Bilinear Approximation Research Articles

Equivalence of bilinear Routh and Schwarz approximation methods for discrete-time systems

Two methods using the bilinear Routh approximations and Schwarz approximations have been proposed in the literature for the order reduction of discrete-time systems. Both methods yield the reduced models which preserve the stability and first few time-moments of the original ones. Recently, improvements have been made so that the impulse response energies of original models are also retained in the reduced models. In this Letter, it is shown that the former two methods and their improved versions are respectively equivalent to each other.

Post-limit stiffness and ductility of end-plate beam-to-column steel joints

A procedure for the evaluation of ductility in steel joints is presented. Using the component method as background, a non-linear analysis for a number of end-plate beam-to-column joints is performed that is capable of identifying the “yield” sequence of the various components and the failure of the joint. Each component is characterised using a bi-linear approximation for the force–displacement relation. Comparing these results with the corresponding experimental results leads to a proposal of the post-limit stiffness of the various components. A component ductility index is proposed for each component as a means of classification with respect to ductility, using the three ductility classes currently proposed in the literature. A joint ductility index is also proposed, which can be used to verify available rotation against the structure required rotation.

A component model for the behaviour of steel joints at elevated temperatures

Recent experimental evidence has shown that steel joints exhibit a distinct change in their moment–rotation response under increasing temperature. In terms of cold design, the component method is currently the widely accepted procedure for the evaluation of the various design values. It is the purpose of the present paper to extend the component method to the prediction of the response of steel joints under fire loading. Using typical mechanical models consisting of extensional springs and rigid links, whereby the springs exhibit a non-linear force deformation response (here taken as a bi-linear approximation), an analytical procedure is proposed capable of predicting the moment–rotation response under fire conditions that incorporates the variation of yield stress and Young’s modulus of the various components as the temperature increases. An application to a cruciform flush end-plate beam-to-column steel joint is presented and compared to the experimental results obtained under various loading conditions.

An analysis of finite element locking in a parameter dependent model problem

We consider the bilinear finite element approximation of smooth solutions to a simple parameter dependent elliptic model problem, the problem of highly anisotropic heat conduction. We show that under favorable circumstances that depend on both the finite element mesh and on the type of boundary conditions, the effect of parametric locking of the standard FEM can be reduced by a simple variational crime. In our analysis we split the error in two orthogonal components, the approximation error and the consistency error, and obtain different bounds for these separate components. Also some numerical results are shown.

Digital modelling and robust digital redesign of sampled-data uncertain systems via the interval tuning bilinear approximation method

Based on the bilinear approximation method, we present an interval tuning bilinear approximation method to do the digital model conversion and digital redesign of sampled-data uncertain systems. The tuning bilinear method together with interval arithmetic operation is employed to obtain the approximate discrete-time model and the digital robust control law from the continuous-time uncertain state equation and the given continuous-time robust control law, respectively. The proposed digitally tuning redesigned control law with one parameter can be adjusted so that the resulting dynamic states of the digitally controlled sampled-data uncertain system are able to closely match those of the original continuous-time well-designed uncertain system for a relatively longer sampling period.

A mixed finite element for plate bending with eight enhanced strain modes

AbstractA low‐order thick and thin plate bending element is derived using bilinear approximations for the transverse deflection, the two rotations and the thickness change. The stress–strain relationships from three‐dimensional elasticity are used without any modifications. In order to avoid locking and to improve the accuracy of the results eight enhanced strain modes are used. For an efficient implementation of the mixed element, orthogonal stress and strain functions are utilized. Although the element is a low‐order finite element the numerical results for a series of standard test problems are excellent. Copyright © 2001 John Wiley & Sons, Ltd.

Optimal model reduction of discrete-time descriptor systems

An optimal model reduction method is presented to obtain stable reduced-order models for discrete-time descriptor systems. A parametrization based on the bilinear Routh approximation of linear normal discrete-time systems is used to parametrize the causal subsystems of the reduced-order models. The expressions for the error and its gradient are explicitly given. They are then employed to solve an unconstrained optimization problem for the model reduction problem. The descriptor system structure is preserved in the reduced-order models.

