Showing papers on "Linearization published in 1990"
TL;DR: In this article, a configuration update procedure for the director (rotation) field is developed, which is singularity free and exact regardless the magnitude of the rotation increment, and the exact linearization of the discrete form of the equilibrium equations is derived in closed form.
Abstract: Computational aspects of a geometrically exact stress resultant model presented in Part I of this work are considered in detail. In particular, by exploiting the underlying geometric structure of the model, a configuration update procedure for the director (rotation) field is developed which is singularity free and exact regardless the magnitude of the director (rotation) increment. Our mixed finite element interpolation for the membrane, shear and bending fields presented in PartII of this work are extended to the finite deformation case. The exact linearization of the discrete form of the equilibrium equations is derived in closed form. The formulation is then illustrated by a comprehensive set of numerical experiments which include bifurcation and post-buckling response, we well as comparisons with closed form solutions and experimental results.
580 citations
TL;DR: In this article, the use of high speed, high capacity vector computers allows the resultant finite-difference equations to be factored in-place, allowing inversions to be generated using data from a very large number of source positions.
Abstract: Frequency-domain methods are well suited to the imaging of wide-aperture cross-hole data. However, although the combination of the frequency domain with the wavenumber domain has facilitated the development of rapid algorithms, such as diffraction tomography, this has also required linearization with respect to homogeneous reference media. This restriction, and association restrictions on source-receiver geometries, are overcome by applying inverse techniques that operate in the frequency-space domain.
In order to incorporate the rigorous modelling technique of finite differences into the inverse procedure a nonlinear approach is used. To reduce computational costs the method of finite differences is applied directly to the frequency-domain wave equation. The use of high speed, high capacity vector computers allow the resultant finite-difference equations to be factored in-place. In this way wavefields can be computed for additional source positions at minimal extra cost, allowing inversions to be generated using data from a very large number of source positions.
Synthetic studies show that where weak scatter approximations are valid, diffraction tomography performs slightly better than a single iteration of non-linear inversion. However, if the background velocities increase systematically with depth, diffraction tomography is ineffective whereas non-linear inversion yields useful images from one frequency component of the data after a single iteration. Further synthetic studies indicate the efficacy of the method in the time-lapse monitoring of injection fluids in tertiary hydrocarbon recovery projects.
567 citations
TL;DR: In this paper, an implicit finite element formulation of the rate tangent modulus method is derived for which the consistent tangents, resulting in quadratic convergence of the equilibrium iterations, can be written out in closed form for arbitrary material models.
Abstract: Some constitutive and computational aspects of finite deformation plasticity are discussed. Attention is restricted to multiplicative theories of plasticity, in which the deformation gradients are assumed to be decomposable into elastic and plastic terms. It is shown by way of consistent linearization of momentum balance that geometric terms arise which are associated with the motion of the intermediate configuration and which in general render the tangent operator non-symmetric even for associated plastic flow. Both explicit (i.e. no equilibrium iteration) and implicit finite element formulations are considered. An assumed strain formulation is used to accommodate the near-incompressibility associated with fully developed isochoric plastic flow. As an example of explicit integration, the rate tangent modulus method is reviewed in some detail. An implicit scheme is derived for which the consistent tangents, resulting in quadratic convergence of the equilibrium iterations, can be written out in closed form for arbitrary material models. All the geometrical terms associated with the motion of the intermediate configuration and the treatment of incompressibility are given explicitly. Examples of application to void growth and coalescence and to crack tip blunting are developed which illustrate the performance of the implicit method.
301 citations
TL;DR: In this paper, the authors developed tools for feedback control and sensitivity analysis of systems modelled by nonlinear differential-algebraic equations, which can handle multiple inputs and outputs without pairing, and the ability to handle input and output constraints without complicated anti-reset windup logic.
273 citations
TL;DR: In this article, an algorithm for the determination of flow control valve settings to minimize leakage in water-distribution networks is presented, where nonlinear network equations describing nodal heads and pipeflows are augmented by terms that explicitly account for pressure-dependent leakage and by models the effect of valve actions.
Abstract: The paper concerns the problems of minimization of leakage in water-distribution networks. It has been reported that leakage from some networks may account for a significant amount of the water put into supply. For some aging urban networks, rates of up to 50% have been quoted, with average rates of 25% being quite typical. These high rates of leakage represent a significant economic loss. An algorithm for the determination of flow control valve settings to minimize leakage is presented. The nonlinear network equations describing nodal heads and pipeflows are augmented by terms that explicitly account for pressure-dependent leakage and by terms that model the effect of valve actions. Successive linearization of these equations using the linear-theory method allows a linear program that minimizes leakage to be formulated and solved. The performance of the method is demonstrated by application to an example network.
