on the implementation of discontinuous galerkin methods for volterra integral equations with highly oscillatory bessel kernels in this paper we consider implementation of discontinuous galerkin methods for weakly singular volterra integral equations of first and second kinds with highly oscillatory bessel kernels which cost the same operations independent of large values of frequencies preliminary numerical results show that the proposed method is efficient for verifying accuracy of approximate solutions

tracking control of a manipulator under uncertainty by fuzzy p id controller one of the most important manipulator operations is the control of the manipulator to track a given trajectory most commercial robot systems currently are equipped with conventional pid controllers due to their simplicity in structure and ease of design using pid control however it is di cult to achieve a desired tracking control performance since the dynamic equations of a mechanical manipulator are tightly coupled highly nonlinear and uncertain in order to improve the tracking control performance under uncertainty this paper presents a new hybrid control scheme for the manipulator which consists of a fuzzy logic proportional controller and a conventional integral and derivative controller fuzzy p id in comparison with a conventional pid controller only one additional parameter has to be adjusted to tune the fuzzy p id controller the proposed control scheme has been implemented on a direct drive two link manipulator the outlined experimental results on tracking control under inertial parameters changes demonstrate the eaectiveness and the robustness of the new fuzzy p id controller c 2001 elsevier science b v all rights reserved

extension of the perron frobenius theorem to homogeneous systems this paper deals with a particular class of positive systems the state components of a positive system are positive or zero for all positive times these systems are often encountered in applied areas such as chemical engineering or biology it is shown that for this particular class the first orthant contains an invariant ray in its interior an invariant ray generalizes the concept of an eigenvector of linear systems to nonlinear homogeneous systems then sufficient conditions for uniqueness of this ray are given the main result states that the vector field on an invariant ray determines the stability properties of the zero solution with respect to initial conditions in the first orthant the asymptotic behavior of the solutions is examined finally we compare our results to the perron frobenius theorem which gives a detailed picture of the dynamical behavior of positive linear systems

on fast multipole methods for volterra integral equations with highly oscillatory kernels abstract this paper explores the fast multipole methods fmms to accelerate the approximation for weakly singular volterra integral equations with highly oscillatory trigonometric kernels by constructing the fast translation path the fmm is utilized to speed up the iterative method which reduces the complexity from o n 2 to o n especially we use the collocation method to discretize the volterra integral equation with constants and linear elements respectively then apply the gmres to solve the dense and non symmetric linear system in addition the highly oscillatory integrals derived from the algorithm are calculated effectively by the steepest descent method the proposed method shows that the numerical solutions become more accurate as the frequency increases both of the optimal convergence rates of truncation and the error bounds analysis are represented in the end

a robust self tuning scheme for pi and pd type fuzzy controllers proposes a simple but robust model independent self tuning scheme for fuzzy logic controllers flcs here the output scaling factor sf is adjusted online by fuzzy rules according to the current trend of the controlled process the rule base for tuning the output sf is defined on error e and change of error spl delta e of the controlled variable using the most natural and unbiased membership functions mfs the proposed self tuning technique is applied to both pi and pd type flcs to conduct simulation analysis for a wide range of different linear and nonlinear second order processes including a marginally stable system where even the well known ziegler nichols tuned conventional pi or pid controllers fail to provide an acceptable performance due to excessively large overshoot performances of the proposed self tuning flcs are compared with those of their corresponding conventional flcs in terms of several performance measures such as peak overshoot settling time rise time integral absolute error and integral of time multiplied absolute error in addition to the responses due to step set point change and load disturbance and in each case the proposed scheme shows a remarkably improved performance over its conventional counterpart

analytical structure of a fuzzy controller with linear control rules abstract we prove that a fuzzy controller with linear control rules and mamdani minimum inference is the sum of a global two dimensional multilevel relay and a local nonlinear proportional integral pi controller the properties of the nonlinear gains of the local pi controller are investigated moreover it is proven that as the number of input fuzzy sets approaches the global multilevel relay approaches a global linear pi controller while the local nonlinear pi controller disappears

analytic formulation of linguistic rules for fuzzy controller in this paper an analytic formulation of a fuzzy controller using rules with two conditions is given this formulation will be exploited in order to prove the nonlinearity of the controller and to establish equivalence between fuzzy controllers and pi ones