Optimal model reduction of discrete-time descriptor systems

An optimal model reduction method is presented to obtain stable reduced-order models for discrete-time descriptor systems. A parametrization based on the bilinear Routh approximation of linear normal discrete-time systems is used to parametrize the causal subsystems of the reduced-order models. The expressions for the error and its gradient are explicitly given. They are then employed to solve an unconstrained optimization problem for the model reduction problem. The descriptor system structure is preserved in the reduced-order models.

The Estimate of The Integral Vortex of Controlled Systems by The Lyapunov Functions Method

Some classes of nonlinear controlled systems are considered. This systems have a linear or bilinear approximation. For different classes of the allowed controls the sufficient conditions of boundedness of the integral vortex are obtained. The dependence of this characteristic on degree of perturbations is discussed. External estimation of the integral vortex by means of the Lyapunov functions is constracted.

Equivalent post-buckling models for the flexural behaviour of steel connections

Analytical solutions for the evaluation of the behaviour of steel connections are presented which are able to reproduce their full non-linear behaviour. Because usual models for the analysis of steel connections consist of translational springs and rigid links whereby the springs exhibit a non-linear force–deformation response, usually taken as a bi-linear approximation, they require an incremental non-linear analysis. Using a substitute elastic post-buckling model where each bi-linear spring is replaced by two equivalent elastic springs in the context of a post-buckling stability analysis using an energy formulation, closed-form solutions are obtained for a connection loaded in bending. Application to a beam-to-column welded connection using the component (spring) characterisation of code regulations yields the same results in terms of moment resistance and initial stiffness, being additionally able to trace the full unstiffening response.

Improved Bilinear Routh Approximation Method for Discrete-Time Systems

Hwang and Shieh proposed a bilinear Routh approximation method for reducing the order of discrete-time systems. A reduced model derived by the method is not only stable whenever an original model is stable, but also fits the first few time-moments of the original one. This paper addresses the possibility of improving the method by letting the impulse response energy of the original model also be conserved in the reduced model without destroying the stability preserving and time-moments matching properties.

Are Bilinear Quadrilaterals Better Than Linear Triangles?

This paper compares the theoretical effectiveness of bilinear approximation over quadrilaterals with linear approximation over triangles. Anisotropic mesh transformation is used to generate asymptotically optimally efficient meshes for piecewise linear interpolation over triangles and bilinear interpolation over quadrilaterals. For approximating a convex function, although bilinear quadrilaterals are more efficient, linear triangles are more accurate and may be preferred in finite element computations; whereas for saddle-shaped functions, quadrilaterals may offer a higher-order approximation on a well-designed mesh. A surprising finding is different grid orientations may yield an order of magnitude improvement in approximation accuracy

Digital Redesign of State-Delayed Systems Counting the Computer-Control Delay via Law of Mean.

This paper presents a new approach for digital redesign of the continuous-time state-delayed system with the state-feedback controller. The bilinear approximation method together with the linear interpolation is utilized to approximate some integration functions, and the concept of the Law of Mean from the integral calculus is applied to establish an equivalent state-feedback digital controller from the available statefeedback analog controller. In addition, for practical implementation of the developed digital controller by a microcomputer, the computation time delay of the digitally redesigned control law is considered and equivalently modeled at the input, and then a digital predictor controller is proposed for the sampled-data state-delayed system with the control delay.

On Optimal Bilinear Quadrilateral Meshes

The novelty of this work is in presenting interesting error properties of two types of asymptotically ‘optimal’ quadrilateral meshes for bilinear approximation. The first type of mesh has an error equidistributing property, where the maximum interpolation error is asymptotically the same over all elements. The second type has faster than expected ‘super-convergence’ property for certain saddle-shaped data functions. The ‘super-convergent’ mesh may be an order of magnitude more accurate than the error equidistributing mesh. Both types of mesh are generated by a coordinate transformation of a regular mesh of squares. The coordinate transformation is derived by interpreting the Hessian matrix of a data function as a metric tensor. The insights in this work may have application in mesh design near known corner or point singularities.

