Showing papers in "Mathematical Problems in Engineering in 2004"

The beta Gumbel distribution

Abstract: The Gumbel distribution is perhaps the most widely applied statistical distribution for problems in engineering. In this paper, we introduce a generalization—referred to as the beta Gumbel distribution—generated from the logit of a beta random variable. We provide a comprehensive treatment of the mathematical properties of this new distribution. We derive the analytical shapes of the corresponding probability density function and the hazard rate function and provide graphical illustrations. We calculate expressions for the nth moment and the asymptotic distribution of the extreme order statistics. We investigate the variation of the skewness and kurtosis measures. We also discuss estimation by the method of maximum likelihood. We hope that this generalization will attract wider applicability in engineering.

Sliding mode control of a magnetic levitation system

Abstract: Sliding mode control schemes of the static and dynamic types are proposed for the control of a magnetic levitation system. The proposed controllers guarantee the asymptotic regulation of the statesof the system to their desired values. Simulation results of the proposed controllers are given to illustrate the effectiveness of them. Robustness of the control schemes to changes in the parameters of the system is also investigated.

Peristaltic transport of an Oldroyd-B fluid in a planar channel

Abstract: The effects of an Oldroyd-B fluid on the peristaltic mechanism are examined under the long wavelength assumption. Analytical expressions for the stream function, the axial velocity, and the pressure rise per wavelength are obtained up to the second order in the dimensionless wave number. The effects of the various parameters of interest on the flow are shown and discussed.

A dynamic programming algorithm for the buffer allocation problem in homogeneous asymptotically reliable serial production lines

Abstract: In this study, the buffer allocation problem (BAP) in homogeneous, asymptotically reliable serial production lines is considered. A known aggregation method, given by Lim, Meerkov, and Top (1990), for the performance evaluation (i.e., estimation of throughput) of this type of production lines when the buffer allocation is known, is used as an evaluative method in conjunction with a newly developed dynamic programming (DP) algorithm for the BAP. The proposed algorithm is applied to production lines where the number of machines is varying from four up to a hundred machines. The proposed algorithm is fast because it reduces the volume of computations by rejecting allocations that do not lead to maximization of the line's throughput. Numerical results are also given for large production lines.

Stability analysis of periodically switched linear systems using Floquet theory

Abstract: Stability of a switched system that consists of a set of linear time invariant subsystems and a periodic switching rule is investigated. Based on the Floquet theory, necessary and sufficient conditions are given for exponential stability. It is shown that there exists a slow switching rule that achieves exponential stability if at least one of these subsystems is asymptotically stable. It is also shown that there exists a fast switching rule that achieves exponential stability if the average of these subsystems is asymptotically stable. The results are illustrated by examples.

Volterra equations with fractional stochastic integrals

Abstract: Some fractional stochastic systems of integral equations are studied. The fractional stochastic Skorohod integrals are also studied. The existence and uniquness of the considered stochastic fractional systems are established. An application of the fractional Black-Scholes is considered.

Sliding mode observers and observability singularity in chaotic synchronization

Abstract: We present a new secured data transmission based on a chaotic synchronization and observability singularity. For this, we adopt an approach based on an inclusion of the message in the system structure and we use a sliding mode observer for system with unknown input in order to recover the information. We end the paper with an example of chaotic system with an observability bifurcation. Moreover, this example highlights some benefits of the so-called step-by-step sliding mode observer.

Reliability for Laplace distributions

Abstract: In the area of stress-strength models there has been a large amount of work as regards estimation of the reliability R=Pr(X2<X1) when X1 and X2 are independent random variables belonging to the same univariate family of distributions. The algebraic form for R=Pr(X2<X1) has been worked out for the majority of the well-known distributions in the standard forms. However, there are still many other distributions (including generalizations of the well-known distributions) for which the form of R has not been derived. In this paper, we consider several Laplace distributions and derive the corresponding forms for the reliability R. The calculations involve the use of special functions.