on an output feedback finite time stabilization problem studies the problem of finite time output feedback stabilization for the double integrator system a class of output feedback controllers that can achieve global finite time stability for the double integrator system are constructed based on a finite time separation principle furthermore it is shown that the closed loop system resulting from our control law can maintain its local finite time stability regardless of some nonlinear perturbations thus our control law actually applies to a large class of nonlinear second order systems

continuous finite time stabilization of the translational and rotational double integrators a class of bounded continuous time invariant finite time stabilizing feedback laws is given for the double integrator lyapunov theory is used to prove finite time convergence for the rotational double integrator these controllers are modified to obtain finite time stabilizing feedback that avoid unwinding

global finite time stability characterized through a local notion of homogeneity in this work a notion of local homogeneity is formally defined important results involving homogeneous functions or vector fields are reformulated in the consequent local context in particular global finite time stability is characterized through the proposed notion of local homogeneity as an application of the developed analytical setting a global finite time stabilization scheme for robot manipulators with bounded inputs is presented the developed framework and results prove to be useful to relax analytical and synthesis constraints imposed or generated by homogeneity requirements

effects of increasing the footprints of uncertainty on analytical structure of the classes of interval type 2 mamdani and ts fuzzy controllers the fundamental difference between an interval type 2 it2 fuzzy controller and a type 1 fuzzy controller is the footprint of uncertainty fou of an it2 fuzzy set in this paper we study how fous affect the analytical structure i e the input output mathematical relationship of a broad class of it2 mamdani and takagi sugeno ts controllers the controllers employ arbitrary fuzzy rules the karnik mendel km or enhanced km type reducer the minimum and operator and the centroid defuzzifier the controllers utilize commonly used it2 fuzzy sets for their input variables and any kind of type 2 fuzzy sets for their output variable we prove that with increase of fous of the input fuzzy sets the mamdani controllers approach constant controllers and the ts controllers approach piecewise linear controllers the resemblance to the constant or pricewise linear controllers increases as the fous increase when all the fous are at their maximum to reflect the highest level of uncertainties the mamdani and ts controllers become the constant controllers and piecewise linear controllers respectively we investigate how change in the resemblance takes place progressively as fous increase we also show that an increase in the resemblance narrows control gain variations for part of the it2 controllers which can worsen control performance these findings implicit controller design too large fous are generally undesirable for the input fuzzy sets because they can make an it2 controller behave like a constant or piecewise linear controller real time control experiment results are provided to illustrate the theoretical analysis

an improved fast multipole algorithm for potential fields on the line a new version of the fast multipole method fmm for the evaluation of potential fields on the line is introduced it uses a new representation of potential fields based on generalized gaussian quadratures j ma v rokhlin and s wandzura siam j numer anal 33 1996 pp 971 996 n yarvin and v rokhlin siam j sci comput 20 1999 pp 699 718 in this representation most translation operators are diagonal to efficiently incorporate this representation into the fmm an apparatus is introduced for transforming between different types of expansions this apparatus is highly general and is based on formulae for the least squares approximation of linear operators the performance of the method is illustrated with several numerical examples it is roughly twice as fast as previously published algorithms

error bounds for approximation in chebyshev points this paper improves error bounds for gauss clenshaw curtis and fejer s first quadrature by using new error estimates for polynomial interpolation in chebyshev points we also derive convergence rates of chebyshev interpolation polynomials of the first and second kind for numerical evaluation of highly oscillatory integrals preliminary numerical results show that the improved error bounds are reasonably sharp

approximation accuracy analysis of fuzzy systems as function approximators this paper establishes the approximation error bounds for various classes of fuzzy systems i e fuzzy systems generated by different inferential and defuzzification methods based on these bounds the approximation accuracy of various classes of fuzzy systems is analyzed and compared it is seen that the class of fuzzy systems generated by the product inference and the center average defuzzifier has better approximation accuracy and properties than the class of fuzzy systems generated by the min inference and the center average defuzzifier and the class of fuzzy systems defuzzified by the mom defuzzifier in addition it is proved that fuzzy systems can represent any linear and multilinear function and explicit expressions of fuzzy systems generated by the mom defuzzified method are given

