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Showing papers on "Linear approximation published in 2000"



Journal ArticleDOI
TL;DR: In this paper, the problem of global asymptotically stabilizing a certain class of uncertain feedforward nonlinear systems is considered and the control law is obtained by nesting saturation functions whose amplitude can be rendered arbitrarily small.

111 citations


Journal ArticleDOI
TL;DR: It is shown that the gridding algorithm can be considered an approximation to the least squares method, and the application of this method to a case of spiral magnetic resonance imaging shows a reduction of more than 4 dB in the average reconstruction error.
Abstract: Gridding reconstruction is a method to reconstruct data onto a Cartesian grid from a set of nonuniformly sampled measurements. This method is appreciated for being robust and computationally fast. However, it lacks solid analysis and design tools to quantify or minimize the reconstruction error. Least squares reconstruction (LSR), on the other hand, is another method which is optimal in the sense that it minimizes the reconstruction error. This method is computationally intensive and, in many cases, sensitive to measurement noise. Hence, it is rarely used in practice. Despite their seemingly different approaches, the gridding and LSR methods are shown to be closely related. The similarity between these two methods is accentuated when they are properly expressed in a common matrix form. It is shown that the gridding algorithm can be considered an approximation to the least squares method. The optimal gridding parameters are defined as the ones which yield the minimum approximation error. These parameters are calculated by minimizing the norm of an approximation error matrix. This problem is studied and solved in the general form of approximation using linearly structured matrices. This method not only supports more general forms of the gridding algorithm, it can also be used to accelerate the reconstruction techniques from incomplete data. The application of this method to a case of two-dimensional (2-D) spiral magnetic resonance imaging shows a reduction of more than 4 dB in the average reconstruction error.

111 citations


Journal ArticleDOI
TL;DR: The a posteriori error analysis of stabilised finite element approximations to linear transport problems via duality arguments is developed and it is shown that the second approach is superior in the sense that it leads to sharper a posterioru error bounds and more economical adaptively refined meshes.

105 citations


Journal ArticleDOI
TL;DR: A generic definition of a nonlinearity measure is presented on the basis of the “best” linear approximation of anonlinear system to be applied both to the analysis of steady state operating points of continuously operated processes as well as to a trajectory dependent analysis of batch or other transient processes.

96 citations


Journal ArticleDOI
01 Apr 2000
TL;DR: The main objective in the present paper is to improve the efficiency of Paxson's method for synthesizing self-similar network traffic by demonstrating that a linear approximation can be used to determine the power spectrum of the FGN.
Abstract: The present paper focuses on self-similar network traffic generation. Network traffic modeling studies the generation of synthetic sequences. The generated sequences must have similar features to the measured traffic. Exact methods for generating self-similar sequences are not appropriate for long traces. Our main objective in the present paper is to improve the efficiency of Paxson's method for synthesizing self-similar network traffic. Paxson's method uses a fast, approximate synthesis for the power spectrum of the FGN and uses the inverse Fourier transform to obtain the time-domain sequences. We demonstrate that a linear approximation can be used to determine the power spectrum of the FGN. This linear approximation reduces the complexity of the computation without compromising the accuracy in synthesizing the power spectrum of the FGN. Our results show that long traces can be generated in much less time. To compare our method with existing ones, we will measure the running time in generating long and short sample paths from the FGN. We will also conduct experiments to show that our method can generate self-similar traffic for specified Hurst parameters with high accuracy.

63 citations


Journal ArticleDOI
TL;DR: A class of broadband pilot test signals is proposed, termed sparse odd multisines, which can be used to establish the system bandwidth and detect nonlinearities, and signals are defined within this class which allow the measurement of the best linear approximation of a nonlinear system.
Abstract: This paper examines the effects of nonlinearities on frequency response function measurements using periodic multifrequency signals. A class of broadband pilot test signals is proposed, termed sparse odd multisines, which can be used to establish the system bandwidth and detect nonlinearities. Signals are then defined within this class which allow the measurement of the best linear approximation of a nonlinear system. A comparison is made with related work in this area.

