Local parameter identifiability for one class of hybrid systems

We consider the problem of local parameter identifiability for a hybrid system with components that are continuous and discrete in time. The set of observations is the vector-solution (depending on the parameter that is continuous in time) of the discrete component. Sufficient conditions of local parameter identifiability have been formulated using the earlier introduced notion of normalized separability of the set of parameters from the kernel of a special functional. An example where the condition of normed separability is reduced to some rank criterion is given.     

Solution of Dynamic Deformation Problems for Prestressed Laminated Plates Based on the Three-Dimensional Theory of Elasticity

A three-dimensional analytical solution describing forced harmonic vibrations of prestressed laminated plates is found for the case of a hinged support. The solution is based on the analytical separation of variables. It is assumed that the prestressed state is homogeneous, subcritical, linear, and momentless and that the vibration amplitudes are small. A solution based on a model with a polynomial approximation of the required displacement functions across the plate thickness is also considered. These functions are found on the front surfaces of the structure. This allows us to solve the problem both in the continuous and discrete structural approaches. In the continuous structural approach, the order of the resolving system of equations is independent of the number of layers. In the discrete structural approach, for rigid contact of layers with similar boundary conditions at the plate end face, an algorithm can be introduced which reduces significantly the number of operations required for realization of the model proposed. In the numerical examples presented, both rigid and sliding contacts of layers and various prestressed conditions are considered. Both approaches give results that agree well.     

Combinatorial analysis of two basic forms of hidden periodicity in categorial sequences

Two kinds of mixing of periodical components according to the Spencer-Brown laws of form are considered. If the identical fragments of a periodic component remain unchanged, then the periodicity is of the penetrant (calling) form; if the identical fragments break up, then it is of the co-penetrant (crossing) form. The hidden periodicity of the penetrant form is studied using the symptom analysis; the co-penetrant form of periodicity is studied using the order asymmetry method. In the symptom analysis, the method of principal components and SSA are modified for finite geometries. The order asymmetry method is the cluster analysis where the distance between two gradations characterizes the deviation from the periodicity in a subsequence over these gradations.     

Multi-harmonic measurements and numerical simulations of nonlinear vibrations of a beam with non-ideal boundary conditions

This study presents a direct comparison of measured and predicted nonlinear vibrations of a clamped–clamped steel beam with non-ideal boundary conditions. A multi-harmonic comparison of simulations with measurements is performed in the vicinity of the primary resonance. First of all, a nonlinear analytical model of the beam is developed taking into account non-ideal boundary conditions. Three simulation methods are implemented to investigate the nonlinear behavior of the clamped–clamped beam. The method of multiple scales is used to compute an analytical expression of the frequency response which enables an easy updating of the model. Then, two numerical methods, the Harmonic Balance Method and a time-integration method with shooting algorithm, are employed and compared one with each other. The Harmonic Balance Method enables to simulate the vibrational stationary response of a nonlinear system projected on several harmonics. This study then proposes a method to compare numerical simulations with measurements of all these harmonics. A signal analysis tool is developed to extract the system harmonics’ frequency responses from the temporal signal of a swept sine experiment. An evolutionary updating algorithm (Covariance Matrix Adaptation Evolution Strategy), coupled with highly selective filters is used to identify both fundamental frequency and harmonic amplitudes in the temporal signal, at every moment. This tool enables to extract the harmonic amplitudes of the output signal as well as the input signal. The input of the Harmonic Balance Method can then be either an ideal mono-harmonic signal or a multi-harmonic experimental signal. Finally, the present work focuses on the comparison of experimental and simulated results. From experimental output harmonics and numerical simulations, it is shown that it is possible to distinguish the nonlinearities of the clamped–clamped beam and the effect of the non-ideal input signal.     

