Showing papers in "Mathematical Problems in Engineering in 2002"
TL;DR: In this article, it is shown that stability of zero dynamics should be taken into account when the regular form consists of blocks of second-order equations, and new theoretical methods are developed in the context of these studies: sliding made nonlinear observers, observers with binary measurements, parameter estimation in systems with sliding mode control.
Abstract: The first sliding mode control application may be found in the papers back in the 1930s in Russia. With its versatile yet simple design procedure the methodology is proven to be one of the most powerful solutions for many practical control designs. For the sake of demonstration this paper is oriented towards application aspects of sliding mode control methodology. First the design approach based on the regularization is generalized for mechanical systems. It is shown that stability of zero dynamics should be taken into account when the regular form consists of blocks of second-order equations. Majority of applications in the paper are related to control and estimation methods of automotive industry. New theoretical methods are developed in the context of these studies: sliding made nonlinear observers, observers with binary
measurements, parameter estimation in systems with sliding mode control.
1,061 citations
TL;DR: In this paper, the problem of sliding mode manifold design for uncertain nonlinear system with discontinuous control is examined, and the original plant is decomposed such that the problem is divided into a number of simpler sub-problems.
Abstract: The paper examines the problem of sliding mode manifold design for uncertain nonlinear system with discontinuous control. The original plant first is decomposed such that the problem is divided into a number of simpler sub-problems. Then the block control recursive procedure is presented in which nonlinear sliding manifold is derived. Finally combined high gain and Lyapunov functions techniques are applied to establish hierarchy of the control gains and to estimate the upper bounds of the sliding mode equation solutions.
91 citations
TL;DR: In this paper, the eigenvalues and eigenfunctions of the Laplacian with Neumann boundary conditions on an equilateral triangle are derived using direct elementary mathematical techniques.
Abstract: Lame's formulas for the eigenvalues and eigenfunctions of the Laplacian with Neumann boundary conditions on an equilateral triangle are derived using direct elementary mathematical techniques. They are shown to form a complete orthonormal system. Various properties of the spectrum and nodal lines are explored. Implications for related geometries are considered.
60 citations
TL;DR: In this paper, the authors have established the properties of weighted distributions, including the preservation and stability of weighted and length-biased distributions, as well as relations between weighted distributions and stochastic orderings.
Abstract: Inequalities, relations and stochastic orderings, as well as useful ageing notions for weighted distributions are established. Also presented are preservation and stability results and comparisons for weighted and length-biased distributions. Relations for length-biased and equilibrium distributions as examples of weighted distributions are also presented.
48 citations
TL;DR: The Stochastic Bass Model (SBM) as mentioned in this paper is a stochastic version of the Bass Model, which allows for stochastically varying the number of adoptions of a new product.
Abstract: For a large variety of new products, the Bass Model (BM) describes the empirical cumulative-adoptions curve extremely well. The BM postulates that the trajectory of cumulative adoptions of a new product follows a deterministic function whose instantaneous growth rate depends on two parameters, one of which captures an individual's intrinsic tendency to purchase, independent of the number of previous adopters, and the other captures a positive force of influence on an individual by previous adopters. In this paper, we formulate a stochastic version of the BM, which we call the Stochastic Bass Model (SBM), where the trajectory of cumulative number of adoptions is governed by a pure birth process. We show that with an appropriately-chosen set of birth rates, the fractions of individuals who have adopted the product by time t in a family of SBMs indexed by the size of the target population converge in probability to the deterministic fraction in a corresponding BM, when the population size approaches infinity. The formulation therefore supports and expands the BM by allowing stochastic trajectories.
39 citations
TL;DR: In this paper, a variational formulation of the power station replacement problem has been proposed, which employs the theory of variations to the highest degree, and the equivalent minimizer is a second-order polynomial with piece-wise constant coefficients.
Abstract: In this paper, we revise the classical formulation of the problem depriving it of the concepts that are superfluous from the mathematical point of view. We observe that a number of power stations can be substituted by a single one that behaves equivalently to the entire set. Proceeding in this way, we obtain a variational formulation in its purest sense (without restrictions). This formulation allows us to employ the theory of calculus of variations to the highest degree. We then calculate the equivalent minimizer in the case where the cost functions are second-order polynomials. We prove that the equivalent minimizer is a second-order polynomial with piece-wise constant coefficients. Moreover, it belongs to the class C1. Finally, we present various examples prompted by real systems and perform the proposed algorithms using Mathematica.
