fuzzy interval optimal control problem abstract this paper proposes the discrete time fuzzy interval optimal control problem via single level constrained fuzzy arithmetic fuzzy interval functions are established and allow us to evaluate the dynamics of the fuzzy interval difference equation moreover fuzzy interval dynamic programming is developed to obtain the optimal fuzzy interval solution helping us to make a realistic decision a numerical example illustrates the theory and provides the practical solution for an user

widely linear sphere decoder in mimo systems by exploiting the conjugate symmetry of linearly modulated signals this paper investigates the widely linear processing wlp for the detection of circular signals such as m ary phase shift keying mpsk signals and m ary quadrature amplitude modulation mqam signals first a unified mathematical model is derived to describe the conjugate symmetry of general mpsk mqam signals in the unified model a phase rotation matrix prm is introduced to partition the constellation of multiple input multiple output mimo signals into subsets signals in a subset share the same prm second a widely linear receiver is proposed in each subset for mimo detection to avoid repetitive wlp in each subset a widely linear sphere decoder wlsd is further proposed for mimo systems wlsd transforms the traditional sphere decoder sd searching for a true transmitted vector into a shrunk one by searching for the corresponding phase rotation vector finally the diversity order of wlsd is proven to be more than n r frac n t 1 2 and less than n r where n t or n r denotes the number of transmitting or receiving antennas additional performance analysis is also conducted to quantify the signal to noise ratio improvement the complexity analysis reveals that the candidate phase rotation vectors of wlsd are no more than frac 1 2 n t of the sd candidates simulation results show that the proposed wlsd can achieve quasi optimal bit error rate performance while the computational complexity is reduced by more than a half compared with the schnorr euchner sphere decoder

invited paper survey of quantitative feedback theory qft abstract qft is an engineering design theory devoted to the practical design of feedback control systems the foundation of qft is that feedback is needed in control only when plant p parameter and or disturbance d uncertainties sets p p d d exceed the acceptable a system performance uncertainty set a a the principal properties of qft are as follows 1 the amount of feedback needed is tuned to the p d a sets if a exceeds p d feedback is not needed at all 2 the simplest modelling is used a command disturbance and sensor noise inputs and b the available sensing points and the defined outputs no special controllability test is needed in either linear or non linear plants it is inherent in the design procedure there is no observability problem because uncertainty is included the number of independent sensors determines the number of independent loop transmissions li the functions which provide the benefits of feedback 3 the simplest mathematical tools h

full mean square performance bounds on quaternion estimators for improper data novel physical insights are provided into the mean square performance bounds of quaternion valued widely linear wl semi widely linear swl and strictly linear estimators for the generality of quaternion valued gaussian data this is achieved by first defining three kinds of complementary mean square errors cmses of these estimators and by further exploiting the corresponding degrees of mathbb h improperness second order noncircularity next the investigation of the bounds of the attainable cmses by these classes of estimators shows that only a joint consideration of the proposed cmse analysis and the standard mse analysis provides enough degrees of freedom for a detailed account of the mse performance the so established framework for the analysis of estimators for mathbb h improper data is shown to be capable of measuring error power distribution for each data channel an important finding which is not possible to obtain through the standard mse analysis only simulations in the system identification setting support the analysis

properness and widely linear processing of quaternion random vectors in this paper the second order circularity of quaternion random vectors is analyzed unlike the case of complex vectors there exist three different kinds of quaternion properness which are based on the vanishing of three different complementary covariance matrices the different kinds of properness have direct implications on the cayley dickson representation of the quaternion vector and also on several well known multivariate statistical analysis methods in particular the quaternion extensions of the partial least squares pls multiple linear regression mlr and canonical correlation analysis cca techniques are analyzed showing that in general the optimal linear processing is full widely linear however in the case of jointly q proper or c proper vectors the optimal processing reduces respectively to the conventional or semi widely linear processing finally a measure for the degree of improperness of a quaternion random vector is proposed which is based on the kullback leibler divergence between two zero mean gaussian distributions one of them with the actual augmented covariance matrix and the other with its closest proper version this measure quantifies the entropy loss due to the improperness of the quaternion vector and it admits an intuitive geometrical interpretation based on kullback leibler projections onto sets of proper augmented covariance matrices

theoretical and semantic distinctions of fuzzy possibilistic and mixed fuzzy possibilistic optimization theoretical semantic and algorithmic distinctions among fuzzy possibilistic and mixed fuzzy possibilistic optimization are presented and illustrated the theory underlying fuzzy possibilistic and mixed fuzzy possibilistic optimization is developed and demonstrated and points to the appropriate use of distinct solution methods associated with each type of optimization dependant on the semantics of the problem semantics is key to both the input where one is obtaining the data and constructing the optimization model in the presence of uncertainty and the output where the meaning of the results is necessary for understanding solutions the case in which the optimization model arises from the data that is a combination of fuzzy and possibilistic distributions is also derived lastly examples illustrate the theory this paper is a modification and an amplification of a presentation made at ifsa 05 w a lodwich k d jamison theory and semantics for fuzzy and possibilistic optimization in fuzzy logic soft computing and computational intelligence eleventh internat fuzzy systems association world congress july 28 31 2005 beijing china vol iii pp 1805 1810 26

