systematic design and stability analysis of a fuzzy logic controller although the fuzzy logic controller has been adopted in many engineering applications its performance is not guaranteed since there is no definite stability analysis it may be the main factor that we hesitate to adopt the flc in critical applications in this paper a guideline of systematic analysis and design of an flc is presented for guaranteeing the stability and or the performance based on the well developed variable structure system theory we account for the relationship between the tracking accuracy of a control system and the i o fuzzy set membership shapes of the flc it is also possible to explain the performance or the stability of commonly used flcs through the proposed guideline for the design and analysis

discrete lagrange s equations and canonical equations based on the principle of least action by assuming that time is discrete discrete lagrange s equations and discrete canonical equations are derived on the basis of the principle of least action the discrete mechanics contains no differentiation and integration but contains difference quotients and summations the time interval is not required to be infinitesimal the discrete canonical equations conserve the hamiltonian exactly

high order schemes for conservative or dissipative systems we propose a new method for designing high order finite difference schemes that inherit conservation or dissipation properties from conservative or dissipative systems such as hamiltonian systems with without damping terms the proposed method has a feature that the computational costs of the resulting schemes do not increase in practice even when the order of accuracy is increased

two step stability analysis for general polynomial fuzzy model based control systems this paper investigates the stability of a polynomial fuzzy model based pfmb control system formed by a nonlinear plant represented by a polynomial fuzzy model and a polynomial fuzzy controller connected in a closed loop three cases of polynomial fuzzy controllers are proposed for the control process with the consideration of a matched mismatched number of rules and or premise membership functions which demonstrate different levels of controller complexity design flexibility and stability analysis results a general polynomial lyapunov function candidate is proposed to investigate the system stability unlike the published work there is no constraint on the polynomial lyapunov function candidate which is independent of the form of the polynomial fuzzy model thus it can be applied to a wider class of pfmb control systems and potentially produces more relaxed stability analysis results two step stability conditions in terms of sum of squares sos are obtained to numerically find a feasible solution to facilitate the stability analysis and relax the stability analysis result the boundary information of membership functions is taken into account in the stability analysis and incorporated into the sos based stability conditions simulation examples are given to illustrate the effectiveness of the proposed approach

new solution family of the jacobi equations characterization invariants and global darboux analysis a new family of skew symmetric solutions of the jacobi partial differential equations for finite dimensional poisson systems is characterized and analyzed such family has some remarkable properties firstly it is defined for arbitrary values of the dimension and the rank secondly it is described in terms of arbitrary differentiable functions namely it is not limited to a given degree of nonlinearity additionally it is possible to determine explicitly the fundamental properties of those solutions such as their casimir invariants and the algorithm for the reduction to the darboux canonical form which have been reported only for a very limited sample of finite dimensional poisson structures moreover such analysis is carried out globally in phase space thus improving the usual local scope of the darboux theorem

design of stable fuzzy control systems using lyapunov s method in fuzzy hypercubes in this paper a systematic procedure is presented to analyze and design a stable fuzzy controller for a class of nonlinear systems based on the lyapunov s directive method we account for the relationship between the stability of the control system and the i o sets of the fuzzy controller first the state space is partitioned into fuzzy hypercubes according to the product space of the input fuzzy sets of the states then all fuzzy hypercubes are considered and made to satisfy the lyapunov s stability condition by choosing proper output centers of the associated fuzzy rules through the proposed design guideline it is also possible to determine the performance and accuracy of fuzzy control systems an example with a fuzzy controller for a chaotic system is presented to illustrate the design procedure

relaxation methods for problems with strictly convex costs and linear constraints consider the problem of minimizing a strictly convex possibly nondifferentiable and nonseparable cost subject to linear constraints we propose a dual coordinate ascent method for this problem that uses inexact line search and either essentially cyclic or gauss southwell order of coordinate relaxation we show under very weak conditions that this method generates a sequence of primal vectors converging to the optimal primal solution under an additional regularity assumption and assuming that the effective domain of the cost function is polyhedral we show that a related sequence of dual vectors converges in cost to the optimal cost if the constraint set has an interior point in the effective domain of the cost function then this sequence of dual vectors is bounded and each of its limit points is an optimal dual solution when the cost function is strongly convex we show that the order of coordinate relaxation can become progressively more chaotic these results significantly improve upon those obtained previously

conservation laws of semidiscrete hamiltonian equations many evolution partial differential equations pdes can be cast into hamiltonian form conservation laws of these equations are related to one parameter hamiltonian symmetries admitted by the pdes p j olver applications of lie groups to differential equations springer new york 1986 in this paper we consider symmetries and noether s theorem for semidiscrete hamiltonian equations which are obtained by space discretization of hamiltonian pdes using symmetries one can find conservation laws of these equations several applications including a transfer equation and the korteweg de vries equation are presented

a symplectic integration algorithm for separable hamiltonian functions abstract we derive an algorithm to numerically integrate differential equations derivable from a separable hamiltonian function this symplectic algorithm is accurate to fourth order in the time step and preserves exactly the poincare cartan integral invariants associated with the topology of the phase flow we compare the efficiency and accuracy of this method to that of existing integrators both symplectic and non symplectic by integrating the equations of motion corresponding to a nonlinear pendulum a particle in the field of a standing wave and a harmonic oscillator perturbed by a plane wave

