entropy based operations on fuzzy sets by using a fuzzy entropy approach three sets of new generalized operators are presented after a general discussion on fuzzy entropy the concept of an elementary entropy function of a fuzzy set is introduced using this mapping the generalized intersections and unions are defined as mappings that assign the least and the most fuzzy membership grade to each of the elements of the domain of the operators respectively it is shown that these operators can be constructed from the conventional min and max operations next two modified sets of operations are introduced the second part of the paper investigates the applicability of the new operators in fuzzy logic controllers simulations have been carried out so as to determine the effects of the operators on the performance of the fuzzy controllers it is concluded that the first set of operators does not provide stable control but the performance of the fuzzy controller can be improved by using the modified operations for a class of plants

design and analysis of a fuzzy proportional integral derivative controller abstract this paper describes the design principle tracking performance and stability analysis of a fuzzy proportional integral pi plus a derivative d controller first the fuzzy pi d controller is derived from the conventional continuous time linear pi d controller then the fuzzification control rule base and defuzzification in the design of the fuzzy controller are discussed in detail the resulting controller is a discrete time fuzzy version of the conventional pi d controller which has the same linear structure in the proportional integral and derivative parts but has nonconstant gains the proportional integral and derivative gains are nonlinear functions of the input signals the new fuzzy pi d controller thus preserves the simple linear structure of its conventional counterpart yet enhances the self tuning control capability computer simulation results have demonstrated the advantages of the fuzzy controller particularly when the process to be controlled is nonlinear after a brief 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 shown to compare with the conventional pi d controller computer simulation results have shown the new fuzzy controller indeed has satisfactory tracking performance

on the application of fuzzy sets theory to the optimal flood control problem arising in water resources systems abstract we first survey the application of fuzzy sets theory to various problems occurring in water resources systems the problem of optimal flood control planning by an appropriate integration of structural and non structural measures with the objective of optimizing the flood damage reduction due to recurrent floods is modeled via fuzzy sets methodologies a two level optimization model appropriate for planning decisions first on a regional or local level and then on a national level is proposed a third phase involving coordination is appended in particular a fuzzy optimization model involving an adroit combination of fuzzy dynamic programming type reasoning and a branch and bound type search procedure is offered as a more utilitarian approach than its crisp equivalent the performance of the algorithm is illustrated with an example

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

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

fuzzy controls under various fuzzy reasoning methods abstract a fuzzy logic controller consists of linguistic control rules tied together by means of two concepts fuzzy implications and a compositional rule of inference most of the existing fuzzy logic controllers are based on the approximate reasoning method due to mamdani this paper introduces other fuzzy implications such as the arithmetic rule and maximin rule for linguistic control rules and compares control results for a plant model with first order delay under various approximate reasoning methods moreover control results are compared when the widths of fuzzy sets of linguistic control rules are changed

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

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

analytical structures and analysis of the simplest fuzzy pi controllers this paper deals with simplest fuzzy pi controllers which employ two fuzzy numbers on the universe of discourse uod of each input variable and three fuzzy numbers on the uod of output variable analytical structures of such controllers are derived using triangular membership functions for fuzzification different combinations of t norms and t conorms different inference methods and center of area coa method for defuzzification properties of these controllers are investigated a comparative study is made on i the fuzzy pi controllers derived and ii on the fuzzy pi controllers and their counterpart conventional pi controller moreover sufficient conditions for bounded input bounded output bibo stability of fuzzy pi control systems are established using the well known small gain theorem

the sphere packing problem in dimension 8the sphere packing problem in dimension 8 in this paper we prove that no packing of unit balls in euclidean space r 8 has density greater than that of the e8 lattice packing

the sphere packing problem in dimension 24 building on viazovska s recent solution of the sphere packing problem in eight dimensions we prove that the leech lattice is the densest packing of congruent spheres in twenty four dimensions and that it is the unique optimal periodic packing in particular we find an optimal auxiliary function for the linear programming bounds which is an analogue of viazovska s function for the eight dimensional case

