Showing papers on "Linearization published in 2004"

Unscented filtering and nonlinear estimation

Abstract: The extended Kalman filter (EKF) is probably the most widely used estimation algorithm for nonlinear systems. However, more than 35 years of experience in the estimation community has shown that is difficult to implement, difficult to tune, and only reliable for systems that are almost linear on the time scale of the updates. Many of these difficulties arise from its use of linearization. To overcome this limitation, the unscented transformation (UT) was developed as a method to propagate mean and covariance information through nonlinear transformations. It is more accurate, easier to implement, and uses the same order of calculations as linearization. This paper reviews the motivation, development, use, and implications of the UT.

A delay-dependent approach to robust H/sub /spl infin// filtering for uncertain discrete-time state-delayed systems

Abstract: A delay-dependent approach to robust H/sub /spl infin// filtering is proposed for linear discrete-time uncertain systems with multiple delays in the state. The uncertain parameters are supposed to reside in a polytope and the attention is focused on the design of robust filters guaranteeing a prescribed H/sub /spl infin// noise attenuation level. The proposed filter design methodology incorporates some recently appeared results, such as Moon's new version of the upper bound for the inner product of two vectors and de Oliveira's idea of parameter-dependent stability, which greatly reduce the overdesign introduced in the derivation process. In addition to the full-order filtering problem, the challenging reduced-order case is also addressed by using different linearization procedures. Both full- and reduced-order filters can be obtained from the solution of convex optimization problems in terms of linear matrix inequalities, which can be solved via efficient interior-point algorithms. Numerical examples have been presented to illustrate the feasibility and advantages of the proposed methodologies.

Development of linear-parameter-varying models for aircraft

Abstract: This paper presents a comparative study of three linear-parameter-varying (LPV) modeling approaches and their application to the longitudinal motion of a Boeing 747 series 100/200. The three approaches used to obtain the quasi-LPV models are Jacobian linearization, state transformation, and function substitution. Development of linear parameter varying models are a key step in applying LPV control synthesis. The models are obtained for the up-and-away flight envelope of the Boeing 747-100/200. Comparisons of the three models in terms of their advantages, drawbacks, and modeling difficulty are presented. Open-loop time responses show the three quasi-LPV models matching the behavior of the nonlinear model when in the trim region. Differences between the models are more apparent as the response of the aircraft deviates from the nominal trim conditions. ¯

Orthogonal polynomials for power amplifier modeling and predistorter design

Abstract: The polynomial model is commonly used in power amplifier (PA) modeling and predistorter design. However, the conventional polynomial model exhibits numerical instabilities when higher order terms are included. In this paper, we introduce a novel set of orthogonal polynomials, which can be used for PA as well as predistorter modeling. Theoretically, the conventional and orthogonal polynomial models are "equivalent" and, thus, should behave similarly. In practice, however, the two approaches can perform quite differently in the presence of finite precision processing. Simulation results show that the orthogonal polynomials can alleviate the numerical instability problem associated with the conventional polynomials and generally yield better PA modeling accuracy as well as predistortion linearization performance.

Interconnection and damping assignment passivity-based control of mechanical systems with underactuation degree one

Abstract: We consider the problem of (asymptotic) stabilization of mechanical systems with underactuation degree one. A state-feedback design is derived applying the interconnection and damping assignment passivity-based control methodology. Its application relies on the possibility of solving a set of partial differential equations that identify the energy functions that can be assigned to the closed-loop. The following results are established: 1) identification - in terms of some algebraic inequalities - of a subclass of these systems for which the partial differential equations are trivially solved; 2) characterization of all systems which are feedback-equivalent to this subclass; and 3) introduction of a suitable parametrization of the assignable energy functions that provides the designer with a handle to address transient performance and robustness issues. An additional feature of our developments is that the open-loop system need not be described by a port-controlled Hamiltonian (or Lagrangian) model, a situation that arises often in applications due to model reductions or preliminary feedbacks that destroy the structure. The new result is applied to obtain an (almost) globally stabilizing controller for the inertia wheel pendulum, a controller for the chariot with pendulum system that can swing-up the pendulum from any position in the upper half plane and stop the chariot at any desired location, and an (almost) globally stabilizing scheme for the vertical takeoff and landing aircraft with strong input coupling. In all cases we obtain very simple and intuitive solutions that do not rely on, rather unnatural and technique-driven, linearization or decoupling procedures but instead endows the closed-loop system with a Hamiltonian structure with desired potential and kinetic energy functions.

Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods

Abstract: We discuss the state of the art in numerical solution methods for large scale polynomial or rational eigenvalue problems. We present the currently available solution methods such as the Jacobi-Davidson, Arnoldi or the rational Krylov method and analyze their properties. We briefly introduce a new linearization technique and demonstrate how it can be used to improve structure preservation and with this the accuracy and efficiency of linearization based methods. We present several recent applications where structured and unstructured nonlinear eigenvalue problems arise and some numerical results. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Estimation of frequency and its rate of change for applications in power systems

Abstract: A new method for estimation of power frequency and its rate of change is presented. Unlike conventional methods which are based on the concept of linearization, the proposed scheme accommodates the inherent nonlinearity of the frequency estimation problem. This makes the method capable of providing a fast and accurate estimate of the frequency when its deviation from the nominal value is incremental or large. The estimator is based on a newly developed quadrature phase-locked loop concept. The method is highly immune to noise and distortions. The estimator performance is robust with respect to the parameters of its structure. Structural simplicity and performance robustness are other salient features of the method.

Selection of the ground state for nonlinear schrödinger equations

Abstract: We prove for a class of nonlinear Schrodinger systems (NLS) having two nonlinear bound states that the (generic) large time behavior is characterized by decay of the excited state, asymptotic approach to the nonlinear ground state and dispersive radiation. Our analysis elucidates the mechanism through which initial conditions which are very near the excited state branch evolve into a (nonlinear) ground state, a phenomenon known as ground state selection. Key steps in the analysis are the introduction of a particular linearization and the derivation of a normal form which reflects the dynamics on all time scales and yields, in particular, nonlinear master equations. Then, a novel multiple time scale dynamic stability theory is developed. Consequently, we give a detailed description of the asymptotic behavior of the two bound state NLS for all small initial data. The methods are general and can be extended to treat NLS with more than two bound states and more general nonlinearities including those of Hartree–Fock type.

Homogeneous observers, iterative design, and global stabilization of high-order nonlinear systems by smooth output feedback

Abstract: We study the problem of global stabilization by smooth output feedback, for a class of n-dimensional homogeneous systems whose Jacobian linearization is neither controllable nor observable. A new output feedback control scheme is proposed for the explicit design of both homogeneous observers and controllers. While the smooth state feedback control law is constructed based on the tool of adding a power integrator, the observer design is new and carried out by developing a machinery, which makes it possible to assign the observer gains one-by-one, in an iterative manner. Such design philosophy is fundamentally different from that of the traditional "Luenberger" observer in which the observer gain is determined by observability. In the case of linear systems, our design method provides not only a new insight but also an alternative solution to the output feedback stabilization problem. For a class of high-order nonhomogeneous systems, we further show how the proposed design method, with an appropriate modification, can still achieve global output feedback stabilization. Examples and simulations are given to demonstrate the main features and effectiveness of the proposed output feedback control schemes.

Convergence of Petviashvili's Iteration Method for Numerical Approximation of Stationary Solutions of Nonlinear Wave Equations

Abstract: We analyze a heuristic numerical method suggested by V. I. Petviashvili in 1976 for approximation of stationary solutions of nonlinear wave equations. The method is used to construct numerically the solitary wave solutions, such as solitons, lumps, and vortices, in a space of one and higher dimensions. Assuming that the stationary solution exists, we find conditions when the iteration method converges to the stationary solution and when the rate of convergence is the fastest. The theory is illustrated with examples of physical interest such as generalized Korteweg--de Vries, Benjamin--Ono, Zakharov--Kuznetsov, Kadomtsev--Petviashvili, and Klein--Gordon equations.

Aeroelastic stability analysis of wind turbines using an eigenvalue approach

Abstract: A design tool for performing aeroelastic stability analysis of wind turbines is presented in this paper. The method behind this tool is described in a general form, as independent of the particular aeroelastic modelling as possible. Here, the structure is modelled by a Finite beam Element Method, and the aerodynamic loads are modelled by the Blade Element Momentum method coupled with a Beddoes-Leishman type dynamic stall model in a state-space formulation. The linearization of the equations of motion is performed about a steady-state equilibrium, where the deterministic forcing of the turbine is neglected. To eliminate the periodic coefficients and avoid using the Floquet Theory, the multi-blade transformation is utilized. From the corresponding eigenvalue problem, the eigenvalues and eigenvectors can be computed at any operation condition to give the aeroelastic modal properties: Natural frequencies, damping and mode shapes. An example shows a good agreement between predicted and measured aeroelastic damping of a stall-regulated 600 kW turbine. Copyright © 2004 John Wiley & Sons, Ltd.

Analysis and Control of Nonlinear Process Systems

Abstract: Basic Notions of Systems and Signals.- State-space Models.- Dynamic Process Models.- Input-output Models and Realization Theory.- Controllability and Observability.- Stability and The Lyapunov Method.- Passivity and the Hamiltonian View.- State Feedback Controllers.- Feedback and Input-output Linearization of Nonlinear Systems.- Passivation by Feedback.- Stabilization and Loop-shaping.