algebraic methods for diagonalization of a quaternion matrix in quaternionic quantum theory by means of complex representation and real representation of a quaternion matrix this paper studies the problem of diagonalization of a quaternion matrix gives two algebraic methods for diagonalization of quaternion matrices in quaternionic quantum theory

fuzzy modeling in terms of surprise this paper presents a new approach to fuzzy modeling based on the concept of surprise the new concept is related to the traditional membership function by an antitone transformation advantages of the surprise approach include 1 it has a consistent semantic interpretation 2 it allows the joint use of quantitative and qualitative knowledge using simple rules of logic 3 it is a direct extension of and allows combination with the least squares approach to reconciling conflicting approximate numerical data 4 it is ideally suited to optimization under imprecise or conflicting goals specified by a combination of soft and hard interval constraints 5 it gives a straightforward approach to constructing families of functions consistent with fuzzy associative memories as used in fuzzy control with tuning parameters reflecting linguistic ambiguity that can be adapted to available performance data

augmented performance bounds on strictly linear and widely linear estimators with complex data novel physical insights are provided into the performances of strictly linear sl and widely linear wl estimators of the generality of complex valued data both proper second order circular and improper second order noncircular this is achieved by first performing a novel complementary mean square error cmse analysis in order to quantify the degrees of improperness second order noncircularity of the sl and wl estimation errors the exact bounds on the cmse difference between the sl and wl estimators are investigated to show that only a joint consideration of the standard mse analysis and the proposed cmse analysis has enough degrees of freedom for a rigorous account of the performance of wl and sl estimators this also makes it possible to rigourously quantify the contributions to the wl performance advantage from the individual real and imaginary channels an important finding not possible to obtain by using the standard mse analysis only

gaussian particle filtering sequential bayesian estimation for nonlinear dynamic state space models involves recursive estimation of filtering and predictive distributions of unobserved time varying signals based on noisy observations this paper introduces a new filter called the gaussian particle filter it is based on the particle filtering concept and it approximates the posterior distributions by single gaussians similar to gaussian filters like the extended kalman filter and its variants it is shown that under the gaussianity assumption the gaussian particle filter is asymptotically optimal in the number of particles and hence has much improved performance and versatility over other gaussian filters especially when nontrivial nonlinearities are present simulation results are presented to demonstrate the versatility and improved performance of the gaussian particle filter over conventional gaussian filters and the lower complexity than known particle filters

approximate series representations of linear operations on second order stochastic processes application to simulation series representations of the more usual linear operations in weak sense on a second order stochastic process are studied the starting point of this analysis is the optimal cambanis expansion of the stochastic process considered likewise the extensions of the approximate series expansions based on the rayleigh ritz method are presented for such linear operations on the process the main advantages of these extensions are that they are computationally feasible and entail a significant reduction in the computational burden finally their applicability as a practical simulation tool is examined

widely linear estimation with complex data mean square estimation of complex and normal data is not linear as in the real case but widely linear the purpose of the paper is to calculate the optimum widely linear mean square estimate and to present its main properties the advantage with respect to the linear procedure is especially analyzed

the interacting multiple model algorithm for systems with markovian switching coefficients an important problem in filtering for linear systems with markovian switching coefficients dynamic multiple model systems is the management of hypotheses which is necessary to limit the computational requirements a novel approach to hypotheses merging is presented for this problem the novelty lies in the timing of hypotheses merging when applied to the problem of filtering for a linear system with markovian coefficients the method is an elegant way to derive the interacting multiple model imm algorithm evaluation of the imm algorithm shows that it performs well at a relatively low computational load these results imply a significant change in the state of the art of approximate bayesian filtering for systems with markovian coefficients

nonfragile h_ infty filtering of continuous time fuzzy systems this paper is concerned with the problem of nonfragile h filtering for a class of continuous time fuzzy systems attention is focused on the design of a filter such that the filtering error system preserves a prescribed h performance where the filter to be designed is assumed to have gain variations by using the quadratic lyapunov function approach and adding slack matrix variables sufficient conditions for designing the nonfragile h filter are proposed in terms of solutions to a set of linear matrix inequalities lmis finally a simulation example will be given to show the efficiency of the proposed design methods

