a novel fuzzy framework for nonlinear system control nonlinear pid and gain scheduling controls have attracted much research interests in recent years these control strategies can accommodate some nonlinear characteristics by allowing the gains varied or rescheduled as a function of system states in this paper a novel fuzzy framework is developed to transfer this type of nonlinear controller to its fuzzy domain representation it is proved that the resulting fuzzy controller is functionally exactly identical to the original control system one of the benefits of the suggested approach is that it provides a well defined prototype for the design of fuzzy control system the resulting linguistic representation can facilitate investigation in linguistic terms into how the controller operates whereas expert knowledge can be effectively implemented to improve control performance the viability of the proposed mapping technique is demonstrated by using simulations corresponding to a flexible link robot

fuzzy control rules and their natural laws abstract we consider the convergence of fuzzy controllers as the number of fuzzy rules tends to infinity we assume that the fuzzy rules are regular and show the existence of limit laws for five different fuzzy control rules

brief paper an analytical study on structure stability and design of general nonlinear takagi sugeno fuzzy control systems in this paper we first study analytical structure of general nonlinear takagi sugeno ts for short fuzzy controllers then establish a condition for analytically determining asymptotic stability of the fuzzy control systems at the equilibrium point and finally use the stability condition in design of the control systems that are at least locally stable the general ts fuzzy controllers use arbitrary input fuzzy sets any types of fuzzy logic and ts fuzzy rules with linear consequent and the generalized defuzzifier which contains the popular centroid defuzzifier as a special case we have mathematically proved that the general ts fuzzy controllers are nonlinear controllers with variable gains continuously changing with controllers input variables using lyapunov s linearization method we have established a necessary and sufficient condition for analytically determining local asymptotic stability of ts fuzzy control systems each of which is made up of a fuzzy controller of the general class and a nonlinear plant we show that the condition can be used in practice even when the plant model is not explicitly known we have utilized the stability condition to design with or without plant model general ts fuzzy control systems that are at least locally stable three numerical examples are given to illustrate in detail how to use our new results our results offer four important practical advantages 1 our stability condition being a necessary and sufficient one is the tightest possible stability condition 2 the condition is simple and easy to use partially because it only needs the fuzzy controller structure around the equilibrium point 3 the condition can be used for determining system local stability and designing fuzzy control systems that are stable at least around the equilibrium point even when the explicit plant models are unavailable and 4 the condition covers a very broad range of nonlinear ts fuzzy control systems for which a meaningful global stability condition seems impossible to establish

new design and stability analysis of fuzzy proportional derivative control systems this paper describes the design principle tracking performance and stability analysis of a fuzzy proportional derivative pd controller first the fuzzy pd controller is derived from the conventional continuous time linear pd controller then the fuzzification control rule base and defuzzification in the design of the fuzzy pd controller are discussed in detail the resulting controller is a discrete time fuzzy version of the conventional pd controller which has the same linear structure in the proportional and the derivative parts but has nonconstant gains both the proportional and derivative gains are nonlinear functions of the input signals the new fuzzy pd controller thus preserves the simple linear structure of the conventional pd controller yet enhances its self tuning control capability computer simulation results have demonstrated this advantage of the fuzzy pd controller particularly when the process to be controlled is nonlinear after a detailed stability analysis where a simple and realistic sufficient condition for the bounded input bounded output stability of the overall feedback control system was derived several computer simulation results are compared with the conventional pd controller although the conventional and fuzzy pd controllers are not exactly comparable the authors compare them in order to have a sense of how well the fuzzy pd controller performs for this reason in the simulations several first order and second order linear systems with or without time delays are first used to test the performance of the fuzzy pd controller for step reference inputs the fuzzy pd control systems show remarkable performance as well as if not better than the conventional pd control systems moreover the fuzzy pd controller is compared to the conventional pd controller for a particular second order linear system showing the advantage of the fuzzy pd controller over the conventional one in the sense that in order to obtain the same control performance the conventional pd controller has to employ an extremely large gain while the fuzzy controller uses a reasonably small gain finally in the case of nonlinear systems the authors provide some examples to show that the fuzzy pd controller can track the set points satisfactorily but the conventional pd controller cannot