fuzzy rule based combination of linear and switching state feedback controllers this paper presents a fuzzy rule base combined controller which is a fuzzy rule based combination of linear and switching state feedback controllers for nonlinear systems subject to parameter uncertainties the switching state feedback controller is employed to drive the system states toward the origin when the system state approaches the origin the linear state feedback controller will gradually replace the switching state feedback controller the smooth transition between the linear and switching state feedback controllers is governed by the fuzzy rules by using the fuzzy rule based combination technique the proposed fuzzy rule base combined controller integrates the advantages of both the linear and switching state feedback controllers but eliminates their disadvantages as a result the proposed fuzzy controller provides good performance during the transient period and the chattering effect is removed when the system state approaches the origin stability conditions will be derived to guarantee the system stability furthermore a saturation function is employed to replace the switching component to alleviate the chattering during the transient period by using the proposed fuzzy rule based combination technique the steady state error introduced by the saturation function can be eliminated application examples will be given to show the merits of the proposed approach

two classes of linearly implicit local energy preserving approach for general multi symplectic hamiltonian pdes abstract two classes of efficient and robust schemes are proposed for the general multi symplectic hamiltonian systems using the invariant energy quadratization ieq approach the schemes are linear second order accurate local energy preserving and preserve the global energy they are not restricted to specific forms of the nonlinear part of the state function and only require solving linear equations at each time step we applied the new schemes to various multi symplectic hamiltonian pdes to demonstrate their effectiveness computational efficiency and accuracy

maximum entropy in data analysis with error carrying constraints the problem of finding the maximum entropy solution which satisfies some poorly defined constraints is central to data processing hitherto all attempts to solve it have been conceptually somewhat incomplete the authors consider the traditional boltzmann gibbs approach to entropy when data are available the various outcomes can be assigned a priori probabilities or weights according to how much each of them agrees with the measured distribution when errors are taken into account the reasoning yields a statistic s chi 2 which is a precise expression for the entropy the maximum s chi 2 solution defines a distribution uniquely even in circumstances when there is no information apart from those provided by the data the inferred distribution differs from the given by the data mean values and bears a closer resemblance to the underlying population than do the raw data where additional constraints are present this maximum entropy procedure is more effective in incorporating them than are traditional maximum likelihood techniques s chi 2 maximisation may also be carried out iteratively up to a certain limit it offers the most advantage in situations when one has a thorough understanding of the inherent noise behaviour

partially finite programming in l_1 and the existence of maximum entropy estimates best entropy estimation is a technique that has been widely applied in many areas of science it consists of estimating an unknown density from some of its moments by maximizing some measure of the entropy of the estimate this problem can be modelled as a partially finite convex program with an integrable function as the variable a complete duality and existence theory is developed for this problem and for an associated extended problem which allows singular measure theoretic solutions this theory explains the appearance of singular components observed in the literature when the burg entropy is used it also provides a unified treatment of existence conditions when the burg boltzmann shannon or some other entropy is used as the objective some examples are discussed

maximum entropy and spectral estimation a review the method of maximum entropy is reviewed with emphasis on its relationship to entropy rate wiener filters autoregressive processes extrapolation the levinson algorithm lattice all pole and all pass filters and stability

multisignal minimum cross entropy spectrum analysis with weighted initial estimates this paper presents a generalization of multisignal minimum cross entropy spectrum analysis multisignal relative entropy spectrum analysis multisignal resa a method for simultaneously estimating a number of power spectra when an initial prior estimate of each is available and new information is obtained in the form of values of the sum of their autocorrelation functions the generalization involves attaching to each initial spectrum estimate a possibly frequency dependent weighting parameter that indicates its relative reliability i e the relative degree of belief associated with the initial estimate mathematical properties of the generalized method are discussed and illustrative numerical examples are given

the maximum entropy configuration in the case of randomly fluctuating constraints an attempt is made to crystallise the meaning of the maximum entropy state when a system is subject to constraints which carry random uncertainties such a state is well defined and there exists a simple statistic which allows it to be determined unambiguously provided it is assumed that one s state of knowledge of the system fluctuates in close resemblance to the quantum uncertainty principle without loss of generality the scenario is verified for a variety of error carrying constraints by results from a large number of random die samples

an extended discrete gradient formula for oscillatory hamiltonian systems in this paper incorporating the idea of the discrete gradient method into the extended runge kutta nystrom integrator we derive and analyze an extended discrete gradient formula for the oscillatory hamiltonian system with the hamiltonian where represents generalized positions represents generalized momenta and is a symmetric and positive semi definite matrix the solution of this system is a nonlinear oscillator basically many nonlinear oscillatory mechanical systems with a partitioned hamiltonian function lend themselves to this approach the extended discrete gradient formula presented in this paper exactly preserves the energy h p q we derive some properties of the new formula the convergence is analyzed for the implicit schemes based on the discrete gradient formula and it turns out that the convergence of the implicit schemes based on the extended discrete gradient formula is independent of m which is a significant property for the oscillatory hamiltonian system thus it transpires that a larger step size can be chosen for the new energy preserving schemes than that for the traditional discrete gradient methods when applied to the oscillatory hamiltonian system illustrative examples show the competence and efficiency of the new schemes in comparison with the traditional discrete gradient methods in the scientific literature

mechanical systems subject to holonomic constraints differential algebraic formulations and conservative integration abstract the numerical integration in time of the equations of motion for mechanical systems subject to holonomic constraints is considered schemes are introduced for the direct treatment of a differential algebraic form of the equations that preserve the constraints the total energy and other integrals such as linear and angular momentum arising from affine symmetries moreover the schemes can be shown to preserve the property of time reversibility in an appropriate sense an example is given to illustrate various aspects of the proposed methods