the partition of unity finite element approach with hp refinement for the stationary fokker planck equation abstract in this paper the stationary fokker planck equation fpe is solved for nonlinear dynamical systems using a local numerical technique based on the meshless partition of unity finite element method pufem the method is applied to stationary fpe for 2 3 and 4 d systems and is argued to be an excellent candidate for higher dimensional problems and the transient problem local refinement is applied by introducing higher order polynomials in selected subdomains local p refinement to keep the problem size small while ensuring high approximation accuracy various local approximations are blended using novel pasting functions that provide any desired order of continuity results are compared with existing global and local techniques local p refinement is touted as an important step towards breaking the curse of dimensionality in numerical solution of fpe

relationships between modality constrained programming problems and various fuzzy mathematical programming problems abstract in this paper we focus on the relationships between modality constrained programming problems and various fuzzy mathematical programming problems i e flexible programming problem fuzzy goal programming problem and fuzzy mathematical programming problems proposed by orlovski tanaka et al inuiguchi et al dubois and sakawa et al fuzzy mathematical programming problems discussed in this paper can be interpreted in the framework of modality constrained programming problems thus modality constrained programming problems can be regarded as a base model for unifying various fuzzy mathematical programming problems

adaptive gaussian sum filter for nonlinear bayesian estimation a nonlinear filter is developed by representing the state probability density function by a finite sum of gaussian density kernels whose mean and covariance are propagated from one time step to the next using linear system theory methods such as extended kalman filter or unscented kalman filter the novelty in the proposed method is that the weights of the gaussian kernels are updated at every time step by solving a convex optimization problem posed by requiring the gaussian sum approximation to satisfy the fokker planck kolmogorov equation for continuous time dynamical systems and the chapman kolmogorov equation for discrete time dynamical systems the numerical simulation results show that updating the weights of different mixture components during propagation mode of the filter not only provides us with better state estimates but also with a more accurate state probability density function

complex valued signal processing the proper way to deal with impropriety complex valued signals occur in many areas of science and engineering and are thus of fundamental interest in the past it has often been assumed usually implicitly that complex random signals are proper or circular a proper complex random variable is uncorrelated with its complex conjugate and a circular complex random variable has a probability distribution that is invariant under rotation in the complex plane while these assumptions are convenient because they simplify computations there are many cases where proper and circular random signals are very poor models of the underlying physics when taking impropriety and noncircularity into account the right type of processing can provide significant performance gains there are two key ingredients in the statistical signal processing of complex valued data 1 utilizing the complete statistical characterization of complex valued random signals and 2 the optimization of real valued cost functions with respect to complex parameters in this overview article we review the necessary tools among which are widely linear transformations augmented statistical descriptions and wirtinger calculus we also present some selected recent developments in the field of complex valued signal processing addressing the topics of model selection filtering and source separation

gaussian filters for nonlinear filtering problems we develop and analyze real time and accurate filters for nonlinear filtering problems based on the gaussian distributions we present the systematic formulation of gaussian filters and develop efficient and accurate numerical integration of the optimal filter we also discuss the mixed gaussian filters in which the conditional probability density is approximated by the sum of gaussian distributions a new update rule of weights for gaussian sum filters is proposed our numerical tests demonstrate that new filters significantly improve the extended kalman filter with no additional cost and the new gaussian sum filter has a nearly optimal performance

particle smoothing in continuous time a fast approach via density estimation we consider the particle smoothing problem for state space models where the transition density is not available in closed form in particular for continuous time nonlinear models expressed via stochastic differential equations sdes conventional forward backward and two filter smoothers for the particle filter require a closed form transition density with the linear gaussian euler maruyama discretization usually applied to the sdes to achieve this we develop a pair of variants using kernel density approximations to relieve the dependence and in doing so enable use of faster and more accurate discretization schemes such as runge kutta in addition the new methods admit arbitrary proposal distributions providing an avenue to deal with degeneracy issues experimental results on a functional magnetic resonance imaging fmri deconvolution task demonstrate comparable accuracy and significantly improved runtime over conventional techniques

a nonlinear stochastic filter for continuous time state estimation nonlinear filters produce a nonparametric estimate of the probability density of state at each point in time currently known nonlinear filters include particle filters and the kushner equation and its un normalized version the zakai equation however these filters have limited measurement models particle filters require measurement at discrete times and the kushner and zakai equations only apply when the measurement can be represented as a function of the state we present a new nonlinear filter for continuous time measurements with a much more general stochastic measurement model it integrates to bayes rule over short time intervals and provides bayes optimal estimates from quantized intermittent or ambiguous sensor measurements the filter has a close link to information theory and we show that the rate of change of entropy of the density estimate is equal to the mutual information between the measurement and the state and thus the maximum achievable this is a fundamentally new class of filter that is widely applicable to nonlinear estimation for continuous time control