a fuzzy dynamic learning controller for chemical process control a fuzzy dynamic learning controller is proposed and applied to control of time delayed non linear and unstable chemical processes the proposed fuzzy dynamic learning controller can self adjust its fuzzy control rules using the external dynamic information from the process during on line control and it can create the new fuzzy control rules autonomously using its learning capability from past control trends the proposed controller shows better performance than the conventional fuzzy logic controller and the fuzzy self organizing controller

fuzzy controller theory limit theorems for linear fuzzy control rules abstract for a general fuzzy controller employing linear fuzzy control rules it is shown that as the number of rules grow the defuzzified output becomes a linear function of the input in particular for a sufficiently large number of rules the defuzzified output is approximately the same as the pi pid controller

model reference adaptive fuzzy control a linguistic space approach in this paper a model reference adaptive fuzzy controller with the adaptivity achieved by tuning both the scaling factors and the control rules is presented the proposed tuning strategy includes two parts 1 by referring a reference trajectory created in the linguistic space to tune the scaling factors and 2 by applying an auxiliary correction matrix generated by specific tuning rules to compensate the control rules since only the control signal the output error and change in output error of the plant are needed in the construction of the reference trajectory and the tuning rules the tuning strategy has the advantage of real time property simulation results with a damping process and an oscillating process as the controlled plants show the performance and adaptivity of the proposed controller

local homogeneity based global continuous control for mechanical systems with constrained inputs finite time and exponential stabilisation abstracta global continuous control scheme for the finite time or local exponential stabilisation of mechanical systems with constrained inputs is proposed the approach is formally developed within the theoretical framework of local homogeneity this has permitted to solve the formulated problem not only guaranteeing input saturation avoidance but also giving a wide range of design flexibility the proposed scheme is characterised by a saturating proportional derivative type term with generalised saturating and locally homogeneous structure that permits multiple design choices on both aspects the work includes a simulation implementation section where the veracity of the so cited argument claiming that finite time stabilisers are faster than asymptotical ones is studied in particular a way to carry out the design so as to indeed guarantee faster stabilisation through finite time controllers beyond their finite time convergence is shown

on the evaluation of cauchy principal value integrals of oscillatory functions the problem of the numerical evaluation of cauchy principal value integrals of oscillatory functions 1 1e i w xf x x tdx where 1 t 1 has been discussed based on analytic continuation if f is analytic in a sufficiently large complex region g containing 1 1 the integrals can be transformed into the problems of integrating two integrals on 0 with the integrand that does not oscillate and that decays exponentially fast which can be efficiently computed by using the gauss laguerre quadrature rule the validity of the method has been demonstrated in the provision of two numerical experiments and their results

homogeneous lyapunov function for homogeneous continuous vector field the goal of this article is to provide a construction of a homogeneous lyapunov function p associated with a system of differential equations j f x x r n 1 under the hypotheses 1 f c r n vanishes at x 0 and is homogeneous 2 the zero solution of this system is locally asymptotically stable moreover the lyapunov function v x tends to infinity with 1 x i and belongs to c r 0 r n cp with p e as large as wanted as application to the theory of homogeneous systems we present two well known results of robustness in a slightly extended form and with simpler proofs

a model reference control structure using a fuzzy neural network in this paper we present a design method for a model reference control structure using a fuzzy neural network we study a simple fuzzy logic based neural network system knowledge of rules is explicitly encoded in the weights of the proposed network and inferences are executed efficiently at high rate two fuzzy neural networks are utilized in the control structure one is a controller called the fuzzy neural network controller fnnc the other is an identifier called the fuzzy neural network identifier fnni adaptive learning rates for both the fnnc and fnni are guaranteed to converge by a lyapunov function the on line control ability robustness learning ability and interpolation ability of the proposed model reference control structure are confirmed by simulation results

laplace transforms for approximation of highly oscillatory volterra integral equations of the first kind abstract this paper focuses on laplace and inverse laplace transforms for approximation of volterra integral equations of the first kind with highly oscillatory bessel kernels where the explicit formulae for the solution of the first kind integral equations are derived from which the integral equations can also be efficiently calculated by the clenshaw curtis filon type methods furthermore by applying the asymptotics of the solution some simpler formulas for approximating the solution for large values of the parameters are deduced preliminary numerical results are presented based on the approximate formulae and the explicit formulae which are compared with the convolution quadrature and numerical inverse laplace transform methods all these methods share that the costs the same independent of large values of frequencies

