Nonlinear control of systems with multiple equilibria and unknown sinusoidal disturbance

In this paper, nonlinear systems having multiple equilibrium points and low order dynamics are investigated. Roll motions of ships are studied by means of modern nonlinear techniques to exemplify the behavior of such nonlinear systems in the case when they are under the influence of external sinusoidal disturbances with unknown amplitudes. The main objective is to analyze the performance of this system at different operating conditions, including those giving rise to chaos, and to design a controller with an overparameterized structure to stabilize the system at the origin. A nonlinear recursive backstepping controller is proposed and the transient performance is investigated. Lyapunov-based techniques are used to force systematic following of a reference model while introducing a nonlinear parameter estimator to guarantee adaptivity. Robustness problems as well as ways to tune the controller parameters are examined. Simulation results are submitted for the uncontrolled and controlled cases, verifying the effectiveness of the proposed controller. Finally, a discussion and conclusions are given with possible future extensions.     

Mixed convection boundary layer flow over a permeable vertical surface with prescribed wall heat flux

The effects of suction/injection on the laminar mixed convection boundary-layer flow on a vertical wall with a prescribed heat flux are considered. The conditions which allow the equations to be reduced to similarity form are derived and numerical solutions of the resulting equations are obtained for a range of values of the suction/injection and buoyancy parameters. Two specific cases, corresponding to a stagnation point flow and uniform wall heat flux, are treated in detail. Results are presented in terms of the skin friction and wall temperature with a selection of velocity and temperature profiles also being given. Dual solutions are found to exist for assisting flow, these are an addition to what has been reported previously for opposing flows. Solutions for some limiting values of the parameters are also derived.      

Equilibrium solutions for multiobjective bimatrix games incorporating fuzzy goals

Equilibrium solutions in terms of the degree of attainment of a fuzzy goal for games in fuzzy and multiobjective environments are examined. We introduce a fuzzy goal for a payoff in order to incorporate ambiguity of human judgments and assume that a player tries to maximize his degree of attainment of the fuzzy goal. A fuzzy goal for a payoff and the equilibrium solution with respect to the degree of attainment of a fuzzy goal are defined. Two basic methods, one by weighting coefficients and the other by a minimum component, are employed to aggregate multiple fuzzy goals. When the membership functions are linear, computational methods for the equilibrium solutions are developed. It is shown that the equilibrium solutions are equal to the optimal solutions of mathematical programming problems in both cases. The relations between the equilibrium solutions for multiobjective bimatrix games incorporating fuzzy goals and the Pareto-optimal equilibrium solutions are considered.     

Wavelet-Based Estimation for Univariate Stable Laws

Stable distributions are characterized by four parameters which can be estimated via a number of methods, and although approximate maximum likelihood estimation techniques have been proposed, they are computationally intensive and difficult to implement. This article describes a fast, wavelet-based, regression-type method for estimating the parameters of a stable distribution. Fourier domain representations, combined with a wavelet multiresolution approach, are shown to be effective and highly efficient tools for inference in stable law families. Our procedures are illustrated and compared with other estimation methods using simulated data, and an application to a real data example is explored. One novel aspect of this work is that here wavelets are being used to solve a parametric problem, rather than a nonparametric one, which is the more typical context in wavelet applications.     

A large deviation analysis of errors in measurement based admission control to buffered and bufferless resources

In measurement based admission control, measured traffic parameters are used to determine the maximum number of connections that can be admitted to a resource within a given quality constraint. The assumption that the measured parameters are the true ones can compromise admission control; measured parameters are random quantities, causing additional variability. This paper analyzes the impact of measurement error within the framework of Large Deviation theory. For a class of admission controls, large deviation principles are established for the number of admitted connections, and for the attained overflow rates. These are applied to admission to bufferless resources, and buffered resources in both the many sources and large buffer asymptotic. The sampling properties of effective bandwidths are presented, together with a discussion the impact of the temporal extent of individual samples on estimator variability. Sample correlations are shown to increase estimator variance; procedures to make admission control robust with respect to these are described.     

