Bilinear Approximation Research Articles

A new Interpolatory Model Reduction Approach for Quadratic Bilinear Descriptor Systems

AbstractWe propose interpolation based projection techniques for model reduction of quadratic bilinear descriptor systems. The approach identifies projection matrices from the bilinear approximation of the original quadratic bilinear descriptor system and applies Petrov‐Galerkin projection to the original system to construct the required reduced quadratic bilinear descriptor system. Two different techniques of computing the projection matrices from the bilinear approximation are proposed. One is based on linear parametric model reduction and the other is the bilinear rational Krylov method. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Digital and analog integer delayed modeling and control for multivariable systems with multiple time delays in states, inputs and outputs

This paper presents a new approximated integer delayed modeling and control technique for Continuous-time Fractional Delay Differential Equations (CFDDEs). Using the impulse response sequences of the CFDDE and the balanced model-reduction method, a delay-free discrete-time state-space model is constructed. Then, an equivalent Discrete-time Integer Delay Difference Equation (DIDDE) is obtained by transforming the obtained discrete-time state-space model into a controller-type block companion form. Furthermore, based on the obtained DIDDE, an equivalent Continuous-time Integer Delay Differential Equation (CIDDE) is determined by means of the newly developed Chebyshev's bilinear approximation method. For digital control of the CFDDE, an optimal Discrete-time Integer Delayed Control Law (DIDCL) is designed using the conventional discrete-time LQR approach together with the obtained delay-free discrete-time state-space model. On the other hand, for continuous-time control of the CFDDE, a Continuous-time Integer Delayed Control Law (CIDCL) is determined from the designed DIDCL by means of the inverse Chebyshev's bilinear approximation method. Finally, digital and analog integer delayed observers are constructed for the implementations of the developed DIDCL and CIDCL, respectively. An illustrative example is given to demonstrate the effectiveness of the proposed method.

Nonlinear tensor product approximation of functions

We are interested in approximation of a multivariate function f(x1,…,xd) by linear combinations of products u1(x1)⋯ud(xd) of univariate functions ui(xi), i=1,…,d. In the case d=2 it is the classical problem of bilinear approximation. In the case of approximation in the L2 space the bilinear approximation problem is closely related to the problem of singular value decomposition (also called Schmidt expansion) of the corresponding integral operator with the kernel f(x1,x2). There are known results on the rate of decay of errors of best bilinear approximation in Lp under different smoothness assumptions on f. The problem of multilinear approximation (nonlinear tensor product approximation) in the case d≥3 is more difficult and much less studied than the bilinear approximation problem. We will present results on best multilinear approximation in Lp under mixed smoothness assumption on f.

Comparative Analysis of the Energy Dissipation of Steel Buildings with Concentric and Eccentric Braces

A comparative study has been done to analyze the behavior of regular steel building structures of 4, 6, 8 and 10 stories, located in seismic zone 5 and soil type S1. The structures were upgraded with different brace configurations according to current Venezuelan codes. A total number of 24 numerical models were analyzed considering non-linear static and incremental dynamic analysis (IDA). The buildings were initially designed as moment resisting frames, and upgraded with six different bracing configurations: concentric braces in “X” and inverted “V”; eccentric braces inverted "V" with horizontal links, inverted “Y” and “X” with vertical links. Short length links were used to ensure a shear failure. The used methodology is based on obtaining the capacity, IDA curves, and bilinear approximations of these curves that allow the determination of yield and ultimate capacity points, in order to estimate important parameters of seismic response: overstrength and ductility; and considering these areas under the curves to estimate elastic deformation energy, energy dissipated by hysteretic damping and equivalent damping. According to the results, the cases with no brace enhancement showed the lowest lateral strength and lateral stiffness and high deformation capacity. On the other hand, the concentric bracing cases, resulted with the highest stiffness and strength and the lowest deformation capacity, therefore they have low ductility and energy dissipation capacity under seismic loading. Structures with links showed intermediate stiffness and strengths, resulting in the best performance in terms of ductility and energy dissipation capacity. The present study provides a better understanding of the benefits of eccentrically braced systems.