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

finite time controllers continuous finite time differential equations are introduced as fast accurate controllers for dynamical systems these have qualities superior to controllers which are currently in use in such appl

fuzzy control theory the linear case abstract a linear fuzzy proportional integral pi controller with one input and one output is defined in terms of piecewise linear membership functions for fuzzification control rules and defuzzification algorithm it is shown that a linear fuzzy controller is not equivalent to a linear non fuzzy pi controller if the rules are evaluated using the zadeh probability or lukasiewicz fuzzy logic alone however the linear fuzzy controller is precisely equivalent to a linear non fuzzy pi controller if mixed fuzzy logic is used to evaluate the control rules when the fuzzy logics used are selected with due regard to prior associations implied by the control rule operands themselves this finding has relevance to the selection of fuzzy logics in fuzzy expert systems and for fuzzy logics in general

a method for deriving the analytical structure of a broad class of typical interval type 2 mamdani fuzzy controllers type 2 t2 fuzzy controllers are emerging as the related t2 fuzzy logic and algorithms have recently been advancing rapidly at present a t2 fuzzy controller is viewed and used as a black box function generator that produces a desired nonlinear mapping between the input and output of the controller we call the mapping analytical structure the mathematical expression of the analytical structure however is not explicitly known to the controller designer this is in sharp contrast with the analytical structure of a conventional controller which is not only always explicitly known but serves as a starting point for system analysis and design obviously the knowledge of a t2 fuzzy controller s analytical structure can have significant benefits they include 1 understanding more precisely how the controller works in the same sense as we understand how a conventional controller e g the pid controller functions 2 making t2 fuzzy control more acceptable to safety critical fields such as biomedicine 3 taking advantage of the well developed nonlinear control theory to develop better analysis and design methods for t2 control systems e g less conservative system stability criteria and 4 permitting rigorous comparative exploration on differences between the t2 and type 1 t1 fuzzy controllers and their relative merits and pitfalls e g performance and structural complexity in this paper we develop an innovative technique which is capable of deriving the analytical structure for a wide class of interval t2 mamdani fuzzy controllers the configuration of the controllers is typical and is substantially more general than the related efforts in the literature it uses any number and types of interval t2 input fuzzy sets any number and types of general or interval t2 output fuzzy sets arbitrary fuzzy rules zadeh and operator karnik mendel center of sets type reducer and the centroid defuzzifier we show in detail how the derivation method works in a general setting and provide the analytical structure of an example t2 controller as well in addition we utilize the method to prove that a subset of the t2 fuzzy controllers are the sum of two nonlinear pi or pd controllers each of which has a variable proportional gain and a variable integral gain or derivative gain plus a variable offset if and only if the input fuzzy sets are piecewise linear e g triangular and or trapezoidal the sum of the two nonlinear pi or pd controllers is a new discovery relative to the current literature

a nonlinear fuzzy controller with linear control rules is the sum of a global two dimensional multilevel relay and a local nonlinear proportional integral controller abstract the author analytically proves that a nonlinear fuzzy controller with linear control rules and n members for input fuzzy sets is the sum of a global two dimensional multilevel relay and a local nonlinear proportional integral pi controller which adjusts the control action generated by the global multilevel relay as n increases the resolution of the global multilevel relay is enhanced but the role of the local nonlinear pi controller in total control action is decreased as n approaches the global multilevel relay approaches a regular linear pi controller while the control action from the local nonlinear pi controller approaches zero the role of the global multilevel relay and the local nonlinear pi controller in total control action is quantitatively described as is the degree of nonlinearity of the fuzzy controllers with different n

a self organizing fuzzy logic controller for dynamic systems using a fuzzy auto regressive moving average farma model the paper proposes a complete design method for an online self organizing fuzzy logic controller without using any plant model by mimicking the human learning process the control algorithm finds control rules of a system for which little knowledge has been known in a conventional fuzzy logic control knowledge on the system supplied by an expert is required in developing control rules however the proposed new fuzzy logic controller needs no expert in making control rules instead rules are generated using the history of input output pairs and new inference and defuzzification methods are developed the generated rules are stored in the fuzzy rule space and updated online by a self organizing procedure the validity of the proposed fuzzy logic control method has been demonstrated numerically in controlling an inverted pendulum