Well-Posedness of a Problem of Propagation of Nonlinear Acoustic-Gravity Waves in the Atmosphere Generated by Surface Pressure Variations

Currently there are many international microbarograph networks for high-resolution recording of wave pressure variations on the Earth’s surface. This arouses interest in wave propagation in the atmosphere generated by atmospheric pressure variations. A full system of nonlinear hydrodynamic equations for atmospheric gases with lower boundary conditions in the form of wavelike pressure variations on the Earth’s surface is considered. Since the wave amplitudes near the Earth’s surface are small, linearized equations are used in the analysis of well-posedness of the problem. With the help of a wave energy functional method, it is shown that in the non-dissipative case the solution to the boundary value problem is uniquely determined by the variable pressure field on the Earth’s surface. The corresponding dissipative problem is well-posed if, in addition to the pressure field, appropriate conditions on the velocity and temperature on the Earth’s surface are given. In the case of an isothermal atmosphere, the problem admits analytical solutions that are harmonic in the variables x and t. A good agreement between the numerical and analytical solutions is obtained. The study shows that the temperature and density can rapidly vary at the lower boundary of the boundary value problem. An example of solving the three-dimensional problem with variable pressure on the Earth’s surface taken from experimental observations is given. The developed algorithms and computer programs can be used to simulate atmospheric waves generated by pressure variations on the Earth’s surface.     

Poisson pencils, algebraic integrability, and separation of variables

In this paper we review a recently introduced method for solving the Hamilton-Jacobi equations by the method of Separation of Variables. This method is based on the notion of pencil of Poisson brackets and on the bihamiltonian approach to integrable systems. We discuss how separability conditions can be intrinsically characterized within such a geometrical set-up, the definition of the separation coordinates being encompassed in the bihamiltonian structure itself. We finally discuss these constructions studying in details a particular example, based on a generalization of the classical Toda Lattice.     

An Open-loop criterion for the solvability of a closed-loop guidance problem with incomplete information: Linear control systems

The method of open-loop control packages is a tool for stating the solvability of guaranteed closed-loop control problems under incomplete information on the observed states. In this paper, a solution method is specified for the problem of guaranteed closed-loop guidance of a linear control system to a convex target set at a prescribed point in time. It is assumed that the observed signal on the system’s states is linear and the set of its admissible initial states is finite. It is proved that the problem under consideration is equivalent to the problem of open-loop guidance of an extended linear control system to an extended convex target set. Using a separation theorem for convex sets, we derive a solvability criterion, which reduces to solving a finite-dimensional optimization problem. An illustrative example is considered.     

Theorems on the existence of separating surfaces

LetR andG be finite of sets inE ^d. This paper presents theorems on the existence of strict linear and spherical separators ofR andG that are similar to the fundamental separation theorem of Kirchberger. Kirchberger's theorem impliet that the strict linear separability of finite setsR andG is determined by the separability of all subsets of up tod+2 points ofR⊃G. This paper shows that under certain conditions, the linear separability ofR andG is determined by the separability of significantly fewer than all subfamilies of up tod+2 members ofR ⊃G. The same treatment is made of Lay's extension of Kirchberger's theorem to separation by hyperspheres. This research was supported by a PGS3 scholarship from NSERC.     

Short-term forecasting of traffic delays in highway construction zones using on-line approximators

Highway construction zones are often the cause of traffic delays. This is a natural consequence of the high congestion and nonuniform traffic flow conditions in construction zones. Most of the current algorithms for computing traffic delays are accurate for low density traffic conditions, and address the estimation of current travel time only. This paper presents a method for short-term forecasting of traffic delays in highway construction zones using data from presence detectors. The method is based on a modular approach wherein data from adjacent detectors is processed for estimating the travel time between the two detectors. The travel time estimates are then considered as time-series data, and the problem of short-term forecasting of traffic delay is formulated as a time-series evolution problem. A generic structure referred to as an on-line approximator is used for the prediction of travel time based on current and past travel time estimates. Simulation examples are used to illustrate the traffic delay forecasting algorithm.     

