Showing papers on "Linearization published in 2001"

Efficient linearization of the augmented plane-wave method

Abstract: The present thesis concerns method development and applications in the field of first principles electronic structure calculations.Augmented planewaves combine the simple planewaves with exact solutions of the Schrodinger equation for a spherical potential. This combination yields a very good set of basis functions for describing the electronic structure everywhere in a crystal potential. In the present work, developments of the original augmented planewave (APW) method are presented. It is shown that the exact APW eigenvalues can be found using information from the eigenvalues of the APW secular matrix. This provides a more efficient scheme to solve the APW eigenvalue problem, than the traditional evaluation of the secular determinant. Further, a new way of linearizing the APW method is presented and compared to the traditional linearized APW method (LAPW). Using a combination of the original APW basis functions and the so called local orbitals (lo), the APW+lo linearization is found to reproduce the results of the LAPW method, but already at a smaller basis set size. Another advantage of the new linearization is a faster convergence of forces, with respect to the basis set size, as compared to the LAPW method.The applications include studies of the non-collinear magnetic configuration in the fcc-based high-temperature phase of iron, γ-Fe. The system is found to be extremely sensitive to volume changes, as well as to a tetragonal distortion of the cubic unit cell. A continuum of degenerate spin spiral configurations, including the global energy minimum, are found for the undistorted crystal. The in-plane anisotropy of the ideal interface between a ferromagnetic layer of bcc Fe and the semiconducting ZnSe crystal is also investigated. In contrast to the four-fold symmetric arrangement of the atoms at the interface, the in-plane magnetic anisotropy displays a large uniaxiality. The calculated easy axes are in agreement with experiments for both Se and Zn terminated interfaces. In addition, calculations of the hyperfine parameters were performed for Li intercalated battery materials.

A fast SHAKE algorithm to solve distance constraint equations for small molecules in molecular dynamics simulations

Abstract: A common method for the application of distance constraints in molecular simulations employing Cartesian coordinates is the SHAKE procedure for determining the Lagrange multipliers regarding the constraints. This method relies on the linearization and decoupling of the equations governing the atomic coordinate resetting corresponding to each constraint in a molecule, and is thus iterative. In the present study, we consider an alternative method, M-SHAKE, which solves the coupled equations simultaneously by matrix inversion. The performances of the two methods are compared in simulations of the pure solvents water, dimethyl sulfoxide, and chloroform. It is concluded that M-SHAKE is significantly faster than SHAKE when either (1) the molecules contain few distance constraints (solvent), or (2) when a high level of accuracy is required in the application of the constraints. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 501–508, 2001

Control Lyapunov-Razumikhin functions and robust stabilization of time delay systems

Abstract: Motivated by control Lyapunov functions and Razumikhin theorems on stability of time delay systems, we introduce the concept of control Lyapunov-Razumikhin functions (CLRF). The main reason for considering CLRFs is construction of robust stabilizing control laws for time delay systems. Most existing universal formulas that apply to CLFs, are not applicable to CLRFs. It turns out that the domination redesign control law applies, achieving global practical stability and, under an additional assumption, global asymptotic stability. This additional assumption is satisfied in the practically important case when the quadratic part of a CLRF is itself a CLRF for the Jacobian linearization of the system. The CLRF based domination redesign possesses robustness to input unmodeled dynamics including an infinite gain margin. While, in general, construction of CLRFs is an open problem, we show that for several classes of time delay systems a CLRF can be constructed in a systematic way.

Enforcing Passivity for Admittance Matrices Approximated by Rational Functions

Abstract: A linear power system component can be included in a transient simulation as a terminal equivalent by approximating its admittance matrix Y by rational functions in the frequency domain. Physical behavior of the resulting model entails that it should absorb active power for any set of applied voltages, at any frequency. This requires the real part of Y to be positive definite (PD). We calculate a correction to the rational approximation of Y which enforces the PD-criterion to be satisfied. The correction is minimal with respect to the fitting error. The method is based on linearization and constrained minimization by quadratic programming. Examples show that models not satisfying the PD-criterion can lead to an unstable simulation, even though the rational approximation has stable poles only. Enforcement of the PD-criterion is demonstrated to give a stable result.