design of a fuzzy controller with fuzzy sliding surface abstract fuzzy logic controllers have succeeded in many control problems that the conventional control theories have difficulties to deal with however the design of the fuzzy logic controllers depends to a large extent on the expert s knowledge or on trial and error moreover due to the linguistic expression of the fuzzy controller it has not been easy to guarantee the stability and robustness of the fuzzy control systems to overcome these drawbacks we propose to design a fuzzy controller with the fuzzy sliding surface the fuzzy rules are defined based on this fuzzy sliding surface and the output of the fuzzy controller is inferred by the proper compositional rule of inference then we show the stability of this fuzzy control system and the boundedness of the tracking error by using the lyapunov theory finally we apply this fuzzy controller to control a nonlinear time varying system and confirm the validity of the controller

brief paper stabilization and h control of nonlinear port controlled hamiltonian systems subject to actuator saturation this paper investigates the stabilization and h control of nonlinear port controlled hamiltonian pch systems subject to actuator saturation and proposes a number of new results first the stabilization problem is studied and a control design method is developed by using both the dissipative hamiltonian structural and saturating actuator properties second for the case that there are external disturbances in the systems an h controller is designed to attenuate the disturbances finally the results obtained for hamiltonian systems are applied to the stabilization and h control of nonlinear affine systems subject to actuator saturation and several interesting results are presented study of an example of power system with simulations shows that the controller proposed in this paper is effective

hamiltonian systems discrete time approximation losslessness passivity and composability abstract in this paper a passive integrator dedicated to input output hamiltonian systems approximation is presented in a first step a discrete hamiltonian framework endowed with a lie derivative like formula is introduced it is shown that the discrete dynamics encodes energy conservation and passivity additionally the characterization of the discrete dynamics in terms of dirac structure is shown to be invariant by interconnection the class is thus composable networked systems belong to the class in a second step the discrete dynamics is considered as a one step integration method the method is shown to be convergent and provides a discrete time approximation of an input output hamiltonian system accordingly the discrete dynamics inherits intrinsic energetic characteristics storage function and dissipation rate from the original system the method is thus tagged as passive integrator as an illustration the closed loop behavior of interconnected subsystems and the stabilization of a rigid body spinning around its center of mass are presented

explicit volume preserving and symplectic integrators for trigonometric polynomial flows we introduce explicit volume preserving and symplectic integrators for the case of generalized trigonometric polynomial flows the method is demonstrated using the arter flow and computational trials are conducted using a 4 dimensional vector field

a stabilization of multistep linearly implicit schemes for dissipative systems we consider numerical integration of dissipative gradient systems for such systems a class of special stable integrators that strictly maintain dissipation is known but they generally yield expensive fully implicit schemes and when the system is large linearization is indispensable for practical efficiency however this can in turn destroy the originally expected stability and so far no effective principle has been formulated for a stable linearization in this note we point out that the behavior of the linearized schemes can be understood from a dynamical systems theory viewpoint and propose a simple principle for a stable linearization

stability analysis of polynomial fuzzy model based control systems using switching polynomial lyapunov function this paper investigates the stability problem of polynomial fuzzy model based control system which is formed by a polynomial fuzzy model and a polynomial fuzzy controller connected in a closed loop a switching polynomial lyapunov function consisting of a number of local polynomial lyapunov functions is proposed to investigate the system stability it demonstrates a nice property in favor of the stability analysis that each local polynomial lyapunov function transits continuously to each other as different local polynomial lyapunov functions are employed to investigate the system stability according to the operating domain relaxed stability conditions compared with the stability analysis result with a common lyapunov function can be developed in order to allow a greater design flexibility for the polynomial fuzzy controller the proposed polynomial fuzzy model based control scheme does not require that both the polynomial fuzzy model and polynomial fuzzy controller share the same premise membership functions stability conditions in terms of sum of squares are obtained to guarantee system stability and facilitate control synthesis simulation examples are given to verify the stability analysis results and demonstrate the effectiveness of the proposed polynomial fuzzy control scheme

on the linear convergence of descent methods for convex essentially smooth minimization consider the problem of minimizing over a polyhedral set the composition of an affine mapping with a strictly convex essentially smooth function a general result on the linear convergence of descent methods for solving this problem is presented by applying this result the linear convergence of both the gradient projection algorithm of goldstein and levitin and polyak and a matrix splitting algorithm using regular splitting is established the results do not require that the cost function be strongly convex or that the optimal solution set be bounded the key to the analysis lies in a new error bound for estimating the distance from a feasible point to the optimal solution set

a lyapunov type theorem for dissipative numerical integrators with adaptive time stepping the asymptotic behavior of continuous dissipative systems and dissipative numerical integrators with fixed time stepping can be fully investigated by a lyapunov type theorem on continuous and discrete dynamical systems respectively however once adaptive time stepping is involved such theories cease to work and usually the dynamics should be investigated from the past instead of the standard forward way such as in terms of pullback attractors in this paper we present a different approach we stick to a forward definition of limit sets and show that still we can establish a lyapunov type theorem which reveals the precise asymptotic behavior of adaptive time stepping integrators in the presence of a discrete lyapunov functional

inherently energy conserving time finite elements for classical mechanics in this paper we develop a finite element method for the temporal discretization of the equations of motion the continuous galerkin method is based upon a weighted residual statement of hamilton s canonical equations we show that the proposed finite element formulation is energy conserving in a natural sense a family of implicit one step algorithms is generated by specifying the polynomial approximation in conjunction with the quadrature formula used for the evaluation of time integrals the numerical implementation of linear quadratic and cubic time finite elements is treated in detail for the model problem of a circular pendulum in addition to that concerning dynamical systems with several degrees of freedom we address the design of nonstandard quadrature rules which retain the energy conservation property our numerical investigations assess the effect of numerical quadrature in time on the accuracy and energy conservation property of the time stepping schemes