numerical solutions to volterra integral equations of the second kind with oscillatory trigonometric kernels this paper explores quadrature methods for weakly singular volterra integral equation of the second kind with highly oscillatory trigonometric kernels and presents piecewise constant and linear collocation methods for approximation of the solution the evaluation of the highly oscillatory kernels of integral equation will give rise to the computation of oscillatory integrals which can be done by using efficient filon type methods moreover by using asymptotic analysis of the solution the convergence rates for piecewise constant and linear collocation methods which show the higher the frequency the more accurate the numerical results the effectiveness and accuracy of the proposed method are tested by numerical examples

finite time stabilisation of simple mechanical systems using continuous feedback stabilisation of simple mechanical systems in finite time with continuous state feedback is considered here the dynamics are represented by generalised local coordinates a general methodology to construct control lyapunov functions that are holder continuous and that can be used to show finite time stability of the feedback controlled system is presented this construction also gives the feedback control law and results in the feedback system being holder continuous as well unlike lipschitz continuous feedback control systems the feedback control scheme given here converges to the desired equilibrium in finite time moreover unlike discontinuous and hybrid control schemes the feedback control law does not lead to chattering in the presence of measurement noise does not excite unmodelled high frequency dynamics and can be implemented with actuators that can only deliver continuous control inputs the advantages of continuous finite time stabilisation over continuous asymptotic stabilisation of

fuzzy control theory a nonlinear case abstract sources of nonlinearity in a fuzzy controller include the fuzzification algorithm used for the controller inputs fuzzy control rules type of fuzzy logic used for evaluating the fuzzy control rules and the defuzzification algorithm used for the controller output we analyze the performance of a simple fuzzy controller with linear and nonlinear defuzzification algorithms we prove theoretically that such a fuzzy controller the smallest possible with two inputs error and rate change of error and a nonlinear defuzzification algorithm is equivalent to a nonfuzzy nonlinear proportional integral pi controller with proportional gain and integral gain changing with error and rate change of error about a setpoint furthermore this fuzzy controller is precisely equivalent to a conventional linear pi controller if a linear defuzzification algorithm is employed computer simulation showed that the performance of the fuzzy controller was almost the same as that of the pi controller when first order and second order linear processes were used furthermore the fuzzy controller was significantly better when a first order with a time delay model was used more importantly the simulated result illustrated that the fuzzy controller was stable when a nonlinear process model was controlled but the pi controller was unstable

adaptive fuzzy relational predictive control abstract the performance of two fuzzy relational controllers namely the self learning predictive fuzzy controller slpfc bourke and fisher proc 5th int conf on fuzzy systems new orleans 1996 pp 464 1470 and the fuzzy relational long range predictive controller frlrpc postlethwaite iee proc d 138 3 1991 199 206 are evaluated experimentally on a laboratory scale non linear interacting tank process both fuzzy controllers gave good closed loop control performance but the slpfc gave slightly better results it was also found that the choice of the on line fuzzy relational identification scheme has a large impact on control performance

on the calculation of highly oscillatory integrals with an algebraic singularity abstract in this paper we explore the evaluation of highly oscillatory integrals containing an algebraic singularity based on the change of variable t x r where r is a positive real number and the analytical continuation robust numerical results demonstrate the accuracy and effectiveness of the proposed approach

analytical structure and characteristics of symmetric karnik mendel type reduced interval type 2 fuzzy pi and pd controllers this paper presents the analytical structure of a class of interval type 2 it2 fuzzy proportional derivative pd and proportional integral pi controllers that have symmetrical rule base and symmetrical consequent sets two assumptions are made 1 the zadeh and operator is employed as the t norm operator 2 type reduction is performed by the karnik mendel km type reduction method the main contributions are the methodology that identifies the input conditions where the km algorithm uses a new switch point to compute the bounds of the type reduced set the closed form expressions that relate the inputs and output of an it2 fuzzy controller and insights into the potential performance improvement because of the inclusion of the footprint of uncertainty fou compared with its t1 counterpart two additional fou parameters generate 31 extra local regions each providing a unique relationship between the inputs and output signals the generation of a relatively large number of local regions at the cost of two extra design parameters indicates that an it2 fuzzy controller may be able to provide better performance furthermore by comparing the analytical structure with the corresponding t1 counterpart the potential advantages to use the it2 over the t1 fuzzy controller are studied four interesting characteristics are identified and they provide insights into why the it2 fuzzy controller may better balance the conflicting aims of fast rise time and small overshoot