28 citations
TL;DR: In this paper, the authors considered stochastic stability and stabilizability of linear discrete-time systems with Markovian jumps and mode-dependent time-delays, and provided sufficient conditions for the stochastically stability and stochnastic stability of this class of systems.
Abstract: This paper considers stochastic stability and stochastic stabilizability of linear discrete-time systems with Markovian jumps and mode-dependent time-delays. Linear matrix inequality (LMI) techniques are used to obtain sufficient conditions for the stochastic stability and stochastic stabilizability of this class of systems. A control design algorithm is also provided. A numerical example is given to demonstrate the effectiveness of the obtained theoretical results.
22 citations
TL;DR: In this paper, the stability and dissipativity of hybrid nonnegative and compartmental dynamical systems were investigated using linear Lyapunov functions with linear hybrid supply rates, and the results were used to develop general stability criteria for feedback interconnections.
Abstract: Nonnegative and Compartmental dynamical systems are governed by conservation laws and are comprised of homogeneous compartments which exchange variable nonnegative quantities of material via intercompartmental flow laws. These systems typically possess hierarchical (and possibly hybrid) structures and are remarkably effective in capturing the phenomenological features of many biological and physiological dynamical systems. In this paper we develop several results on stability and dissipativity of hybrid nonnegative and Compartmental dynamical systems. Specifically, using linear Lyapunov functions we develop sufficient conditions for Lyapunov and asymptotic stability for hybrid nonnegative dynamical systems. In addition, using linear and nonlinear storage functions with linear hybrid supply rates we develop new notions of dissipativity theory for hybrid nonnegative dynamical systems. Finally, these results are used to develop general stability criteria for feedback interconnections of hybrid nonnegative dynamical
systems.
20 citations
TL;DR: A novel way to implement a color image filter using the quantum-dot cellular automata framework due to its adequacy to implement highly parallel systems is introduced here.
Abstract: A novel way to implement a color image filter is introduced here. The quantum-dot cellular automata (QCA) framework is adopted due to its adequacy to implement highly parallel systems. The principle behind the new design is explained in detail, and simulation results are included to demonstrate the effectiveness of the proposed method.
19 citations
TL;DR: In this paper, an interior point algorithm is proposed to solve the multi-period hydrothermal economic dispatch (HTED) problem, which exploits the special bordered block diagonal structure and sparsity of the Newton system for the first order necessary conditions.
Abstract: This paper presents an interior point algorithm to solve the multiperiod hydrothermal economic dispatch (HTED). The multiperiod HTED is a large scale nonlinear programming problem. Various optimization methods have been applied to the multiperiod HTED, but most neglect important network characteristics or require decomposition into thermal and hydro subproblems. The algorithm described here exploits the special bordered block diagonal structure and sparsity of the Newton system for the first order necessary conditions to result in a fast efficient algorithm that can account for all network aspects. Applying this new algorithm challenges a conventional method for the use of available hydro resources known as the peak shaving heuristic.
19 citations
TL;DR: In this paper, the authors considered the problem of designing a decentralized detection filter for a large homogeneous collection of LTI systems, where the collection of systems considered here draws inspiration from platoons of vehicles and the considered interactions amongst systems in the collection are banded and lower triangular, mimicking the typical "look-ahead" nature of interactions in a platoon of vehicles.
Abstract: In this paper, the problem of designing a decentralized detection filter for a large homogeneous collection of LTI systems is considered. The collection of systems considered here draws inspiration from platoons of vehicles, and the considered interactions amongst systems in the collection are banded and lower triangular, mimicking the typical “look-ahead” nature of interactions in a platoon of vehicles. A fault in a system propagates to other systems in the collection via such interactions.The decentralized detection filter for the collection is composed of interacting detection filters, one for each system. The feasibility of communicating the state estimates to other systems in the collection is assumed here. An important concern is the propagation of state estimation errors. In order that the state estimation errors not amplify as they propagate, a ℋ ∞ constraint on the state estimation error propagation dynamics is imposed. A sufficient condition for constructing a decentralized detection filter for the collection is presented. An example is provided to illustrate the design procedure.