hamiltonian conserving discrete canonical equations based on variational difference quotients abstract new discrete mechanics based on the assumption of the discrete time is proposed the discrete mechanics does not contain any continuous differentiation but contains only difference quotients resulting discrete hamiltonian s canonical equations are single time step difference equations and exactly conserve the hamiltonian the canonical equations give the numerical results more accurately than the heun scheme and the 4th order runge kutta scheme

minimum cross entropy spectral analysis the principle of minimum cross entropy minimum directed divergence minimum discrimination information minimum relative entropy is summarized discussed and applied to the classical problem of estimating power spectra given values of the autocorrelation function this new method differs from previous methods in its explicit inclusion of a prior estimate of the power spectrum and it reduces to maximum entropy spectral analysis as a special case the prior estimate can be viewed as a means of shaping the spectral estimator cross entropy minimization yields a family of shaped spectral estimators consistent with known autocorrelations results are derived in two equivalent ways once by minimizing the cross entropy of underlying probability densities and once by arguments concerning the cross entropy between the input and output of linear filters several example minimum cross entropy spectra are included

a new look at entropy for solving linear inverse problems entropy based methods are widely used for solving inverse problems particularly when the solution is known to be positive here we address linear ill posed and noisy inverse problems of the form z ax n with a general convex constraint x spl isin x where x is a convex set although projective methods are well adapted to this context we study alternative methods which rely highly on some information based criteria our goal is to clarify the role played by entropy in this field and to present a new point of view on entropy using general tools and results coming from convex analysis we present then a new and broad scheme for entropic based inversion of linear noisy inverse problems this scheme was introduced by navaza in 1985 in connection with a physical modeling for crystallographic applications and further studied by dacunha castelle and gamboa 1990 important features of this paper are i a unified presentation of many well known reconstruction criteria ii proposal of new criteria for reconstruction under various prior knowledge and with various noise statistics iii a description of practical inversion of data using the aforementioned criteria and iv a presentation of some reconstruction results

stable adaptive fuzzy control of nonlinear systems a direct adaptive fuzzy controller that does not require an accurate mathematical model of the system under control is capable of incorporating fuzzy if then control rules directly into the controllers and guarantees the global stability of the resulting closed loop system in the sense that all signals involved are uniformly bounded is developed the specific formula for the bounds is provided so that controller designers can determine the bounds based on their requirements the direct adaptive fuzzy controller is used to regulate an unstable system to the origin and to control the duffing chaotic system to track a trajectory the simulation results show that the controller worked without using any fuzzy control rules and that after fuzzy control rules were incorporated the adaptation speed became much faster it is shown explicitly how the supervisory control forces the state to remain within the constraint set and how the adaptive fuzzy controller learns to regain control

fuzzy hierarchical swing up and sliding position controller for the inverted pendulum cart system in this paper a fuzzy hierarchical swing up and sliding position controller is proposed for the swing up and position controls of an inverted pendulum cart system the proposed fuzzy hierarchical swing up and sliding position controller includes a fuzzy switching controller a fuzzy swing up controller and a twin fuzzy sliding position controller the fuzzy switching controller is designed to smoothly switch between the swing up and position controls with the inverted pendulum cart system decomposed into a pendulum subsystem and a cart subsystem fuzzy sliding position controllers are presented to robustly balance the pendulum and cart at the desired positions the sliding mode and stability of the fuzzy sliding position control systems are guaranteed moreover an adaptive mechanism is provided to on line adjust the parameters in the fuzzy sliding position controllers simulation results are included to illustrate the effectiveness of the proposed fuzzy hierarchical swing up and sliding position controller

energy preserving numerical schemes of high accuracy for one dimensional hamiltonian systems we present a class of non standard numerical schemes which are modifications of the discrete gradient method they preserve the energy integral exactly up to the round off error the considered class contains locally exact discrete gradient schemes and integrators of arbitrary high order in numerical experiments we compare our integrators with some other numerical schemes including the standard discrete gradient method the leap frog scheme and a symplectic scheme of fourth order we study the error accumulation for a very long time and the conservation of the energy integral

on the equivalence between sor type methods for linear systems and the discrete gradient methods for gradient systems abstract inspired by the iterative nature of many discretization methods for continuous dynamical systems connections between iterative numerical methods in numerical linear algebra and continuous dynamical systems have been studied since 1970s for stationary iterative methods solving linear systems chu 1988 2008 discussed a connection to continuous dynamical systems by using the explicit euler method however further understanding of stationary iterative methods might be limited due to the use of the explicit euler method this paper presents a new connection based on the so called discrete gradient methods between sor type methods and gradient systems there the key of the discussion is the equivalence between sor type methods and the discrete gradient methods applied to gradient systems the discussion leads to new interpretations for sor type methods for example a new derivation of sor type methods is found these methods monotonically decrease a certain quadratic function and a new interpretation of the relaxation parameter is obtained besides while studying the new connection a new discrete gradient is also obtained

new four dimensional solutions of the jacobi equations for poisson structures a four dimensional family of skew symmetric solutions of the jacobi equations for poisson structures is characterized as a consequence previously known types of poisson structures found in a diversity of physical situations appear to be obtainable as particular cases of new family of solutions additionally it is possible to apply constructive methods for the explicit determination of fundamental properties of those solutions such as their casimir invariants symplectic structure and the algorithm for the reduction to the darboux canonical form which have been reported only for a limited sample of known finite dimensional poisson structures moreover the results developed are valid globally in phase space thus ameliorating the usual scope of darboux theorem which is of local nature