analytical structure of a two input two output fuzzy controller and its relation to pi and multilevel relay controllers abstract this study investigates the analytical structure of a two input two output fuzzy controller which employs triangular shaped input fuzzy sets trapezoidal shaped output fuzzy sets linear control rules probabilistic and fuzzy logic ukasiewicz or fuzzy logic mamdani s minimum inference method and the center of gravity defuzzification algorithm we analytically prove that the structure of the fuzzy controller is the sum of global four dimensional multilevel relays and local nonlinear proportional integral pi controllers with variable gains continuously changing with process outputs the global multilevel relays play a major role in determining control action of the fuzzy controller while the local pi controllers locally fine tune the control action of the relays properties of the fuzzy controller structure are analytically and quantitatively investigated moreover it is proved that as the number of control rules approaches each global four dimensional multilevel relay becomes the sum of two global linear pi controllers while the local pi controllers disappear

local solutions to high frequency 2d scattering problems we consider the solution of high frequency scattering problems in two dimensions modeled by an integral equation on the boundary of a smooth scattering object we devise a numerical method to obtain solutions on only parts of the boundary with little computational effort the method incorporates asymptotic properties of the solution and can therefore attain particularly good results for high frequencies we show that the integral equation in this approach reduces to an ordinary differential equation

global finite time stabilization by output feedback for planar systems without observable linearization this note considers the problem of global finite time stabilization by output feedback for a class of planar systems without controllable observable linearization a sufficient condition for the solvability of the problem is established by developing a nonsmooth observer and modifying the adding a power integrator technique we show that an output feedback controller can be explicitly constructed to globally stabilize the systems in finite time as a direct application of the main result global output feedback finite time stabilization is achieved for the double linear integrator systems perturbed by some nonlinear functions which are not necessarily homogeneous

modified asymptotic orders of the direct filon method for a class of volterra integral equations the direct filon method provides efficient approximation for solutions of volterra integral equations with highly oscillatory kernels in this paper we study the rate of convergence of this method for a class of equations with trigonometric kernels the asymptotic order is developed by examining the property of the solution numerical experiments are included to verify the given order

fast algorithms for polynomial interpolation integration and differentiation for functions tabulated at chebyshev nodes on an interval spectral interpolation and indefinite integration can be performed stably and efficiently via the fast fourier transform for many other sets of nodes such as the gaussian nodes on an interval the classical interpolation and indefinite integration schemes are stable but slow requiring o n 2 arithmetic operations with n the number of nodes in the discretization of the interval in this paper a group of algorithms is presented for the efficient evaluation of lagrange polynomial interpolants at multiple points on the line and for the rapid indefinite integration and differentiation of functions tabulated at nodes other than chebyshev the interpolation scheme requires o n cdot log 1 varepsilon arithmetic operations and o n cdot log n n cdot log 1 varepsilon operations are required for the integration and differentiation schemes where varepsilon is the precision of computations and n is the number of nodes t

on the evaluation of highly oscillatory integrals by analytic continuation we consider the integration of one dimensional highly oscillatory functions based on analytic continuation rapidly converging quadrature rules are derived for a general class of oscillatory integrals with an analytic integrand the accuracy of the quadrature increases both for the case of a fixed number of points and increasing frequency and for the case of an increasing number of points and fixed frequency these results are then used to obtain quadrature rules for more general oscillatory integrals i e for functions that exhibit some smoothness but that are not analytic the approach described in this paper is related to the steepest descent method but it does not employ asymptotic expansions it can be used for small or moderate frequencies as well as for very high frequencies the approach is compared with the oscillatory integration techniques recently developed by iserles and norsett

