Showing papers on "Linearization published in 1983"

Engineering Optimization: Methods and Applications

Abstract: Functions of a Single Variable. Functions of Several Variables. Linear Programming. Constrained Optimality Criteria. Transformation Methods. Constrained Direct Search. Linearization Methods for Constrained Problems. Direction--Generation Methods Based on Linearization. Quadratic Approximation Methods for Constrained Problems. Structured Problems and Algorithms. Comparison of Constrained Optimization Methods. Strategies for Optimization Studies. Engineering Case Studies. Appendixes. Author and Subject Indexes.

Formulation and Solution of Nonlinear Integer Production Planning Problems for Flexible Manufacturing Systems

Abstract: A flexible manufacturing system FMS is an integrated, computer-controlled complex of automated material handling devices and numerically controlled machine tools that can simultaneously process medium-sized volumes of a variety of part types. FMSs are becoming an attractive substitute for the conventional means of batch manufacturing, especially in the metal-cutting industry. This new production technology has been designed to attain the efficiency of well-balanced, machine-paced transfer lines, while utilizing the flexibility that job shops have to simultaneously machine multiple part types. Some properties and constraints of these systems are similar to those of flow and job shops, while others are different. This technology creates the need to develop new and appropriate planning and control procedures that take advantage of the system's capabilities for higher production rates. This paper defines a set of five production planning problems that must be solved for efficient use of an FMS, and addresses specifically the grouping and loading problems. These two problems are first formulated in detail as nonlinear 0-1 mixed integer programs. In an effort to develop solution methodologies for these two planning problems, several linearization methods are examined and applied to data from an existing FMS. To decrease computational time, the constraint size of the linearized integer problems is reduced according to various methods. Several real world problems are solved in very reasonable time using the linearization that results in the fewest additional constraints and/or variables. The problem characteristics that determine which linearization to use, and the application of the linearized models in the solution of actual planning problems, are also discussed.

Canonical form observer design for non-linear time-variable systems

Abstract: An observer of canonical (phase-variable) form for non-linear time-variable systems is introduced. The development of this non-linear time-variable form requires regularity of the non-linear time-variable- observability matrix of the system. From the relationships derived during the development, it follows that a non-linear time-variable observer can be dimensioned by an eigenvalue assignment with respect to the canonical state coordinates if a linearization of system non-linearities about the reconstructed state trajectories is permissible. This is an assumption similar to that of the extended Kalman filter based on a linearization about the current estimate.

The painleve property for partial differential equations. ii. backlund transformation, lax pairs, and the schwarzian derivative

Abstract: In this paper we investigate the Painleve property for partial differential equations. By application to several well‐known partial differential equations (Burgers, KdV, MKdV, Bousinesq, higher‐order KdV and KP equations) it is shown that consideration of the ‘‘singular manifold’’ leads to a formulation of these equations in terms of the ‘‘Schwarzian derivative.’’ This formulation is invariant under the Moebius group (acting on dependent variables) and is shown to obtain the appropriate Lax pair (linearization) for the underlying nonlinear pde.

Adaptive linearization of power amplifiers in digital radio systems

Abstract: High-frequency power amplifiers operate most efficiently at saturation, i.e., in the nonlinear range of their input/output characteristics. This phenomenon has traditionally dictated the use of constant envelope modulation methods for data transmission, resulting in circular signal constellations. This approach has inherently limited the admissible data rates in digital radio. In this paper we present a method for solving this problem without sacrificing amplifier power efficiency. We describe and analyze an adaptive linearizer that can automatically compensate for amplifier nonlinearity and thus make it possible to transmit multilevel quadrature amplitude modulated signals without incurring intolerable constellation distortions. The linearizer utilizes a real-time, data-directed, recursive algorithm for predistorting the signal constellation. Our analysis and computer simulations indicate that the algorithm is robust and converges rapidly from a blind start. Furthermore, the signal constellation and the average transmitted power can both be changed through software.

Nonlinear Kalman filtering techniques for terrain-aided navigation

Abstract: The application of nonlinear Kalman filtering techniques to the continuous updating of an inertial navigation system using individual radar terrain-clearance measurements has been investigated. During this investigation, three different approaches for handling the highly nonlinear terrain measurement function were developed and their performance was established. These were 1) a simple first-order extended Kalman filter using local derivatives of the terrain surface, 2) a modified stochastic linearization technique which adaptively fits a least squares plane to the terrain surface and treats the associated fit error as an additional noise source, 3) a parallel Kalman filter technique utilizing a bank of reduced-order filters that was especially important in applications with large initial position uncertainties. Theoretical and simulation results are presented.

