Nonsingular decoupled terminal sliding-mode control for a class of fourth-order nonlinear systems

This paper presents a nonsingular decoupled terminal sliding mode control (NDTSMC) method for a class of fourth-order nonlinear systems. First, the nonlinear fourth-order system is decoupled into two second-order subsystems which are referred to as the primary and secondary subsystems. The sliding surface of each subsystem was designed by utilizing time-varying coefficients which are computed by linear functions derived from the input–output mapping of the one-dimensional fuzzy rule base. Then, the control target of the secondary subsystem was embedded to the primary subsystem by the help of an intermediate signal. Thereafter, a nonsingular terminal sliding mode control (NTSMC) method was utilized to make both subsystems converge to their equilibrium points in finite time. The simulation results on the inverted pendulum system are given to show the effectiveness of the proposed method. It is seen that the proposed method exhibits a considerable improvement in terms of a faster dynamic response and lower IAE and ITAE values as compared with the existing decoupled control methods.     

Normal Linear Regression Models With Recursive Graphical Markov Structure

A multivariate normal statistical model defined by the Markov properties determined by an acyclic digraph admits a recursive factorization of its likelihood function (LF) into the product of conditional LFs, each factor having the form of a classical multivariate linear regression model (≡WMANOVA model). Here these models are extended in a natural way to normal linear regression models whose LFs continue to admit such recursive factorizations, from which maximum likelihood estimators and likelihood ratio (LR) test statistics can be derived by classical linear methods. The central distribution of the LR test statistic for testing one such multivariate normal linear regression model against another is derived, and the relation of these regression models to block-recursive normal linear systems is established. It is shown how a collection of nonnested dependent normal linear regression models (≡Wseemingly unrelated regressions) can be combined into a single multivariate normal linear regression model by imposing a parsimonious set of graphical Markov (≡Wconditional independence) restrictions.     

Regular splittings and monotone iteration functions

Summary In this paper we introduce the set of so-called monotone iteration functions (MI-functions) belonging to a given function. We prove necessary and sufficient conditions in order that a given MI-function is (in a precisely defined sense) at least as fast as a second one.Regular splittings of a function which were initially introduced for linear functions by R.S. Varga in 1960 are generating MI-functions in a natural manner.For linear functions every MI-function is generated by a regular splitting. For nonlinear functions, however, this is generally not the case.     

Capturing the multivariate extremal index: bounds and interconnections

The multivariate extremal index function is a direction specific extension of the well-known univariate extremal index. Since statistical inference on this function is difficult it is desirable to have a broad characterization of its attributes. We extend the set of common properties of the multivariate extremal index function and derive sharp bounds for the entire function given only marginal dependence. Our results correspond to certain restrictions on the two dependence functions defining the multivariate extremal index, which are opposed to Smith and Weissman’s (1996) conjecture on arbitrary dependence functions. We show further how another popular dependence measure, the extremal coefficient, is closely related to the multivariate extremal index. Thus, given the value of the former it turns out that the above bounds may be improved substantially. Conversely, we specify improved bounds for the extremal coefficient itself that capitalize on marginal dependence, thereby approximating two views of dependence that have frequently been treated separately. Our results are completed with example processes.      

A piecewise ellipsoidal reachable set estimation method for continuous bimodal piecewise affine systems

In this work, the issue of estimation of reachable sets in continuous bimodal piecewise affine systems is studied. A new method is proposed, in the framework of ellipsoidal bounding, using piecewise quadratic Lyapunov functions. Although bimodal piecewise affine systems can be seen as a special class of affine hybrid systems, reachability methods developed for affine hybrid systems might be inappropriately complex for bimodal dynamics. This work goes in the direction of exploiting the dynamical structure of the system to propose a simpler approach. More specifically, because of the piecewise nature of the Lyapunov function, we first derive conditions to ensure that a given quadratic function is positive on half spaces. Then, we exploit the property of bimodal piecewise quadratic functions being continuous on a given hyperplane. Finally, linear matrix characterizations of the estimate of the reachable set are derived.     