an augmented extended kalman filter algorithm for complex valued recurrent neural networks an augmented complex valued extended kalman filter acekf algorithm for the class of nonlinear adaptive filters realized as fully connected recurrent neural networks is introduced this is achieved based on some recent developments in the so called augmented complex statistics and the use of general fully complex nonlinear activation functions within the neurons this makes the acekf suitable for processing general complex valued nonlinear and nonstationary signals and also bivariate signals with strong component correlations simulations on benchmark and real world complex valued signals support the approach

numerical integration of stochastic differential equations several numerical methods for treating stochastic differential equations are considered both the convergence in the mean square limit and the convergence of the moments is discussed and the generation of appropriate random numbers is treated the necessity of simulations at various time steps with an extrapolation to time step zero is emphasized and demonstrated by a simple example

complex linear quadratic systems for detection and array processing linear quadratic lq filters for detection and estimation are widely used in the real case we investigate their extension in the complex case which introduces various new questions in particular we calculate the optimum lq array receiver in a non gaussian environment by using the deflection criterion and evaluate some of its performance

a quaternion widely linear adaptive filter a quaternion widely linear qwl model for quaternion valued mean square error mse estimation is proposed the augmented statistics are first introduced into the field of quaternions and it is demonstrated that this allows for capturing the complete second order statistics available the qwl model is next incorporated into the quaternion least mean square qlms algorithm to yield the widely linear qlms wl qlms this allows for a unified approach to adaptive filtering of both q proper and q improper signals leading to improved accuracies compared to the qlms class of algorithms simulations on both benchmark and real world data support the analysis

second order statistics of complex signals the second order statistical properties of complex signals are usually characterized by the covariance function however this is not sufficient for a complete second order description and it is necessary to introduce another moment called the relation function its properties and especially the conditions that it must satisfy are analyzed both for stationary and nonstationary signals this leads to a new perspective concerning the concept of complex white noise as well as the modeling of any signal as the output of a linear system driven by a white noise finally this is applied to complex autoregressive signals and it is shown that the classical prediction problem must be reformulated when the relation function is taken into consideration

class of widely linear complex kalman filters recently a class of widely linear augmented complex valued kalman filters kfs that make use of augmented complex statistics have been proposed for sequential state space estimation of the generality of complex signals this was achieved in the context of neural network training and has allowed for a unified treatment of both second order circular and noncircular signals that is both those with rotation invariant and rotation dependent distributions in this paper we revisit the augmented complex kf augmented complex extended kf and augmented complex unscented kf in a more general context and analyze their performances for different degrees of noncircularity of input and the state and measurement noises for rigor a theoretical bound for the performance advantage of widely linear kfs over their strictly linear counterparts is provided the analysis also addresses the duality with bivariate real valued kfs together with several issues of implementation simulations using both synthetic and real world proper and improper signals support the analysis

numerical methods for stochastic differential equations stochastic differential equations sde s play an important role in physics but existing numerical methods for solving such equations are of low accuracy and poor stability a general strategy for developing accurate and efficient schemes for solving stochastic equations is outlined here high order numerical methods are developed for the integration of stochastic differential equations with strong solutions we demonstrate the accuracy of the resulting integration schemes by computing the errors in approximate solutions for sde s which have known exact solutions

fuzzy optimization an appraisal abstract this paper takes a general look at core ideas that make up the burgeoning body of fuzzy mathematical programming emphasizing the methodological view although fuzzy mathematical programming has enjoyed a rapidly increasing acceptance within the scientific community some technical hurdles exist to hinder a unanimity reasons for this as well as possible ways for improvement are also discussed

functionals of complex ornstein uhlenbeck processes abstract the exact distribution of the sufficient statistics and distribution of the maximum likelihood estimator of the drift damping parameter in a stationary complex ornstein uhlenbeck process given by 1 1 is investigated complete tables of the distribution function for different levels are given by the help of matlab the comparison with the earlier calculations are discussed the relation of the famous model of chandler wobble proposed by kolmogorov is investigated 1

a quaternion widely linear model for nonlinear gaussian estimation this paper deals with the nonlinear minimum mean squared error estimation problem by using a quaternion widely linear model on the basis of the information supplied by a gaussian signal and its square a quaternion observation process is defined and then by applying a widely linear processing an optimal estimator for the continuous time setting is provided the special structure of the estimator proves its superiority over the complex valued widely linear solution the continuous discrete version of the problem is also studied where the solution takes the form of a suboptimal estimator useful in practical applications in addition the particular case of signal plus noise is considered in which the suboptimal solution and its associated error can be implemented through an iterative algorithm two numerical simulation examples are presented showing the advantages of the proposed approach

a general model for fuzzy linear programming abstract in the current literature there are several models of fuzzy linear programming problems the aim of this paper is first to describe the more important problems in fuzzy linear programming and second to give a general model of fuzzy linear programming problems involving all of the above ones finally a resolution method for that general model is proposed