Smooth Dynamical Systems

Abstract: Some Simple Examples Equivalent Systems Integration of Vector Fields Linear Systems, Linearization, Stable Manifolds Stable Systems Appendices.

Recent Improvements in Aerodynamic Design Optimization on Unstructured Meshes

Abstract: Recent improvements in an unstructured-grid method for large-scale aerodynamic design are presented. Previous work had shown such computations to be prohibitively long in a sequential processing environment. Also, robust adjoint solutions and mesh movement procedures were difficult to realize, particularly for viscous flows. To overcome these limiting factors, a set of design codes based on a discrete adjoint method is extended to a multiprocessor environment using a shared memory approach. A nearly linear speedup is demonstrated, and the consistency of the linearizations is shown to remain valid. The full linearization of the residual is used to precondition the adjoint system, and a significantly improved convergence rate is obtained. A new mesh movement algorithm is implemented and several advantages over an existing technique are presented. Several design cases are shown for turbulent flows in two and three dimensions.

Dimensional model reduction in non-linear finite element dynamics of solids and structures

Abstract: A general approach to the dimensional reduction of non-linear finite element models of solid dynamics is presented. For the Newmark implicit time-discretization, the computationally most expensive phase is the repeated solution of the system of linear equations for displacement increments. To deal with this, it is shown how the problem can be formulated in an approximation (Ritz) basis of much smaller dimension. Similarly, the explicit Newmark algorithm can be also written in a reduced-dimension basis, and the computation time savings in that case follow from an increase in the stable time step length. In addition, the empirical eigenvectors are proposed as the basis in which to expand the incremental problem. This basis achieves approximation optimality by using computational data for the response of the full model in time to construct a reduced basis which reproduces the full system in a statistical sense. Because of this ‘global’ time viewpoint, the basis need not be updated as with reduced bases computed from a linearization of the full finite element model. If the dynamics of a finite element model is expressed in terms of a small number of basis vectors, the asymptotic cost of the solution with the reduced model is lowered and optimal scalability of the computational algorithm with the size of the model is achieved. At the same time, numerical experiments indicate that by using reduced models, substantial savings can be achieved even in the pre-asymptotic range. Furthermore, the algorithm parallelizes very efficiently. The method we present is expected to become a useful tool in applications requiring a large number of repeated non-linear solid dynamics simulations, such as convergence studies, design optimization, and design of controllers of mechanical systems.

Linearization: reducing distortion in power amplifiers

Abstract: This article discusses techniques for the cancellation of distortion (linearization) in power amplifiers. Different methods of linearization are introduced and compared. The linearization of solid-state power amplifiers (SSPAs), traveling-wave tube amplifiers (TWTAs) and klystron power amplifiers (KPAs) are considered. Although the focus of this article is on power amplifiers, many of the techniques are applicable to other components such as mixers, low-noise amplifiers, and even photonic components, such as lasers and optical modulators.

Exact linearization and noninteracting control of a 4 rotors helicopter via dynamic feedback

Abstract: Presents a nonlinear dynamic model for a four rotors helicopter in a form suited for control design. We show that the input-output decoupling problem is not solvable for this model by means of a static state feedback control law. Then, a dynamic feedback controller is developed which renders the closed-loop system linear, controllable and noninteractive after a change of coordinates in the state-space. Finally, the stability and the robustness of the proposed control law in the presence of wind, turbulences and parametric uncertainties is analyzed through a simulated case study.

Inverse feedforward controller for complex hysteretic nonlinearities in smart-material systems

Abstract: Undesired complex hysteretic nonlinearities are present to a varying degree in virtually all smart-material-based sensors and actuators provided they are driven with sufficiently high amplitudes. In motion and active vibration control applications, for example, these nonlinearities can excite unwanted dynamics, which leads in the best case to reduced system performance and in the worst case to unstable system operation. This necessitates the development of purely phenomenological models that characterize these nonlinearities in a way that is sufficiently accurate, amenable to a compensator design for actuator linearization, and efficient enough for use in real-time applications. To fulfil these demanding requirements, this article describes a new compensator design method for invertible complex hysteretic nonlinearities that is based on the so-called Prandtl-Ishlinskii hysteresis operator. The parameter identification of this model can be formulated as a quadratic optimization problem, which produces the best L 2 2 -norm approximation for the measured output-input data of the real hysteretic nonlinearity. Special linear inequality constraints for the parameters guarantee the unique solvability of the identification problem and the invertability of the identified model. This leads to a robustness of the identification procedure against unknown measurement errors, unknown model errors, and unknown model orders. The corresponding compensator can be directly calculated and thus efficiently implemented from the model by analytical transformation laws. Finally, the compensator design method is used to generate an inverse feedforward controller for the linearization of a magnetostrictive actuator. In comparision to the conventionally controlled magnetostrictive actuator, the nonlinearity error of the inverse controlled magnetostrictive actuator is lowered from about 30% to about 3%.