stability issues on takagi sugeno fuzzy model parametric approach the main objective of this research is to provide a systematic stability analysis technique for fuzzy systems using well established results from robust stability of a family of polynomials the model based takagi sugeno 1985 fuzzy control system is considered and shown to have a dependent coefficient structure involving firing strengths two kharitonov regions via overbounding are established to facilitate the extreme results for stability test in the space of polynomials as well as the complex plane the criterion obtained is a sufficient condition for a fuzzy logic controller three nonlinear examples are demonstrated to show that the systematic approach proposed here can be used to solve fuzzy stability problems fuzzy users concerned over stability issues may find these analytic tools useful

which is the better entropy expression for speech processing s log s or log s in maximum entropy spectral analysis mesa one maximizes the integral of logs f where s f is a power spectrum the resulting spectral estimate which is equivalent to that obtained by linear prediction and other methods is popular in speech processing applications an alternative expression s f logs f is used in optical processing and elsewhere this paper considers whether the alternative expression leads to spectral estimates useful in speech processing we investigate the question both theoretically and empirically the theoretical investigation is based on generalizations of file two estimates the generalizations take into account prior estimates of the unknown power spectrum it is shown that both estimates result from applying a generalized version of the principle of maximum entropy but they differ concerning the quantities that are treated as random variables the empirical investigation is based on speech synthesized using the different spectral estimates although both estimates lead to intelligible speech speech based on the mesa estimate is qualitatively superior

exponential integrators with quadratic energy preservation for linear poisson systems abstract motivated by the recent paper wang and wu 2018 12 where the authors proposed a class of functionally fitted energy preserving integrators for poisson systems in this paper we aim at developing exponential integrators with quadratic energy preservation for the important class of linear poisson systems by rewriting the linear poisson system in a new formulation we apply the symplectic exponential runge kutta methods and thus obtain a novel class of exponential integrators for the linear poisson system it is proved that the obtained exponential integrator can exactly preserve the quadratic energy in comparison with the functionally fitted energy preserving methods whose numerical performance depends on the fitted frequency the new exponential integrators are independent of the fitted frequency and convenient to use furthermore we investigate the convergence of the fixed point iteration used in the implementation of the proposed exponential integrator and show that the convergence region of the stepsize is independent of the norm of the skew symmetric matrix finally numerical results with the euler equations of a free rigid body soundly support the remarkable performance of the exponential integrator in both the global errors and the energy preservation

on foundation of the generalized nambu mechanics we outline basic principles of a canonical formalism for the nambu mechanics a generalization of hamiltonian mechanics proposed by yoichiro nambu in 1973 it is based on the notion of a nambu bracket which generalizes the poisson bracket a binary operation on classical observables on the phase space to the multiple operation of higher ordern 3 nambu dynamics is described by the phase flow given by nambu hamilton equations of motion a system of ode s which involvesn 1 hamiltonians we introduce the fundamental identity for the nambu bracket a generalization of the jacobi identity as a consistency condition for the dynamics we show that nambu bracket structure defines a hierarchy of infinite families of subordinated structures of lower order including poisson bracket structure which satisfy certain matching conditions the notion of nambu bracket enables us to define nambu poisson manifolds phase spaces for the nambu mechanics which turn out to be more rigid than poisson manifolds phase spaces for the hamiltonian mechanics we introduce the analog of the action form and the action principle for the nambu mechanics in its formulation dynamics of loops n 2 dimensional chains for the generaln ary case naturally appears we discuss several approaches to the quantization of nambu mechanics based on the deformation theory path integral formulation and on nambu heisenberg commutation relations in the latter formalism we present an explicit representation of the nambu heisenberg relation in then 3 case we emphasize the role ternary and higher order algebraic operations and mathematical structures related to them play in passing from hamilton s to nambu s dynamical picture

control by interconnection and standard passivity based control of port hamiltonian systems the dynamics of many physical processes can be suitably described by port hamiltonian ph models where the importance of the energy function the interconnection pattern and the dissipation of the system is underscored to regulate the behavior of ph systems it is natural to adopt a passivity based control pbc perspective where the control objectives are achieved shaping the energy function and adding dissipation in this paper we consider the pbc techniques of control by interconnection cbi and standard pbc in cbi the controller is another ph system connected to the plant through a power preserving interconnection to add up their energy functions while in standard pbc energy shaping is achieved via static state feedback in spite of the conceptual appeal of formulating the control problem as the interaction of dynamical systems the current version of cbi imposes a severe restriction on the plant dissipation structure that stymies its practical application on the other hand standard pbc which is usually derived from a uninspiring and non intuitive ldquopassive output generationrdquo viewpoint is one of the most successful controller design techniques the main objectives of this paper are 1 to extend the cbi method to make it more widely applicable in particular to overcome the aforementioned dissipation obstacle 2 to show that various popular variants of standard pbc can be derived proceeding from a unified perspective 3 to establish the connections between cbi and standard pbc proving that the latter is obtained restricting the former to a suitable subset providing a nice geometric interpretation to standard pbc and comparing the size of the set of ph plants for which they are applicable

brief series expansions for analytic systems linear in control this paper presents a series expansion for the evolution of a class of nonlinear systems characterized by constant input vector fields we present a series expansion that can be computed via explicit recursive expressions and we derive sufficient conditions for uniform convergence over the finite and infinite time horizon furthermore we present a simplified series and convergence analysis for the setting of second order polynomial vector fields the treatment only relies on elementary notions on analytic functions number theory and operator norms