Asymptotics for $M$-Type Smoothing Splines

Abstract: Limit theorems giving rates of convergence of nonparametric regression estimates obtained from smoothing splines are proved. The main emphasis is on nonlinear, robust smoothing splines, but new results are obtained for the usual (linear) case. It is assumed that the knots become asymptotically uniform in a vague sense. Convergence of derivatives is also investigated. The main mathematical tools are a linearization of the robust smoothing spline, and an approximation of the linear smoothing spline utilizing the Green's function of an associated boundary value problem.

Werner deconvolution for automated magnetic interpretation and its refinement using Marquardt's inverse modeling

Abstract: We present a deconvolution for automated magnetic interpretation based on Werner’s (1953) simplified thin‐dike assumption which leads to the linearization of complex nonlinear magnetic problems. The usefulness of the method is expanded by the fact that the horizontal gradient of the total field caused by the edge of a thick interface body is equivalent to the total field from a thin dike. Statistical decision, numerical iteration, and a seven‐point operator are used to improve approximations of susceptibility, dip, depth, and horizontal location of the source. Marquardt’s nonlinear least‐squares method for inverse modeling is then used to refine automatically the first approximation provided by the deconvolution. Synthetic and real total‐field data are used to demonstrate the process.

A Probabilistic Approach to Power System Stability Analysis

Abstract: Power system stability analysis is usually performed in a deterministic framework in which the time domain response of the power system is studied for certain specific disturbances to determine the adequacy of the system. However, the occurrence of disturbances and their attendant protective switching sequences are random processes and it would be more meaningful to determine the probability of stability for a power system. An approach for such a determination is presented in this paper. The probability of steady state stability is relatively easier to determine because of the linearization of the system equations. The probability of transient stability, on the other hand, is much more difficult to obtain analytically because of the nonlinear transformations required. However, a Monte Carlo simulation method to determine transient stability probability is shown to be feasible.

Self-consistency iterations in electronic-structure calculations

Abstract: The convergence of self-consistency iterations in electronic-structure calculations based on density-functional theory is examined by the linearization of the self-consistency equations around the exact solution. In particular, we study the convergence of the usual procedure employing a mixture of the input and output of the last iteration. We show that this procedure converges for a suitably chosen mixture. However, the convergence is necessarily slow in certain cases. These problems are connected either with large charge oscillations or with the onset of magnetism. We discuss physical situations where such problems occur. Moreover, we propose some improved iteration schemes which are illustrated in calculations for $3d$ impurities in Cu.

Partial and robust linearization by feedback

Abstract: It is argued that if the nonlinearities in a system are mild, and the controller is sufficiently stabilizing, the inaccuracies of a linear model, which is often taken to be sufficient for a controller design, can be safely neglected. For systems with severe nonlinearity a linearizing technique is described, based on the change of state coordinates and nonlinear feedback; in the total context of a stable feedback design the linearization technique is considered robust. Furthermore, attention is paid to partial linearization using the same transformations. It is found that there always exist maximally linearizing transformations which are not necessarily unique.

Linearization of second-order nonlinear oscillation theorems

Abstract: The problem of oscillation of super- and sublinear Emden-Fowler equations is studied. Established are a number of oscillation theorems involving comparison with related linear equations. Recent results on linear oscillation can thus be used to obtain interesting oscillation criteria for nonlinear equations. 25 references.

Optimal control of robotic manipulators

Abstract: This dissertation addresses the problem of finding the desired optimal trajectories and the optimal control inputs for a rigid multi-degrees-of-freedom robotic manipulator in a positioning mode. For a general, highly coupled and nonlinear dynamic model an analytical control law is derived and structures of controllers are suggested. The equations of motion of a multilink mechanism and its actuators are unified into a common dynamic model. A nonlinear feedback transformation which globally linearizes, decouples and places the poles of the closed loop system is derived. Each of the decoupled subsystems represents a single degree of freedom of the manipulator motion. This transformation is realized by on-line computations based on the measurements of the manipulator's state variables and reference to a look-up table, and requires no matrix inversion. The set of uncoupled subsystems in a form of triple integrators allows a solution of various optimal control problems for each degree of freedom independently. The time optimal positioning problem of a multi-degrees-of-freedom robotic manipulator with bounded velocity, acceleration and jerk is posed in a form which guarantees uniqueness of the control and state trajectories. The solution is demonstrated via single and double link examples. The look-up table implementing the feedback transformation introduces non-analytic (quantization) errors. The time optimal trajectories of a double link manipulator are shown to be rather insensitive to this type of error. For a class of errors an existence theorem of the linear feedback controller is proven which guarantees closed loop asymptotic stability at the state space origin. Finally, outlines of various controller structures are discussed. This dissertation is the first application of the following principles: (1) Instead of optimizing the real system, a fictitous system should be introduced which is more convenient for optimization and can be obtained from the original system, for example, by a state space transformation. (2) The state space feedback transformation can be realized for decoupling, linearization and pole-placement simultaneously. (3) The programmed open loop control for a deterministic system can be devised with the assumption that deviations are compensated by a separated subsystem for "control in-the-small."