Covering symmetric semi-monotone functions

We define a new set of functions called semi-monotone, a subclass of skew-supermodular functions. We show that the problem of augmenting a given graph to cover a symmetric semi-monotone function is NP-complete if all the values of the function are in {0,1} and we provide a minimax theorem if all the values of the function are different from 1. Our problem is equivalent to the node to area augmentation problem. Our contribution is to provide a significantly simpler and shorter proof.     

Multivariate count data generalized linear models: Three approaches based on the Sarmanov distribution

Starting from the question: What is the accident risk of an insured individual?, we consider that the customer has contracted policies in different insurance lines: motor and home. Three models based on the multivariate Sarmanov distribution are analyzed. Driven by a real data set that takes into account three types of accident risks, two for motor and one for home, three trivariate Sarmanov distributions with generalized linear models (GLMs) for marginals are considered and fitted to the data. To estimate the parameters of these three models, we discuss a method for approaching the maximum likelihood (ML) estimators. Finally, the three models are compared numerically with the simpler trivariate Negative Binomial GLM and with elliptical copula based models.     

Nonadditive Set Functions on a Finite Set and Linear Inequalities

A set function is a function whose domain is the power set of a set, which is assumed to be finite in this paper. We treat a possibly nonadditive set function, i.e., a set function which does not satisfy necessarily additivity, ?(A) + ?(B) = ?(A ∪ B) forA ∩ B = ∅, as an element of the linear space on the power set. Then some of the famous classes of set functions are polyhedral in that linear space, i.e., expressed by a finite number of linear inequalities. We specify the sets of the coefficients of the linear inequalities for some classes of set functions. Then we consider the following three problems: (a) the domain extension problem for nonadditive set functions, (b) the sandwich problem for nonadditive set functions, and (c) the representation problem of a binary relation by a nonadditive set function, i.e., the problem of nonadditive comparative probabilities.     

Stein iterations for the coupled discrete-time Riccati equations

We consider a set of discrete-time coupled algebraic Riccati equations that arise in quadratic optimal control of Markovian jump linear systems. Two iterations for computing a symmetric (maximal) solution of this system are investigated. We construct sequences of the solutions of the decoupled Stein equations and show that these sequences converge to a solution of the considered system. Numerical experiments are given.     

Weight Functions Method in Stability Study of Systems

A new method for systems stability analysis is presented. This method is called weight functions method and it replaces the problem of Liapunov function finding with a problem of finding a number of functions (weight functions) equal to the number of first order differential equations describing the system. It is known that there are not general methods for finding Liapunov functions. The weight functions method is simpler than the classical method since one function at a time has to found. Conditions of solution stability for linear and nonlinear systems and some examples are presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the constructive characterization of threshold functions

In this paper, the structure of the set of threshold functions and complexity problems are considered. The notion of the signature of a threshold function is defined. It is shown that if a threshold function essentially depends on all of its variables, then the signature of this function is unique. The set of threshold functions is partitioned into classes with equal signatures. A theorem characterizing this partition is proved. The importance of the class of monotone threshold functions is emphasized. The complexity of transferring one threshold function specified by a linear form into another is examined. It is shown that in the worst case this transfer would take exponential time. The structure of the set of linear forms specifying the same threshold function is also examined. It is proved that for any threshold function this set of linear forms has a unique basis in terms of the operation of addition of linear forms. It is also shown that this basis is countable.     

Dominating Sets for Convex Functions with Some Applications

A number of optimization methods require as a first step the construction of a dominating set (a set containing an optimal solution) enjoying properties such as compactness or convexity. In this paper, we address the problem of constructing dominating sets for problems whose objective is a componentwise nondecreasing function of (possibly an infinite number of) convex functions, and we show how to obtain a convex dominating set in terms of dominating sets of simpler problems. The applicability of the results obtained is illustrated with the statement of new localization results in the fields of linear regression and location.     

Approximate decoupling of multivariate polynomials using weighted tensor decomposition

Many scientific and engineering disciplines use multivariate polynomials. Decomposing a multivariate polynomial vector function into a sandwiched structure of univariate polynomials surrounded by linear transformations can provide useful insight into the function while reducing the number of parameters. Such a decoupled representation can be realized with techniques based on tensor decomposition methods, but these techniques have only been studied in the exact case. Generalizing the existing techniques to the noisy case is an important next step for the decoupling problem. For this extension, we have considered a weight factor during the tensor decomposition process, leading to an alternating weighted least squares scheme. In addition, we applied the proposed weighted decoupling algorithm in the area of system identification, and we observed smaller model errors with the weighted decoupling algorithm than those with the unweighted decoupling algorithm.     