A feedback control scheme for reversing a truck and trailer vehicle

Abstract: A control scheme is proposed for stabilization of backward driving along simple paths for a miniaturized vehicle composed of a truck and a two-axle trailer. The paths chosen are straight lines and arcs of circles. When reversing, the truck and trailer under examination can be modeled as an unstable nonlinear system with state and input saturations. The simplified goal of stabilizing along a trajectory (instead of a point) allows us to consider a system with controllable linearization. Still, the combination of instability and saturations makes the task impossible with a single controller. In fact, the system cannot be driven backward from all initial states because of the jack-knife effects between the parts of the multibody vehicle; it is sometimes necessary to drive forward to enter into a specific region of attraction. This leads to the use of hybrid controllers. The scheme has been implemented and successfully used to reverse the radio-controlled vehicle.

A linear MOS transconductor using source degeneration and adaptive biasing

Abstract: This paper presents a new configuration for linear MOS voltage-to-current conversion (transconductance). The proposed circuit combines two previously reported linearization methods. The topology achieves 60-dB linearity for a fully balanced input dynamic range up to 1 V/sub pp/ at a 3.3-V supply voltage, with slightly decreasing performance in the unbalanced case. The linearity is preserved during the tuning process for a moderate range of transconductance values. The approach is validated by both computer simulations and experiments.

Flight control design using extended state observer and non-smooth feedback

Abstract: This paper proposes a novel nonlinear approach for high performance flight control design. The dynamic linearization is accomplished via a kind of unknown input observer, called extended state observer. A nonsmooth feedback law is employed to achieve the desirable dynamic performances. A Lyapunov function is constructed for the proposed method.

Combination theory and equilibrium evaporation

Abstract: This paper is an analysis of equilibrium evaporation and its role in the energy balance of a terrestrial surface, as described by combination theory. Three themes are covered: first, a brief historical review identifies multiple definitions of the concept of equilibrium evaporation. Second, these are formalized by developing the basic principles of combination theory with minimum approximation. Several measures are utilized to do this: linearization is avoided, radiative and storage coupling are incorporated systematically, and actual and linearized saturation deficits are distinguished. The formalism is used to analyse several algebraically defined states and limits for the surface energy balance. Third, the thermodynamic foundation of equilibrium evaporation is analysed by studying surface-atmosphere feedbacks in arbitrary closed and open evaporating systems. It is shown that under steady energy supply any closed evaporating system evolves towards a quasi-steady state in which the Bowen ratio takes the equilibrium value 1/ev, where ev is the ratio of the latent- and sensible-heat contents of saturated air with temperature, evaluated at the volume-averaged temperature in the closed system. This applies whether the system is well-mixed or imperfectly mixed, and whatever the internal distribution of surface fluxes and surface and aerodynamic resistances. In contrast, open systems cannot reach such an equilibrium. This evolutionary definition of equilibrium evaporation differs from an alternative algebraic definition, the fully decoupled limit. The differences between the two definitions are identified, and the evolutionary definition is shown to be more fundamental. Thus, the correct temperature for evaluating e in determining equilibrium evaporation is the volume-averaged temperature in a closed region, which in the case of a convective boundary layer is well approximated by the mixed-layer temperature.

Nonlinear stability analysis for a class of TCP/AQM networks

Abstract: Recent work has shown the benefit of using proportional feedback in TCP/AQM (transmission control protocol/active queue management) networks. By proportional feedback we mean the marking probability is proportional to the instantaneous queue length. Our earlier work (2001) relied on linearization of nonlinear fluid-flow models of TCP. In this work we address these nonlinearities directly and establish some stability results when the marking is proportional. In the case of delay-free marking, we show the system's equilibrium point to be asymptotically stable for all proportional gains. In the more realistic case of delayed feedback, we establish local asymptotic stability and quantify a region of attraction.