Effects of even-order nonlinear terms on power amplifier modeling and predistortion linearization

Abstract: Power amplifier (PA) is an essential component in communication systems and is nonlinear in nature. Digital baseband predistortion is an emerging cost effective approach to linearize a PA. To study PA nonlinear characteristics and to construct a predistorter, accurate nonlinear models are often necessary. Polynomials have been used extensively for modeling the behavior of the PA or the predistorter. For bandpass communication signals, attention has been paid mainly to odd-order nonlinear terms. In this paper, we reveal the benefit of including even-order nonlinear terms in baseband modeling of the PA and in enhancing predistortion performance.

Effective optimization for fuzzy model predictive control

Abstract: This paper addresses the optimization in fuzzy model predictive control. When the prediction model is a nonlinear fuzzy model, nonconvex, time-consuming optimization is necessary, with no guarantee of finding an optimal solution. A possible way around this problem is to linearize the fuzzy model at the current operating point and use linear predictive control (i.e., quadratic programming). For long-range predictive control, however, the influence of the linearization error may significantly deteriorate the performance. In our approach, this is remedied by linearizing the fuzzy model along the predicted input and output trajectories. One can further improve the model prediction by iteratively applying the optimized control sequence to the fuzzy model and linearizing along the so obtained simulated trajectories. Four different methods for the construction of the optimization problem are proposed, making difference between the cases when a single linear model or a set of linear models are used. By choosing an appropriate method, the user can achieve a desired tradeoff between the control performance and the computational load. The proposed techniques have been tested and evaluated using two simulated industrial benchmarks: pH control in a continuous stirred tank reactor and a high-purity distillation column.

Nonlinear evolution and runup–rundown of long waves over a sloping beach

Abstract: The initial value problem of the nonlinear evolution, shoreline motion and flow velocities of long waves climbing sloping beaches is solved analytically for different initial waveforms. A major difficulty in earlier work utilizing hodograph-type transformation when solving either boundary value or initial value problems has been the specification of equivalent boundary or initial condition in the transformed space. Here, in solving the initial value problem, the transformation is linearized in space at $t{=}0$, then the full nonlinear transformation is used to solve the initial value problem of the nonlinear shallow-water wave equations. A solution method is presented to describe the most physically realistic initial waveforms and simplified equations for the runup–rundown motions and shoreline velocities. This linearization of the initial condition does not appear to affect the subsequent nonlinear evolution, as shown through comparisons with earlier studies. Comparisons with runup results from solutions of the boundary value problem suggest the same variation with the runup laws. The methodology presented here appears simpler than earlier work as it does not involve the numerical calculation of singular elliptic integrals.

Discrete solution of the electrokinetic equations

Abstract: We present a robust scheme for solving the electrokinetic equations. This goal is achieved by combining the lattice-Boltzmann method with a discrete solution of the convection-diffusion equation for the different charged and neutral species that compose the fluid. The method is based on identifying the elementary fluxes between nodes, which ensures the absence of spurious fluxes in equilibrium. We show how the model is suitable to study electro-osmotic flows. As an illustration, we show that, by introducing appropriate dynamic rules in the presence of solid interfaces, we can compute the sedimentation velocity (and hence the sedimentation potential) of a charged sphere. Our approach does not assume linearization of the Poisson–Boltzmann equation and allows us for a wide variation of the Peclet number.

Input-output feedback linearization of time-delay systems

Abstract: In this note, the input-output linearization problem (IOLP) for a class of single-input-single-output nonlinear systems with multiple delays in the input, the output, and the state is studied. The problem is solved by means of various static or dynamic compensators, including state and output feedback. The mathematical setting is based on some noncommutative algebraic tools and the introduction of a nonlinear version of the so-called Roesser models for this class of systems. These are claimed to be the cornerstones for studying nonlinear time-delay systems. Necessary and sufficient conditions are given for the existence of a static or pure shift output feedback which solves the IOLP. Sufficient conditions for the existence of a dynamic state feedback solution are included as well.

Touch sensor with linearized response

Abstract: A field linearization pattern and a touch sensor incorporating same are disclosed. The touch sensor includes a polygonal field linearization pattern disposed around a touch sensitive area. The field linearization pattern includes a first side and a second side that intersect at a first corner. The field linearization pattern further includes an inner row and an outer row of discrete conductive segments. The inner row includes a conductive corner segment at the first corner. The conductive corner segment extends along a portion of the first and second sides of the linearization pattern. The touch sensor further includes electronics configured to detect a location of an input touch applied to the touch sensitive area by generating an electrical current in the linearization pattern. A current flowing from the first side to the second side of the linearization pattern is substantially confined within the linearization pattern.

Nonlinear observer design by dynamic observer error linearization

Abstract: In this note, we address the problem of transforming a nonlinear system into nonlinear observer canonical form in the extended state-space with the aid of dynamic system extension and introduction of virtual outputs. As an intermediate step for the general problem, we consider a restricted structure of dynamic extension, which is obtained, roughly speaking, by adding the chains of integrators to the outputs of original system. We propose sufficient conditions which can be verified using the system dynamics expressed in their original coordinates. An illustrative example is included that demonstrates the advantage of the proposed method over the conventional method.