Showing papers on "Linearization published in 2005"

Invariant Manifolds for Physical and Chemical Kinetics

Abstract: Introduction.- The Source of Examples.- Invariance Equation in the Differential Form.- Film Extension of the Dynamics: Slowness as Stability.- Entropy, Quasi-Equilibrium and Projector Field.- Newton Method with Incomplete Linearization.- Quasi-chemical Representation.- Hydrodynamics from Grad's Equations: Exact Solutions.- Relaxation Methods.- Method of Invariant Grids.- Method of Natural Projector.- Geometry of Irreversibility: The Film of Nonequilibrium States.- Slow Invariant Manifolds for Open Systems.- Estimation of Dimension of Attractors.- Accuracy Estimation and Post-Processing.- Conclusion.

Dimension Reduction of Large-Scale Second-Order Dynamical Systems via a Second-Order Arnoldi Method

Abstract: A structure-preserving dimension reduction algorithm for large-scale second-order dynamical systems is presented. It is a projection method based on a second-order Krylov subspace. A second-order Arnoldi (SOAR) method is used to generate an orthonormal basis of the projection subspace. The reduced system not only preserves the second-order structure but also has the same order of approximation as the standard Arnoldi-based Krylov subspace method via linearization. The superior numerical properties of the SOAR-based method are demonstrated by examples from structural dynamics and microelectromechanical systems.

Methods in nonlinear analysis

Abstract: Linearization.- Fixed-Point Theorems.- Degree Theory and Applications.- Minimization Methods.- Topological and Variational Methods.

Structure of stochastic dynamics near fixed points.

Abstract: We analyze the structure of stochastic dynamics near either a stable or unstable fixed point, where the force can be approximated by linearization. We find that a cost function that determines a Boltzmann-like stationary distribution can always be defined near it. Such a stationary distribution does not need to satisfy the usual detailed balance condition but might have instead a divergence-free probability current. In the linear case, the force can be split into two parts, one of which gives detailed balance with the diffusive motion, whereas the other induces cyclic motion on surfaces of constant cost function. By using the Jordan transformation for the force matrix, we find an explicit construction of the cost function. We discuss singularities of the transformation and their consequences for the stationary distribution. This Boltzmann-like distribution may be not unique, and nonlinear effects and boundary conditions may change the distribution and induce additional currents even in the neighborhood of a fixed point.

Global finite-time stabilization by output feedback for planar systems without observable linearization

Abstract: This note considers the problem of global finite-time stabilization by output feedback for a class of planar systems without controllable/observable linearization. A sufficient condition for the solvability of the problem is established. By developing a nonsmooth observer and modifying the adding a power integrator technique, we show that an output feedback controller can be explicitly constructed to globally stabilize the systems in finite time. As a direct application of the main result, global output feedback finite-time stabilization is achieved for the double linear integrator systems perturbed by some nonlinear functions which are not necessarily homogeneous.

Standing Waves on an Infinitely Deep Perfect Fluid Under Gravity

Abstract: The existence of two-dimensional standing waves on the surface of an infinitely deep perfect fluid under gravity is established. When formulated as a second-order equation for a real-valued function w on the 2-torus and a positive parameter μ, the problem is fully nonlinear (the highest order x-derivative appears in the nonlinear term but not in the linearization at 0) and completely resonant (there are infinitely many linearly independent eigenmodes of the linearization at 0 for all rational values of the parameter μ). Moreover, for any prescribed order of accuracy there exists an explicit approximate solution of the nonlinear problem in the form of a trigonometric polynomial. Using a Nash-Moser method to seek solutions of the nonlinear problem as perturbations of the approximate solutions, the existence question can be reduced to one of estimating the inverses of linearized operators at non-zero points. After changing coordinates these operators become first order and non-local in space and second order in time. After further changes of variables the main parts become diagonal with constant coefficients and the remainder is regularizing, or quasi-one-dimensional in the sense of [22]. The operator can then be inverted for two reasons. First, the explicit formula for the approximate solution means that, restricted to the infinite-dimensional kernel of the linearization at zero, the inverse exists and can be estimated. Second, the small-divisor problems that arise on the complement of this kernel can be overcome by considering only particular parameter values selected according to their Diophantine properties. A parameter-dependent version of the Nash-Moser implicit function theorem now yields the existence of a set of unimodal standing waves on flows of infinite depth, corresponding to a set of values of the parameter μ>1 which is dense at 1. Unimodal means that the term of smallest order in the amplitude is cos x cos t, which is one of many eigenfunctions of the completely resonant linearized problem.