theory of the fuzzy controller abstract we first introduce notation in an attempt to standardize the description of a basic fuzzy controller if r 1 r 2 r n are the real number inputs to the fuzzy controller and is its defuzzified output then f r 1 r n for some function f by the theory of the fuzzy controller we mean in this paper the study of the structure of f for elementary fuzzy controllers using this standardized notation we first survey the known results on the theory of the fuzzy controller and then discuss important future research topics in this area

numerical calculation of integrals involving oscillatory and singular kernels and some applications of quadratures numerical methods for strongly oscillatory and singular functions are given in this paper beside a summary of standard methods and product integration rules we consider a class of complex integration methods several applications of quadrature processes in problems in telecom munications and physics are also presented 1998 elsevier science ltd all rights reserved integration oscillatory kernel singular kernel orthogonal polynomials product rules gaussian quaclratures error function bessel functions legendre functions

on graded meshes for weakly singular volterra integral equations with oscillatory trigonometric kernels we compare the collocation methods on graded meshes with that on uniform meshes for the solution of the weakly singular volterra integral equation of the second kind with oscillatory trigonometric kernels due to in general unbounded derivatives at the left endpoint of the interval of integration we should approximate the solution of the integral equation by collocation methods on graded meshes however we show that this non smooth behavior of the solution has little effect on the approximate solution when the kernels of integral equation are highly oscillatory numerical examples are given to confirm the proposed results

efficient quadrature of highly oscillatory integrals using derivatives in this paper we explore quadrature methods for highly oscillatory integrals generalizing the method of stationary phase we expand such integrals into asymptotic series in inverse powers of the frequency the outcome is two families of methods one based on a truncation of the asymptotic series and the other extending an approach implicit in the work of filon filon 1928 proc r soc edinb 49 38 47 1 both kinds of methods approximate the integral as a linear combination of function values and derivatives with coefficients that may depend on frequency we determine asymptotic properties of these methods proving perhaps counterintuitively that their performance drastically improves as frequency grows the paper is accompanied by numerical results that demonstrate the potential of this set of ideas 1 ref 4

a wavenumber independent boundary element method for an acoustic scattering problem in this paper we consider the impedance boundary value problem for the helmholtz equation in a half plane with piecewise constant boundary data a problem which models for example outdoor sound propagation over inhomogeneous flat terrain to achieve good approximation at high frequencies with a relatively low number of degrees of freedom we propose a novel galerkin boundary element method using a graded mesh with smaller elements adjacent to discontinuities in impedance and a special set of basis functions so that on each element the approximation space contains polynomials of degree nu multiplied by traces of plane waves on the boundary we prove stability and convergence and show that the error in computing the total acoustic field is cal o n nu 1 log 1 2 n where the number of degrees of freedom is proportional to n log n this error estimate is independent of the wavenumber and thus the number of degrees of freedom required to achieve a prescribed level of accuracy does not increase as the wavenumber tends to infinity

efficient filon type methods for int_a bf x rm e rm i omega g x rm d x based on the transformation y g x some new efficient filon type methods for integration of highly oscillatory function int a bf x rm e rm i omega g x rm d x with an irregular oscillator are presented one is a moment free filon type method for the case that g x has no stationary points in a b the others are based on the filon type method or the asymptotic method together with filon type method for the case that g x has stationary points the effectiveness and accuracy are tested by numerical examples

on the use of the singular value decomposition in the fast multipole method the authors present a new matrix compression algorithm to improve the efficiency of the fast multipole method fmm the method is based on the application of the singular value decomposition svd to the plane wave fmm aggregation matrices these matrices are low ranked which is exploited to provide alternative sets of orthonormal singular basis functions obtained as linear combinations of the original basis by choosing only the most relevant singular functions a much more compact representation is obtained to accurately handle the interactions between the fmm groups the new formulation provides a reduction close to one order of magnitude both in computational cost and memory requirements with a moderate impact on the accuracy of the solution

finite time stability and robust control synthesis of uncertain switched systems stability analysis is developed for uncertain nonlinear switched systems while being asymptotically stable and homogeneous of degree q 0 these systems are shown to approach the equilibrium point in finite time restricted to second order systems this feature is additionally demonstrated to persist regardless of inhomogeneous perturbations based on this fundamental property switched control algorithms are then developed to globally stabilize uncertain minimum phase systems of uniform m vector relative degree 2 2 t the controllers constructed do not rely on the generation of sliding motions while providing robustness features similar to those possessed by their sliding mode counterparts the proposed synthesis procedure is illustrated via application to a friction servo motor