Bootstrapping bilinear models of Simple Vehicles

Learning and adaptivity will play a large role in robotics in the future. Two questions are open: (1) in principle, how much it is possible to learn; and (2) in practice, how much should an agent be able to learn. The bootstrapping scenario describes the extreme case in which agents need to learn “everything” from scratch, including a torque-to-pixels model for its robotic body. This paper considers the bootstrapping problem for a subset of the set of all robots. The Simple Vehicles are an idealization of mobile robots equipped with a set of “canonical” exteroceptive sensors: the camera, the range finder and the field sampler. The sensorimotor dynamics of these sensors are derived and shown to be surprising similar. These sensorimotor dynamics are well approximated by a class of nonlinear systems that assume an instantaneous bilinear relation among observations, commands, and changes in the observations. The bilinear approximation is sufficient to guarantee success in the task of generalized “servoing”: driving the observations to a given goal snapshot. Simulations and experiments substantiate the theoretical results. This is the first instance of a bootstrapping agent that can learn the model of the dynamics of a relatively large universe of systems and use the models to solve well-defined tasks, with no parameter tuning or hand-designed features.

Safe Approximations for Distributionally Robust Joint Chance Constrained Program

In this paper, we present a bilinear second-order cone programming safe approximation for the distributionally robust chance constrained program (DRCCP), assuming that the support of the uncertain parameters, and the first and second marginal moments of the probability with respect to the interval constraint imposed on the sum of the uncertain parameters are given. If we further know the covariance matrix, we can obtain a bilinear semi-definite programming safe approximation. Preliminary numerical tests indicate that the proposed models are competitive.

Bilinear approximations to the mixed-mode I–II delamination cohesive law using an inverse method

This paper presents an experimental and numerical study of the mixed-mode I–II delamination of unidirectional carbon/epoxy laminates. Double cantilever beam, end-notched flexure and mixed-mode bending tests were conducted in order to cover the full range of mode-mix combinations. Finite element cohesive zone models with a bilinear softening cohesive law were subsequently employed to fit experimental load–displacement curves. This required a genetic algorithm to find optimal bilinear softening parameters. Good agreement was achieved between predicted and experimental curves. Furthermore, the bilinear softening parameters showed consistent variations with the global mode-mix that could be interpreted in terms of local damage mechanisms. Nevertheless, further research is needed to define mixed-mode tractions and separations leading to a unique cohesive law.

Comparative assessment of linear and bilinear prism-based strategies for terrain correction computations

We address the evaluation of the potential and of the gravitational attraction of mass distributions, assigned by means of a Digital Terrain Model (DTM), for terrain correction computations. In particular, we improve a recent analytical formulation based on the approximation of topographic masses by vertical prisms with linear top surfaces and compare the computational features of the enhanced formulation with alternative ones based on polyhedral modeling on bilinear approximation of the surface relief or on numerical approaches. The numerical tests carried out on the DTM of an area near Cassino (Italy) proved that the proposed enhanced analytical formulation, besides providing exact values of the potential and of the gravitational attraction, turns out to be faster than alternative approaches.

Application of a Refined Multi-Field Beam Model for the Analysis of Complex Configurations

This article proposes a complex application of a refined electro-mechanical beam formulation. Lagrange-type polynomials are used to interpolate the unknowns over the beam cross section. Three- (L3), four- (L4), and nine-point (L9) polynomials, which lead to linear, bi-linear, and quadratic displacement field approximations over the beam cross-section, are considered. Finite elements are obtained by employing the principle of virtual displacement in conjunction with the Carrera unified formulation (CUF). With the CUF application, the finite element matrices and vectors are expressed in terms of fundamental nuclei whose forms do not depend on the assumption made (L3, L4, or L9). Additional refined beam models are implemented by introducing further discretizations over the beam cross-section. Some assessments from the bibliography have been considered in order to validate the electro-mechanical formulation. Complex three-dimensional geometries have been studied in order to demonstrate the capabilities of the present formulation.

