a review of some methods for ranking fuzzy subsets this paper deals with the problem of ranking n fuzzy subsets of the unit interval a number of methods suggested in the literature is reviewed and tested on a group of selected examples where the fuzzy sets can be nonnormal and or nonconvex the ranking is obtained from i the index of strict preference defined by watson ii three indexes proposed by yager iii the algorithm used by chang iv three versions of the a preference index suggested by adamo v the index defined by baas and kwakernaak vi three modified versions used by baldwin and guild vii the method proposed by kerre viii three forms of the index suggested by jain ix the four grades of dominance studied by dubois and prade in simple cases the results are good for all the methods with some exceptions in questionable cases where the decision must be probably modelled in accordance with the context in which it is imbedded the best indexes seem to be the dominances suggested by dubois and prade these indexes do not force any particular choice but clearly describe the situation hence allowing the decision maker himself to make his best choice

the duality principle in fuzzy set theory abstract a rigorous mathematical approach to extending nonfuzzy structures to fuzzy sets is presented this approach is based on classical ideas of duality in analysis and inverse image in set theory we apply these new techniques to characterizing indistinguishability relations fuzzy interval orders and fuzzy operations zadeh s extension principle is characterized in terms of the duality principle

reasonable properties for the ordering of fuzzy quantities ii as a continuation of the first part related to the first and second class of ordering approaches this paper deals with the fulfilment of reasonable properties in the third class of ordering approaches to do so we briefly introduce fuzzy relations on which the third class of approaches is based then we recall some transitivity related concepts and an ordering procedure based on a acyclic fuzzy relation acyclicity is a very weak restriction on a fuzzy relation we prove that many fuzzy relations used for the comparison of fuzzy quantities satisfy some conditions stronger than acyclicity so we give a widely applicable formulation to derive a total ranking order from a fuzzy relation with our formulation we examine all the ordering indices in the third class with respect to the proposed axioms in part i

consistency and asymptotic normality of some subspace algorithms for systems without observed inputs asymptotic normality for a class of subspace algorithms which estimate the state in a first step is derived expressions for the asymptotic variance are given linear systems with unobserved white noise inputs are considered a class of subspace estimates for the system matrices obtained by estimating the state in the first step is analyzed the main result presented here states asymptotic normality of subspace estimates in addition a consistency result for the system matrix estimates is given an algorithm to compute the asymptotic variances of the estimates is presented in a final section the implications of the result are discussed

ranking of fuzzy sets based on the concept of existence abstract various approaches have been proposed for the comparison or ranking of fuzzy sets however due to the complexity of the problem a general method which can be used for any situation still does not exist this paper formalizes the concept of existence for the ranking of fuzzy sets many of the existing fuzzy ranking methods are shown to be some application of this concept an improved fuzzy ranking method is then introduced based on this concept this newly introduced method is expanded for treating both normal and nonnormal convex and nonconvex fuzzy sets emphasis is placed on the use of the subjectivity of the decision maker such as the optimistic or the pessimistic view points an improved procedure for obtaining linguistic conclusions is also developed finally some numerical examples are given to illustrate the approach

structure and order estimation of multivariable stochastic processes an easy to implement numerically efficient algorithm which estimates the kronecker invariants is presented a procedure allowing estimation of the structure of a state space representation for a multivariable stationary stochastic process from measured output data is presented it is assumed that the observed vector time series is a realization of a process with rational spectrum or the output of a stable time invariant linear system driven by white noise an algorithm is proposed which selects a maximal set of linearly independent rows of the hankel matrix built upon the estimated covariance sequence and thus yields estimates of the kronecker invariants when applied to simulated examples it systematically yielded the good structure without any ambiguity i e with a surprising robustness with respect to the choice of the probability of false alarm the numerical efficiency of the procedure is remarkable and no exhaustive search over the set of all possible kronecker indexes has to be performed

a realization approach to stochastic model reduction the problem of discrete time stochastic model reduction approximation is considered using the canonical correlation analysis approach of akaike 1975 a new order reduction algorithm is developed furthermore it is shown that the inverse of the reduced order realization is asymptotically stable next an explicit relationship between canonical variables and the linear least squares estimate of the state vector is established using this a more direct approach for order reduction is presented and also a new design for reduced order kalman filters is developed finally the uniqueness and symmetry properties for the new realization the balanced stochastic realization along with a simulation result are presented