On the linearization of nonlinear langevin-type stochastic differential equations

Abstract: We show that a logical extension of the piecewise optimal linearization procedure leads to the Gaussian decoupling scheme, where no iteration is required. The scheme is equivalent to solving a few coupled equations. The method is applied to models which represent (a) a single steady state, (b) passage from an initial unstable state to a final preferred stable state by virtue of a finite displacement from the unstable state, and (c) a bivariate case of passage from an unstable state to a final stable state. The results are shown to be in very good agreement with the Monte Carlo calculations carried out for these cases. The method should be of much value in multidimensional cases.

Systems of nonlinear partial differential equations

Abstract: I Expository Lectures.- Algebraic and Topological Invariants for Reaction-Diffusion Equations.- Hyperbolic Systems of Conservation Laws.- Ill-Posed Problems in Thermoelasticity Theory Lecture 1 Twinning of Thermoelastic Materials.- Lecture 2 Problems for Infinite Elastic Prisms.- Lecture 3 St.-Venantr-s Problem for Elastic Prisms.- Nonlinear Systems in Optimal Control Theory and Related Topics.- The Regularity Problem of Extremals of Variational Integrals.- Some Aspects of the Regularity Theory for Nonlinear Elliptic Systems.- Quasilinear Elliptic Systems in Diagonal Form.- Topics in Bifurcation Theory Lecture 1 A Brief Introduction to Bifurcation Theory.- Lecture 2 Unfoldings.- Lecture 3 Symmetry in Bifurcation Theory.- The Compensated Compactness Method Applied to Systems of Conservation Laws.- II Special Sessions.- a. Problems in Nonlinear Elasticity, organized by S. S. Antman (University of Maryland).- Coercivity Conditions in Nonlinear Elasticity.- Constitutive Inequalities and Dynamic Stability in the Linear Theories of Elasticity, Thermoelasticity and Viscoelasticity.- Generalized Solutions to Conservation Laws.- Stability of the Elastica.- Group Theoretic Classification of Conservation Laws in Elasticity.- b. Applications of Bifurcation Theory to Mechanics, organized by J. E. Marsden (University of California, Berkeley).- Phase Transitions Via Bifurcation from Heteroclinic Orbits.- Bifurcation under Continuous Groups of Symmetries.- Morse Decompositions and Global Continuation of Periodic Solutions for Singularly Perturbed Delay Equations.- Bifurcation and Linearization Stability in the Traction Problem.- Singular Elliptic Eigenvalue Problems for Equations and Systems.- c. Nonelliptic Problems and Phase Transitions, organized by J. M. Ball (Heriot-Watt University).- Regularization of Non Elliptic Variational Problems.- Remarks on the Relaxation of Integrals of the Calculus of Variations.- A Diffusion Equation with a Nonmonovone Constitutive Function.- An Admissibility Criterion for Fluids Exhibiting Phase Transitions.- d. Dynamical Systems and Partial Differential Equations, organized by J. K. Hale (Brown University).- Stabilization of Solutions for a System with a Continuum of Equilibria and Distinct Diffusion Coefficients.- Relation between the Trapped Rays and the Distribution of the Eigenfrequencies of the Wave Equation in the Exterior of an Obstacle.- Stabilization Properties for Nonlinear Degenerate Parabolic Equations with Cut-Off Diffusivity.- Dynamics in Parabolic Equations - An Example.

Bifurcation from the Essential Spectrum

Abstract: This report surveys some of the results that have been obtained during the past twenty years concerning bifurcation from a point in the essential spectrum of the linearization of a nonlinear equation. This means that the linearization is not a Fredholm operator and so, even locally, the problem cannot be reduced to an equivalent finite dimensional situation by the Lyapunov-Schmidt method. Nonetheless, the aim is to obtain conclusions, local or global, about bifurcation in the same spirit as the classical results which are well-known in the Fredholm case. A vast survey of recent progress in bifurcation theory for the Fredholm situation has been given by Ize [30] in an article in the previous volume of this series. In the present context all of the standard techniques of nonlinear analysis (variational methods, topological degree, implicit function theorems) have been brought to bear on the problem, but so far only the variational approach has yielded general results in abstract spaces. The other methods have been confined to the context of elliptic equations on unbounded domains or to integral equations involving convolution. This state of affairs is reflected in the presentation of this survey which concentrates on the general results obtained by variational methods and their application to elliptic equations on ℜ N . However in the last section there are a few remarks covering what is known about connected sets, or even curves, of solutions to differential equations and it is to be hoped that in the near future substantial progress will be made in establishing conclusions of this kind in an abstract setting similar to that used in the variational case.