gaussian smoothers for nonlinear systems with one step randomly delayed measurements this technical note is concerned with the nonlinear state smoothing problem in the case that the measurements are randomly delayed by one sampling time under gaussian domain two general gaussian approximation ga and gaussian mixture approximation gma smoothers are proposed in minimum mean square error mmse sense the smoothing implementation is transformed into computing some special posterior covariances which triggers the development of the new unscented kalman smoother uks by applying unscented transformation ut simulation results demonstrate the superior performance of the proposed ga uks and gma uks algorithms

a constraint fuzzy interval analysis approach to fuzzy optimization abstract this article illustrates the efficacy of using constraint fuzzy analysis in fuzzy optimization we present the theoretical approach and concretize this theory via examples

a procedure for ordering fuzzy subsets of the unit interval abstract we introduce a function to help in the ordering of fuzzy subsets of the unit interval this function is the integral of the mean of the level sets associated with the fuzzy subsets various properties of this function are studied

gaussian sum particle filtering we use the gaussian particle filter to build several types of gaussian sum particle filters these filters approximate the filtering and predictive distributions by weighted gaussian mixtures and are basically banks of gaussian particle filters then we extend the use of gaussian particle filters and gaussian sum particle filters to dynamic state space dss models with non gaussian noise with non gaussian noise approximated by gaussian mixtures the non gaussian noise models are approximated by banks of gaussian noise models and gaussian mixture filters are developed using algorithms developed for gaussian noise dss models as a result problems involving heavy tailed densities can be conveniently addressed simulations are presented to exhibit the application of the framework developed herein and the performance of the algorithms is examined

the quaternion lms algorithm for adaptive filtering of hypercomplex processes the quaternion least mean square qlms algorithm is introduced for adaptive filtering of three and four dimensional processes such as those observed in atmospheric modeling wind vector fields these processes exhibit complex nonlinear dynamics and coupling between the dimensions which make their component wise processing by multiple univariate lms bivariate complex lms clms or multichannel lms mlms algorithms inadequate the qlms accounts for these problems naturally as it is derived directly in the quaternion domain the analysis shows that qlms operates inherently based on the so called ldquoaugmentedrdquo statistics that is both the covariance e xx h and pseudocovariance e xx t of the tap input vector x are taken into account in addition the operation in the quaternion domain facilitates fusion of heterogeneous data sources for instance the three vector dimensions of the wind field and air temperature simulations on both benchmark and real world data support the approach

possibilistic linear programming a brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem in this paper we review some fuzzy linear programming methods and techniques from a practical point of view in the rst part the general history and the approach of fuzzy mathematical programming are introduced using a numerical example some models of fuzzy linear programming are described in the second part of the paper fuzzy mathematical programming approaches are compared to stochastic programming ones the advantages and disadvantages of fuzzy mathematical programming approaches are exemplied in the setting of an optimal portfolio selection problem finally some newly developed ideas and techniques in fuzzy mathematical programming are briey reviewed c 2000 elsevier science b v all rights reserved

optimal linear quadratic systems for detection and estimation the problem of linear quadratic systems for detection has long been solved by assuming the deflection criterion and gaussian noise it is shown here that the gaussian assumption can be removed and a complete solution is presented for an arbitrary probability distribution with finite fourth order moments the optimal solution can always be obtained by solving a linear system of equations some properties of the optimal systems are developed for particular examples of nongaussian noise it is shown that there is a strong relationship between linear quadratic optimal detection and optimal estimation which extends results known for the purely linear case

gaussian mixture nonlinear filtering with resampling for mixand narrowing a new gaussian mixture filter has been developed one that uses a resampling step in order to limit the covariances of its individual gaussian components the new filter has been designed to produce accurate solutions of difficult nonlinear non gaussian estimation problems it uses static multiple model filter calculations and extended kalman filter ekf approximations for each gaussian mixand in order to perform dynamic propagation and measurement update the resampling step uses a newly designed algorithm that employs matrix inequalities in order to bound each mixand s covariance resampling occurs between the dynamic propagation and the measurement update in order to ensure bounded covariance in both of these operations the resampling algorithm also attempts to economize on the number of new mixands the resulting filter has been tested on a difficult seven state nonlinear filtering problem it achieves significantly better accuracy than a simple ekf an unscented kalman filter a moving horizon estimator backwards smoothing ekf and a regularized particle filter

proper quaternion gaussian graphical models in this paper we extend gaussian graphical models to proper quaternion gaussian distributions the properness assumption reduces the number of unknowns by a factor of four while graphical models reduce the number of degrees of freedom via sparsity each of the methods allows accurate estimation using a small number of samples to enjoy both gains we show that the proper quaternion gaussian inverse covariance estimation problem is convex and has a closed form solution we proceed to demonstrate that the additional sparsity constraints on the inverse covariance matrix also lead to a convex problem and the optimizations can be efficiently solved by standard numerical methods in the special but practical case of a chordal graph we provide a closed form solution we demonstrate the improved performance of our suggested estimators on both synthetic and real data