63 citations


Journal ArticleDOI
TL;DR: In this paper, a new CAM algorithm is proposed which approximates the surface to be machined by a piecewise curved approximation, which is further optimized by formulating the tool path computation as the generation of a grid based on a variational smoothness penalty function.
Abstract: Current tool path computation in the CAM algorithms approximates the surface by piecewise linear interpolation. In the case of three-axis machining on a CNC machine the tool will exactly reproduce this computed tool path. However in the case of five-axis simultaneous machining the real tool path on the CNC machine will not follow the linear approximation computed by the conventional CAM algorithm. A new CAM algorithm is proposed which approximates the surface to be machined by a piecewise curved approximation. This curve represents the real tool path followed on the five-axis machine. This piecewise curved approximation is further optimized by formulating the tool path computation as the generation of a grid based on a variational smoothness penalty function. This new algorithm considerably improves the accuracy and reduces the number of blocks and machining time.

50 citations


Journal ArticleDOI
TL;DR: In this paper, an algorithm based on the continuation method and on a primal-dual interior point optimization method is proposed to calculate a sequence of optimal power flow (OPF) solutions under variable load conditions.
Abstract: This work presents a methodology to calculate a sequence of optimal power flow (OPF) solutions under variable load conditions. The aim is to obtain a set of optimal operating points in the neighborhood of the bounds of the region defined by the load flow equations and a set of operational limits. For this, an algorithm based on the continuation method and on a primal-dual interior point optimization method is proposed. Such an algorithm consists of two main steps: the predictor step, which uses a linear approximation of the Karush-Kuhn-Tucker (KKT) conditions to estimate a new operating point for an increment in the system load; and the corrector step, which calculates the optimum corresponding to the new load level via a nonlinear primal-dual interior point method. Indices for critical buses and inequality constraints are a byproduct of the methodology. In addition, sensitivity analysis is performed to calculate the amount of reactive compensation which allows for a pre-specified increase in the system load. Results for realistic test systems are presented.

48 citations


Journal ArticleDOI
TL;DR: This paper compares the theoretical effectiveness of bilinear approximation over quadrilaterals with linear approximation over triangles and a surprising finding is different grid orientations may yield an order of magnitude improvement in approximation accuracy.
Abstract: 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

44 citations


Journal ArticleDOI
TL;DR: This work shows how to transform the optimal portfolio selection problem with transaction costs into a quadratic programming model, applicable by establishing a linear approximation on the utility function of return and variance.
Abstract: We study the optimal portfolio selection problem with transaction costs. In general, the efficient frontier can be determined by solving a parametric non-quadratic programming problem. In a general setting, the transaction cost is a V-shaped function of difference between the existing and the new portfolio. We show how to transform this problem into a quadratic programming model. Hence a linear programming algorithm is applicable by establishing a linear approximation on the utility function of return and variance.

Journal ArticleDOI
TL;DR: A space-time Galerkin finite element discretization of the linear quasistatic compressible viscoelasticity problem as described by an elliptic partial differential equation with a Volterra (memory) term is given.
Abstract: We give a space-time Galerkin finite element discretization of the linear quasistatic compressible viscoelasticity problem as described by an elliptic partial differential equation with a Volterra (memory) term. The discretization consists of a continuous piecewise linear approximation in space with a discontinuous piecewise constant or linear approximation in time. We derive an a priori maximum energy-error estimate by exploiting Galerkin "orthogonality" and the data-stability of a related discrete backward problem. Illustrative numerical experiments are also included, as also is a brief description of our first results on a posteriori error estimation. This allows for adaptive control of the space mesh but not of the time step.

Proceedings ArticleDOI
01 Jun 2000
TL;DR: An approximation algorithm by linear programming for floorplan sizing problem that can handle any topological constraints as well as soft/hard/preplaced blocks, and timing constraints and shows one order of improvement over previous methods both in run time and capability to handle a larger problem size even on a very limited computing resource PC.
Abstract: In this paper, we present an approximation algorithm by linear programming (LP) for floorplan sizing problem. Given any topological constraints between blocks, we can formulate it as an LP problem with a cost function for the minimum bounding box area. Unlike slicing structures, this approach can handle any topological constraints as well as soft/hard/preplaced blocks, and timing constraints. Empirically, our method needs few iterations to find the optimum solution and shows one order of improvement over previous methods both in run time and capability to handle a larger problem size even on a very limited computing resource PC.

Journal ArticleDOI
TL;DR: It is shown that the determined correlation coefficient is large enough for applying a fast correlation attack to the output sequence to reconstruct the initial states of the input linear feedback shift registers.
Abstract: The linear sequential circuit approximation method for combiners with memory is used to find mutually correlated linear transforms of the input and output sequences in the well-known summation generator with any number of inputs. It is shown that the determined correlation coefficient is large enough for applying a fast correlation attack to the output sequence to reconstruct the initial states of the input linear feedback shift registers. The proposed attack is based on iterative probabilistic decoding and appropriately generated low-weight parity-checks. The required output sequence length and the computational complexity are both derived. Successful experimental results for the summation generators with three and five inputs are obtained.