206 citations
TL;DR: In this paper, the authors used a bootstrap resampling scheme to find empirically the distribution of uncertainties in the results of the measurements, and the confidence intervals for the eigenvalues were found directly from their empirical distributions.
Abstract: In studies of the anisotropy of susceptibility or remanence of paleomagnetic samples it is conventional to specify the anisotropy in terms of the parameters of the anisotropy ellipsoids, namely the directions of the principal axes of the ellipsoid and their associated eigenvalues. Confidence intervals for these parameters have in the past often been estimated by using a linearization scheme to propagate the effect of small changes through the eigenvalue decomposition. The validity of these approximations is explored using a Monte-Carlo simulation from measurements that are presumed normally distributed, showing that there are circumstances in which the linearization scheme gives confidence intervals that are much too small. Q-Q plots indicate that the common assumption that the noise in the measurements is Gaussian does not always hold. Because of these shortcomings in the conventional technique we propose using a bootstrap resampling scheme to find empirically the distribution of uncertainties in the results. Confidence intervals for the eigenvalues are found directly from their empirical distributions. For the principal axes, approximate elliptical regions of confidence on the unit sphere are parameterized in terms of the Kent or FB5 distribution. The number of modes observed in the distribution of eigenvalues obtained by bootstrapping is used to classify the shape of the susceptibility ellipsoid as spherical, oblate, prolate or triaxial. The empirical nature of the bootstrap technique allows the extension of the analysis of uncertainties to parameters derived from the principal susceptibilities, such as percentage anisotropy or shape factor.
176 citations
23 May 1990
TL;DR: In this article, the authors present an approach to controller design based on finding a linearizable nonlinear system that well approximates the true system over a desirable region, and demonstrate a nonlinear controller for a simple mechanical system patterned after a gymnast performing on a single parallel bar.
Abstract: Recent developments in the theory of geometric nonlinear control provide powerful methods for controller design for a large class of nonlinear systems. Many systems, however, do not satisfy the restrictive conditions necessary for either full state linearization [6, 5] or input-output linearization with internal stability [2]. In this paper, we present an approach to controller design based on finding a linearizable nonlinear system that well approximates the true system over a desirable region.
We outline an engineering procedure for constructing
the approximating nonlinear system given the true system. We
demonstrate this approach by designing a nonlinear controller for a simple mechanical system patterned after a gymnast performing on a single parallel bar.
169 citations
TL;DR: In this article, two approaches are considered to separate the output from measured and unmeasured disturbances in a chemical reactor model, in which the manipulated coolant flow rate appears nonlinearly.
Abstract: Two approaches are considered. The first approach is based on the original process model, while the second is based on an extended model that is control linear. Decoupling the output from measured and unmeasured disturbances is also investigated. The two approaches are evaluated via simulation for a chemical reactor model, in which the manipulated coolant flow rate appears nonlinearly.
168 citations
TL;DR: In this paper, a general approach for determining the lateral phase or group velocity distribution, which is a standard 2D tomography problem, involves linearization, representation of the unknown function as a series in some basis functions, and evaluation of the coefficients by the methods of linear algebra.
Abstract: SUMMARY We discuss and develop further the methods of surface wave tomography in the frame of the geometric ray approximation. The general approach for determining the lateral phase or group velocity distribution, which is a standard 2-D tomography problem, involves linearization, representation of the unknown function as a series in some basis functions, and evaluation of the coefficients by the methods of linear algebra. If the wave paths cover the area under investigation non-uniformly, the basis functions should not be chosen a priori, but constructed proceeding from the pattern of paths. Different criteria for constructing the basis functions are compared, and a relation between them is considered. A more preferable approach is joint interpretation of phase and group velocity data for different periods, because it allows the information about phase velocity variations to be enlarged due to the use of the group velocity data. Both the phase and group traveltimes are represented as linear functionals of the unknown phase slowness corrections. A specific form of the data kernels allows the basis functions to be represented as a product of two functions, one depending on the horizontal coordinates, and the other on frequency.
167 citations
TL;DR: The internal model control (IMC) and globally linearizing control (GMC) structures are reviewed and interpreted in the context of input/output linearization.