on the variety of a highest weight module 1 2 let u g be the enveloping algebra of g detailed information on the primitive spectrum prim u g of u g was summarized by conjecture 7 4 of 181 here a main question was to relate prim u g with the springer correspondence parts i ii of this conjecture were established in i ii and a partial solution to iii was given in 21 sufficient e g to treat g of type a meanwhile borho and brylinski 5 have established iii for induced ideals while barbasch and vogan 1 2 have established a version of iii in which gr i i e prim u g is replaced by a certain wavefront set known to be contained in y gr 1 this last work involves some case by case analysis and gives little indication as to why the springer correspondence should arise in the study of prim u g

the simplest fuzzy controllers using different inference methods are different nonlinear proportional integral controllers with variable gains abstract the author analytically proves that the simplest fuzzy controllers using different inference methods are different nonlinear proportional integral pi controllers with proportional gains and integral gains changing with inputs of the controllers the inference methods involved are mamdani s minimum inference method larsen s product inference method the drastic product inference method and the bounded product inference method configuration of the fuzzy controllers is minimal which includes two input fuzzy sets three output fuzzy sets four control rules zadeh fuzzy logic and lukasiewicz fuzzy logic or and a center of gravity defuzzification algorithm after analytically investigating properties of the nonlinear pi controllers the author reveals that the bounded product inference method is inappropriate for the control purpose while the other three inference methods are appropriate dynamic and static control behaviors of the fuzzy controllers with the appropriate inference methods are analytically compared with each other and are also compared with those of the linear pi controller finally it is analytically proven that the fuzzy control systems have the same local stability at the equilibrium point as the corresponding linear pi control system does

theory and application of a novel fuzzy pid controller using a simplifier takagi sugeno rule scheme we first present a novel general purpose nonlinear pid controller realized via fuzzy pid control that uses our newly introduced simplified takagi sugeno ts rule scheme analytical structure of the fuzzy pid controller is derived and its structure is analyzed in relation to the linear pid controller the unique features of the fuzzy controller are as follows first the proportional integral and derivative gains constantly vary with the output of the system under control the gain variation leads to a shorter rise time a less overshoot and a smaller settling time as compared to a comparable linear pid controller second the characteristics of the gain variation are determined by the fuzzy rules and can intuitively be designed we have also investigated the local stability of the fuzzy pid control systems as an application demonstration we have developed a fuzzy pid control system to regulate in computer simulation blood pressure in postsurgical patients we have chosen this particular control problem because the studies in the literature have established that in order to achieve satisfactory control results using a nonlinear controller with variable gains is necessary the simulation results show that the fuzzy pid controller significantly outperforms its linear counterpart and is safer and more robust over a wide range of patient condition 2000 published by elsevier

a formula connecting sample and moment stability of linear stochastic systems given the linear stochastic system dot x a xi t x x in mathbb r d xi t some nice diffusion sample and moment stability of x equiv 0 are studied via the sample lyapunov exponent lambda and the lyapunov exponent of the pth mean g p of a solution generalizing a result of molchanov for the undamped linear oscillator it is shown that g p is a smooth convex function of p in mathbb r with g 0 0 and g 0 lambda for trace a 0 a second zero of g at p d is detected this allows one to draw simple but quite complete conclusions about the relation between sample and moment stability

characteristic varieties and vanishing cycles on presente une etude systematique des varietes caracteristiques des systemes holonomes a singularites regulieres ce sont des sous varietes lagrangiennes coniques dans le fibre cotangent t x