On the structure of spectra of modulated travelling waves

Abstract: Modulated travelling waves are solutions to reaction-diffusion equations that are time-periodic in an appropriate moving coordinate frame. They may arise through Hopf bifurcations or essential instabilities from pulses o fronts. In this article, a framework for the stability analysis of such solutions is presented: the reaction-diffusion equation is cast as an ill-posed elliptic dynamical system in the spatial variable acting upon time-periodic functions. Using this formulation, points in the esolvent set, the point spectrum, and the essential spectrum of the linearization about a modulated travelling wave are related to the existence of exponential dichotomies on appopriate intervals for the associated spatial elliptic eigenvalue problem. Fredholm properties of the linearized operator are characterized by a relative Morse-Floe index of the elliptic equation. These results are proved without assumptions on the asymptotic shape of the wave. Analogous results are true for the spectra of travelling waves to parabolic equations on unbounded cylinders. As an application, we study the existence and stability of modulated spatially-periodic patterns with long-wavelength that accompany modulated pulses.

An explicit solution of coupled viscous Burgers' equation by the decomposition method

Abstract: We consider a coupled system of viscous Burgers' equations with appropriate initial values using the decomposition method. In this method, the solution is calculated in the form of a convergent power series with easily computable components. The method does not need linearization, weak nonlinearity assumptions or perturbation theory. The decomposition series solution of the problem is quickly obtained by observing the exis- tence of the self-canceling "noise" terms where the sum of components vanishes in the limit.

Fitting transducer characteristics to measured data

Abstract: There are no rules to select the best curve-fitting method for a given set of data. This problem is of great importance in measurement applications. Optimizing analog and digital methods for a transducer's characteristic interpolation or linearization is a field where constant research is being done, particularly since auto-calibration and self-test of intelligent transducers is a topic of major interest. We present an overview of classical methods for data interpolation and least mean squares regression. We make a comparative evaluation of the relative performance of polynomial and artificial neural networks approximations to measurement data with particular attention paid to the reduction of the required calibration set dimension to obtain a given accuracy.

An LQR/LQG theory for systems with saturating actuators

Abstract: An extension of the LQR/LQG methodology to systems with saturating actuators, referred to as SLQR/SLQG, where S stands for saturating, is obtained. The development is based on the method of stochastic linearization, whereby the saturation is replaced by a gain, calculated as a function of the variance of the signal at its input. Using the stochastically linearized system and the Lagrange multiplier technique, solutions of the SLQR/SLQG problems are derived. These solutions are given by standard Riccati and Lyapunov equations coupled with two transcendental equations, which characterize both the variance of the signal at the saturation input and the Lagrange multiplier associated with the constrained minimization problem. It is shown that, under standard stabilizability and detectability conditions, these equations have a unique solution, which can be found by a simple bisection algorithm. When the level of saturation tends to infinity, these equations reduce to their standard LQR/LQG counterparts. In addition, the paper investigates the properties of closed-loop systems with SLQR/SLQG controllers and saturating actuators. In this regard, it is shown that SLQR/SLQG controllers ensure semi-global stability by appropriate choice of a parameter in the performance criterion. Finally, the paper illustrates the techniques developed by a ship roll damping problem.

An assessment of linear versus non-linear multigrid methods for unstructured mesh solvers

Abstract: The relative performance of a nonlinear full approximation storage multigrid algorithm and an equivalent linear multigrid algorithm for solving two different nonlinear problems is investigated. The first case consists of a transient radiation diffusion problem for which an exact linearization is available, while the second problem involves the solution of the steady-state Navier-Stokes equations, where a first-order discrete Jacobian is employed as an approximation to the Jacobian of a second-order-accurate discretization. When an exact linearization is employed, the linear and nonlinear multigrid methods asymptotically converge at identical rates and the linear method is found to be more efficient due to its lower cost per cycle. When an approximate linearization is employed, as in the Navier-Stokes cases, the relative efficiency of the linear approach versus the nonlinear approach depends both on the degree to which the linear system approximates the full Jacobian as well as on the relative cost of linear versus nonlinear multigrid cycles. For cases where convergence is limited by a poor Jacobian approximation, substantial speedup can be obtained using either multigrid method as a preconditioner to a Newton-Krylov method.