Time‐dependent finite element simulations of a shear‐thinning viscoelastic fluid with application to blood flow

SUMMARYA new finite element method is developed to simulate time‐dependent viscoelastic shear‐thinning flows characterized by the generalized Oldroyd‐B model. The focus of the algorithm is improved stability through a free‐energy dissipative scheme by using low‐order piecewise‐constant finite element approximations for stress. The algorithm is further modified by incorporating a pressure‐projection method, a DG‐upwinding scheme, a symmetric interior penalty DG method to solve the elliptic pressure‐update equation and a geometric multigrid preconditioner. The improved stability and cost to accuracy is compared when using higher order discontinuous bilinear approximation, where in addition, we consider the influence of a slope limiter for these elements. The algorithm is applied to the 2D start‐up‐driven cavity problem, and the stability of the free energy is illustrated and compared between element choices. An application of the model to modelling blood in small arterioles and channels is considered by simulating pulsatile blood flow through a stenotic arteriole. The individual influences of viscoelasticity and shear‐thinning within the generalized Oldroyd‐B model are investigated by comparing results to the Newtonian, generalized Newtonian and Oldroyd‐B models. Copyright © 2014 John Wiley & Sons, Ltd.

Observer-based sampled-data control for nonlinear systems: Robust intelligent digital redesign approach

This paper suggests a solution for a robust digital implementation of the output-feedback control system by utilizing intelligent digital redesign (IDR). The general IDR method involves designing a digital fuzzy controller which is comparable to a pre-designed analog one via state-matching. However, under containing the parametric uncertainties, the new problem about the structural property of the uncertainties is inevitably occurred in the procedure of discretization. For solving that, we use the bilinear and inverse-bilinear approximation method and the concerned IDR is viewed as a convex minimization problem of the norm distances between linear operators to be matched. Also, we consider the state estimation error in the plant dynamics so that we attempt to improve the control performance. The robust stability property is preserved by the proposed IDR method and its condition is represented in terms of linear matrix inequalities (LMIs). The simulation results for permanent magnet synchronous motor (PMSM) system are demonstrated to visualize the feasibility of the proposed method.

Best Bilinear Approximations for the Classes of Functions of Many Variables

We obtain upper bounds for the values of the best bilinear approximations in the Lebesgue spaces of periodic functions of many variables from the Besov-type classes. In special cases, it is shown that these bounds are order exact.

Nonlinear amplitude approximation for bilinear systems

Abstract An efficient method to predict vibration amplitudes at the resonant frequencies of dynamical systems with piecewise-linear nonlinearity is developed. This technique is referred to as bilinear amplitude approximation (BAA). BAA constructs a single vibration cycle at each resonant frequency to approximate the periodic steady-state response of the system. It is postulated that the steady-state response is piece-wise linear and can be approximated by analyzing the response over two time intervals during which the system behaves linearly. Overall the dynamics is nonlinear, but the system is in a distinct linear state during each of the two time intervals. Thus, the approximated vibration cycle is constructed using linear analyses. The equation of motion for analyzing the vibration of each state is projected along the overlapping space spanned by the linear mode shapes active in each of the states. This overlapping space is where the vibratory energy is transferred from one state to the other when the system switches from one state to the other. The overlapping space can be obtained using singular value decomposition. The space where the energy is transferred is used together with transition conditions of displacement and velocity compatibility to construct a single vibration cycle and to compute the amplitude of the dynamics. Since the BAA method does not require numerical integration of nonlinear models, computational costs are very low. In this paper, the BAA method is first applied to a single-degree-of-freedom system. Then, a three-degree-of-freedom system is introduced to demonstrate a more general application of BAA. Finally, the BAA method is applied to a full bladed disk with a crack. Results comparing numerical solutions from full-order nonlinear analysis and results obtained using BAA are presented for all systems.

Approximation rates for the hierarchical tensor format in periodic Sobolev spaces

In this note we estimate the asymptotic rates for the L2-error decay and the storage cost when approximating 2π-periodic, d-variate functions from isotropic and mixed Sobolev classes by the recent hierarchical tensor format as introduced by Hackbusch and Kühn. To this end, we survey some results on bilinear approximation due to Temlyakov. The approach taken in this paper improves and generalizes recent results of Griebel and Harbrecht for the bi-variate case.

Best trigonometric and bilinear approximations of classes of functions of several variables

Order-sharp estimates of the best orthogonal trigonometric approximations of the Nikol’skii-Besov classes Bp,θr of periodic functions of several variables in the space Lq are obtained. Also the orders of the best approximations of functions of 2d variables of the form g(x, y) = f(x−y), x, y ∈ \(\mathbb{T}\)d = Πj=1d[−π, π], f(x) ∈ Bp,θr, by linear combinations of products of functions of d variables are established.