differential operators on homogeneous spaces iii characteristic varieties of harish chandra modules and of primitive ideals letg be a semi simple complex algebraic group with lie algebra and flag varietyx g b for each primitive idealj with trivial central character in the enveloping algebrau we define a characteristic variety in the cotangent bundle ofx which projects under the springer resolution mapt x onto the closure of a nilpotent orbit we prove that this characteristic variety is theg saturation of the characteristic variety of a highest weight module with annihilatorj we conjecture that it is irreducible forg sl n our conjecture would provide a geometrical explanation for the classification of primitive ideals in terms of weyl group representations as achieved by a joseph the presentation of these ideas here is simultaneously used to some extent as an opportunity to continue our more general systematic discussion of differential operators on a complete homogeneous space and to study more generally characteristic varieties of harish chandra modules

constructing nonlinear variable gain controllers via the takagi sugeno fuzzy control we investigated the analytical structure of the takagi sugeno ts type of fuzzy controllers which was unavailable in the literature the ts fuzzy controllers we studied employ a new and simplified ts control rule scheme in which all the rule consequent use a common function and are proportional to one another greatly reducing the number of parameters needed in the rules other components of the fuzzy controllers are general arbitrary input fuzzy sets any type of fuzzy logic and the generalized defuzzifier which contains the popular centroid defuzzifier as a special case we proved that all these ts fuzzy controllers were nonlinear variable gain controllers and characteristics of the gain variation were parametrized and governed by the rule proportionality we conducted an in depth analysis on a class of nonlinear variable gain proportional derivative pd controllers we present the results to show 1 how to analyze the characteristics of the variable gains in the context of control 2 why the nonlinear variable gain pd controllers can outperform their linear counterpart and 3 how to generate various gain variation characteristics through the manipulation of the rule proportionality

pad analysis of fuzzy control stability abstract by means of an index represented rule mapping the response g of a fuzzy controller is decomposed into two parts f the classical component and g f the nonlinear component called pad we intend to analyse the pad in order to know what the relation is between fuzzy and classical pid controllers fuzzy controllers mentioned here are defuzzified by the typical center of gravity method in the 1 input case we get that f g 0 which means that the fuzzy controller is exactly the same as the classical p controller whenever the rule mapping is simple definition 1 2 and the linguistic values are a binary family definition 4 1 in the 2 input case of error and error change for simple rule mapping and for simple linguistic values family 4 7 and perfectly for the wxyz linguistic values family definition 6 1 the fuzzy controller can be viewed as a series connection of a classical pd controller and a 1 pad element the describing function of a 1 pad element is very simple and fine and ensures that such fuzzy controller will be stable if the pd controller is the similar consequences are also obtained for the case of three inputs

an approach to fuzzy control of nonlinear systems stability and design issues presents a design methodology for stabilization of a class of nonlinear systems first the authors represent a nonlinear plant with a takagi sugeno fuzzy model then a model based fuzzy controller design utilizing the concept of the so called parallel distributed compensation is employed the main idea of the controller design is to derive each control rule so as to compensate each rule of a fuzzy system the design procedure is conceptually simple and natural moreover the stability analysis and control design problems can be reduced to linear matrix inequality lmi problems therefore they can be solved efficiently in practice by convex programming techniques for lmis the design methodology is illustrated by application to the problem of balancing and swing up of an inverted pendulum on a cart

typical takagi sugeno pi and pd fuzzy controllers analytical structures and stability analysis in the first part of this paper we investigate explicit structure of the typical takagi sugeno ts fuzzy pi and pd controllers by typical we mean the use of at least three trapezoidal or triangular input fuzzy sets for each input variable fuzzy rules with linear consequent zadeh fuzzy logic and operator and the centroid defuzzifier this configuration covers many ts fuzzy controllers in the literature we mathematically prove these fuzzy controllers to be nonlinear pi or pd controllers with proportional gain and integral gain or derivative gain changing with output of the controlled system we reveal these fuzzy controllers to be inherently nonlinear gain scheduling pi or pd controllers with different variable gains in different regions of input space the gains constantly change even in the same region and switch continuously and smoothly between adjacent regions the explicit structural expressions of the fuzzy controllers are derived and their characteristics including the bounds and geometrical shape of the gain variation are studied in the second part of the paper we apply the small gain theorem to the derived structures of the fuzzy controllers and establish a sufficient global bibo stability criterion for nonlinear systems controlled by the fuzzy controllers local stability is also examined