Unscented SLAM for large-scale outdoor environments

Abstract: This paper presents an experimentally validated alternative to the classical extended Kalman filter approach to the solution of the probabilistic state-space simultaneous localization and mapping (SLAM) problem. Several authors have reported the divergence of this classical approach due to the linearization of the inherent nonlinear nature of the SLAM problem. Hence, the approach described in this work aims to avoid the analytical linearization based on Taylor-series expansion of both the model and measurement equations by using the unscented filter. An innovation-based consistency checking validates the feasibility and applicability of the unscented SLAM approach to a real large-scale outdoor exploration mission.

Robust output feedback stabilization of uncertain nonlinear systems with uncontrollable and unobservable linearization

Abstract: This paper investigates the problem of robust output feedback stabilization for a family of uncertain nonlinear systems with uncontrollable/unobservable linearization. To achieve global robust stabilization via smooth output feedback, we introduce a rescaling transformation with an appropriate dilation, which turns out to be very effective in dealing with uncertainty of the system. Using this rescaling technique combined with the nonseparation principle based design method, we develop a robust output feedback control scheme for uncertain nonlinear systems in the p-normal form, under a homogeneous growth condition. The construction of smooth state feedback controllers and homogeneous observers uses only the knowledge of the bounding homogeneous system rather than the uncertain system itself. The robust output feedback design approach is then extended to a class of uncertain cascade systems beyond a strict-triangular structure. Examples are provided to illustrate the results of the paper.

Fundamentals Of Multibody Dynamics. Theory And Applications

Abstract: Preface Particle Dynamics: The Principle fo Newton's Second Law Rigid-Body Kinematics Kinematics for General Multibody Systems Modeling of Forces in Multibody Systems Equations of Motion of Multibody Systems Hamilton-Lagrange and Gibbs-Appel Equations Handling of Constraints in Multibody Systems Dynamics Numerical Stability of Constrained Multibody Systems Linearization and Vibration Analysis of Multibody Systems Dynamics of Multibody Systems with Terminal Flexible Links Dynamic Analysis of Multiple Flexible-Body Systems Modeling of Flexibility Effects Using the Boundary-Element Method Appendix A: Multibody Dynamics Flowchart for the Construction of the Equations of Motion with Constraints Appendix B: Centroid Location and Area Moment of Inertia Appendix C: Center of Gravity and Mass Moment of Inertia of Homogeneous Solids Appendix D: Symbols Description Appendix E: Units and Conversion References Index

Current-source converter based STATCOM: modeling and control

Abstract: STATCOM is a FACTS controller that is used in power systems to regulate the line voltage, enhance the power transmission capacity and extend the transient stability margin. STATCOM is conventionally realized by a voltage-source converter; however, being a current injection device, its performance can be improved when realized by a current-source converter (CSC) that can generate a controllable current directly at its output terminals. In this paper, a STATCOM based on the current-source converter topology is proposed. The nonlinear model of the current-source converter, which is the source of the difficulties in the controller design, has been modified to a linear model through a novel modeling technique. The proposed modeling technique is not based on the linearization of a set of nonlinear equations around an operating point. Instead, the power balance equation and a nonlinear input transformation are used to derive a linear model independent of the operating point. This model acts as the basis for the design of a decoupled state-feedback controller. The proposed STATCOM has been simulated using the PSCAD/EMTDC package. The simulation results show that a CSC-based STATCOM can result in excellent current and voltage waveforms as well as very short response time while operating at a low switching frequency. This makes the proposed scheme suitable for high power applications.

Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green's function

Abstract: Two discretizations, linear and nonlinear, of basic notions of the complex analysis are considered. The underlying lattice is an arbitrary quasicrystallic rhombic tiling of a plane. The linear theory is based on the discrete Cauchy-Riemann equations, the nonlinear one is based on the notion of circle patterns. We clarify the role of the rhombic condition in both theories: under this condition the corresponding equations are integrable (in the sense of 3D consistency, which yields also the existense of zero curvature representations, Backlund transformations etc.). We demonstrate that in some precise sense the linear theory is a linearization of the nonlinear one: the tangent space to a set of integrable circle patterns at an isoradial point consists of discrete holomorphic functions which take real (imaginary) values on two sublattices. We extend solutions of the basic equations of both theories to Z d , where d is the number of different edge slopes of the quasicrystallic tiling. In the linear theory, we give an integral representation of an arbitrary discrete holomorphic function, thus proving the density of discrete exponential functions. We introduce the d-dimensional discrete logarithmic function which is a generalization of Kenyon's discrete Green's function, and uncover several new properties of this function. We prove that it is an isomonodromic solution of the discrete Cauchy-Riemann equations, and that it is a tangent vector to the space of integrable circle patterns along the family of isomonodromic discrete power functions.