TL;DR: In this paper, finite difference approximations are developed for an inverse problem of determining an unknown parameter p(t) which is a coefficient of the solution u in a semi-linear parabolic partial differential equation subject to a boundary integral overspecification.
Abstract: Some finite difference approximations are developed for an inverse problem of determining an unknown parameter p(t) which is a coefficient of the solution u in a semi-linear parabolic partial differential equation subject to a boundary integral overspecification. The accuracy and efficiency of the procedures are discussed. Some computational results using the newly proposed numerical techniques are presented. CPU times needed for this problem are reported.
TL;DR: A new dynamic model for the Token Bucket (TB) algorithm used in computer networks is constructed and an algorithm using feedback control to improve QoS and network utilization is proposed.
Abstract: In this paper, we construct a new dynamic model for the Token Bucket (TB) algorithm used in computer networks and use systems approach for its analysis. This model is then augmented by adding a dynamic model for a multiplexor at an access node where the TB exercises a policing function. In the model, traffic policing, multiplexing and network utilization are formally defined. Based on the model, we study such issues as (quality of service) QoS, traffic sizing and network dimensioning. Also we propose an algorithm using feedback control to improve QoS and network utilization. Applying MPEG video traces as the input traffic to the model, we verify the usefulness and effectiveness of our model.
TL;DR: In this article, a new treatment of kinematical constraints and potential energies arising in the dynamics of systems of rigid bodies and particles is presented which is suited to Newtonian and Lagrangian formulations.
Abstract: A new treatment of kinematical constraints and potential energies arising in the dynamics of systems of rigid bodies and particles is presented which is suited to Newtonian and Lagrangian formulations. Its novel feature is the imposing of invariance requirements on the constraint functions and potential energy functions. These requirements are extensively used in continuum mechanics and, in the present context, one finds certain generalizations of Newton's third law of motion and an elucidation of the nature of constraint forces and moments. One motivation for such a treatment can be found by considering approaches where invariance
requirements are ignored. In contrast to the treatment presented in this paper, it is shown that this may lead to a difficulty in formulating the equations governing the motion of the system.
TL;DR: In this paper, the authors obtained dispersion curves for the first four symmetric modes for a magnesium plate immersed in water and showed that the first mode is highly damped and switches over to the second mode when the normalized frequency exceeds 12.
Abstract: Dispersion relations are obtained for the propagation of symmetric and antisymmetric modes in a free transversely isotropic plate. Dispersion curves are plotted for the first four symmetric modes for a magnesium plate immersed in water. The first mode is highly damped and switches over to the second mode when the normalized frequency exceeds 12.
TL;DR: In this paper, the issues involved in the reconstruction of a quadratic curve in 3D space from arbitrary perspective projections are described, and the relationship between the projections of the curve on the image planes is assumed to be established.
Abstract: The issues involved in the reconstruction of a quadratic curve in 3-D space from arbitrary perspective projections are described in this paper. Correspondence between the projections of the curve on the image planes is assumed to be established. Equations for reconstruction of the 3-D curve, which give the parameters of the 3-D quadratic curve are determined. Uniqueness of the solution in the process of reconstruction is addressed and solved using additional constraints. As practical examples, reconstruction of circles, parabolas and pair of straight lines in 3-D space are demonstrated.
TL;DR: In this article, existence results for boundary value problems on infinite intervals are given for the theory of colloids, which is related to the problem we consider in this paper. But they do not consider the problem in this article.
Abstract: We present existence results for some boundary value problems defined on infinite intervals. In particular our discussion includes a problem which arises in the theory of colloids.
TL;DR: In this paper, a n + 1 -dimensional cylinder Q global solvability of the mixed problem for the nonhomogeneous Carrier equation was studied in a 2D manifold and the existence, uniqueness and exponential decay of the energy for global generalized solutions were proved.
Abstract: We study in a n + 1 -dimensional cylinder Q global solvability of the mixed problem for the nonhomogeneous Carrier equation u t t − M ( x , t , || u ( t ) || 2 ) Δ u + g ( x , t , u t ) = f ( x , t )
without restrictions on a size of initial data and f ( x , t ) . For any natural n, we prove existence, uniqueness and the exponential decay of the energy for global generalized solutions. When n=2 , we prove C ∞ ( Q ) -regularity of solutions.