Corrective Control Computations for Large Power Systems

Abstract: This paper presents a novel methodology for optimal corrective and/or emergency controls computations. It is based on a model reduction procedure, and linearization leading to a linear program which is solved via a dual simplex algorithm with upper bounds developed specifically for this problem. The methodology computes optimal adjustments of generation schedules, VAR source allocations, transformer tap settings and if necessary load shedding. In case of problem infeasibility, it computes the best possible adjustments which will move the operating state of the system closer to a secure state. In addition an analysis of the impact of each individual control action on the emergency conditions of the system is generated. Capacitors and reactors are switched in/out in discrete steps. Such discrete controls are handled with an efficient suboptimal procedure based on a partial enumeration of linear programming relaxations. A computer program has been developed. The software are sparsity coded. Efficiency evaluation of the various components of the program is given. The methodology is illustrated with sample results from two test systems: (a) the IEEE 30 bus system, and (b) the Georgia Power Company's 500 kV/230 kV/115 kV system, which is a 981 bus system. The second test system demonstrates the applicability of the method to practical large scale systems.

A Recursive Free-Body Approach to Computer Simulation of Human Postural Dynamics

Abstract: A recursive, free-body approach to the estimation of joint torques associated with observed motion in linkage mechanisms has recently been shown to be computationally more efficient than any other known approach to this problem. This paper applies this method to the analysis of human postural dynamics and shows how it can also be used to compute accelerations for specified joint torques. The latter calculation, referred to here as the direct dynamics problem, has until now involved symbolic complexity to such an extent as to generally limit computer simulation studies of postural control to very simple models. The model presented in this paper is both straightforward and general, and removes this obstacle to the investigation of possible neural control mechanisms by means of computer simulation. A computationally oriented linearization procedure for the direct dynamics problem is also included in the paper. Finally, example simulation results and corresponding measured body motions for human subjects are presented to validate the method.

Equilibrium states of an elastic conductor in a magnetic field: a paradigm of bifurcation theory

Abstract: In this paper we study the equilibrium states of a nonlinearly elastic conducting wire in a magnetic field. The wire is perfectly flexible and is suspended between fixed supports. The wire carries an electric current and is subjected to a constant magnetic field whose direction is parallel to the line between the supports. We solve this problem exactly and show that the set of solutions gives rise to a paradigmatic bifurcation diagram. We then carry out a study of the equations obtained by linearization about the nontrivial solutions in order to gain some insight into the stability of the various solution branches. Introduction. In this paper we study the equilibrium states of a nonlinearly elastic conducting wire in a magnetic field. The wire is assumed to be perfectly flexible and is suspended between fixed supports. The distance between the supports is assumed to be greater than the natural length of the wire and therefore the wire will always be in tension. The magnetic field is assumed to be constant and is directed parallel to the line between the supports. We show that this problem can be solved exactly. The set of solutions exhibits the classic bifurcation phenomenon. The bifurcation parameter is X = IB where I is the current in the wire and B the strength of the magnetic field. For all values of X > 0 there is a trivial solution in which the wire remains straight. However at the eigenvalues of the problem obtained by linearizing the equilibrium equation about the trivial solution, bifurcation occurs and we obtain branches of nontrivial solutions. The situation is quite similar to the classical problem of the buckling of a beam. Actually in our case the branches of solutions are two-dimensional manifolds because the problem is invariant under rotation about the axis of the supports. This work was motivated by the discussion in [1] (see ?4.7, especially Figure 29 and the reference quoted therein). There are two cases to consider as the wire is (elastically) homogeneous or not. Of course it is of interest to study the stability of the equilibria with respect to the (hyperbolic) equations of motion. This is in general a very difficult problem and in the present case it is complicated even further by the multiplicity of equilibria arising from the rotational invariance. In the case of the homogeneous wire we can study the operator obtained by linearization about a nontrivial solution. We are able to locate Received by the editors June 16, 1982. 1980 Mathematics Subject Classification. Primary 73C50; Secondary 58E07. (D1983 American Matheniatical Society 0002-9947/82/0000-1374/$03.50 377 This content downloaded from 157.55.39.180 on Mon, 25 Apr 2016 06:09:45 UTC All use subject to http://about.jstor.org/terms

Introduction: Differential Equations and Dynamical Systems

Abstract: In this introductory chapter we review some basic topics in the theory of ordinary differential equations from the viewpoint of the global geometrical approach which we develop in this book. After recalling the basic existence and uniqueness theorems, we consider the linear, homogeneous, constant coefficient system and then introduce nonlinear and time-dependent systems and concepts such as the Poincare map and structural stability. We then review some of the better-known results on two-dimensional autonomous systems and end with a statement and sketch of the proof of Peixoto’s theorem, an important result which summarizes much of our knowledge of two-dimensional flows.