further advancements on the output feedback global continuous control for the finite time and exponential stabilisation of bounded input mechanical systems desired conservative force compensation and experiments global saturating proportional saturating derivative sp sd type continuous control for the finite time or local exponential stabilisation of mechanical systems with bounded inputs is achieved a

a fuzzy variable structure approach to feedback regulation of uncertain dynamical systems with application to robotics this paper develops a fuzzy logic based control design methodology for a large class of nonlinear uncertain dynamical systems the proposed control design approach combines the powerful tools of fuzzy logic and approximate reasoning with the advanced mathematical synthesis techniques used in variable structure control systems theory the rationale for the proposed control design approach is motivated by the results of a recent study that has rigorously established some connections between fuzzy logic and variable structure control systems the results of this paper represent a step in the right direction for systematic design of a less expert dependent fuzzy logic control to illustrate the merits of the new control approach the latter is applied to a 2 degree of freedom robot manipulator preliminary simulation results suggest that the proposed control design deals effectively with the chattering problem encountered when the classical variable structure control is used alone

exponential stabilization of driftless nonlinear control systems using homogeneous feedback this paper focuses on the problem of exponential stabilization of controllable driftless systems using time varying homogeneous feedback the analysis is performed with respect to a homogeneous norm in a nonstandard dilation that is compatible with the algebraic structure of the control lie algebra it can be shown that any continuous time varying controller that achieves exponential stability relative to the euclidean norm is necessarily non lipschitz despite these restrictions we provide a set of constructive sufficient conditions for extending smooth asymptotic stabilizers to homogeneous exponential stabilizers the modified feedbacks are everywhere continuous smooth away from the origin and can be extended to a large class of systems with torque inputs the feedback laws are applied to an experimental mobile robot and show significant improvement in convergence rate over smooth stabilizers

a general technique for deriving analytical structure of fuzzy controllers using arbitrary trapezoidal input fuzzy sets and zadeh and operator deriving the analytical structure of fuzzy controllers is very important as it creates a solid foundation for better understanding insightful analysis and more effective design of fuzzy control systems we previously developed a technique for deriving the analytical structure of the fuzzy controllers that use zadeh fuzzy and operator and the symmetric identical trapezoidal or triangular input fuzzy sets many fuzzy controllers use arbitrary trapezoidal triangular input fuzzy sets that are asymmetric at present there exists no technique capable of deriving the analytical structure of these fuzzy controllers extending our original technique we now present a novel method that can accomplish rigorously the structure derivation for any fuzzy controller mamdani type or ts type that employs the arbitrary trapezoidal input fuzzy sets and zadeh fuzzy and operator the new technique contains our original technique as a special case given the importance of pid control we focus on mamdani fuzzy pi and pd controllers in this paper and show in detail how to use the new technique for different configurations of the fuzzy pi pd controllers the controllers use two arbitrary trapezoidal fuzzy sets for each input variable four arbitrary singleton output fuzzy sets four fuzzy rules zadeh fuzzy and operator and the centroid defuzzifier this configuration is more general and complicated than the mamdani fuzzy pi pd controllers in the current literature it actually contains them as special cases we call this configuration the generalized fuzzy pi pd controller

decomposition property of fuzzy systems and its applications this paper presents the decomposition property of fuzzy systems using a simple constructive decomposition procedure that is by properly dividing the input space into sub input spaces a general fuzzy system is decomposed into several sub fuzzy systems which are the simplest fuzzy systems in the sub input spaces based on the decomposition property of fuzzy systems the analysis of fuzzy systems can be divided into two steps first analyze the properties of the simplest fuzzy systems and then use the decomposition property to extend the results to general fuzzy systems using this idea two applications of the decomposition property are given the first is the application to the representation capability analysis of fuzzy systems the second is the application to the analysis of a class of nonlinear control systems then based on the piecewise affine fuzzy system model the existence condition and the design of a stable control for a class of single input single output siso nonlinear systems are presented