabilization result for bilinear systems in presence of perturbations using integral inequalities. Moreover, we prove a new extension of Gamidov’s Lemma that allows us to stabilize a class of affine systems. An example and an application are given to verify the effectiveness of the proposed approach.

Analytical Nonlinear Modeling of Rail and Fastener Longitudinal Response

Comparatively little attention has been given to the quantification of fastener demands, especially in the longitudinal direction. Research quantifying fastener demands is justified by the more than 250 FRA reportable derailments on mainlines and sidings in the United States caused by “defective or missing spikes or rail fasteners” over the last 20 years. Failed fasteners are rarely caused by loads acting from a single direction (vertical, lateral, or longitudinal); they occur from a combination of these loads. A literature review identified that though multiple models have been developed for analyzing track, they were not designed to quantify fastener demands, especially those in the longitudinal direction, and some make assumptions that could be improved on based on more recent research into the mechanics of fastening system behavior. This paper advances the mechanistic–empirical (M-E) track analysis and design approach through the development, validation, and application of a 3D nonlinear parametric track model that quantifies longitudinal fastener demands. Key research findings include: bilinear approximations, in combination with considering the interaction between vertical loads and slip, were necessary to accurately quantify fastener forces; ballast and fastener stiffness had a direct logarithmic relationship on fastener load; and for well-supported sleepers, changes in component resistance to slip produced minimal changes in fastener demands because the vertical applied load increased the required load to produce slip. Going forward, this validated model could be used to quantify fastener and track demands for additional loading and operational scenarios to further optimize component design for improved track safety and reliability.

Study of the volumetric behavior of a residual tropical clay

This paper focuses on the volumetric behavior of residual tropical soils. Two residual tropical soils belonging to the same geological formation and located in the coastal sedimentary basin of Gabon are studied. These soils are known as pelitic soils. As the hydromechanical behavior of residual tropical soils is still relatively unknown, it is interesting to study the influence of suction on the volumetric behavior of these soils through shrinkage curves. The purpose of this study is twofold. First, to study the volume variations in the two studied residual tropical soils with suction changes. Secondly, to propose a modelling of soil-water retention curves (SWRCs) of these soils. For this purpose, shrinkage tests are carried out to study the volume variations of the studied soils. These tests are supported by determining a bilinear approximation model to enable the variation in suction during drying to be estimated. Tests to determine SWRCs at different initial dry densities are used to calibrate the bilinear approximation model between void ratio and suction. The modified Kovács model (MK model) has been adapted to model SWRCs of the studied soils.

Bilinear Systems With Initial Gaps Involving Inelastic Collision: Forced Response Experiments and Simulations

Abstract In this paper, the forced response of a two degrees-of-freedom (DOF) bilinear oscillator with initial gaps involving inelastic collision is discussed. In particular, a focus is placed upon the experimental verification of the generalized bilinear amplitude approximation (BAA) method, which can be used for the accurate estimation of forced responses for bilinear systems with initial gaps. Both experimental and numerical investigations on the system have been carried out. An experimental setup that is capable of representing the dynamics of a 2DOF oscillator has been developed, and forced response tests have been conducted under swept-sine base excitation for different initial gap sizes. The steady-state response of the system under base excitation was computed by both traditional time integration and BAA. It is shown that the results of experiments and numerical predictions are in good agreement especially at resonance. However, slight differences in the responses obtained from both numerical methods are observed. It was found that the time duration where the DOFs are in contact with each other predicted by BAA is longer than that predicted by time integration. Spectral analyses have also been conducted on both experimental and numerical results. It was observed that in a frequency range where intermittent contact between the masses occurs, super-harmonic components of the excitation frequency are present in the spectra. Moreover, as the initial gap size increases, the frequency band where the super-harmonic components are observed decreases.

ДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ РАВНОВЕСИЯ ИДЕАЛЬНО УПРУГОПЛАСТИЧЕСКОЙ СПЛОШНОЙ СРЕДЫ ДЛЯ ПЛОСКОЙ ДЕФОРМАЦИИ В ЦИЛИНДРИЧЕСКИХ КООРДИНАТАХ ПРИ БИЛИНЕЙНОЙ АППРОКСИМАЦИИ ЗАМЫКАЮЩИХ УРАВНЕНИЙ

Abstract. The article considers the construction of differential equations of equilibrium in displacements for plane deformation of elastic perfectly plastic regarding shear deformations continuous medium and nonlinearly elastic continuous medium with respect to volumetric deformations with bilinear approximation of the closing equations, both regarding and regardless geometrical nonlinearity in a cylindrical coordinate system. Nonlinear diagrams of volumetric and shear deformation are approximated by bilinear functions. Proceeding from the assumption of independence, generally speaking, of volume and shear deformation from each other, five main cases of physical dependencies are considered, depending on the relative position of the break points of bilinear diagrams of volume and shear deformation. The construction of bilinear physical dependencies is based on the calculation of the secant moduli of volumetric and shear deformation. In this case, in the first section of the diagrams, the secant modulus of both volumetric and shear deformation is constant, while in the second section of the diagrams, the secant modulus of volumetric deformation is a function of volumetric deformation, and the secant shear modulus is a function of the intensity of shear deformations. Substituting the corresponding bilinear physical relations into the differential equations of equilibrium of a continuous medium, written both regardless and regarding geometrical nonlinearity, the resolving differential equations of equilibrium in displacements for plane deformation in a cylindrical coordinate system are received. The received differential equations of equilibrium in displacements in cylindrical coordinates can be applied in determining the stress-strain state of elastic perfectly plastic with respect to shear deformations continuous medium and nonlinearly elastic with respect to volumetric deformations continuous medium under conditions of plane deformation, both regarding and regardless geometrical nonlinearity, physical relations for which are approximated by bilinear functions.

ДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ РАВНОВЕСИЯ СПЛОШНОЙ СРЕДЫ ДЛЯ ПЛОСКОЙ ДЕФОРМАЦИИ В ЦИЛИНДРИЧЕСКИХ КООРДИНАТАХ ПРИ БИЛИНЕЙНОЙ АППРОКСИМАЦИИ ЗАМЫКАЮЩИХ УРАВНЕНИЙ

Problems of the formulation of differential equations of equilibrium in terms of displacements for a plane strain of continuous media at bilinear approximation of closing equations are considered leaving out of account geometric nonlinearity in the cylindrical coordinate system. Based on the assumption that the curves of volumetric and shear strain are independent from each other, six main cases of physical dependencies are considered, which are the functions of the relative position of break points on the bilinear curves of the volumetric and shear strain. Obtaining of bilinear physical dependencies is based on the calculation of secant moduli of the volumetric and shear strain. On the first line of the curves, secant moduli are constant for both volumetric and shear strain, while on the second line, the secant modulus of the volumetric strain is a function of the volumetric strain, and the secant modulus of the shear strain is a function of the shear strain intensity. Putting the corresponding bilinear physical equations into differential equations of continuum equilibrium, which disregard geometrical nonlinearity, the resulting differential equations of equilibrium are obtained in terms of displacements for a one-dimensional plane strain of continuum in the cylindrical coordinate system. These equations can be used when determining stress-strain state of continuous media under one-dimensional plane strains with no regard for geometrical nonlinearity, and whose physical relations are approximated by bilinear functions.