locally exact modifications of numerical schemes we present a new class of exponential integrators for ordinary differential equations locally exact modifications of known numerical schemes local exactness means that they preserve the linearization of the original system at every point in particular locally exact integrators preserve all fixed points and are a stable we apply this approach to popular schemes including euler schemes the implicit midpoint rule and the trapezoidal rule we found locally exact modifications of discrete gradient schemes for symmetric discrete gradients and coordinate increment discrete gradients preserving their main geometric property exact conservation of the energy integral for arbitrary multidimensional hamiltonian systems in canonical coordinates numerical experiments for a two dimensional anharmonic oscillator show that locally exact schemes have very good accuracy in the neighbourhood of stable equilibrium much higher than suggested by the order of new schemes locally exact modification sometimes increases the order but in many cases leaves it unchanged

local dissipative hamiltonian description of reversible reaction networks in this letter we show that closed reversible chemical reaction networks with independent elementary reactions admit a global pseudo hamiltonian structure which is at least locally dissipative around any equilibrium point the structure matrix of the hamiltonian description reflects the graph topology of the reaction network and it is a smooth function of the concentrations of the chemical species in the positive orthant the physical interpretation of the description is briefly explained and two illustrative examples are presented for global and local dissipative hamiltonian description respectively

preserving multiple first integrals by discrete gradients we consider systems of ordinary differential equations with known first integrals the notion of a discrete tangent space is introduced as the orthogonal complement of an arbitrary set of discrete gradients integrators which exactly conserve all the first integrals simultaneously are then defined in both cases we start from an arbitrary method of a prescribed order say a runge kutta scheme and modify it using two approaches one is based on projection and the other on local coordinates the methods are tested on the kepler problem

decoupled fuzzy sliding mode control a decoupled fuzzy sliding mode controller design is proposed the decoupled method provides a simple way to achieve asymptotic stability for a class of fourth order nonlinear systems with only five fuzzy control rules the ideas behind the controller are as follows first decouple the whole system into two second order systems such that each subsystem has a separate control target expressed in terms of a sliding surface then information from the secondary target conditions the main target which in turn generates a control action to make both subsystems move toward their sliding surface respectively a closely related fuzzy controller to the sliding mode controller is also presented to show the theoretical aspect of the fuzzy approach in which the characteristics of fuzzy sets are determined analytically to ensure the stability and robustness of the fuzzy controller finally the decoupled sliding mode control smc is used to control three highly nonlinear systems and confirms the validity of the proposed approach

exponential collocation methods for conservative or dissipative systems abstract in this paper we propose and analyse a novel class of exponential collocation methods for solving conservative or dissipative systems based on exponential integrators and collocation methods it is shown that these novel methods can be of arbitrarily high order and exactly or approximately preserve first integrals or lyapunov functions we also consider order estimates of the new methods furthermore we explore and discuss the application of our methods in important stiff gradient systems and it turns out that our methods are unconditionally energy diminishing and strongly damped even for very stiff gradient systems practical examples of the new methods are derived and the efficiency and superiority are confirmed and demonstrated by three numerical experiments including a nonlinear schrodinger equation as a byproduct of this paper arbitrary order trigonometric rkn collocation methods are also presented and analysed for second order highly oscillatory general systems the paper is accompanied by numerical results that demonstrate the great potential of this work

nonlinear control via approximate input output linearization the ball and beam example approximate input output linearization of nonlinear systems which fail to have a well defined relative degree is studied for such systems a method for constructing approximate systems that are input output linearizable is provided the analysis presented is motivated through its application to a common undergraduate control laboratory experiment the ball and beam where it is shown to be more effective for trajectory tracking than the standard jacobian linearization

on a class of discretizations of hamiltonian nonlinear partial differential equations abstract we present a new class of discretizations of partial differential equations pdes that preserve a momentum like integral of the motion this results in an effective translational invariance for the dynamical problem and the absence of a peierls nabarro barrier that is usually present in discretizations of hamiltonian pdes a general method to construct such discretizations for any nonlinearity is given and the properties of the resulting differential difference equations are analyzed in a number of different cases for nonlinear klein gordon as well for nonlinear schrodinger type systems while for the former nonintegrability of the dynamical problem is evident in the latter case numerical evidence suggests that the behavior is close to the one of integrable systems we also show how static solutions of the equations can be constructed for these discretizations and discuss the similarities and differences of the method with previously reported ones

stability properties of a general class of nonlinear dynamical systems we establish sufficient conditions for the boundedness of the trajectories and the stability of the fixed points in a class of general nonlinear systems the so called quasi polynomial vector fields with the help of a natural embedding of such systems in a family of generalized lotka volterra lv equations a purely algebraic procedure is developed to determine such conditions we apply our method to obtain new results for lv systems by a reparametrization in time variable and to study general nonlinear vector fields originally far from the lv format

unified approach to hamiltonian systems poisson systems gradient systems and systems with lyapunov functions or first integrals systems with a first integral i e constant of motion or a lyapunov function can be written as linear gradient systems dot x l x nabla v x for an appropriate matrix function l with a generalization to several integrals or lyapunov functions the discrete time analogue delta x delta t l bar nabla v where bar nabla is a discrete gradient preserves v as an integral or lyapunov function respectively

minimum cross entropy inversion spectrum analysis and classification introduction and discussion of possible geophysical applications abstract the principle of minimum cross entropy mce is a general method of inference about unknown probability densities given information in the form of constraints on expected values the background justification and properties of mce inversion are summarized briefly some recent applications in spectrum analysis and pattern classification are then summarized and some speculations on the application of mce to general inversion problems in geophysics are offered