Two analogous methods for estimating the compressive strength of fibrous composites

In this paper, an analogy is made between the solution of micro-buckling of fibrous composites using a three-dimensional model and that of biaxial bending of reinforced concrete short columns. Two approaches are used; the first one uses a reciprocal formula and the second one uses a bilinear approximation to the non-dimensional stresses interaction equation, to estimate the compressive stress in a fibrous composite. The initial misalignment angles of composite fibers, which are the main parameters in the determination of the compressive strength of fibrous composites, are analogous to load eccentricities in concrete columns. The initial misalignment angles in both directions perpendicular to the axis of the fibers are defined by sinusoidal curves. The compressive strength of different fibrous composites, which also depends on the nonlinear shear stress–strain relationship of the matrix material, is estimated using the present approaches. The results obtained in this study agree well with experimental and analytical results available in the literature.

Cohesive zone model and quasibrittle failure of wood: A new light on the adapted specimen geometries for fracture tests

The present study focuses on the cohesive zone modeling of quasibrittle failure of wood. The outline of a general estimation procedure of the cohesive zone parameters based on the one-to-one correspondence which exists between the R-curve and the softening function is presented for any kind of specimen geometry and for the bilinear approximation of the softening function. The connection between the R-curve characteristics and the cohesive parameters provides a new light on the effect of boundary conditions issued from specimen geometries used for wood fracture tests. More particularly, the steady state propagation of the main crack with its fracture process zone is required to estimate relevant fracture properties in wood. Due to the consequent size of the fracture process zone (FPZ) in wood, the steady state propagation requires specimen geometries exhibiting a sufficiently long ligament length to avoid confinement of FPZ during its development and its propagation. Finally, the variability of the cohesive parameters as well as their dependences with respect to the specimen geometry are discussed in the case of Norway Spruce.

Biomechanical time dependency of the periodontal ligament: a combined experimental and numerical approach

The analysis of the non-linear and time-dependent viscoelasticity of the periodontal ligament (PDL) enables a better understanding of the biomechanical features of the key regulator tissue for tooth movement. This is of great significance in the field of orthodontics as targeted tooth movement remains still one of the main goals to accomplish. The investigation of biomechanical aspects of the PDL function, a difficult area of research, helps towards this direction. After analysing the time-dependent biomechanical properties of pig PDL specimens in an in vitro experimental study, it was possible to confirm that PDL has a viscoelastic anisotropic behaviour. Three-dimensional finite element models of mini-pig mandibular premolars with surrounding tissues were developed, based on micro-computed tomography (μCT) data of the experimental specimens. Tooth mobility was numerically analysed under the same force systems as used in the experiment. A bilinear material parameter set was assumed to simulate tooth displacements. The numerical force/displacement curves were fitted to the experimental curves by repeatedly calculating tooth displacements of 0.2mm varying the loading velocities and the parameters, which describe the nonlinearity. The experimental results showed a good agreement with the numerical calculations. Mean values of Young's moduli E1, E2 and ultimate strain ε12 were derived for the elastic behaviour of the PDL for all loading velocities. E1 and E2 values increased with increasing the velocity, while ε12 remained relatively stable. A bilinear approximation of material properties of the PDL is a suitable description of measured force/displacement diagrams. The numerical results can be used to describe mechanical processes, especially stress-strain distributions in the PDL, accurately. Further development of suitable modelling assumptions for the response of PDL under load would be instrumental to orthodontists and engineers for designing more predictable orthodontic force systems and appliances.

Estimates for bilinear approximations of the classes $ S_{{p,\theta}}^{\varOmega }B $ of periodic functions of two variables

We obtain exact-order estimates for the best bilinear approximations of the classes $ S_{{p,\theta}}^{\varOmega }B $ of periodic functions of two variables in the space L q for some relations between the parameters p, q, and θ.

Identification of a Benchmark Wiener–Hammerstein: A bilinear and Hammerstein–Bilinear model approach

In this paper the Wiener–Hammerstein Benchmark is identified as a bilinear discrete system. The bilinear approximation relies on both facts that the Wiener–Hammerstein system can be described by a Volterra series which can be approximated by bilinear systems. The identification is performed with an iterative bilinear subspace identification algorithm previously proposed by the authors. In order to increase accuracy, polynomial static nonlinearities were added to the bilinear model input. These Hammerstein type bilinear models are then identified using the same iterative subspace identification algorithm.