Optimal Design of Periodic Diffractive Structures

The problem of designing a periodic interface between two different materials, which gives rise to a specified far-field diffraction pattern for a given incoming plane wave, is considered. The time harmonic waves are assumed to be TM (transverse magnetic) polarized. The diffraction problem is modeled by a generalized Helmholtz equation with transparent boundary conditions. In this paper the design problem is relaxed to include highly oscillatory profiles. Existence of an optimal design is established. The principal method is based on the theory of homogenization for the model equation. Accepted 31 May 2000. Online publication 26 February 2001.     

Optimal observation of controlled elastic vibrations of a beam in the presence of measurement errors

The problem of constructing of an optimal operation for restoring the state of controlled elastic vibrations of a beam in the presence of measurement errors is investigated. By the method of separation of variables, the problem is reduced to an observation problem with an actual output signal for an infinite system of ordinary differential equations. For each harmonic, a universal optimal operation that restores the deflection of the beam from equilibrium and the velocities of all points of the beam is constructed by amplifying the ideal part of the signal produced by the system.     

Some evidence on forecasting time-series with support vector machines

The importance of predicting future values of a time-series transcends a range of disciplines. Economic and business time-series are typically characterized by trend, cycle, seasonal, and random components. Powerful methods have been developed to capture these components by specifying and estimating statistical models. These methods include exponential smoothing, autoregressive integrated moving average (ARIMA), and partially adaptive estimated ARIMA models. New research in pattern recognition through machine learning offers innovative methodologies that can improve forecasting performance. This paper presents a study of the comparative results of time-series analysis on nine problem domains, each of which exhibits differing time-series characteristics. Comparative analyses use ARIMA selection employing an intelligent agent, ARIMA estimation through partially adaptive methods, and support vector machines. The results find that support vector machines weakly dominate the other methods and achieve the best results in eight of nine different data sets.     

Convex Optimization approach to signals with fast varying instantaneous frequency

Motivated by the limitation of analyzing oscillatory signals composed of multiple components with fast-varying instantaneous frequency, we approach the time-frequency analysis problem by optimization. Based on the proposed adaptive harmonic model, the time-frequency representation of a signal is obtained by directly minimizing a functional, which involves few properties an “ideal time-frequency representation” should satisfy, for example, the signal reconstruction and concentrative time-frequency representation. FISTA (Fast Iterative Shrinkage-Thresholding Algorithm) is applied to achieve an efficient numerical approximation of the functional. We coin the algorithm as Time-frequency bY COnvex OptimizatioN (Tycoon). The numerical results confirm the potential of the Tycoon algorithm.     

Fragments of the theory of nanotransistors: self-switching of a plane transverse hypersonic wave in nonlinearly elastic nanocomposite materials

We present fragments of the theory of nanotransistors concerning general information about transistors, models of nanocomposite materials, effects of nonlinear interaction of waves, and a theoretical analysis of the interaction of cubically nonlinear elastic plane harmonic waves in materials whose nonlinear properties are described by the Murnaghan potential. Using the method of slowly varying amplitudes, we investigate the interaction of two harmonic vertical transverse plane waves. Shortened and evolution equations, and Manley-Rowe relations, are obtained. We analyze analytically and numerically the mechanism of repumping of the energy of a strong pumping wave that propagates at a frequency of ω to a weak signal wave propagating at a frequency of 3ω . The described mechanism of switching of hypersonic waves in a nonlinearly elastic nanomaterial is similar to the mechanism of switching observed in optical and other transistors.     

Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool

The EMD algorithm is a technique that aims to decompose into their building blocks functions that are the superposition of a (reasonably) small number of components, well separated in the time–frequency plane, each of which can be viewed as approximately harmonic locally, with slowly varying amplitudes and frequencies. The EMD has already shown its usefulness in a wide range of applications including meteorology, structural stability analysis, medical studies. On the other hand, the EMD algorithm contains heuristic and ad hoc elements that make it hard to analyze mathematically.In this paper we describe a method that captures the flavor and philosophy of the EMD approach, albeit using a different approach in constructing the components. The proposed method is a combination of wavelet analysis and reallocation method. We introduce a precise mathematical definition for a class of functions that can be viewed as a superposition of a reasonably small number of approximately harmonic components, and we prove that our method does indeed succeed in decomposing arbitrary functions in this class. We provide several examples, for simulated as well as real data.     