The flow of a non-Newtonian fluid induced due to the oscillations of a porous plate

Abstract: An analytic solution of the flow of a third-grade fluid on a porous plate is constructed. The porous plate is executing oscillations in its own plane with superimposed injection or suction. An increasing or decreasing velocity amplitude of the oscillating porous plate is also examined. It is also shown that in case of third-grade fluid, a combination of suction/injection and decreasing/increasing velocity amplitude is possible as well. Several limiting situations with their implications are given and discussed.

New filtering technique for the impulsive noise reduction in color images

Abstract: We present a novel approach to the problem of impulsive noise reduction for colorimages. The new image-filtering technique is based on the maximization of the similarities between pixels in the filtering window. Themethod is able to remove the noise component, while adapting itself to the local image structure. In this way, the proposed algorithm eliminates impulsive noise while preserving edges and fine image details. Since the algorithm can be considered as a modification of the vector median filter driven by fuzzy membership functions, it is fast, computationally efficient, and easy to implement. Experimental results indicate that the new method is superior, in terms of performance, to algorithms commonly used for impulsive noise reduction.

Maximizing the determinant for a special class of block-partitioned matrices

Abstract: An analytical solution is found for the maximum determinant of a block-partitioned class of matrices with constant trace for each block. As an immediate application of this result, the maximum determinant of a sum of Kronecker products is derived.

Optimal guaranteed cost filtering for Markovian jump discrete-time systems

Abstract: This paper develops a result on the design of robust steady-state estimator for a class of uncertain discrete-time systems with Markovian jump parameters. This result extends the steady-state Kalman filter to the case of norm-bounded time-varying uncertainties in the state and measurement equations as well as jumping parameters. We derive a linear state estimator such that the estimation-error covariance is guaranteed to lie within a certain bound for all admissible uncertainties. The solution is given in terms of a family of linear matrix inequalities (LMIs). A numerical example is included to illustrate the theory.

Higher-order Melnikov functions for single-dof mechanical oscillators: Theoretical treatment and applications

Abstract: A Melnikov analysis of single-degree-of-freedom (DOF) oscillators is performed by taking into account the first (classical) and higher-order Melnikov functions, by considering Poincare sections nonorthogonal to the flux, and by explicitly determining both the distance between perturbed and unperturbed manifolds (“one-half” Melnikov functions) and the distance between perturbed stable and unstable manifolds (“full” Melnikov function). The analysis is developed in an abstract framework, and a recursive formula for computing the Melnikov functions is obtained. These results are then applied to various mechanical systems. Softening versus hardening stiffness and homoclinic versus heteroclinic bifurcations are considered, and the influence of higher-order terms is investigated in depth. It is shown that the classical (first-order) Melnikov analysis is practically inaccurate at least for small and large excitation frequencies, in correspondence to degenerate homo/heteroclinic bifurcations, and in the case of generic periodic excitations.

Vector dissipativity theory for large-scale impulsive dynamical systems

Abstract: Modern complex large-scale impulsive systems involve multiple modes of operation placing stringent demands on controller analysis of increasing complexity. In analyzing these large-scale systems, it is often desirable to treat the overall impulsive system as a collection of interconnected impulsive subsystems. Solution properties of the large-scale impulsive system are then deduced from the solution properties of the individual impulsive subsystems and the nature of the impulsive system interconnections. In this paper, we develop vector dissipativity theory for large-scale impulsive dynamical systems. Specifically, using vector storage functions and vector hybrid supply rates, dissipativity properties of the composite large-scale impulsive systems are shown to be determined from the dissipativity properties of the impulsive subsystems and their interconnections. Furthermore, extended Kalman-Yakubovich-Popov conditions, in terms of the impulsive subsystem dynamics and interconnection constraints, characterizing vector dissipativeness via vector system storage functions, are derived. Finally, these results are used to develop feedback interconnection stability results for large-scale impulsive dynamical systems using vector Lyapunov functions.