structural analysis of fuzzy controllers with nonlinear input fuzzy sets in relation to nonlinear pid control with variable gains the popular linear pid controller is mostly effective for linear or nearly linear control problems nonlinear pid controllers however are needed in order to satisfactorily control highly nonlinear plants time varying plants or plants with significant time delay this paper extends our previous papers in which we show rigorously that some fuzzy controllers are actually nonlinear pi pd and pid controllers with variable gains that can outperform their linear counterparts in the present paper we study the analytical structure of an important class of two and three dimensional fuzzy controllers we link the entire class as opposed to one controller at a time to nonlinear pi pd and pid controllers with variable gains by establishing the conditions for the former to structurally become the latter unlike the results in the literature which are exclusively for the fuzzy controllers using linear fuzzy sets for the input variables this class of fuzzy controllers employs nonlinear input fuzzy sets of arbitrary types our structural results are thus more general and contain the existing ones as special cases two concrete examples are provided to illustrate the usefulness of the new results

lagrangian construction for representations of hecke algebras 1 let k be a non archimedean local field with the residue field consisting of q elements according to langlands philosophy irreducible complex representations of a reductive k group g should roughly speaking correspond to homomorphisms of the weil deligne group into the socalled l group let zc g be the iwahori subgroup the category of admissible gkmodules generated by z fixed vectors is known to be equivalent to the category of finite dimensional representations of the hecke algebra h z g z langlands philosophy suggests via that equivalence the following classification of irreducible hecke algebra modules stated in purely complex terms and not involving any local held

comments on robust stabilization of a class of uncertain nonlinear systems via fuzzy control quadratic stabilizability h sup spl infin control theory and linear matrix inequalities this paper presents stability analysis for a class of uncertain nonlinear systems and a method for designing robust fuzzy controllers to stabilize the uncertain nonlinear systems first a stability condition for takagi and sugeno s fuzzy model is given in terms of lyapunov stability theory next new stability conditions for a generalized class of uncertain systems are derived from robust control techniques such as quadratic stabilization h sup spl infin control theory and linear matrix inequalities the derived stability conditions are used to analyze the stability of takagi and sugeno s fuzzy control systems with uncertainty which can be regarded as a generalized class of uncertain nonlinear systems the design method employs the so called parallel distributed compensation important issues for the stability analysis and design are remarked finally three design examples of fuzzy controllers for stabilizing nonlinear systems and uncertain nonlinear systems are presented

studies on the output of fuzzy controller with multiple inputs abstract in this paper we discuss the output of a fuzzy controller in which an arbitrary number of fuzzy control inputs and an arbitrary number of fuzzy sets employed to fuzzify every linguistic variable are considered in the fuzzy control rules a rule base with a rule mapping concept is considered a defuzzifier with the general t norm and t conorm operators is formulated a triangular family is adopted so that the defuzzifier of the complete rule mapping is represented by a formula consisting of only some rules using a specific logic pair to evaluate values we derive that the output of the fuzzy controller can be simply represented by a linear parametric function of the inputs to the fuzzy controller