Evaluation and Comparison of Relative Motion Theories

Abstract: A modeling error index is introduced for evaluating and comparing the accuracy of various theories of the relative motion of satellites to determine the effect of modeling errors on the various theories. The derived index does not require linearization of the equations of motion, and so nonlinear theories can also be evaluated. The index can be thought of as proportional to the percentage error; consequently, the smaller the index, the more accurate the theory. The results show that not including the reference orbit eccentricity and differential gravitational perturbations has a major effect on the accuracy of the theory, and the nonlinear effects are much smaller except for very large relative motion orbits. The two key parameters in the evaluation are the eccentricity of the reference orbit and the relative motion orbit size. The theories compared are Hill’s equations, a small eccentricity state transition matrix, a non-J2 state transition matrix, the Gim–Alfriend state transition matrix, the unit sphere approach, and the Yan–Alfriend nonlinear method. The numerical results show the sequence of the index from high to low should be Hill’s equation, non-J2, small eccentricity, Gim–Alfriend state transition matrix index, with the unit sphere approach and the Yan–Alfriend nonlinear method having the lowest index and equivalent performance.

Linearization of vector radiative transfer with respect to aerosol properties and its use in satellite remote sensing

Abstract: [1] We present a plane parallel radiative transfer model for polarized light that provides the intensity vector field as well as analytical derivatives of the four Stokes parameters at the top of the atmosphere with respect to physical properties of spherical aerosols. The linearization consists of two steps: (1) calculation of the derivatives of the four Stokes parameters at the top of the atmosphere with respect to scattering coefficient, extinction coefficient, and expansion coefficients of the scattering phase matrix. These derivatives are calculated employing the forward-adjoint perturbation theory. General expressions are presented that can be applied for the linearization of any vector radiative transfer model that calculates the internal radiation field in the model atmosphere. (2) The second step is calculation of the derivatives of the scattering coefficient, extinction coefficient, and the expansion coefficients of the scattering phase matrix, with respect to the real and imaginary part of the refractive index, and parameters describing the size distribution (e.g., effective radius, effective variance). These derivatives are analytically calculated following Mie theory. The use of the developed linearized radiative transfer model for the retrieval of aerosol properties is demonstrated using synthetic measurements of intensity and polarization of the Global Ozone Monitoring Experiment-2 (GOME-2). Here it is shown that an iterative retrieval approach based on the linearized radiative transfer model is suited to solve the nonlinear aerosol retrieval problem, and additionally allows a solid error analysis.

Fuzzy modeling and synchronization of hyperchaotic systems

Abstract: This paper presents fuzzy model-based designs for synchronization of hyperchaotic systems. The T–S fuzzy models for hyperchaotic systems are exactly derived. Based on the T–S fuzzy hyperchaotic models, the fuzzy controllers for hyperchaotic synchronization are designed via the exact linearization techniques. Numerical examples are given to demonstrate the effectiveness of the proposed method.

A hybrid digital/RF envelope predistortion linearization system for power amplifiers

Abstract: This paper presents an adaptive wide-band digitally controlled RF envelope predistortion linearization system for power amplifiers (PAs). A field-programmable gate-array-based lookup table is indexed by a digitized envelope power signal, and instantaneously adjusts the input signal amplitude and phase via an RF vector modulator to compensate for the AM-AM and AM-PM distortion. The advantages of this predistortion architecture over conventional baseband digital approaches are that a 20%-33% wider correction bandwidth is achievable at the same clock speeds, and linearization can be performed without the need for a digital baseband input signal. The timing match between the input RF signal and predistorting signal, which is one of the critical factors for performance, was investigated and adjusted to obtain optimum performance. Using three-carrier cdmaOne and wide-band multitone signals, the linearization performances for a 0.5-W GaAs heterostructure field-effect transistor, a 90-W peak-envelope-power (PEP) silicon LDMOS PA, and a 680-W PEP LDMOS PA were examined. In addition, the predistortion performance variation for different signals was studied in terms of signal envelope statistics, output powers, and PA power capacities.