Journal ArticleDOI
TL;DR: A further interesting conclusion from the results is that no loss of generality is suffered using networks with positive hidden-to-output weights, which is sufficient to guarantee the established convergence rates.
Abstract: The problem of approximating functions by neural networks using incremental algorithms is studied. For functions belonging to a rather general class, characterized by certain smoothness properties with respect to the L/sub 2/ norm, we compute upper bounds on the approximation error where error is measured by the L/sub q/ norm, 1/spl les/q/spl les//spl infin/. These results extend previous work, applicable in the case q=2, and provide an explicit algorithm to achieve the derived approximation error rate. In the range q/spl les/2 near-optimal rates of convergence are demonstrated. A gap remains, however, with respect to a recently established lower bound in the case q>2, although the rates achieved are provably better than those obtained by optimal linear approximation. Extensions of the results from the L/sub 2/ norm to L/sub p/ are also discussed. A further interesting conclusion from our results is that no loss of generality is suffered using networks with positive hidden-to-output weights. Moreover, explicit bounds on the size of the hidden-to-output weights are established, which are sufficient to guarantee the established convergence rates.

Journal ArticleDOI
TL;DR: An account is given of the development of methods for approximation by functions which are nonlinear in the free parameters, and special attention is paid to some particular nonlinear approximating families.

Proceedings ArticleDOI
28 Jun 2000
TL;DR: An approach to augment a linear compensator with an online neural network is presented, which provides the benefits of adaptation with only minor modification to the existing control architecture, which is a substantial advantage over other approaches that require complete redesign.
Abstract: An approach to augment a linear compensator with an online neural network is presented. This scheme provides the benefits of adaptation with only minor modification to the existing control architecture, which is a substantial advantage over other approaches that require complete redesign. A neural network update law that guarantees bounded tracking for the augmented architecture is outlined. The advantages of the proposed technique are demonstrated through an application to an autonomous underwater vehicle. The design requirement is for attitude control such that robust trajectory following is achieved. A detailed nonlinear model of the AUV is given, and an operating point for nominal design is selected, about which a linear approximation is obtained. Structured uncertainties due to errors in the computation of hydrodynamic coefficients, linearization and truncation of plant dynamics, as well as effects of unknown disturbances are included in the control synthesis and compensated for by the neural network.

Journal ArticleDOI
TL;DR: In this article, the solutions of the field equations for a mass point are also the exterior solutions for an arbitrary spherically symmetric mass distribution, and the exterior solution is always the Schwarzschild solution.
Abstract: The solutions of a class of theories obtained when we apply the first order formalism are studied. In the linear approximation we obtain the Green function and we prove that the field is independent of the size and internal stresses of the source. We show that the solutions of the field equations for a mass point are also the exterior solutions for an arbitrary spherically symmetric mass distribution. We construct the solutions of the field equation, without any approximation, for the spherically symmetric matter distribution, and prove that the exterior solutions match correctly with the interior solutions. We also prove that one of the exterior solutions is always the Schwarzschild solution. Finally, in the same case, we show that Birkhoff's theorem is satisfied. All the above results are quite similar to general relativity but are very different from the results of the fourth order theories; then we have shown that the first order formalism theories have better classical properties than fourth order theories.

Journal ArticleDOI
TL;DR: In this paper, the boundary effects on a quantum system by examining the problem of a hydrogen atom in a spherical well were discussed. And the boundary corrections to the ground-state energy and wave function were calculated using an approximation method which is linear in energy.
Abstract: We discuss the boundary effects on a quantum system by examining the problem of a hydrogen atom in a spherical well. Using an approximation method which is linear in energy we calculate the boundary corrections to the ground-state energy and wavefunction. We obtain the asymptotic dependence of the ground-state energy on the radius of the well.

ReportDOI
TL;DR: In this article, the authors introduce a new class of instrumental variable (IV) estimators of causal treatment effects for linear and nonlinear models with covariates, and show how to estimate a well-defined approximation to a nonlinear causal response function of unknown functional form using simple parametric models.
Abstract: This article introduces a new class of instrumental variable (IV) estimators of causal treatment effects for linear and nonlinear models with covariates. The rationale for focusing on nonlinear models is to improve the approximation to the causal response function of interest. For example, if the dependent variable is binary or limited, or if the effect of the treatment varies with covariates, a nonlinear model is likely to be appropriate. However, identification is not attained through functional form restrictions. This paper shows how to estimate a well-defined approximation to a nonlinear causal response function of unknown functional form using simple parametric models. As an important special case, I introduce a linear model that provides the best linear approximation to an underlying causal relation. It is shown that Two Stage Least Squares (2SLS) does not always have this property and some possible interpretations of 2SLS coefficients are brie y studied. The ideas and estimators in this paper are illustrated using instrumental variables to estimate the effects of 401(k) retirement programs on savings.