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

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

regular paper the takagi sugeno fuzzy controllers using the simplified linear control rules are nonlinear variable gain controllers takagi sugeno ts for short fuzzy controllers have been used and treated as black box controllers and there exists no explicit structure of any ts fuzzy controller in the literature in this paper we investigate the analytical structure of the ts fuzzy controllers that use our newly introduced simplified ts control rule scheme our scheme requires that all the rule consequent employ a common linear function and be proportional to one another the scheme drastically reduces the number of adjustable parameters required by the original ts rule scheme product fuzzy logic and was employed in the rule antecedents we theoretically proved that the ts controllers using the simplified rule scheme were nonlinear variable gain controllers the proportionality of the rule consequent parameterized the gain variation characteristics as special cases the ts controllers with three or two input variables and the simplified linear rule consequent were nonlinear variable gain pid pi or pd controllers as an illustration we mathematically analyzed the nonlinear variable gain pi controllers and revealed that the gain variation empowered the controllers to outperform their linear counterparts stability of the nonlinear pi control systems was also addressed a practical example is given to show how to achieve the desired characteristics of the gain variation by properly designing the simplified rules

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

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

deriving analytical input output relationship for fuzzy controllers using arbitrary input fuzzy sets and zadeh fuzzy and operator a fuzzy controller uses either zadeh or product fuzzy and operator with the former being more frequently used than the latter we have recently published a novel technique for deriving analytical input output relation for the fuzzy controllers that use zadeh and operator and arbitrary trapezoidal input fuzzy sets including triangular ones as special cases in this paper we have developed a general technique based on that technique to cover arbitrary types of input fuzzy sets moreover we have established some necessary and sufficient conditions to characterize general relationship between shape of input fuzzy sets and shape of input space divisions an important and integral issue because analytical relationship differs in different regions of input space the new technique and the shape relations are applicable to any type of fuzzy controllers e g mamdani type or takagi sugeno type the analytical structures that we have derived provide an unprecedented opportunity to insightfully and rigorously examine the advantages and shortcomings of different design choices available for various components of the fuzzy controllers we have focused on type selection for input fuzzy sets of mamdani fuzzy controllers our preliminary analysis indicates that the fuzzy controllers using trapezoidal fuzzy sets may be understood and possibly analyzed and designed more sensibly and easily in the context of conventional control theory than the fuzzy controllers using any other types of fuzzy sets our proposition is that trapezoidal fuzzy sets should be the first choice and used most of time possible implication for automatic learning of input fuzzy sets via neural networks or genetic algorithms is briefly discussed

minimal two spheres and the topology of manifolds with positive curvature on totally isotropic two planes there has been much interest among differential geometers in finding relationships between curvature and topology of riemannian manifolds for the most part efforts have been directed towards understanding the implications of positive or negative sectional curvature ricci curvature or scalar curvature but there are other hypotheses on the curvature which also deserve investigation in this article we will consider the topological implications of a new curvature assumption positive curvature on totally isotropic two planes and we will prove via the sacks uhlenbeck theory of minimal two spheres that a compact simply connected riemannian manifold of dimension at least four with positive curvature on totally isotropic two planes is homeomorphic to a sphere as corollaries we will obtain a proof via minimal two spheres for riemannian manifolds of dimension at least four of the berger klingenberg rauch toponogov sphere theorem strengthened to pointwise quarter pinching as well as a proof that every simply connected compact riemannian manifold with positive curvature operators is homeomorphic to a sphere let m be an n dimensional riemannian manifold with tangent space tpm at the point p e m recall that the curvature operator at p is the self adjoint

general siso takagi sugeno fuzzy systems with linear rule consequent are universal approximators takagi sugeno ts fuzzy systems have been employed as fuzzy controllers and fuzzy models in successfully solving difficult control and modeling problems in practice virtually all the ts fuzzy systems use linear rule consequent at present there exist no results qualitative or quantitative to answer the fundamentally important question that is especially critical to ts fuzzy systems as fuzzy controllers and models are ts fuzzy systems with linear rule consequent universal approximators if the answer is yes then how can they be constructed to achieve prespecified approximation accuracy and what are the sufficient renditions on systems configuration in this paper we provide answers to these questions for a general class of single input single output siso fuzzy systems that use any type of continuous input fuzzy sets ts fuzzy rules with linear consequent and a generalized defuzzifier containing the widely used centroid defuzzifier as a special case we first constructively prove that this general class of siso ts fuzzy systems can uniformly approximate any polynomial arbitrarily well and then prove by utilizing the weierstrass approximation theorem that the general ts fuzzy systems can uniformly approximate any continuous function with arbitrarily high precision furthermore we have derived a formula as part of sufficient conditions for the fuzzy approximation that can compute the minimal upper bound on the number of input fuzzy sets and rules needed for any given continuous function and prespecified approximation error bound an illustrative numerical example is provided