Global optimization of nonconvex factorable programming problems

Abstract: In this paper, we consider a special class of nonconvex programming problems for which the objective function and constraints are defined in terms of general nonconvex factorable functions. We propose a branch-and-bound approach based on linear programming relaxations generated through various approximation schemes that utilize, for example, the Mean-Value Theorem and Chebyshev interpolation polynomials coordinated with a Reformulation-Linearization Technique (RLT). A suitable partitioning process is proposed that induces convergence to a global optimum. The algorithm has been implemented in C++ and some preliminary computational results are reported on a set of fifteen engineering process control and design test problems from various sources in the literature. The results indicate that the proposed procedure generates tight relaxations, even via the initial node linear program itself. Furthermore, for nine of these fifteen problems, the application of a local search method that is initialized at the LP relaxation solution produced the actual global optimum at the initial node of the enumeration tree. Moreover, for two test cases, the global optimum found improves upon the solutions previously reported in the source literature.

Nonlinear decentralized disturbance attenuation excitation control via new recursive design for multi-machine power systems

Abstract: Summary form only given,as follows. In this paper a new nonlinear decentralized disturbance attenuation excitation control for multi-machine power systems is proposed based on recursive design without linearization treatment. The proposed controller improves system robustness to dynamic uncertainties and also attenuates bounded exogenous disturbances on the system in the sense of L/sub 2/-gain. Computer test results on a 6-machine system show clearly that the proposed excitation control strategy can enhance transient stability of power systems more effectively than other excitation controllers.

Two-grid finite-element schemes for the steady Navier-Stokes problem in polyhedra.

Abstract: We discretize a steady Navier–Stokes system on a three-dimensional polyhedron by finite-elements schemes defined on two grids. In the first step, the fully nonlinear problem is solved on a coarse grid, with mesh-size H. In the second step, the problem is linearized by substituting into the nonlinear term, the velocity uH computed at step one, and the linearized problem is solved on a fine grid with mesh-size h. This approach is motivated by the fact that the contribution of uH to the error analysis is measured in the L norm, and thus, for the lowest-degree elements on a Lipschitz polyhedron, is of the order of H. Hence, an error of the order of h can be recovered at the second step, provided h = H. When the domain is convex, a similar result can be obtained with h = H. Both results are valid in two dimensions. 0 – Introduction Let us consider the following equation in a bounded domain Ω of R3: −∆u+ u = f in Ω , (0.1) u = 0 on ∂Ω . (0.2) For any f in L2(Ω), it admits a unique solution in the Sobolev space H1 0 (Ω) (i.e. the space of functions φ such that φ and ∂φ ∂xi belong to L 2(Ω) for i = 1, 2, 3, and φ = 0 on ∂Ω). Its variational formulation writes: ∀ v ∈ H 0 (Ω) , a(u, v) + (u, v) = (f, v) , (0.3) Received : May 23, 2000. AMS Subject Classification: 76D05, 65N15, 65N30, 65N55.

Linearization criteria for a system of second-order ordinary differential equations

Abstract: Firstly, we prove two linearization criteria for a system of two second-order ordinary differential equations (odes). The first states that a system of two non-linear second-order odes is reducible via a point transformation to a linear system if and only if it admits the four-dimensional abelian Lie algebra L 4,1 : [X i ,X j ]=0, i, j=1,…, 4 . The second states that in order for a system of two non-linear second-order odes to be linearizable, it is necessary and sufficient that it admits the four-dimensional Lie algebra L 4,2 : [X i ,X j ]=0, [X i ,X 4 ]=X i , i, j=1, 2, 3 . The approach used is constructive and enables one to explicitly work out the transformation that leads to linearization. Secondly, we give conditions under which a system of two second-order non-linear odes is reducible to the free particle equations x″=0, y″=0 . These linearization criteria are then generalized to a system of n(n>2) second-order odes. Finally, we give examples of how one can effect linearization for a system.