decoupled local global energy preserving schemes for the n coupled nonlinear schrodinger equations abstract we develop two local energy preserving integrators and a global energy preserving integrator for the general multisymplectic hamiltonian system when applied to the 1d and multi dimensional n coupled nonlinear schrodinger equations the given schemes have the exact preservation of the local global conservation law and are decoupled in the components n n 1 2 n i e each of the components can be solved independently the decoupled feature is significant and helpful for overcoming the computational difficulty of the n coupled n 3 nonlinear schrodinger equations especially of the multi dimensional case the composition method is employed to improve the accuracy of the schemes in time and the discrete fast fourier transform is used to reduce the computational complexity several numerical experiments are carried out to exhibit the behaviors of the wave solutions numerical results confirm the theoretical results

maximum entropy reconstruction using derivative information part 2 computational results summary maximum entropy density estimation a technique for reconstructing an unknown density function on the basis of certain measurements has applications in various areas of applied physical sciences and engineering here we present numerical results for the maximum entropy inversion program based on a new class of information measures which are designed to control derivative values of the unknown densities

on the energy theorem for the lagrange equations in the discrete case a method for discretizing lagrange equations is proposed in the conservative case the solutions of the system obtained this way verify the energy integral exactly

integrals of quadratic ordinary differential equations in r3 the lotka volterra system abstract a method already introduced by the last two authors for finding the integrable cases of three dimensional autonomous ordinary differential equations based on the frobenius integrability theorem is described in detail using this method and computer algebra the so called three dimensional lotka volterra system is studied many cases of integrability are thus found the study of this system is completed by the application of painleve analysis and the jacobi last multiplier method the methods used are of general interest and can be applied to many other systems

decoupled fuzzy controller design with single input fuzzy logic a decoupled fuzzy controller design with single input fuzzy logic is proposed we will utilize the approach of the single input fuzzy logic to explore the fourth order nonlinear systems in addition five fuzzy rules are given to control a class of fourth order nonlinear systems using this approach the system can achieve asymptotic stability and the response of system will converge faster than that of previous reports two simulation studies of a cart pole system and a ball beam system are presented to demonstrate the effectiveness of the method

maximum entropy data analysis another derivation of s χ2 an earlier work suggested a statistic s 2 as the key to maximum entropy data analysis in the simplest experimental situations the present letter reinforces such a claim by offering an alternative proof slightly more elaborate inference procedures are also treated together with a critique of currently available techniques

improving the accuracy of the discrete gradient method in the one dimensional case we present two numerical schemes of high accuracy for one dimensional dynamical systems they are modifications of the discrete gradient method and keep its advantages including stability and conservation of the energy integral however their accuracy is higher by several orders of magnitude

properties of cross entropy minimization the principle of minimum cross entropy minimum directed divergence minimum discrimination information is a general method of inference about an unknown probability density when there exists a prior estimate of the density and new information in the form of constraints on expected values various fundamental properties of cross entropy minimization are proven and collected in one place cross entropy s well known properties as an information measure are extended and strengthened when one of the densities involved is the result of cross entropy minimization the interplay between properties of cross entropy minimization as an inference procedure and properties of cross entropy as an information measure is pointed out examples are included and general analytic and computational methods of finding minimum cross entropy probability densities are discussed

robust control by fuzzy sliding mode abstract most fuzzy controllers fcs for nonlinear second order systems are designed with a two dimensional phase plane in mind we show that the performance and the robustness of this kind of fc stems from their property of driving the system into the sliding mode sm in which the controlled system is invariant to parameter fluctuations and disturbances additionally the continuous distribution of the control values in the phase plane causes a behavior similar to that of a sliding mode controller smc with a boundary layer bl near the switching line this gives assured tracking quality even in the presence of high model uncertainties tracing the fc back to the principle of an smc one obtains evidence about the stability of the closed loop system the choice of the scaling factors for the crisp inputs and outputs can be guided by the comparison of the fc with the smc and with the modified smc respectively at the end of the paper an fc for a higher order system is proposed simulation results show the practicability of the method

rate distortion speech coding with a minimum discrimination information distortion measure an information theory approach to the theory and practice of linear predictive coded lpc speech compression systems is developed it is shown that a traditional lpc system can be viewed as a minimum distortion or nearest neighbor system where the distortion measure is a minimum discrimination information between a speech process model and an observed frame of actual speech this distortion measure is used in an algorithm for computer aided design of block source codes subject to a fidelity criterion to obtain a 750 bits s speech compression system that resembles an lpc system but has a much lower rate a larger memory requirement and requires no on line lpc analysis quantitative and informal subjective comparisons are made among our system and lpc systems

approximation theory of fuzzy systems spl minus siso case in this paper the approximation properties of mimo fuzzy systems generated by the product inference are discussed we first give an analysis of fuzzy basic functions fbf s and present several properties of fbf s based on these properties of fbf s we obtain several basic approximation properties of fuzzy systems 1 basic approximation property which reveals the basic approximation mechanism of fuzzy systems 2 uniform approximation bounds which give the uniform approximation bounds between the desired control or decision functions and fuzzy systems 3 uniform convergent property which shows that fuzzy systems with defined approximation accuracy can always be obtained by dividing the input space into finer fuzzy regions and 4 universal approximation property which shows that fuzzy systems are universal approximators and extends some previous results on this aspect the similarity between fuzzy systems and mathematical approximation is discussed and an idea to improve approximation accuracy is suggested based on uniform approximation bounds