TL;DR: In this article, Schwartz' distributions theory in nonlinear setting is presented for nonlinear systems which admit both standard and impulsive inputs, and generalized solutions of these differential equations are defined via closure, in a certain topology, of the set of the conventional solutions corresponding to standard integrable inputs.
Abstract: The paper is intended to be of tutorial value for Schwartz' distributions theory in nonlinear setting. Mathematical models are presented for nonlinear systems which admit both standard and impulsive inputs. These models are governed by differential equations in distributions whose meaning is generalized to involve nonlinear, non single-valued operating over distributions. The set of generalized solutions of these differential equations is defined via closure, in a certain topology, of the set of the conventional solutions corresponding to standard integrable inputs. The theory is exemplified by mechanical systems with impulsive phenomena, optimal impulsive feedback synthesis, sampled-data filtering of stochastic and deterministic dynamic systems.
TL;DR: A uniform approach that has been developed during the last decade for asymptotic analysis of large capacity networks with product form of the stationary probability distribution with explicit form up to the normalization constant is described.
Abstract: Loss and closed queueing network models have long been of interest to telephone and computer engineers and becoming increasingly important as models of data transmission networks. This paper describes a uniform approach that has been developed during the last decade for asymptotic analysis of large capacity networks with product form of the stationary probability distribution. Such a distribution has an explicit form up to the normalization constant, or the partition function. The approach is based on representing the partition function as a contour integral in complex space and evaluating the integral using the saddle point method and theory of residues. This paper provides an introduction to the area and a review of recent work.
TL;DR: This paper extends the LQG design methodology to systems with saturating actuators and shows that resulting linear controllers do not require an anti-windup implementation.
Abstract: This paper extends the LQG design methodology to systems with saturating actuators and shows that resulting linear controllers do not require an anti-windup implementation.
TL;DR: An upper and lower solution theory for boundary value problems modelled from the Thomas-Fermi equation subject to a boundary condition corresponding to the neutral atom with Bohr radius is presented in this paper.
Abstract: This paper presents an upper and lower solution theory for boundary value problems modelled from the Thomas–Fermi equation subject to a boundary condition corresponding to the neutral atom with Bohr radius.
TL;DR: In this paper, the authors introduce Modified Altitude and Dive-Guidance laws for escaping a microburst with turbulence, and develop a procedure to estimate the highest altitude at which an aircraft can fly through the microburst without running into stall.
Abstract: This paper introduces Modified Altitude- and Dive-Guidance laws for escaping a microburst with turbulence. The goal is to develop a procedure to estimate the highest altitude at which an aircraft can fly through a microburst without running into stall. First, a new metric is constructed that quantifies the aircraft upward force capability in a microburst encounter. In the absence of turbulence, the metric is shown to be a decreasing function of altitude. This suggests that descending to a low altitude may improve safety in the sense that the aircraft will have more upward force capability to maintain its altitude. In the presence of stochastic turbulence, the metric is treated as a random variable and its probability distribution function is analytically approximated as a function of altitude. This approximation allows us to determine the highest safe altitude at which the aircraft may descend, hence avoiding to descend too low. This highest safe altitude is used as the commanded altitude in Modified Altitude- and Dive-Guidance. Monte Carlo simulations show that these Modified Altitude- and Dive-Guidance strategies can decrease the probability of minimum altitude being lower than a given value without significantly increasing the probability of crash.
TL;DR: In this paper, formalism for cluster analysis, based on the “Rank of Links”-theory, is suggested and an algorithm of clustering has been designed that shows nice workability and reliability.
Abstract: In this paper, formalism for cluster analysis, based on the “Rank of Links”-theory, is suggested. It tackles resemble measures, cross-distance matrices, “rank of links”-metric and some other cluster characteristics. Using these notions, an algorithm of clustering has been designed. Its application to estimation and prognosis of decision-making process shows nice workability and reliability.
TL;DR: A polynomial-time path-finding algorithm in AND-OR graphs given p arcs and n nodes, which is superior to the complexity of previously known algorithms.