Abstract: We focuse on exact linearization methods including Su-Hunt-Meyer, input/output, and full linearization. The internal model control (IMC) and globally linearizing control (GMC) structures are reviewed and interpreted in the context of input/output linearization. Further topics of current research interest are also identified
157 citations
TL;DR: Partial differential equations that conserve energy can often be written as infinite-dimensional Hamiltonian systems of the following general form: du/dt=JE'(u), where J:X * → X is a symplectic matrix (i.e., JJ * =−1) and E: X→R is a C 2 functional defined on some Hilbert space X as discussed by the authors.
Abstract: Partial differential equations that conserve energy can often be written as infinite-dimensional Hamiltonian systems of the following general form: du/dt=JE'(u), where J:X * →X is a symplectic matrix (i.e., JJ * =−1) and E: X→R is a C 2 functional defined on some Hilbert space X
TL;DR: In this article, the synthesis of nonlinear controllers for multivariable nonlinear processes that make the closed-loop system linear in an input/output sense is discussed, and necessary and sufficient conditions for linearizability via static state feedback are derived.
Abstract: This work concerns the synthesis of nonlinear controllers for multivariable nonlinear processes that make the closed-loop system linear in an input/output sense. Necessary and sufficient conditions for input/output linearizability via static state feedback are derived as well as formulas for the feedback law. Once such a static state feedback is applied to the process, an external multivariable linear controller with integral action can control it to set point. The proposed control methodology is tested through simulations in a semibatch copolymerization reactor example.
TL;DR: This linearization scheme provides an equivalent mixed integer linear programming problem which yields a tighter continuous relaxation than that obtainable via the alternative linearization techniques available in the literature.
Abstract: This paper is concerned with a new linearization strategy for a class of zero-one mixed integer programming problems that contains quadratic cross-product terms between continuous and binary variables, and between the binary variables themselves. This linearization scheme provides an equivalent mixed integer linear programming problem which yields a tighter continuous relaxation than that obtainable via the alternative linearization techniques available in the literature. Moreover, the proposed technique provides a unifying framework in the sense that all the alternate methods lead to formulations that are accessible through appropriate surrogates of the constraints of the new linearized formulation. Extensions to various other types of mixed integer nonlinear programming problems are also discussed.
TL;DR: In this paper, the problem of synthesizing nonlinear state feedback controllers for second-order non-minimum-phase nonlinear systems is addressed, where the controller must explicitly or implicitly generate a process inverse (Garcia and Morari, 1982).
TL;DR: In this paper, the authors proposed a generalization of Kane's equations for multibody codes to simulate the behavior of elastic structures undergoing large rotation and translation with small vibrations, which does not suffer from this defect and is valid for an arbifnary structure, and illustrative examples are given to demonstrate the validity and generality of the formulation.
Abstract: Conventional theories underlying my multibody codes used for simulating the behavior of elastic structures undergoing large rotation and translation with small vibrations fail to predict dynamic stiffening of the structures. This can lead to significantly incorrect simulations in many practical situations. A theory that does not suffer from this defect and is valid for an arbifnary structure is given here. The formulation is based on Kane's equations and consists of two steps: First, generalized inertia forces are written for an arbitrary structure for which one is forced to linearize prematurely in the modal coordinates; next, this defect in linearization is compensated for by the introduction of contributions to the generalized active forces from the "motion stiffness" of the stnrctwe. The stress associated with the motion stiffness is identified as due to 12 sets of inertia forces and 9 sets of inertia couples distributed throughout the body during ihe most general motion of its flying reference frame. An algorithm is set for a reader wishing to implement the theory, and illustrative examples are given to demonstrate tbe validity and generality of the formulation.
Book•
25 Oct 1990TL;DR: Introduction to dynamical systems first order linear dynamcial systems introduction to nonlinear Dynamical systems complex behaviour for nonlinear dynamicalSystems of several equations.
Abstract: Introduction to dynamical systems first order linear dynamcial systems introduction to nonlinear dynamical systems complex behaviour for nonlinear dynamical systems higher order linear dynamical systems dynamical systems of several equations nonlinear systems of several equations.
TL;DR: An algorithm is developed that meets the requirement to obtain solutions where all or some of the design variables take their values from a given set of discrete values, while finding global solutions for the mixed-discrete problem.
TL;DR: In this article, a general framework for the analysis of curved shells is developed using a simple quadrilateral C0 model (HMSH5), where the governing equations are derived based on a consistent linearization of an incremental mixed variational principle of the modified Hellinger/Reissner type with independent assumptions for displacement and strain fields.