a new class of energy preserving numerical integration methods the first ever energy preserving b series numerical integration method for ordinary differential equations is presented and applied to several hamiltonian systems related novel lie algebraic results are also discussed

discrete gradient methods for solving odes numerically while preserving a first integral we show that all direct methods for preserving a first integral during the numerical integration of an ordinary differential equation fit into the unified framework of discrete gradient methods using this framework we construct several new integral preserving schemes

preserving algebraic invariants with runge kutta methods abstract we study runge kutta methods for the integration of ordinary differential equations and the retention of algebraic invariants as a general rule we derive two conditions for the retention of such invariants the first is a condition on the coefficients of the methods the second is a pair of partial differential equations that otherwise must be obeyed by the invariant this paper extends previous work on multistep methods in iserles technical report na1997 13 damtp university of cambridge 1997 the cases related to the retention of quadratic and cubic invariants perhaps of greatest relevance in applications are thoroughly discussed we conclude recommending a generalized class of runge kutta schemes namely lie group type runge kutta methods these are schemes for the solution of odes on lie groups but can be employed together with group actions to preserve a larger class of algebraic invariants without restrictions on the coefficients

maximum entropy image reconstruction in x ray and diffraction tomography the authors propose a bayesian approach with maximum entropy me priors to reconstruct an object from either the fourier domain data the fourier transform of diffracted field measurements in the case of diffraction tomography or directly from the original projection data in the case of x ray tomography the objective function obtained is composed of a quadratic term resulting from chi sup 2 statistics and an entropy term that is minimized using variational techniques and a conjugate gradient iterative method the computational cost and practical implementation of the algorithm are discussed some simulated results in x ray and diffraction tomography are given to compare this method to the classical ones

a numerical integrator for the two fixed centres problem conserving all constants of motion the two fixed centres problem describes the motion of a particle influenced by the gravitational pull of two fixed particles this is a well known integrable hamiltonian system having three constants of motion this paper presents a discrete two dimensional two fixed centres problem which conserves two of these constants of motion moreover this discrete system preserves the discrete analogue of the third constant of motion a canonical transformation is introduced to separate the variables and remove singularities of the system a combination of this transformation and energy preserving method plays an important role in deriving the discrete system with a variable time step

integral preserving integrators ordinary differential equations having a first integral may be solved numerically using one of several methods with the integral preserved to machine accuracy one such method is the discrete gradient method it is shown here that the order of the method can be bootstrapped repeatedly to higher orders of accuracy the method is illustrated using the henon heiles system

a stability approach to fuzzy control design for nonlinear systems abstract by the application of fuzzy set theory a method for feedback control of nonlinear systems based on the principle of sliding mode control is developed called fuzzy sliding mode control the control method provides a simple way to achieve asymptotic stability of the systems other attractive features of the method include a minimal realization of the fuzzy controller and robustness against model uncertainties and external disturbances in addition the method is capable of handling the chattering problem inherent to sliding mode control simply and effectively a simulation study of an inverted pendulum system is presented to demonstrate these features of the method also it is indicated that the method is easy to use for practicing control engineers

reconstruction from partial information with applications to tomography the problem of reconstruction from projections in hilbert space is treated an axiomatic basis is considered which leads to techniques which provide improved reconstructions by incorporating prior knowledge to tailor the hilbert space to the problem at hand when applied to reconstruction of the fourier transform of a function sampled at finitely many discrete points the procedures lead to previously derived optimal estimation techniques when applied to x ray tomography the procedures lead to new reconstruction techniques which are shown to include as a special case the minimum energy reconstruction of logan and shepp duke math j 42 1975 pp 645 659

mechanical integrators derived from a discrete variational principle many numerical integrators for mechanical system simulation are created by using discrete algorithms to approximate the continuous equations of motion in this paper we present a procedure to construct time stepping algorithms that approximate the flow of continuous odes for mechanical systems by discretizing hamilton s principle rather than the equations of motion the discrete equations share similarities to the continuous equations by preserving invariants including the symplectic form and the momentum map we first present a formulation of discrete mechanics along with a discrete variational principle we then show that the resulting equations of motion preserve the symplectic form and that this formulation of mechanics leads to conservation laws from a discrete version of noether s theorem we then use the discrete mechanics formulation to develop a procedure for constructing symplectic momentum mechanical integrators for lagrangian systems with holonomic constraints we apply the construction procedure to the rigid body and the double spherical pendulum to demonstrate numerical properties of the integrators

numerical simulation for a nonlinear partial differential equation with variable coefficients by means of the discrete variational derivative method partial differential equations with possibly discontinuous coefficients play an important part in engineering physics and ecology in this paper we will study nonlinear partial differential equations with variable coefficients arising from population models generally speaking it is difficult to analyze the behavior of nonlinear partial differential equations therefore we usually rely on the numerical approximation currently there is an increasing interest in designing numerical schemes that preserve energy properties for differential equations we will design the numerical schemes that preserve discrete energy property and show numerical experiments for a nonlinear partial differential equation with variable coefficients

long time behaviour of discretizations of the simple pendulum equation we compare several discretizations of the simple pendulum equation in a series of numerical experiments the stress is put on the long time behaviour the chosen numerical schemes are either symplectic maps or integrable energy preserving maps or both therefore they preserve qualitative features of solutions such as periodicity we describe characteristic periodic time dependences of numerical estimates of the period and the amplitude and explain them as systematic numerical by effects produced by any method finally we propose a new numerical scheme which is a modification of the discrete gradient method this modified discrete gradient method preserves almost exactly the period of small oscillations for any time step