Multivariate arrangement increasing functions with applications in probability and statistics

A real valued function of s vector arguments in Rⁿ is said to be arrangement increasing if the function increases in value as the components of the vector arguments become more similarly arranged. Various examples of arrangement increasing functions are given including many joint multivariate densities, measures of concordance between judges and the permanent of a matrix with nonnegative components. Preservation properties of the class of arrangement increasing functions are examined, and applications are given including useful probabilistic inequalities for linear combinations of exchangeable random vectors.     

An Algorithm for Solving Multicriterion Linear Programming Problems with Examples

The purpose of this paper is to develop a useful technique for solving linear programmes involving more than one objective function. Motivation for solving multicriterion linear programmes is given along with the inherent difficulty associated with obtaining a satisfactory solution set. By applying a linear programming approach for the solution of two person–zero sum games with mixed strategies, it is shown that a linear optimization problem with multiple objective functions can be formulated in this fashion in order to obtain a solution set satisfying all the requirements for an efficient solution of the problem. The solution method is then refined to take into account disparities between the magnitude of the values generated by each of the objective functions and solution preferences as determined by a decision-maker. A summary of the technique is then given along with several examples in order to demonstrate its applicability.     

On a class of quadratic programs

This paper considers a class of quadratic programs where the constraints ae linear and the objective is a product of two linear functions. Assuming the two linear factors to be non-negative, maximization and minimization cases are considered. Each case is analyzed with the help of a bicriteria linear program obtained by replacing the quadratic objective with the two linear functions. Global minimum (maximum) is attained at an efficient extreme point (efficient point) of the feasible set in the solution space and corresponds to an efficient extreme point (efficient point) of the feasible set in the bicriteria space. Utilizing this fact and certain other properties, two finite algorithms, including validations are given for solving the respective problems. Each of these, essentially, consists of solving a sequence of linear programs. Finally, a method is provided for relaxing the non-negativity assumption on the two linear factors of the objective function.     

Characterization of Linear Structures

We study the notionof linear structure of a function defined from F ^mto F ⁿ, and in particular of a Boolean function.We characterize the existence of linear structures by means ofthe Fourier transform of the function. For Boolean functions,this characterization can be stated in a simpler way. Finally,we give some constructions of resilient Boolean functions whichhave no linear structure.     

Approximations of pseudo-Boolean functions; applications to game theory

This paper studies the approximation of pseudo-Boolean functions by linear functions and more generally by functions of (at most) a specified degree. Here a pseudo-Boolean function means a real valued function defined on {0,1} ⁿ , and its degree is that of the unique multilinear polynomial that expresses it; linear functions are those of degree at most one. The approximation consists in choosing among all linear functions the one which is closest to a given function, where distance is measured by the Euclidean metric onR ²ⁿ . A characterization of the best linear approximation is obtained in terms of the average value of the function and its first derivatives. This leads to an explicit formula for computing the approximation from the polynomial expression of the given function. These results are later generalized to handle approximations of higher degrees, and further results are obtained regarding the interaction of approximations of different degrees. For the linear case, a certain constrained version of the approximation problem is also studied. Special attention is given to some important properties of pseudo-Boolean functions and the extent to which they are preserved in the approximation. A separate section points out the relevance of linear approximations to game theory and shows that the well known Banzhaf power index and Shapley value are obtained as best linear approximations of the game (each in a suitably defined sense).Supported by the Air Force Office of Scientific Research (under grant number AFOSR 89-0512 and AFOSR 90-0008 to Rutgers University), as well as the National Science Foundation (under grant number DMS 89-06870).     

An approximation method to model multivariate interpolation problems: Indexing HDMR

Dealing with univariate or bivariate data sets instead of a multivariate data set is an important concern in interpolation problems and computer-based applications. This paper presents a new data partitioning method that partitions the given multivariate data set into univariate and bivariate data sets and constructs an approximate analytical structure that interpolates function values at arbitrarily distributed points of the given grid. A number of numerical implementations are also given to show the performance of this new method.