Extended scenario analysis

Scenario analysis, as proposed by Rockafellar and Wets, is a stochastic programming technique employing discrete scenarios with known probabilities, usually covering several time periods. The requirement of nonaticipativity (not using future information to make present decisions) is enforced during the computational solution by using Spingarn's method of partial inverses. The scenario analysis method as proposed relies on separability (with respect to scenarios) of all problem elements except the nonanticipativity constraint.We show how, by making a little more use of the partial inverse technique, one can include nonseparable convex constraints in such a procedure. As an illustrative example, we show how to analyze a portfolio optimization problem of Markowitz type (minimize variance for a given return) using scenarios. This offers the prospect of extending classical portfolio analysis from models based on historical behavior to models incorporating future scenarios of any desired type.The research reported here was sponsored by the National Science Foundation under Grant CCR-8801489, and by the Air Force Systems Command, USAF, under Grant No. AFOSR-89-0058. The US Government has certain rights in this material, and is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.     

On the problem of frequencies separation

In assumption that the input signal consists of several harmonic oscillations, the paper considers the problem of construction of a linearly-causal filter that preserves some components of the signal and rejects the others. Some new estimates for the strengthening coefficient of linearly-causal filters are obtained, which solve the problem.     

Study of non-linear dynamic behavior of open cracked functionally graded Timoshenko beam under forced excitation using harmonic balance method in conjunction with an iterative technique

The nonlinear modeling and subsequent dynamic analysis of cracked Timoshenko beams with functionally graded material (FGM) properties is investigated for the first time using harmonic balance method followed by an iterative technique. Crack is assumed to be open throughout. During modeling, nonlinear strain–displacement relation is considered. Rotational spring model is adopted in order to model the open cracks. Energy formulations are established using Timoshenko beam theory. Nonlinear governing differential equations of motion are derived using Lagrange's equation. In order to incorporate the influence of higher order harmonics, harmonic balance method is employed. This reduces the governing differential equations into nonlinear set of algebraic equations. These equations are solved using two different iterative techniques. Methodology is computationally easier and efficient as well. This is observed that although assumption of simple harmonic motion (SHM) simplifies the problem, it yields to erroneous results at higher amplitude of motion. However, accuracy of the solution is improved considerably when the contribution of higher order harmonic terms are considered in the analysis. Results are compared with the available results, which confirm the validity of the methodology. Subsequent to that a parametric study on influence of forcing term, material indices and crack parameters on large amplitude vibration of Timoshenko beams is performed for two different boundary conditions.     

Sublinear operators with rough kernel generated by Calderón–Zygmund operators and their commutators on generalized Morrey spaces

The aim of this paper is to establish the boundedness of certain sublinear operators with rough kernel generated by Calderón–Zygmund operators and their commutators on generalized Morrey spaces under generic size conditions which are satisfied by most of the operators in harmonic analysis. The Marcinkiewicz operator which satisfies the conditions of these theorems can be considered as an example.     

A Linear Separability Criterion for Sets of Euclidean Space

We prove new theorems which describe a necessary and sufficient condition for linear (strong and non-strong) separability and inseparability of the sets in a finite-dimensional Euclidean space. We propose a universal measure for the thickness of the geometric margin (both the strong separation margin (separator) and the margin of unseparated points (pseudo-separator)) formed between the parallel generalized supporting hyperplanes of the two sets which are separated. The introduced measure allows comparing results of linear separation obtained by different techniques for both linearly separable and inseparable sets. An optimization program whose formulation provides a maximum thickness of the separator for the separable sets is considered. When the sets are inseparable, the same solver is guaranteed to construct a pseudo-separator with a minimum thickness. We estimate the distance between the convex and closed sets. We construct a cone of generalized support vectors for hyperplanes, each one of which linearly separates the considered sets. The interconnection of the problem of different types of linear separation of sets with some related problems is studied.