Semiconjugates of One-dimensional Maps

R -semiconjugate maps are defined as a natural means of relating a map F of R m to a mapping { of the interval via a link map H . Invariants are seen to be special types of semiconjugate links where { is the identity. Basic relationships between the dynamical behaviors of { and F are established, and conditions under which a link map H is a Liapunov function are obtained. Examples and applications involving concepts from stability to chaos are discussed.     

Capacitated location allocation problem with stochastic location and fuzzy demand: A hybrid algorithm

In this article, a capacitated location allocation problem is considered in which the demands and the locations of the customers are uncertain. The demands are assumed fuzzy, the locations follow a normal probability distribution, and the distances between the locations and the customers are taken Euclidean and squared Euclidean. The fuzzy expected cost programming, the fuzzy β-cost minimization model, and the credibility maximization model are three types of fuzzy programming that are developed to model the problem. Moreover, two closed-form Euclidean and squared Euclidean expressions are used to evaluate the expected distance between customers and facilities. In order to solve the problem at hand, a hybrid intelligent algorithm is applied in which the simplex algorithm, fuzzy simulation, and a modified genetic algorithm are integrated. Finally, in order to illustrate the efficiency of the proposed hybrid algorithm, some numerical examples are presented.     

Transverse curves and chain curvature in the Heisenberg group

A chain is the intersection of a complex totally geodesic subspace in complex hyperbolic 2-space with the boundary. The boundary admits a canonical contact structure, and chains are distiguished curves transverse to this structure. The space of chains is analyzed both as a quotient of the contact bundle, and as a subset of ℂP². The space of chains admits a canonical, indefinite Hermitian metric, and curves in the space of chains with null tangent vectors are shown to correspond to a path of chains tangent to a curve in the boundary transverse to the contact structure. A family of local differential chain curvature operators are introduced which exactly characterize when a transverse curve is a chain. In particular, operators that are invariant under the stabilizer of a point in the interior of complex hyperbolic space, or a point on the boundary, are developed in detail. Finally, these chain curvature operators are used to prove a generalization of Louiville's theorem: a sufficiently smooth mapping from the boundary of complex hyperbolic 2-space to itself which preserves chains must be the restriction of a global automorphism.     

Semi-B-Preinvex Functions

In this note, a class of functions, called semi-B-preinvex function, which are a generalization of the semipreinvex functions and the B-vex functions, is introduced. Examples are given to show that there exist functions which are semi-B-preinvex functions, but are neither semipreinvex nor B-vex. A property of the semi-B-preinvex functions is obtained.Communicated by F. GiannessiThis research was partially supported by a Science Committee Project, Research Foundation of Chongqing, Grant 8409. The authors are thankful to the referees and Prof. F. Giannessi for suggestions and comments which helped to give the present form to this paper.     

Fitting mixed-effects models when data are left truncated

Damage sizes, i.e. all damages occurring to a policy and not only those that are reported to an insurance company, are modelled as a linear mixed model. Only those damages that are larger than their deductibles are reported to the company, and this fact should be taken into account when analyzing such data. In statistical terms, the problem is to make inference in a linear mixed model with left truncated data. Estimation methods based on a Monte Carlo simulation of the likelihood are proposed, and extensive simulations to evaluate the quality of the methods are reported. The proposed methods are then used to analyze claimsizes for some marine insurance data, where shipowners represent random effects and technical data about the ships represent fixed effects.     

Mixed convection boundary layer flow over a permeable vertical surface with prescribed wall heat flux

The effects of suction/injection on the laminar mixed convection boundary-layer flow on a vertical wall with a prescribed heat flux are considered. The conditions which allow the equations to be reduced to similarity form are derived and numerical solutions of the resulting equations are obtained for a range of values of the suction/injection and buoyancy parameters. Two specific cases, corresponding to a stagnation point flow and uniform wall heat flux, are treated in detail. Results are presented in terms of the skin friction and wall temperature with a selection of velocity and temperature profiles also being given. Dual solutions are found to exist for assisting flow, these are an addition to what has been reported previously for opposing flows. Solutions for some limiting values of the parameters are also derived.     