quaternions and matrices of quaternions abstract we give a brief survey on quaternions and matrices of quaternions present new proofs for certain known results and discuss the quaternionic analogues of complex matrices the methods of converting a quaternion matrix to a pair of complex matrices and homotopy theory are emphasized

a continuous time linear system identification method for slowly sampled data both direct and indirect methods exist for identifying continuous time linear systems a direct method estimates continuous time input and output signals from their samples and then use them to obtain a continuous time model whereas an indirect method estimates a discrete time model first both methods rely on fast sampling to ensure good accuracy in this paper we propose a more direct method where a continuous time linear model is directly fitted to the available samples this method produces an exact model asymptotically modulo some possible aliasing ambiguity even when the sampling rate is relatively slow we also state conditions under which the aliasing ambiguity can be resolved and we provide experiments showing that the proposed method is a valid option when a slow sampling frequency must be used but a large number of samples is available

quantitative synthesis of uncertain cascaded multiple input multiple output feedback systems there is given an n n multiple input multiple output mimo linear time invariant plant of m cascaded sections p pa pb pm each an n n matrix of transfer functions the m vector outputs y ya ym 1 y pa ya pa pb yb pa pb pmu pu can be measured and the data used for feedback purposes each pi is known only to belong to a given set x1d4ab i due to uncertainty in the plant parameters p is embedded in a feedback structure with an n 1 command input vector r r1 rn and m free n n loop compensation matrices to be chosen by the designer there are assigned specifications on the n2 system transfer functions fuv s s rcirc v s to be satisfied for all pi in i i a b m the basic problem is how to divide the feedback burden among the m available loop matrices the technique presented in this paper is based upon two synthesis techniques previously devised for the following two problems 1 the same as above but for single input single output siso systems with each pl

quaternion valued nonlinear adaptive filtering a class of nonlinear quaternion valued adaptive filtering algorithms is proposed based on locally analytic nonlinear activation functions to circumvent the stringent standard analyticity conditions which are prohibitive to the development of nonlinear adaptive quaternion valued estimation models we use the fact that stochastic gradient learning algorithms require only local analyticity at the operating point in the estimation space it is shown that the quaternion valued exponential function is locally analytic and since local analyticity extends to polynomials products and ratios we show that a class of transcendental nonlinear functions can serve as activation functions in nonlinear and neural adaptive models this provides a unifying framework for the derivation of gradient based learning algorithms in the quaternion domain and the derived algorithms are shown to have the same generic form as their real and complex valued counterparts to make such models second order optimal for the generality of quaternion signals both circular and noncircular we use recent developments in augmented quaternion statistics to introduce widely linear versions of the proposed nonlinear adaptive quaternion valued filters this allows full exploitation of second order information in the data contained both in the covariance and pseudocovariances to cater rigorously for second order noncircularity improperness and the corresponding power mismatch in the signal components simulations over a range of circular and noncircular synthetic processes and a real world 3 d noncircular wind signal support the approach

a tutorial on particle filters for online nonlinear non gaussian bayesian tracking increasingly for many application areas it is becoming important to include elements of nonlinearity and non gaussianity in order to model accurately the underlying dynamics of a physical system moreover it is typically crucial to process data on line as it arrives both from the point of view of storage costs as well as for rapid adaptation to changing signal characteristics in this paper we review both optimal and suboptimal bayesian algorithms for nonlinear non gaussian tracking problems with a focus on particle filters particle filters are sequential monte carlo methods based on point mass or particle representations of probability densities which can be applied to any state space model and which generalize the traditional kalman filtering methods several variants of the particle filter such as sir asir and rpf are introduced within a generic framework of the sequential importance sampling sis algorithm these are discussed and compared with the standard ekf through an illustrative example

quantitative cascaded multiple input multiple output synthesis by an improved method there is given an n n multiple input multiple output mimo linear time invariant plant of m cascaded sections p pa pb pm each an n n matrix of transfer functions the m vector outputs y ya ym 1 y pa ya pa pb yb pa pb pmu pu can be measured and the data used for feedback purposes each pi is known only to belong to a given set x1d4ab i due to uncertainty in the plant parameters p is embedded in a feedback structure with an n 1 command input vector r r1 rn and m free n n loop compensation matrices to be chosen by the designer there are assigned specifications on the n2 system transfer functions fuv s s rcirc v s to be satisfied for all pi in i i a b m the basic problem is how to divide the feedback burden among the m available loop matrices the paper presents a synthesis procedure which is easier to implement in some respects and is much more economical in loop gains and bandwidths than the previous one however the relations and trade offs betw