geometric and spectral estimates based on spectral ricci curvature assumptions we obtain a bonnet myers theorem under a spectral condition a closed riemannian manifold mn g for which the lowest eigenvalue of the ricci tensor is such that the schr odinger operator n 2 is positive has finite fundamental group as a continuation of our earlier results we obtain isoperimetric inequalities from a kato condition on the ricci curvature furthermore we obtain the kato condition for the ricci curvature under purely geometric assumptions

intersection homology ii in 19 20 we introduced topological invariants ih x called intersection homology groups for the study of singular spaces x these groups depend on the choice of a perversity p a perversity is a function from 2 3 to the non negative integers such that both c and c 2 c are positive and increasing functions of c 2 1 the group ihr is defined for spaces x called pseudomanifolds a pseudomanifold of dimension n is a space that admits a stratification x x n x n z x n 3 x 1 x 0

on the lp independence of the spectrum of the hodge laplacian on non compact manifolds abstract the central aim of this paper is the study of the spectrum of the hodge laplacian on differential forms of any order k in l p the underlying space is a c smooth open manifold m n with ricci curvature bounded below and uniformly subexponential volume growth it will be demonstrated that on such manifolds the l p spectrum of the hodge laplacian on differential k forms is independent of p for 1 p whenever the weitzenbock tensor on k forms is also bounded below it follows as a corollary that the isolated eigenvalues of finite multiplicity are l p independent the proof relies on the existence of a gaussian upper bound for the heat kernel of the hodge laplacian by considering the l p spectra on the hyperbolic space h n 1 we conclude that the subexponential volume growth condition is necessary in the case of one forms as an application we will show that the spectrum of the laplacian on one forms has no gaps on certain manifolds with a pole or that are in a warped product form this will be done under less strict curvature restrictions than what has been known so far and it was achieved by finding the l 1 spectrum of the laplacian

compact manifolds with a little negative curvature on etend le theoreme de bochner et le theoreme de myers pour des p formes en donnant des obstructions topologiques pour des varietes admettant des metriques avec une petite courbure negative

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

towards a paradigm for fuzzy logic control the purpose of this paper is to lay a foundation for fuzzy logic control flc design constructing a class of fuzzy logic controllers that is suitable for a broad range of controls applications some scalar definitions in flc are extended to the n dimensional case including vector fuzzy number and membership vector a rigorous mathematical expression is given for the function g x called the reasoning surface manufactured by a fuzzy associative memory fam for the existence and uniqueness of solutions in any closed loop system with a fuzzy logic controller it is shown that the fam function g x must be lipschitz this places restrictions on the allowed rules in the flc it is argued that for effective control design the membership functions should not be uniformly spaced the triangular membership functions generally suffice and product inferencing should be used under these conditions the flc manufactures a piecewise multilinear lipschitz fam function g x that is a generalized proportional plus derivative plus offset controller the scalar and two dimensional cases are studied in detail