A Lagrangean based branch-and-cut algorithm for global optimization of nonconvex mixed-integer nonlinear programs with decomposable structures

In this work we present a global optimization algorithm for solving a class of large-scale nonconvex optimization models that have a decomposable structure. Such models, which are very expensive to solve to global optimality, are frequently encountered in two-stage stochastic programming problems, engineering design, and also in planning and scheduling. A generic formulation and reformulation of the decomposable models is given. We propose a specialized deterministic branch-and-cut algorithm to solve these models to global optimality, wherein bounds on the global optimum are obtained by solving convex relaxations of these models with certain cuts added to them in order to tighten the relaxations. These cuts are based on the solutions of the sub-problems obtained by applying Lagrangean decomposition to the original nonconvex model. Numerical examples are presented to illustrate the effectiveness of the proposed method compared to available commercial global optimization solvers that are based on branch and bound methods.     

Tests for dimensionality and interactions of mean vectors under general and reducible covariance structures

Likelihood ratio tests are derived for testing the structure of mean values in a two-way classification. The most general hypothesis considered is when the mean values are subject to row and column effects and interaction has a given complexity. The observations corresponding to a row or a column classification are assumed to have an unknown dispersion (variance covariance) matrix. Two types of dispersion matrices are considered, one with a general and another with a reducible structure. Some special cases are considered. The results of the paper provide generalizations of tests on dimensionality and interactions in a two-way array of mean values considered by Fisher, Anderson, Fujikoshi, Mandel, and Rao.     

Root Systems for Asymmetric Geometric Representations of Coxeter Groups

Results are obtained concerning root systems for asymmetric geometric representations of Coxeter groups. These representations were independently introduced by Vinberg and Eriksson, and generalize the standard geometric representation of a Coxeter group in such a way as to include certain restrictions of all Kac–Moody Weyl groups. In particular, a characterization of when a nontrivial multiple of a root may also be a root is given in the general context. Characterizations of when the number of such multiples of a root is finite and when the number of positive roots sent to negative roots by a group element is finite are also given. These characterizations are stated in terms of combinatorial conditions on a graph closely related to the Coxeter graph for the group. Other finiteness results for the symmetric case which are connected to the Tits cone and to a natural partial order on positive roots are extended to this asymmetric setting.     

Finite element analysis of laminar and turbulent flows using LES and subgrid-scale models

Numerical simulations of laminar and turbulent flows in a lid driven cavity and over a backward-facing step are presented in this work. The main objectives of this research are to know more about the structure of turbulent flows, to identify their three-dimensional characteristic and to study physical effects due to heat transfer. The filtered Navier–Stokes equations are used to simulate large scales, however they are supplemented by subgrid-scale (SGS) models to simulate the energy transfer from large scales toward subgrid-scales, where this energy will be dissipated by molecular viscosity. Two SGS models are applied: the classical Smagorinsky’s model and the Dynamic model for large eddy simulation (LES). Both models are implemented in a three-dimensional finite element code using linear tetrahedral elements. Qualitative and quantitative aspects of two and three-dimensional flows in a lid-driven cavity and over a backward-facing step, using LES, are analyzed comparing numerical and experimental results obtained by other authors.     

Allocation of dependent divisional resources

A resource allocation problem is considered with resources that are dependent in the sense that an allocation to an activity requires the application of several resources, except for certain activities which are divisional in the sense that an allocation to such an activity requires the use of only a single resource. Return and cost functions are assumed to be continuous and increasing, and the allocation variables are continuous. Conditions are given for the replacement of the continuous problem by an associated problem with discrete variables and a single constraint, and to a given degree of accuracy. The associated problem can be efficiently solved by dynamic programming. Certain divisional resource allocation problems with discrete variables and several linear constraints are shown to be equivalent to a discrete problem with a single constraint. A numerical example is given.     