Abstract: Adopting an updated Lagrange approach, the general framework for the fully non-linear analysis of curved shells is developed using a simple quadrilateral C0 model (HMSH5). The governing equations are derived based on a consistent linearization of an incremental mixed variational principle of the modified Hellinger/Reissner type with independent assumptions for displacement and strain fields. Emphasis is placed on devising effective solution procedures to deal with large rotations in space, finite stretches and generalized rate-type material models. In particular, a geometrically exact scheme for configuration update is developed by making use of the so-called exponential mapping algorithm, and the resulting element was shown to exhibit a quadratic rate of (asymptotic) convergence in solving practical shell problems with Newton–Raphson type iterative schemes. For the purpose of updating the spatial stress field of the element, an ‘objective’ generalized midpoint integration rule is utilized, which relies crucially on the concept of polar decomposition for the deformation gradient, and is in keeping with the underlying mixed method. Finally, the effectiveness and practical usefulness of the HMSH5 element are demonstrated through a number of test cases involving beams, plates and shells undergoing very large displacements and rotations.
TL;DR: The compressible Navier-Stokes equations are solved in thin-layer form for a variety of two-dimension al inviscid and viscous problems by preconditione d conjugate gradient-like algorithms, which is found to be competitive with the best current schemes, but has wide applications in parallel computing and unstructured mesh computations.
Abstract: The compressible Navier-Stokes equations are solved for a variety of two-dimensional inviscid and viscous problems by preconditioned conjugate gradient-like algorithms. Roe's flux difference splitting technique is used to discretize the inviscid fluxes. The viscous terms are discretized by using central differences. An algebraic turbulence model is also incorporated. The system of linear equations which arises out of the linearization of a fully implicit scheme is solved iteratively by the well known methods of GMRES (Generalized Minimum Residual technique) and Chebyschev iteration. Incomplete LU factorization and block diagonal factorization are used as preconditioners. The resulting algorithm is competitive with the best current schemes, but has wide applications in parallel computing and unstructured mesh computations.
05 Dec 1990
TL;DR: In this paper, a nonlinear adaptive control of the torque-ripple in hybrid step motors and its cancellation using adaptive linearization control is discussed, where the adaptive system is robust to a class of state and parameter-dependent modeling errors and disturbances even when the adaptation gain and convergence rate of the unperturbed system become small.
Abstract: The modeling of torque-ripple in hybrid step motors and its cancellation using adaptive linearization control are discussed. Although the nonlinear adaptive control of this problem can fit into a general framework, a representation of the torque-ripple which reduces the number of adapted parameters per torque-ripple harmonic by half is used. By doing so, it is possible to prove conditions on exogenous signals to guarantee the persistency of excitation of the regressor, and hence the exponential stability of the unperturbed system. It is shown that the adaptive system is robust to a class of state- and parameter-dependent modeling errors and disturbances even when the adaptation gain and convergence rate of the unperturbed system become small. The adapted parameter errors are proved to converge to a neighborhood of zero whose radius can be made small by slow adaptation. The proposed control scheme is verified in an experiment in which a 32-dB reduction in torque-ripple component at the rotor pole frequency is observed. >
TL;DR: In this paper, a second-order closure method is presented for determining the response of non-linear systems to random excitations, where the excitation is taken to be the sum of a deterministic harmonic component and a random component.
Abstract: A second-order closure method is presented for determining the response of non-linear systems to random excitations. The excitation is taken to be the sum of a deterministic harmonic component and a random component. The latter may be white noise or harmonic with separable non-stationary random amplitude and phase. The method of multiple scales is used to determine the equations describing the modulation of the amplitude and phase. Neglecting the third-order central moments, we use these equations to determine the stationary mean and mean-square response. The effect of the system parameters on the response statistics is investigated. The presence of the nonlinearity causes multi-valued regions where more than one mean-square value of the response is possible. The local stability of the stationary mean and mean-square responses is analysed. Alternatively, assuming the random component of the response to be small compared with the mean response, we determine steady-state periodic responses to the deterministic part of the excitation. The effect of the random part of the excitation on the stable periodic responses is analysed as a perturbation and a closed-form expression for the mean-square response is obtained. Away from the transition zone separating stable and unstable periodic responses, the results of these two approaches are in good agreement. Comparisons of the results of these methods with that obtained by the method of equivalent linearization are presented.
Book•
01 Jan 1990
TL;DR: In this paper, Cartan's results on absolute equivalence of differential systems are shown to imply a generalization of a recent result on dynamic feedback linearization of scalar control systems.