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

traces of fuzzy binary relations abstract in this paper the right and left trace of a fuzzy binary relation is defined and investigated basic properties of a fuzzy relation such as reflexivity different kinds of transitivity linearity etc are characterized by means of its traces

consistent impulse response estimation and system realization from noisy data a novel method for the estimation of a finite part of the impulse response is presented the method is consistent in case the measured output data are prone to colored noise the combination of kung s realization algorithm yields a consistent system identification algorithm that is particularly useful for the identification of systems with uncoupled deterministic and stochastic dynamics this algorithm is called irear a detailed linear sensitivity analysis of the impulse response estimation and kung s realization algorithms yields novel error and covariance formulae for the estimated impulse response and system matrices and the poles and frequency response functions that result from the identified system description

unified approach fuzzy graph problems we present a taxonomy of fuzzy graphs that treats fuzziness in vertex existence edge existence edge connectivity and edge weight within that framework we formulate some standard graph theoretic problems shortest paths and minimum cut for fuzzy graphs using a unified approach distinguished by its uniform application of guiding principles such as the construction of membership grades via the ranking of fuzzy numbers the preservation of membership grade normalization and the collapsing of fuzzy sets of graphs into fuzzy graphs finally we provide algorithmic solutions to these problems with examples

identification of the deterministic part of mimo state space models given in innovations form from input output data the problem of linear multivariable state space model identification from input output data can under the presence of process and measurement noise be solved in a non iterative way when incorporating instrumental variables constructed from both input and output sequences in the recently developed class of multivariable output error state space model class of subspace model identification schemes abstract in this paper we describe two algorithms to identify a linear time invariant finite dimensional state space model from input output data the system to be identified is assumed to be excited by a measurable input and an unknown process tloise and the measurements are disturbed by unknown measurement noise both noise sequences are discrete zero mean white noise the first algorithm gives consistent estimates only for the case where the input also is zero mean white noise while the same result is obtained with the second algorithm without this constraint for the special case where the input signal is discrete zero mean white noise it is explicitly shown that this second algorithm is a special case of the recently developed multivariable output error state space moesp class of algorithms based on instrumental variables the usefulness of the presented schemes is highlighted in a realistic simulation study

transitive fuzzy orderings of fuzzy numbers abstract in this paper we introduce a generalization of the baas kwakernaak index by replacing the min operation in their definition by a t norm some properties of the thus defined induced fuzzy ordering are established in particular it is shown that restrictions of the induced fuzzy ordering on some special classes of fuzzy numbers are reflexive fuzzy orders

global exponential stability for stochastic coupled systems on networks with markovian switching abstract this paper investigates the global exponential stability of stochastic coupled systems on networks with markovian switching scsnms sufficient conditions are established to guarantee p th moment exponential stability and almost surely exponential stability of the scsnms by using concepts from graph theory and m matrix into lyapunov method consequently the upper boundaries of the p th moment lyapunov exponent and the sample lyapunov exponent are given finally to illustrate the capabilities of the stability principle easy verifiable sufficient conditions of global exponential stability for a stochastic coupled oscillator on network with markovian switching are obtained

indicators of fuzzy relations in this paper first we give the definitions of various indicators of fuzzy relations and their basic properties then we investigate the relationships between these indicators particularly between those of t transitivity negative s transitivity t s semitransitivity and t s ferrers property the investigation is divided into three parts results in the general case results under a condition proposed by peng et al and results under special t norms and t conorms on account of our research certain results on fuzzy relations in the literature are extended to the degree description

on fuzzy number lattice r abstract we give a systematic development of fuzzy number lattice theory many of our results generalize to number lattice over the lattice algebra our first main result is that the real number chain r is a homomorphic image of the fuzzy number lattice r that is r r and the congruence class r is a component of r thus we can be specialized to describe the structure of the fuzzy number lattice r next we study the structure and representation of the congruence class

consistency and relative efficiency of subspace methods we give a consistency proof for two subspace methods we then show the asymptotic equivalence of a special subspace method and the initial estimate proposed by hannan and rissanen finally a simulation study comparing two subspace methods and the maximum likelihood method is performed

n4sid subspace algorithms for the identification of combined deterministic stochastic systems abstract recently a great deal of attention has been given to numerical algorithms for subspace state space system identification n4sid in this paper we derive two new n4sid algorithms to identify mixed deterministic stochastic systems both algorithms determine state sequences through the projection of input and output data these state sequences are shown to be outputs of non steady state kalman filter banks from these it is easy to determine the state space system matrices the n4sid algorithms are always convergent non iterative and numerically stable since they only make use of qr and singular value decompositions both n4sid algorithms are similar but the second one trades off accuracy for simplicity these new algorithms are compared with existing subspace algorithms in theory and in practice