semi widely simulation and estimation of continuous time bbc eta proper quaternion random signals the repercussions of quaternion bbc eta properness on the continuous time simulation and estimation problems are studied as a first step a semi widely linear quaternion karhunen loeve expansion is derived then this series representation is used in both problems to give the solutions the simulation technique proposed is useful for transformations of nonstationary gaussian quaternion signals the estimation problem is also analyzed in a continuous discrete setting and by considering two different formulations the cambanis formulation and the signal plus noise model initially the estimators are developed on the basis of a semi widely linear processing afterwards the solutions are improved by incorporating the information supplied by the square quaternion observations and by using a semi widely nonlinear processing in this last scenario we assume gaussian observations and unknown probability distributions of the signal of interest

geometrical properties of optimal volterra filters for signal detection linear quadratic filters are a special example of volterra filters that are limited to the second order it is shown that all the results recently published which are valid in the linear quadratic case can be extended with the appropriate notations to volterra filters of arbitrary order particularly the optimum volterra filter giving the maximum of the deflection for detecting a signal in noise is wholly calculated in addition several geometrical properties of optimal volterra filters are investigated by introducing appropriate scalar products in particular the concept of space orthogonal to the signal and the noise alone reference nar property are introduced allowing a decomposition of the optimal filter that exhibits a relation between detection and estimation extensions to the infinite case and relations with the likelihood ratio are also investigated

testing for quaternion propriety we consider the problem of testing whether a quaternion valued gaussian random vector is proper the quaternion covariance matrix fully describes the second order properties of a quaternion random vector only if the distribution is proper the exact distribution of the likelihood ratio test under the hypothesis of propriety is derived for general sample size n and vector dimensionality p as this is in terms of meijer s g function various approximations are considered including box type and saddlepoint approximations we find in particular that a new approach matching the first three cumulants is easy to implement and extremely accurate

performance analysis of deficient length quaternion least mean square adaptive filters quaternion adaptive filters have been widely used for processing 3d and 4d phenomena deficient length quaternion adaptive filters are explicitly or implicitly used in many practical applications where the length of system impulse response is large or unknown however their statistical behaviors are yet to be fully understood as theoretical results on the class of full length quaternion least mean square qlms algorithms do not necessarily apply to their deficient length versions this article fills this void and analyses the mean and mean square convergence of the deficient length qlms algorithms both for the strictly linear and widely linear cases transient and steady state performance is characterised by exploiting the augmented statistics of noncircular quaternion random vectors a novel decorrelation technique in the quaternion domain is shown to allow for the development of intuitive closed form solutions for correlated quaternion gaussian inputs thus unveiling the relationship between the algorithm behaviour and the noncircularity of quaternion input data the analysis also provides a general framework whereby the strictly linear and semi widely linear qlms algorithms can be seen as deficient length versions of the widely linear qlms numerical simulations validate the accuracy of the theoretical results and support the behaviour of the considered algorithms

string submodular functions with curvature constraints consider the problem of choosing a string of actions to optimize an objective function that is string submodular it was shown in previous papers that the greedy strategy consisting of a string of actions that only locally maximizes the step wise gain in the objective function achieves at least a 1 e 1 approximation to the optimal strategy this paper improves this approximation by introducing additional constraints on curvature namely total backward curvature total forward curvature and elemental forward curvature we show that if the objective function has total backward curvature sigma then the greedy strategy achieves at least a 1 sigma 1 e sigma approximation of the optimal strategy if the objective function has total forward curvature epsilon then the greedy strategy achieves at least a 1 epsilon approximation of the optimal strategy moreover we consider a generalization of the diminishing return property by defining the elemental forward curvature we also introduce the notion of string matroid and consider the problem of maximizing the objective function subject to a string matroid constraint we investigate two applications of string submodular functions with curvature constraints 1 choosing a string of actions to maximize the expected fraction of accomplished tasks and 2 designing a string of measurement matrices such that the information gain is maximized

the mean value of a fuzzy number abstract the notion of expectation is defined for fuzzy numbers of intervals viewing them as consonant random sets this expectation is interval valued and remains additive in the sense of fuzzy interval addition as a by product some light is shed on the debate between probability and possibility theory and the problem of how to carry measure theoretic notions into the field of possibility theory

a numerical solution for multichannel detection a numerical approach to the vector signal detection problem is proposed by using an approximate series representation for vector complex valued random processes this technique allows us to provide computationally feasible suboptimum receivers for the problem of detecting an improper or proper complex signal in additive white noise its implementation requires the knowledge of the correlation matrix of the vector process involved the performance of the technique devised is assessed by some communication examples