general takagi sugeno fuzzy systems with simplified linear rule consequent are universal controllers models and filters abstract takagi sugeno ts fuzzy systems have successfully been employed mainly in a trial and error manner to solve many control and modeling problems but their applications as signal filters remain to be fully explored compared to their nonfuzzy counterparts ts fuzzy controllers and models are difficult to be efficiently constructed because there is a large number of design parameters in the rule consequent the number grows dramatically with the increase of the number of input fuzzy sets and input variables furthermore there exists little published result on relationship between ts fuzzy controllers models filters and their nonfuzzy counterparts in this paper we investigate in relation to some popular nonfuzzy controllers models and filters analytical structure of a general class of multi input single output miso ts fuzzy systems that use arbitrary fuzzy rules with our recently introduced simplified linear rule consequent other components of the fuzzy systems in this study are general arbitrary continuous input fuzzy sets any type of fuzzy logic and and the generalized defuzzifier containing the widely used centroid defuzzifier as a special case we prove that the general miso ts fuzzy systems are 1 nonlinear variable gain controllers when implemented as controllers or 2 nonlinear time varying auto regressive with the extra input arx models when implemented as models or 3 nonlinear infinite impulse response iir or finite impulse response fir filters when implemented as filters furthermore we constructively prove that the general ts fuzzy systems with the simplified linear rule consequent are universal approximators and can approximate any continuous function in closed domain arbitrarily well the practical implication of these results is that these fuzzy systems with much less design parameters are always able to produce solutions to various control modeling and filtering problems we also establish sufficient conditions that can be used to calculate the number of input fuzzy sets and rules needed for achieving prespecified approximation accuracy

fuzzy controllers lifting the linear nonlinear frontier abstract this paper brings back some ideas related with the linear analysis of fuzzy controllers a fuzzy control algorithm does not go further than a non linear function described by its inference map firstly it is possible to obtain the inference map corresponding to a classical pid controller by choosing adequately the linguistic terms the membership functions and the table of rules then the inverse problem is presented to obtain the closer pid to a given fuzzy controller from these ideas the paper focuses its attention on the obtainment of a useful tool to design fuzzy controllers at least as good as the pid that allows the system to follow a specified behaviour the next step is to improve the fuzzy controller parameters finally an industrial application of these ideas is shown the design of a fuzzy controller over a clinker cooler with grill improving p and pi controllers already functioning is discussed

characteristic cycles for the loop grassmannian and nilpotent orbits x which is a linear combination of closures of conormal bundles to submanifolds of x intuitively the microlocal multiplicities c me asurethesingularity of at in settings related to representation theory a group g acts on x is g equivariant and the microlocal multiplicities play a significant but only partially understood role in representation theory see ro sv abv and kasa e g in this paper we compute microlocal multiplicities for certain cases of interest in representation theory let g be a connected reductive group with loop group lg and let p bethesubgroup of lg consisting of loops with positive fourier coefficients then p orbits on theloop grassmannian lg p correspond to the irreducible representations l of thedual re ductivegroup g for dominant weights and of a torus of

explicit formulas for fuzzy controller abstract in this paper we present some explicit formulas of fuzzy control in the simple case of one condition these formulas are then used on the one hand to study the effects of some design factors inference operators number of partitions etc and on the other hand to carry out simulations

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

fuzzy chaos hybrid controller for controlling of nonlinear systems a novel concept for controlling of nonlinear systems using chaos and fuzzy model based regulators is presented in the control of such systems we employ two phases the first of which uses open loop control forming a chaotic attractor or using chaotic inherent features in a system itself once the system states reach a predefined convex domain open loop control is cut off and a fuzzy model based controller is employed under state feedback control in the second phase the relaxed stability conditions and linear matrix inequalities lmis based design for a fuzzy regulator is introduced to construct a fuzzy attractive domain in which a global solution is obtained so as to achieve the desired stability condition of the closed loop system the proposed controller architecture has been tested using three nonlinear systems the henon map the lorenz attractor and a two link manipulator the simulation results show the effectiveness of the proposed controller

a little microlocal morse theory if a complex analytic function f has a stratified isolated critical point then it is known that the cohomology of the milnor fibre of f has a direct sum decomposition in terms of the normal morse data to the strata we use microlocal morse theory to obtain the same result under the weakened hypothesis that the vanishing cycles along f have isolated support we also investigate an index theoretic proof of this fact