Stress-Strain State of Steam Turbine Lock Joint Under Plastic Deformation

The stress-strain state problem for the lock joint of the rotor blades of the first stage of the medium-pressure cylinder under plastic deformation is solved. When solving the problem, the theory of elastic-plastic deformations is used. The problem is solved using two different approaches to specifying plastic deformation curves. The applicability of using a simpler bilinear approximation instead of the classical multilinear one is estimated. Based on the example of solving this problem, the time required to perform the calculation with the use of the bilinear and multilinear approximations is shown. Comparison of the results obtained in the form of the distribution of plastic deformations, equivalent stresses, and contact stresses over support pads made it possible to assess the difference when the two types of approximation are used. The obtained result error value when using the bilinear approximation made it possible to draw conclusions about the applicability of this approach to the processing of plastic deformation curves for solving problems of this kind. The problem is solved using the finite element method. To objectively assess the effect of plastic deformation on the redistribution of loads in the lock joint, a finite element model is used, obtained when solving the problem of the thermally stressed state of the rotor blade lock joint. The distribution of contact stresses in the lock joint is shown. The results are compared with those obtained earlier when solving the problem of thermoelasticity. Significant differences in the level of contact stresses are noted. Results of the computational assessment of the stress-strain state of the lock joint of the rotor blades of the first stage of the medium-pressure cylinder of a steam turbine are presented, which allow characterizing the degree of relaxation and redistribution of stresses in the structure in comparison with the results obtained earlier when solving the problem of thermoelasticity. Conclusions are made about the economic viability of using the calculation methods presented.

Nonlinear predictive control employing Carleman bilinearization and fixed search directions

AbstractThis work is concerned with the predictive control of nonlinear systems using the Carleman bilinearization technique. Given a nonlinear system model with sufficient regularity, a bilinear approximation can be obtained by means of Carleman's technique. After discretizing the continuous time model, several algorithms exist to tackle the bilinear model predictive control (BMPC) problem. More specifically, the present work exploits the combination of Carleman's approximation with a fixed search directions algorithm previously proposed within the context of BMPC. Simulations are carried out in order to compare predictive controllers designed using bilinear and linear plant models, but acting on the original nonlinear system. Moreover, the fixed search directions algorithm is compared with the use of a standard interior point optimizer. As a result, the proposed method is shown to provide improvements in terms of either stability, constraint satisfaction or reduced computational effort.

Prediction of the Bilinear Stress-Strain Curve of Aluminum Alloys Using Artificial Intelligence and Big Data

Aluminum alloys are among the most widely used materials in demanding industries such as aerospace, automotive or food packaging and, therefore, it is essential to predict the behavior and properties of each component. Tools based on artificial intelligence can be used to face this complex problem. In this work, a computer-aided tool is developed to predict relevant mechanical properties of aluminum alloys—Young’s modulus, yield stress, ultimate tensile strength and elongation at break. These predictions are based on the alloy chemical composition and tempers, and are employed to estimate the bilinear approximation of the stress-strain curve, very useful as a decision tool that helps in the selection of materials. The system is based on the use of artificial neural networks supported by a big data collection about technological characteristics of thousands of commercial materials. Thus, the volume of data exceeds 5 k entries. Once the relevant data have been retrieved, filtered and organized, an artificial neural network is defined and, after the training, the system is able to make predictions about the material properties with an average confidence greater than 95 % . Finally, the trained network is employed to show how it can be used to support decisions about engineering applications.

Дифференциальные уравнения равновесия идеально упругопластической сплошной среды для плоской деформации в декартовых координатах при билинейной аппроксимации замыкающих уравнений

Дифференциальные уравнения равновесия идеально упругопластической сплошной среды для плоской деформации в декартовых координатах при билинейной аппроксимации замыкающих уравнений

Benchmark Analyses on the Elastic–Plastic Behavior of Carbon Steel Pipes Under Seismic Load

Abstract A series of inelastic benchmark and parametric analyses was conducted on the tests of a pipe elbow and a piping system model made from carbon steel to investigate variabilities of elastic–plastic analysis results and clarify the factors affecting the analytical results. The analysis on the pipe elbow was the inelastic static analysis and the analysis on the piping system model included the inelastic dynamic analysis under the random input motion. From the benchmark analysis results, we found that setting the yield stress in the material property approximation had a significant influence on the inelastic analytical results, while the work hardening modulus in the bilinear approximation of the stress–strain curve had little influence. The parametric analysis was performed with the viewpoint to examine the influence of the yield stress and the work hardening modulus when the material property was approximated as the bilinear relation. The results also showed that the setting of the yield stress had larger impact on the analytical results than the setting of the work hardening modules. The parametric analysis results confirmed that the variation in the analytical results among different analysts would be reduced by using the same material property approximation.