on the behavior of the continuous time spectrogram for arbitrarily narrow windows the behavior of the continuous time ct spectrogram in particular of its first frequency moment ffm and instantaneous bandwidth ib in the limit of infinitely thin windows is revisited new insight is gained and novel quantitative results are presented power series in the width of the short time fourier transform stft window are derived for the ct spectrogram and its moments which are valid for any reasonably regular narrow window and which universally yield the instantaneous frequency if as the first term in the ffm expansion the well known result for ct signals according to which the spectrogram ffm tends to the if when the square of the stft window approaches a dirac delta delta function is thus extended indeed upon introduction of functional sequences depending on a small width parameter it is shown that the spectrogram ffm always tends to the if when the stft window becomes arbitrarily narrow with no actual need for the latter or any of its powers to converge to a delta function it is also checked within the framework here developed that the spectrogram ib behaves as expected going to infinity when the ffm approaches the if it is argued that when estimating the if of ct signals with the spectrogram it may be eventually more convenient to work with functions that become arbitrarily close to a delta function than to its square root whatever mathematical object this might be in addition the results of the analysis on the if and the ib are illustrated with a few examples of standard signal and window forms

quaternion var modelling and estimation quaternion vector autoregression var modeling is a natural extension of real and complex var we demonstrate how a quaternion var can be treated as a special case of structured real var we show that generalized least squares and under gaussianity maximum likelihood estimation of the parameters reduces to simple least squares estimation if the innovations are quaternion proper

generalized likelihood ratios for testing the properness of quaternion gaussian vectors in a recent paper the second order statistical analysis of quaternion random vectors has shown that there exist two different kinds of quaternion widely linear processing which are associated with the two main types of quaternion properness in this paper we consider the problem of determining from a finite number of independent vector observations whether a quaternion gaussian vector is proper or not specifically we derive three generalized likelihood ratio tests glrts for testing the two main kinds of quaternion properness and show that the glrts reduce to the estimation of three previously proposed quaternion improperness measures interestingly the three glrt statistics improperness measures can be interpreted as an estimate of the entropy loss due to the quaternion improperness additionally we analyze the case in which the orthogonal basis for the representation of the quaternion vector is unknown which results in the problem of estimating the principal bbc properness direction i e the pure unit quaternion minimizing the bbc improperness measure although this estimation problem is not convex we propose a technique based on successive convex approximations which can be solved in closed form finally some simulation examples illustrate the performance and practical application of the proposed tests

complex autoregressive model for shape recognition a complex autoregressive model for invariant feature extraction to recognize arbitrary shapes on a plane is presented a fast algorithm to calculate complex autoregressive coefficients and complex parcor coefficients of the model is also shown the coefficients are invariant to rotation around the origin and to choice of the starting point in tracing a boundary it is possible to make them invariant to scale and translation experimental results that the complicated shapes like nonconvex boundaries can be recognized in high accuracy even in the low order model it is seen that the complex parcor coefficients tend to provide more accurate classification than the complex ar coefficients

adaptive continuous time linear quadratic gaussian control the adaptive linear quadratic gaussian control problem where the linear transformation of the state a and the linear transformation of the control b are unknown is solved assuming only that a b is controllable and a q sub 1 sup 1 2 is observable where q sub 1 determines the quadratic form for the state in the integrand of the cost functional a weighted least squares algorithm is modified by using a random regularization to ensure that the family of estimated models is uniformly controllable and observable a diminishing excitation is used with the adaptive control to ensure that the family of estimates is strongly consistent a lagged certainty equivalence control using this family of estimates is shown to be self optimizing for an ergodic quadratic cost functional

fuzzy linear programming problems with fuzzy numbers abstract linear programming problems with fuzzy parameters are formulated by fuzzy functions the ambiguity considered here is not randomness but fuzziness which is associated with the lack of a sharp transition from membership to nonmembership parameters on constraint and objective functions are given by fuzzy numbers in this paper our object is the formulation of a fuzzy linear programming problem to obtain a reasonable solution under consideration of the ambiguity of parameters this fuzzy linear programming problem with fuzzy numbers can be regarded as a model of decision problems where human estimation is influential

mixture kalman filters in treating dynamic systems sequential monte carlo methods use discrete samples to represent a complicated probability distribution and use rejection sampling importance sampling and weighted resampling to complete the on line filtering task we propose a special sequential monte carlo method the mixture kalman filter which uses a random mixture of the gaussian distributions to approximate a target distribution it is designed for on line estimation and prediction of conditional and partial conditional dynamic linear models which are themselves a class of widely used non linear systems and also serve to approximate many others compared with a few available filtering methods including monte carlo methods the gain in efficiency that is provided by the mixture kalman filter can be very substantial another contribution of the paper is the formulation of many non linear systems into conditional or partial conditional linear form to which the mixture kalman filter can be applied examples in target tracking and digital communications are given to demonstrate the procedures proposed