Access control for MPEG video applications using neural network and simulated annealing

Abstract: We present a dynamic modelfor access control mechanism used in computer communication network applied to MPEG video transmission over Internet. This modelis different fromthosedeveloped inthe previous works related to this topic. In our model, token buckets supported by data buffersare used to shape incoming traffic and one multiplexor, serving all the token pools, multiplexes all theconforming traffic. The model is governed by a system of discrete nonlinear difference equations. Weuse neural network as the feedback controller which receives at its input (measurable) available information and provides at its output the optimal control. The simulated annealing algorithm isusedto optimize the system performance by adjusting the weights. For illustration, we presentnumerical results which show that the system performance of MPEG video server can be improved by using neural network and simulated annealing approach.

An analytical solution of the generalized equation of energy transport in one-dimensional semi-infinite domains.

Abstract: This paper presents an integral solution of the generalized one-dimensional equation of energy transport with the convective term.The solution of the problem has been achieved by the use of a novel technique that involves generalized derivatives (in particular, derivatives of noninteger orders). Confluent hypergeometric functions, known as Whittaker's functions, appear in the course of the solution procedure upon applying the Laplace transform to the original transport equation.The analytical solution of the problem is written in the integral form and provides a relationship between the local values of the transported property (e.g., temperature, mass, momentum, etc.) and its flux.The solution is valid everywhere within the domain, including the domain boundary.

Optimum feedback strategy for access control mechanism modelled as stochastic differential equation in computer network

Abstract: We consider optimum feedback control strategy for computer communication network, in particular, the access control mechanism. The dynamic model representing the source and the access control system is described by a system of stochastic differential equations developed in our previous works. Simulated annealing (SA) was used to optimize the parameters of the control law based on neural network. This technique was found to be computationally intensive. In this paper, we have proposed to use a more powerful algorithm known as recursive random search (RRS). By using this technique, we have been able to reduce the computation time by a factor of five without compromising the optimality. This is very important for optimization of high-dimensional systems serving a large number of aggregate users. The results show that the proposed control law can improve the network performance by improving throughput, reducing multiplexor and TB losses, and relaxing, not avoiding, congestion.

Direct current electric potential in an anisotropic half-space with vertical contact containing a conductive 3D body

Abstract: Detailed studies of anomalous conductors in otherwise homogeneous media have been modelled. Vertical contacts form common geometries in galvanic studies when describing geological formations with different electrical conductivities on either side. However, previous studies of vertical discontinuities have been mainly concerned with isotropic environments. In this paper, we deal with the effect on the electric potentials, such as mise-a-la-masse anomalies, due to a conductor near a vertical contact between two anisotropic regions. We also demonstrate the interactive effects when the conductive body is placed across the vertical contact. This problem is normally very difficult to solve by the traditional numerical methods. The integral equations for the electric potential in anisotropic half-spaces are established. Green's function is obtained using the reflection and transmission image method in which five images are needed to fit the boundary conditions on the vertical interface and the air-earth surface. The effects of the anisotropy of the environments and the conductive body on the electric potential are illustrated with the aid of several numerical examples.

A hybrid neural network model for the dynamics of the Kuramoto-Sivashinsky equation

Abstract: A hybrid approach consisting of two neural networks is used to model the oscillatory dynamical behavior of the Kuramoto-Sivashinsky (KS) equation at a bifurcation parameter α=84.25. This oscillatory behavior results from a fixed point that occurs at α=72 having a shape of two-humped curve that becomes unstable and undergoes a Hopf bifurcation at α=83.75. First, Karhunen-Loeve (KL) decomposition was used to extract five coherent structures of the oscillatory behavior capturing almost 100% of the energy. Based on the five coherent structures, a system offive ordinary differential equations (ODEs) whose dynamics is similar to the original dynamics of the KS equation was derived via KL Galerkin projection. Then, an autoassociative neural network was utilized on the amplitudes of the ODEs system with the task of reducing the dimension of the dynamical behavior to its intrinsic dimension, and a feedforward neural network was usedto model the dynamics at a future time. We show that by combining KL decomposition and neural networks, a reduced dynamical model of the KS equation is obtained.

Localization of periodic orbits of autonomous systems based on high-order extremum conditions

Abstract: This paper gives localization and nonexistence conditions of periodic orbits in some subsets of the state space. Mainly, our approach is based on high-order extremum conditions, on high-order tangency conditions of a nonsingular solution of a polynomial system with an algebraic surface, and on some ideas related to algebraically-dependent polynomials. Examples of the localization analysis of periodic orbits are presented including the Blasius equations, the generalized mass action (GMA) system, and the mathematical model of the chemical reaction with autocatalytic step.