multivariable harmonic balance for central pattern generators the central pattern generator cpg is a nonlinear oscillator formed by a group of neurons providing a fundamental control mechanism underlying rhythmic movements in animal locomotion we consider a class of cpgs modeled by a set of interconnected identical neurons based on the idea of multivariable harmonic balance we show how the oscillation profile is related to the connectivity matrix that specifies the architecture and strengths of the interconnections specifically the frequency amplitudes and phases are essentially encoded in terms of a pair of eigenvalue and eigenvector this basic principle is used to estimate the oscillation profile of a given cpg model moreover a systematic method is proposed for designing a cpg based nonlinear oscillator that achieves a prescribed oscillation profile

matrix perturbation analysis for weakly coupled oscillators the central pattern generator for controlling animal locomotion is generally not a single oscillator but a group of interconnected oscillators although the whole group can be analyzed as a bulky single oscillator it is difficult with such an approach to reveal the essential mechanism of phase coordination among oscillators in this paper a new approach based on the multivariable harmonic balance is used to analyze a group of identical oscillators that are weakly coupled to each other specifically a lower dimensional matrix representing the coupling structure is abstracted from the overall neuronal connectivity matrix of large dimension so that the coupling matrix captures the essential mechanism for phase coordination of the oscillators moreover this framework provides an effective way to design the neuronal interconnections to achieve prescribed coordination among the oscillators

on ranking fuzzy sets abstract a mathematical model describing a fuzzy ordering of fuzzy sets is suggested in this model an induced fuzzy ordering is defined as the inverse image of the natural linear ordering on real numbers under natural fuzzy correspondence between fuzzy sets and real numbers the transitivity property of the strict fuzzy ordering associated with the induced fuzzy ordering is established

realization theory for multivariate stationary gaussian processes this paper collects in one place a comprehensive theory of stochastic realization for continuous time stationary gaussian vector processes which in various pieces has appeared in a number of our earlier papers it begins with an abstract state space theory based on the concept of splitting subspace these results are then carried over to the spectral domain and described in terms of hardy functions finally differential equations type stochastic realizations are constructed the theory is coordinate free and it accommodates infinite dimensional representations minimality and other systems theoretical concepts being defined by subspace inclusion rather than by dimension we have strived for conceptual completeness rather than generality and the same framework can be used for other types of stochastic realization problems

ranking fuzzy numbers in the setting of possibility theory the arithmetic manipulation of fuzzy numbers or fuzzy intervals is now well understood equally important for application purposes is the problem of ranking fuzzy numbers or fuzzy intervals which is addressed in this paper a complete set of comparison indices is proposed in the framework of zadeh s possibility theory it is shown that generally four indices enable one to completely describe the respective locations of two fuzzy numbers moreover this approach is related to previous ones and its possible extension to the ranking of n fuzzy numbers is discussed at length finally it is shown that all the information necessary and sufficient to characterize the respective locations of two fuzzy numbers can be recovered from the knowledge of their mutual compatibilities

fuzzy reasoning for solving fuzzy mathematical programming problems abstract we interpret fuzzy linear programming flp problems with fuzzy coefficients and fuzzy inequality relations as multiple fuzzy reasoning schemes mfr where the antecedents of the scheme correspond to the constraints of the flp problem and the fact of the scheme is the objective of the flp problem then the solution process consists of two steps first for every decision variable x r n we compute the fuzzy value of the objective function max x via sup min convolution of the antecedents constraints and the fact objective then an optimal solution to flp problem is any point which produces a maximal element of the set max x x r n in the sense of the given inequality relation we show that our solution process for a classical crisp lp problem results in a solution in the classical sense and under well chosen inequality relations and objective function coincides with those suggested by buckley fuzzy sets and systems 31 1989 329 341 delgado et al control and cybernetics 16 1987 114 121 fuzzy sets and systems 26 1988 49 62 negoita fuzzy sets and systems 6 1981 261 269 ramik and rimanek fuzzy sets and systems 16 1985 123 138 verdegay fuzzy information and decision processes north holland amsterdam 1982 control and cybernetics 13 1984 230 239 and zimmermann int j general systems 2 1976 209 215 furthermore we show how to extend the proposed solution principle to non linear programming problems with fuzzy coefficients we illustrate our approach by some simple examples

a gauss newton method for the synthesis of periodic outputs with central pattern generators it is assumed that a central pattern generator possesses an exponentially stable limit cycle which originates a periodic output signal we propose a method based on a gauss newton iteration to determine the values of the neural coupling parameters that allows to approximate a given reference output signal we present two applications the first is a ring network of morris lecar neurons where the output of the system is the sum of the membrane potential of all neurons the second is a network of six neural cells for the generation of the leg movements of a hexapod