TL;DR: In this paper, the methods of harmonic and stochastic linearization are discussed with respect to applications in shock-absorber dynamics, and it is shown that the parameters of an equivalent linear system can be obtained directly from experimental data.
Abstract: In this paper the methods of harmonic and stochastic linearization are discussed with respect to applications in shock-absorber dynamics. It is shown that the parameters of an equivalent linear system can be obtained directly from experimental data. In a second step an attempt is made to give a simple physical interpretation of the experimental results, which were obtained for a typical passenger car's shock-absorber.
TL;DR: In this article, the authors investigated the effect of sampling on linearization for continuous time systems and showed that linearizability via digital feedback imposes highly nongeneric constraints on the structure of the plant, even if this is known to be linearizable with continuous-time feedback.
TL;DR: In this paper, a nonlinear feedback control for a flexible joint manipulator is investigated, and it is shown that if the elastic (parasitic) modes are weakly observable from the output of the system, a state-space coordinate transformation and a static state feedback and control space transformation will turn the flexible system into a linear controllable and observable system.
Abstract: A nonlinear feedback control for a flexible joint manipulator is investigated. It is shown that if the elastic (parasitic) modes are weakly observable from the output of the system, a state-space coordinate transformation and a static state feedback and control space transformation will turn the flexible system into a linear controllable and observable system. In contrast, if the parasitics are strongly observable, a dynamic state feedback is required for input-output linearization. Numerical simulations for a single link flexible joint manipulator are reported, illustrating the application of the methodology. >
TL;DR: In this paper, consistent tangent moduli for the generalized Duvaut-Lions viscoplasticity model are derived based on consistent linearization of the residual functions associated with two alternative unconditionally stable constitutive integration algorithms; namely, the implicit backward Euler and the full integration.
Abstract: Consistent (algorithmic) tangent moduli for the generalized Duvaut-Lions viscoplasticity model are derived in this work. The derivations are based on consistent linearization of the residual functions associated with two alternative unconditionally stable constitutive integration algorithms; namely, the implicit backward Euler and the “full integration” algorithms. This “consistent linearization” procedure is equally applicable to the Perzyna-type viscoplasticity formulations. In particular, the von Mises isotropic/kinematic hardening viscoplasticity model is chosen as a model problem for demonstration. Consistent viscoplastic tangent moduli for other choices of (single or multiple) loading surfaces can be derived in a similar fashion provided that consistent elastoplastic (inviscid) tangent moduli are available. It is noted that since continuum tangent moduli do not exist at all for viscoplasticity, use of the proposed consistent tangent modul is not only desirable but necessary in the Newton-type finite-element computations. In addition, due to the difference in the two constitutive integration algorithms used, the corresponding consistent tangent moduli are not the same even when time steps are small. Numerical examples are also presented to illustrate the remarkable quadratic performance of the proposed consistent tangent moduli for the generalized Duvaut-Lions viscoplasticity model.
TL;DR: In this article, an algorithm for processing inertial data in an earth-fixed Cartesian frame is developed, which is compared with the standard algorithm that uses the local-level frame and the geographic coordinate system for the model formulation.
Abstract: An algorithm for processing strapdown inertial data in an earth-fixed Cartesian frame is developed in this paper. It is compared with the standard algorithm that uses the local-level frame and the geographic coordinate system for the model formulation. A general formulation of the modeling equations for the two approaches is given, and the linearization of the equations and the formulation of the appropriate Kalman filter are outlined. The derivation of the reference gravity model for both frames is briefly discussed, and numerically efficient formulas for the model's computation are given. In the case of the Cartesian algorithm, this leads to new formulas for all three components of the gravity vector. Real data are used to compare the two algorithms and to show that the accuracy is the same in both cases, but that the Cartesian formulation is about 30 percent more efficient.
13 May 1990
TL;DR: A discussion is presented of the control of legged-robot body orientation in flight using the internal motion of the leg using the angular momentum constraint and the concept of holonomy is introduced for constructing an optimal path.
Abstract: A discussion is presented of the control of legged-robot body orientation in flight using the internal motion of the leg. The angular momentum constraint (nonholonomic) is used to recast the problem into a nonholonomic motion planning problem. Chow's theorem is then applied to verify that the system is controllable, and the concept of holonomy is introduced for constructing an optimal path. Finally, linearization control is used in the internal motion space to realize the planned path. An additional degree of control to dynamically balance a legged robot that runs is provided with this strategy. >