Proceedings ArticleDOI
01 Dec 2000
TL;DR: It is shown that the general least squares problem for estimating a projective transformation can be analytically reduced to a 2-dimensional nonquadratic minimization problem and provided both analytical and experimental evidence that the minimization of this function is computationally attractive.
Abstract: The estimation of the parameters of a projective transformation that relates the coordinates of two image planes is a standard problem that arises in image and video mosaicking, virtual video, and problems in computer vision This problem is often posed as a least squares minimization problem based on a finite set of noisy point samples of the underlying transformation While in some special cases this problem can be solved using a linear approximation, in general, it results in an 8-dimensional nonquadratic minimization problem that is solved numerically using an 'off-the-shelf' procedure such as the Levenberg-Marquardt algorithm We show that the general least squares problem for estimating a projective transformation can be analytically reduced to a 2-dimensional nonquadratic minimization problem Moreover, we provide both analytical and experimental evidence that the minimization of this function is computationally attractive We propose a particular algorithm that is a combination of a projection and an approximate Gauss-Newton scheme, and experimentally verify that this algorithm efficiently solves the least squares problem

Journal ArticleDOI
TL;DR: Comparing computer simulations of a model of a two-joint arm with six monarticular and biarticular muscles shows that ignoring the time delay resulted in large errors in the estimation of the hand equilibrium trajectory, which could explain why experimentally estimated hand equilibrium trajectories may be complex, even during a simple reaching movement.
Abstract: It has been widely claimed that linear models of the neuromuscular apparatus give very inaccurate approximations of human arm reaching movements. The present paper examines this claim by quantifying the contributions of the various non-linear effects of muscle force generation on the accuracy of linear approximation. We performed computer simulations of a model of a two-joint arm with six monarticular and biarticular muscles. The global actions of individual muscles resulted in a linear dependence of the joint torques on the joint angles and angular velocities, despite the great non-linearity of the muscle properties. The effect of time delay in force generation is much more important for model accuracy than all the non-linear effects, while ignoring this time delay in linear approximation results in large errors. Thus, the viscosity coefficients are rather underestimated and some of them can even be paradoxically estimated to be negative. Similarly, our computation showed that ignoring the time delay resulted in large errors in the estimation of the hand equilibrium trajectory. This could explain why experimentally estimated hand equilibrium trajectories may be complex, even during a simple reaching movement. The hand equilibrium trajectory estimated by a linear model becomes simple when the time delay is taken into account, and it is close to that actually used in the non-linear model. The results therefore provide a theoretical basis for estimating the hand equilibrium trajectory during arm reaching movements and hence for estimating the time course of the motor control signals associated with this trajectory, as set out in the equilibrium point hypothesis.

Journal ArticleDOI
TL;DR: This paper studies algorithms for decomposition, reconstruction, and approximation based on piecewise linear prewavelets on bounded triangulations of arbitrary topology and shows that the Schur complement of the associated two scale matrix is symmetric, positive definite, and well conditioned.

Journal ArticleDOI
TL;DR: In this paper, an exact equation describing the finite displacements of assemblies consisting of rigid bars and pin joints is presented, and a numerical procedure is proposed for carrying out the calculations, and some examples are shown in order to illustrate the procedure.

Posted Content
TL;DR: In this article, the authors introduce a new class of instrumental variable (IV) estimators of causal treatment effects for linear and nonlinear models with covariates, and show how to estimate a well-defined approximation to a nonlinear causal response function of unknown functional form using simple parametric models.
Abstract: This article introduces a new class of instrumental variable (IV) estimators of causal treatment effects for linear and nonlinear models with covariates. The rationale for focusing on nonlinear models is to improve the approximation to the causal response function of interest. For example, if the dependent variable is binary or limited, or if the effect of the treatment varies with covariates, a nonlinear model is likely to be appropriate. However, identification is not attained through functional form restrictions. This paper shows how to estimate a well-defined approximation to a nonlinear causal response function of unknown functional form using simple parametric models. As an important special case, I introduce a linear model that provides the best linear approximation to an underlying causal relation. It is shown that Two Stage Least Squares (2SLS) does not always have this property and some possible interpretations of 2SLS coefficients are brie y studied. The ideas and estimators in this paper are illustrated using instrumental variables to estimate the effects of 401(k) retirement programs on savings.