Abstract: We present a polynomial-time path-finding algorithm in AND-OR graphs Given p arcs and n nodes, the complexity of the algorithm is O(np), which is superior to the complexity of previously known algorithms.
TL;DR: The attractors are estimated with the help of the comparison principle, using Vector Norms, which ensures the robustness with respect to uncertainties and perturbations of the open loop system.
Abstract: A fuzzy controller with singleton defuzzification can be considered as the association of a regionwise constant term and of a regionwise non linear term, the latter being bounded by a linear controller. Based on the regionwise structure of fuzzy controller, the state space is partitioned into a series of disjoint sets. The fuzzy controller parameters are tuned in order to ensure that the ith set is included into the domain of attraction of the preceding sets of the series. If the first set of the series is included into the region of attraction of the equilibrium point, the overall fuzzy controlled system is stable. The attractors are estimated with the help of the comparison principle, using Vector Norms, which ensures the robustness with respect to uncertainties and perturbations of the open loop system.
TL;DR: In this paper, a version of robust stochastic maximum principle (RSMP) is applied to the Minimax Mayer Problem (MMP) with the control-dependent diffusion term.
Abstract: This paper develops a version of Robust Stochastic Maximum Principle (RSMP) applied to the Minimax Mayer Problem formulated for stochastic differential equations with the control-dependent diffusion term. The parametric families of first and second order adjoint stochastic processes are introduced to construct the corresponding Hamiltonian formalism. The Hamiltonian function used for the construction of the robust optimal control is shown to be equal to the Lebesque integral over a parametric set of the standard stochastic Hamiltonians corresponding to a fixed value of the uncertain parameter. The paper deals with a cost function given at finite horizon and containing the mathematical expectation of a terminal term. A terminal condition, covered by a vector function, is also considered. The optimal control strategies, adapted for available information, for the wide class of uncertain systems given by an stochastic differential equation with unknown parameters from a given compact set, are constructed. This problem
belongs to the class of minimax stochastic optimization problems. The proof is based on the recent results obtained for Minimax Mayer Problem with a finite uncertainty set [14,43-45] as well as on the variation results of [53] derived for Stochastic Maximum Principle for nonlinear stochastic systems under complete information. The corresponding discussion of the obtain results concludes this study.
TL;DR: In this article, a numerical procedure for an inverse problem of identification of an unknown source in a heat equation is presented, where polygons linear pieces are determined consecutively from the solution of minimization problem based on the overspecified data.
Abstract: A numerical procedure for an inverse problem of identification of an unknown source in a heat equation is presented. Approach of proposed method is to approximate unknown function by polygons linear pieces which are determined consecutively from the solution of minimization problem based on the overspecified data. Numerical examples are presented.
TL;DR: In this paper, the problem of heat transfer in a PTC thermistor, which may form one element of an electric circuit, is solved numerically by a finite element method, based on Galerkin finite element using quadratic splines as shape functions.
Abstract: The problem of heat transfer in a Positive Temperature Coefficient (PTC) thermistor, which may form one element of an electric circuit, is solved numerically by a finite element method. The approach used is based on Galerkin finite element using quadratic splines as shape functions. The resulting system of ordinary differential equations is solved by the finite difference method. Comparison is made with numerical and analytical solutions and the accuracy of the computed solutions indicates that the method is well suited for the solution of the PTC thermistor problem.
TL;DR: In this article, necessary and sufficient observability conditions for linear time-varying systems with coefficients being time polynomials were derived from the Gabrielov-Khovansky theorem on multiplicity of a zero of a Noetherian function.
Abstract: We propose necessary and sufficient Observability conditions for linear time-varying systems with coefficients being time polynomials. These conditions are deduced from the Gabrielov–Khovansky theorem on multiplicity of a zero of a Noetherian function and the Wei–Norman formula for the representation of a solution of a linear time-varying system as a product of matrix exponentials. We define a Noetherian chain consisted of some finite number of usual exponentials corresponding to this system. Our results are formulated in terms of a Noetherian chain generated by these exponential functions and an upper bound of multiplicity of zero of one locally analytic function which is defined with help of the Wei–Norman formula. Relations with Observability conditions of bilinear systems are discussed. The case of two-dimensional systems is examined.