Phenomenological modeling of the non-linear electro-mechanical coupling in ferroelectrics

Our objective is the construction of a phenomenological model of ferroelectricity for general electro-mechanical loading histories, which is simple enough to be implemented in an FE-code with realistic effort. In this paper we motivate and verify the one-dimensional formulation of such a model. It relies upon introducing the remanent polarization and the remanent strain as internal variables besides stress, strain, electric field and polarization. The internal variables are governed by ordinary differential equations. Each of these evolution equations is subjected to two loading conditions of different nature. The first one indicates the onset of changes of the remanent quantities by domain switching, while the second one characterizes the saturation value of a remanent quantity corresponding to a totally switched domain structure. Polarization induced anisotropy is taken into account as far as it seemed necessary. For simplicity, no rate effects are included. The model response to uniaxial electro-mechanical loading histories will be discussed in comparison to known experimental results. By means of a bilinear approximation the following characteristic phenomena of macroscopic ferroelectricity are represented : dielectric hysteresis, polarization induced piezoelectricity, butterfly hysteresis, ferroelastic hysteresis, mechanical depolarization.

Model Conversions of Uncertain Linear System via the Interval Tuning Bilinear Approximation Method.

This paper presents a new tuning bilinear approximation method together with the interval arithmetic operation to convert a continuous-time (descrete-time) uncertain system to an equivalent discrete-time (continuous-time) uncertain model.Based on the bilinear approximation method, we induct one parameter for tuning the error bound between the exact and approximate uncertain system matrices.The bounds of interval system matrices via the proposed interval tuning bilinear approximation method are narrower than those of the existing interval bilinear and the interval pade approximation methods(1).The resulting interval models (the enclosing interval models) are able to tightly enclose the exact uncertian models.Also the approximate discrete-time interval solution is able to tightly enclose the exact interval solution of the continuous-time uncertain state-space equation.The proposed method will be helpful for analysis and synthesis of uncertain systems.

Nonperturbative SUSY correlators at finite temperature

We calculate finite temperature effects on a correlation function in the two dimensional supersymmetric nonlinear O(3) sigma model. The correlation function violates chiral symmetry and at zero temperature it has been shown to be a constant, which gives rise to a double-valued condensate. Within the bilinear approximation we find an exact result in a one-instanton background at finite temperature. In contrast with the result at zero temperature we find that the correlation function decays exponentially at large distances.

Digital modelling and robust digital redesign of sampled-data uncertain system with input time delay

Using the bilinear and tuning bilinear approximation together with the interval arithmetic operation, the discrete-time uncertain model and robust digital redesign of sampled-data uncertain system with input time delay are presented. The system matrices characterizing the stale-space representation of the original uncertain system are assumed to be interval matrices. The developed discrete-time interval model tightly encloses the exactly discretized continuous-time input time-delay uncertain system, and it can be utilized for digital simulation. Based on bilinear approximation, a tuning bilinear approximation is presented to find the digitally redesign controller of the digitally controlled sampled-data uncertain system with input time delay. The digitally redesigned predictor controller is able to closely match the states of the digitally redesigned uncertain sampled-data system with those of the original continuous-time predictor controlled uncertain system.

Output feedback stabilization of hydraulic drives based on bilinear approximations and canonical observers

This paper deals with the control of hydrostatic drives on the basis of bilinear models. It is shown that by using bilinear models a considerably better approximation of the non-linear behaviour of hydraulic drives can be achieved compared with common linear models. The bilinear model approach gives rise to control results valid not only for fixed operating points but also for the complete operation range of the drives. In particular, an output feedback using a canonical observer and quadratic state feedback is proposed. A separation theorem for this non-linear control scheme similar to that for linear systems is proved, i.e. the dynamics of the observer and of the controlled plant are adjustable separately.

Visualizing nonlinear vector field topology

We present our results on the visualization of nonlinear vector field topology. The underlying mathematics is done in Clifford algebra, a system describing geometry by extending the usual vector space by a multiplication of vectors. We started with the observation that all known algorithms for vector field topology are based on piecewise linear or bilinear approximation, and that these methods destroy the local topology if nonlinear behavior is present. Our algorithm looks for such situations, chooses an appropriate polynomial approximation in these areas, and, finally, visualizes the topology. This overcomes the problem, and the algorithm is still very fast because we are using linear approximation outside these small but important areas. The paper contains a detailed description of the algorithm and a basic introduction to Clifford algebra.