Journal ArticleDOI
TL;DR: In this paper, a nonlinear observer based on suitably processed optical information is proposed for the localization of nonholonomic mobile robots by using an estimated expression of the extended output Jacobian, local practical stabilization of the observation error dynamics is also ensured.

Journal ArticleDOI
TL;DR: In this paper, a new method is presented for approximating the stationary probability density function of the response of a general class of non-linear single-degree-of-freedom dynamical systems subjected to additive stochastic white noise excitation.
Abstract: A new method is presented for approximating the stationary probability density function of the response of a general class of non-linear single-degree-of-freedom dynamical systems subjected to additive stochastic white noise excitation. The method is based on finding the best probability density function (PDF) from a parameterized class of trial non-Gaussian PDFs by minimizing a weighted norm of the Fokker–Planck–Kolmogorov equation error. The proposed procedure yields simple expressions in terms of one-dimensional integrals for determining desired probabilistic characteristics of the system response, such as moments and mean outcrossing rates. Examples illustrating the applicability and accuracy of the method include a system modeling the rolling motion of a ship and a Duffing oscillator with non-linear damping. Comparisons are made with some other approximate methods, including equivalent linearization, partial linearization, equivalent non-linearization, and dissipation energy balancing methods, that show that the new method yields substantially improved estimates for the expected outcrossing rates of the response. These outcrossing rates are crucial for evaluating the reliability of the system. In contrast, the equivalent non-linearization and the dissipation energy balancing methods, known to provide the most accurate estimates for the mean-square response, give very poor estimates of the mean outcrossing rates as the number of level outcrossings decreases.

Journal ArticleDOI
TL;DR: A sequential approximation algorithm is presented here that is particularly suited for problems in engineering design and structural optimization, where the number of variables is very large and function and sensitivity evaluations are computationally expensive.
Abstract: A sequential approximation algorithm is presented here that is particularly suited for problems in engineering design and structural optimization, where the number of variables is very large and function and sensitivity evaluations are computationally expensive. A sequence of sub-problems are generated using a linear approximation for the objective function and setting move limits on the variables using a barrier method. These sub-problems are strictly convex and computation per iteration is significantly reduced by not solving the sub-problems exactly. Instead a few Newton-steps are taken for each subproblem generated. A criterion, for setting the move limit, is described that reduces or eliminates step size reduction during line search. The method was found to perform well for unconstrained and linearly constrained optimization problems. It is particularly suitable for application to design of optimal shape and topology of structures by minimizing their compliance since it requires very few function evaluations, does not require the hessian of the objective function and evaluates its gradient only once for every subproblem generated.

Proceedings ArticleDOI
01 Jan 2000
TL;DR: The theory of bisimulation is extended to the case when the reachable-state mappings can only be approximated and the approximation error is analyzed for the general case.
Abstract: This paper concerns the theory for computational methods to construct finite-state approximations for hybrid systems for verification and supervisory control synthesis. The theory of bisimulation is extended to the case when the reachable-state mappings can only be approximated. New results for the flow pipe approximation method for computing approximations to reachable states for continuous dynamics are also presented. A previous technique for reducing the computational burden for linear systems is extended to the affine case and the approximation error is analyzed for the general case. In the concluding section, the reader is pointed to a Web site containing a complete implementation of the techniques described in this paper.

01 Sep 2000
TL;DR: In this paper, the impact of nonlinear friction on frequency response function (FRF) measurements of mechanical systems with non-linear friction is discussed and a linear approximation of the system and a reliable estimate of the level of systematic and stochastic nonlinear distortions are derived using a single special odd multisine excitation.
Abstract: This paper discusses the impact of nonlinear friction on frequency response function (FRF) measurements of mechanical systems with nonlinear friction. Having an idea of the level of nonlinear distortion is valuable information for robust linear control design: it allows to estimate the level of accuracy of a linear approximation of the system under consideration. The impact of nonlinear distortions is measured using the multisine excitation approach presented in [5]. A linear approximation of the system and a reliable estimate of the level of systematic and stochastic nonlinear distortions are derived using a single special odd multisine excitation. This is shown by means of simulations of mechanical system with friction. The friction model is nonlinear and integrates sliding and presliding behaviour [7].