ℋ∞ constant gain state feedback stabilization of stochastic hybrid systems with Wiener process

Abstract: This paper considers the stabilization problem of the class of continuous-time linear stochastic hybrid systems with Wiener process. The ℋ∞ state feedback stabilization problem is treated. A state feedback controller with constant gain that does not require access to the system mode is designed. LMI-based conditions are developed to design the state feedback controller with constant gain that stochastically stabilizes the studied class of systems and, at the same time, achieve the disturbance rejection of a desired level. The minimum disturbance rejection is also determined. Numerical examples are given to show the usefulness of the proposed results.

Adaptive stabilization of continuous-time systems through a controllable modified estimation model

Abstract: This paper presents an indirect adaptive control scheme of continuous-time systems. The estimated plant model is controllable and then the adaptive scheme is free from singularities. Such singularities are avoided through a modification of the estimated plant parameter vector so that its associated Sylvester matrix is guaranteed to be nonsingular. That property is achieved by ensuring that the absolute value of its determinant does not lie below a positive threshold. An alternative modification scheme based on the achievement of a modifieddiagonally dominant Sylvester matrix of the parameter estimates is also proposed. This diagonal dominance is achieved through estimates modification as a way to guarantee the controllability of the modified estimated model when a controllability measure of the estimation model without modification fails. In both schemes, the use of an explicit hysteresis switching function for the modification of the estimates is not required to ensure the controllability of the modified estimated model. Both schemes ensure that chattering due to switches associated with the modification is not present.

Robust control of discrete-time hybrid systems with uncertain modal dynamics

Abstract: We study systems that are subject to sudden structural changes due to either changes in the operational mode of the system or failure. We consider linear dynamicalsystems that depend on a modal variable which is either modeled as a finite-state Markov chain or generated by an automaton that is subject to an external disturbance. In the Markov chain case, the objective of the control is to minimize a risk-sensitive cost functional. The risk-sensitive cost functional measures the risk sensitivity of the system to transitions caused by the random modal variable. In the case when a disturbed automaton describes the modal variable, the objective of the control is to make the system as robust to changes in the external disturbance as possible. Optimality conditions for both problems are derived and it is shown that the disturbance rejection problem is closely related to a certain risk-sensitive control problem for the hybrid system.

Asymptotic-group analysis of algebraic equations

Abstract: Both the method of asymptotic analysis and the theory of extension group are applied to study the Descates equation. The proposed algorithm allows to obtain various variants of simplification and can be easily generalized to their algebraic and differential equations.

On the economical solution method for a system of linear algebraic equations

Abstract: The present work proposes a novel optimal and exact method of solving large systems of linear algebraic equations. In the approach under consideration, the solution of a system of algebraic linear equations is found as a point of intersection of hyperplanes, which needs a minimal amount of computer operating storage. Two examples are given. In the first example, the boundary value problem for a three-dimensional stationary heat transfer equation in a parallelepiped in ℝ3 is considered, where boundary value problems of first, second, or third order, or their combinations, are taken into account. The governing differential equations are reduced to algebraic ones with the help of the finite element and boundary element methods for different meshes applied. The obtained results are compared with known analytical solutions. The second example concerns computation of a nonhomogeneous shallow physically and geometrically nonlinear shell subject to transversal uniformly distributed load. The partial differential equations are reduced to a system of nonlinear algebraic equations with the error of O(hx12

Fluid queues driven by a birth and death process with alternating flow rates

Abstract: Fluid queue driven by a birth and death process (BDP) with only one negative effective input rate has been considered in the literature. As an alternative, here we consider a fluid queue in which the input is characterized by a BDP with alternating positive and negative flow rates on a finite state space. Also, the BDP has two alternating arrival rates and two alternating service rates. Explicit expression for the distribution function of the buffer occupancy is obtained. The case where the state space is infinite is also discussed. Graphs are presented to visualize the buffer content distribution.