Control of an Electrostatic Microelectromechanical System Using Static and Dynamic Output Feedback

Abstract: This paper examines control strategies for electrostatically actuated microelectromechanical systems (MEMS), with the goals of using feasible measurements to eliminate the pull-in bifurcation, robustly stabilize any desired operating point in the capacitive gap, decrease settling time, and reduce overshoot. We show that input-output linearization, passivity-based design, and the theory of port-controlled Hamiltonian systems lead naturally to static output feedback of device charge. This formalizes and extends previously reported results from the MEMS literature. Further analysis suggests that significantly improving transient behavior in lightly damped MEMS requires dynamic estimation of electrode velocity. We implement output-feedback control using a reduced-order nonlinear observer. Simulations predict greatly improved transient behavior, and large reductions in control voltage.

Robust feedback linearization and gh∞ controller for a quadrotor unmanned aerial vehicle

Abstract: In this paper, a mixed robust feedback linearization with linear GH1 controller is applied to a nonlinear quadrotor unmanned aerial vehicle. An actuator saturation and constrain on state space output are introduced to analyse the worst case of control law design.The results show that the overall system becomes robust when weighting functions are chosen judiciously. Performance issues of the controller are illustrated in a simulation study that takes into account parameter uncertainties and external disturbances as well as measurement noise. K e y w o r d s: UAV, GH, sensitivity, robust linearization

An energy-based control for an n-H-bridges multilevel active rectifier

Abstract: This paper deals with the control of a multilevel n-H-bridges front-end rectifier. This topology allows n distinct dc buses to be fed by the same ac source offering a high loading flexibility suitable for traction applications as well as for industrial automation plants. However, this flexibility can lead the system to instability if the dc buses operate at different voltage levels and with unbalanced loads. Thus, linear controllers, designed on the basis of the small-signal linearization, are not effective any longer and stability can not be ensured as large-signal disturbances occur. The use of a passivity-based control (PBC) designed via energy considerations and without small-signal linearization properly fits stability problems related to this type of converter. The system has been split into n subsystems via energy considerations in order to achieve the separate control of each dc bus and its stability in case of load changes or disturbances generated by other buses. Then, a set of n passivity-based controllers (one for each subsystem) is adopted: the controllers are linked using dynamical parameters computed through energy balance equations. Hence, the system dc buses are independent and stable as experimental results demonstrate.

An Adaptive Semiactive Control Algorithm for Magnetorheological Suspension Systems

Abstract: In this paper, we will present a nonlinear-model-based adaptive semiactive control algorithm developed for magnetorheological (MR) suspension systems exposed to broadband nonstationary random vibration sources that are assumed to be unknown or not measurable. If there exist unknown and/or varying parameters of the dynamic system such as mass and stiffness, then the adaptive algorithm can include on-line system identification such as a recursive least-squares method. Based on a nonparametric MR damper model, the adaptive system stability is proved by converting the hysteresis inherent with MR dampers to a memoryless nonlinearity with sector conditions. The convergence of the adaptive system, however, is investigated through a linearization approach including further numerical illustration of specific cases. Finally the simulation results for a magnetorheological seat suspension system with the suggested adaptive control are presented. The results are compared with low-damping and high-damping cases, and such comparison further shows the effectiveness of the proposed nonlinear model-based adaptive control algorithm for damping tuning.

Existence of Compressible Current-Vortex Sheets: Variable Coefficients Linear Analysis

Abstract: We study the initial-boundary value problem resulting from the linearization of the equations of ideal compressible magnetohydrodynamics and the Rankine-Hugoniot relations about an unsteady piecewise smooth solution. This solution is supposed to be a classical solution of the system of magnetohydrodynamics on either side of a surface of tangential discontinuity (current-vortex sheet). Under some assumptions on the unperturbed flow, we prove an energy a priori estimate for the linearized problem. Since the tangential discontinuity is characteristic, the functional setting is provided by the anisotropic weighted Sobolev space W21,σ. Despite the fact that the constant coefficients linearized problem does not meet the uniform Kreiss-Lopatinskii condition, the estimate we obtain is without loss of smoothness even for the variable coefficients problem and nonplanar current-vortex sheets. The result of this paper is a necessary step in proving the local-in-time existence of current-vortex sheet solutions of the nonlinear equations of magnetohydrodynamics.