Identification of Non-Varying Coefficients in Varying- Coefficient Models

A partially varying-coefficient model is one of the useful modelling tools. In this model, some coefficients of a linear model are kept to be constant whilst the others are allowed to vary with another factor. However, rarely can the analysts know a priori which coefficients can be assumed to be constant and which ones are varying with the given factor. Therefore, the identification problem of the constant coefficients should be solved before the partially varying-coefficient model is used to analyze a real-world data set. In this article, a simple test method is proposed to achieve this task, in which the test statistic is constructed as the sample variance of the estimates of each coefficient function in a well-known varying-coefficient model. Moreover two procedures, called F-approximation and three-moment X~2 approximation, are employed to derive the p-value of the test. Furthermore, some simulations are conducted to examine the performance of the test and the results are satisfactory.     

A search method for optimal control of a flow shop system of traditional machines

We consider a convex and nondifferentiable optimization problem for deterministic flow shop systems in which the arrival times of the jobs are known and jobs are processed in the order they arrive. The decision variables are the service times that are to be set only once before processing the first job, and cannot be altered between processes. The cost objective is the sum of regular costs on job completion times and service costs inversely proportional to the controllable service times. A finite set of subproblems, which can be solved by trust-region methods, are defined and their solutions are related to the optimal solution of the optimization problem under consideration. Exploiting these relationships, we introduce a two-phase search method which converges in a finite number of iterations. A numerical study is held to demonstrate the solution performance of the search method compared to a subgradient method proposed in earlier work.     

Computing Aviation Sparing Policies: Solving a Large Nonlinear Integer Program

Deployed US Navy aircraft carriers must stock a large number of spare parts to support the various types of aircraft embarked on the ship. The sparing policy determines the spares that will be stocked on the ship to keep the embarked aircraft ready to fly. Given a fleet of ten or more aircraft carriers and a cost of approximately 50 million dollars per carrier plus the cost of spares maintained in warehouses in the United States, the sparing problem constitutes a significant portion of the Navy’s resources. The objective of this work is to find a minimum-cost sparing policy that meets the readiness requirements of the embarked aircraft. This is a very large, nonlinear, integer optimization problem. The cost function is piecewise linear and convex while the constraint mapping is highly nonlinear. The distinguishing characteristics of this problem from an optimization viewpoint are that a large number of decision variables are required to be integer and that the nonlinear constraint functions are essentially “black box” functions; that is, they are very difficult (and expensive) to evaluate and their derivatives are not available. Moreover, they are not convex. Integer programming problems with a large number of variables are difficult to solve in general and most successful approaches to solving nonlinear integer problems have involved linear approximation and relaxation techniques that, because of the complexity of the constraint functions, are inappropriate for attacking this problem. We instead employ a pattern search method to each iteration of an interior point-type algorithm to solve the relaxed version of the problem. From the solution found by the pattern search on each interior point iteration, we begin another pattern search on the integer lattice to find a good integer solution. The best integer solution found across all interations is returned as the optimal solution. The pattern searches are distributed across a local area network of non-dedicated, heterogeneous computers in an office environment, thus, drastically reducing the time required to find the solution.     

Weighted least squares solutions to general coupled Sylvester matrix equations

This paper is concerned with weighted least squares solutions to general coupled Sylvester matrix equations. Gradient based iterative algorithms are proposed to solve this problem. This type of iterative algorithm includes a wide class of iterative algorithms, and two special cases of them are studied in detail in this paper. Necessary and sufficient conditions guaranteeing the convergence of the proposed algorithms are presented. Sufficient conditions that are easy to compute are also given. The optimal step sizes such that the convergence rates of the algorithms, which are properly defined in this paper, are maximized and established. Several special cases of the weighted least squares problem, such as a least squares solution to the coupled Sylvester matrix equations problem, solutions to the general coupled Sylvester matrix equations problem, and a weighted least squares solution to the linear matrix equation problem are simultaneously solved. Several numerical examples are given to illustrate the effectiveness of the proposed algorithms.