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

parametric classes of generalized conjunction and disjunction operations for fuzzy modeling it is argued that inference procedures of fuzzy models do not always require commutativity and associativity of the operations used this raises the possibility of considering nonassociative and noncommutative conjunction and disjunction operations such operations are investigated in this paper and different methods for their generation are proposed a number of new types of conjunction operations that are simpler than the known parametric classes of t norms are given and as an application example the approximation of a function by a fuzzy inference system is considered

sphere packing bounds via spherical codes the sphere packing problem asks for the greatest density of a packing of congruent balls in euclidean space the current best upper bound in all suciently high dimensions is due to kabatiansky and levenshtein in 1978 we revisit their argument and improve their bound by a constant factor using a simple geometric argument and we extend the argument to packings in hyperbolic space for which it gives an exponential improvement over the previously known bounds additionally we show that the cohn elkies linear programming bound is always at least as strong as the kabatiansky levenshtein bound this result is analogous to rodemich s theorem in coding theory finally we develop hyperbolic linear programming bounds and prove the analogue of rodemich s theorem there as well

analytical structures and analysis of fuzzy pd controllers with multifuzzy sets having variable cross point level this paper deals with fuzzy pd controllers which employ n 3 input fuzzy sets with variable cross point level and 2n 1 output fuzzy sets analytical structures of fuzzy pd controllers are derived using triangular membership functions for fuzzification rule base with four linear control rules two different triangular conorms lukasiewicz or and zadeh or different inference methods such as mamdani s minimum larsen s product and drastic product and center of area method for defuzzification properties of such fuzzy pd controllers are studied influence of variable cross point level on controller performance is demonstrated with an example

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

analytical structure of the typical fuzzy controllers employing trapezoidal input fuzzy sets and nonlinear control rules fuzzy controllers are generally nonlinear controllers and as such it is important toreveal analytical structure of fuzzy controllers relative to conventional nonlinear controllers so that the well developed nonlinear control theory can be utilized to effectively analyze and design fuzzy control systems in our previous papers we have shown that the analytical structure of the fuzzy controllers using triangular input fuzzy sets and linear fuzzy control rules is the sum of a global two dimensional multilevel relay and a local nonlinear pi controller in this paper we present the analytical structure of much more general and typical fuzzy controllers the ones that employ trapezoidal input fuzzy sets and arbitrary nonlinear control rules we analytically prove that the structure of such controllers is the sum of a global nonlinear controller and a local nonlinear pi like controller whose proportional gain and integral gain vary both globally and locally with change of the controllers inputs this result significantly generalizes our previous ones and contains them only as special cases analytical results are presented which reveal the characteristics and properties of the fuzzy controllers we show that the global nonlinear controller is characterized solely by fuzzy control rules that is different nonlinear control rules result in different types of global nonlinear controllers when the number of input fuzzy sets approaches oo the global nonlinear controller approaches a control rules dependent nonlinear controller whereas the local nonlinear pi like controller disappears the results presented in this paper can directly be used to generate analytical structure of any specific fuzzy controllers and we demonstrate how by illustrative examples

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

three dimensional min max gravity based fuzzy pid inference analysis and tuning this paper proposes a novel algorithm to produce an analytical solution for three input fuzzy pid systems the analysis is based on the most common inference method zadeh mamdani s min max gravity fuzzy reasoning two types of fuzzy systems are investigated both types use unevenly distributed triangular shaped fuzzy input memberships in the first type the output fuzzy sets are considered to be uniformly distributed symmetrical triangular type memberships and in the second type the output fuzzy sets are considered unevenly distributed singleton functions a new input transformation technique is developed to reduce the number of input conditions required in defining the fuzzy output the multi phase complexity that exists in conventional fuzzy reasoning analysis is thus avoided an efficient solution algorithm is outlined to produce the general fuzzy output solution using a minimum number of nonlinear expressions with the proposed method the fuzzy output solution for a three input fuzzy system having uniform output membership functions requires only two nonlinear expressions when the output fuzzy sets are unevenly distributed the general three input based solution requires only four nonlinear expressions the fuzzy controller output is finally decomposed into linear and nonlinear parts the output decomposition is useful in defining tuning gains for fuzzy pid systems

convolution roots of radial positive definite functions with compact support a classical theorem of boas kac and krein states that a characteristic function with x 0 for x t admits a representation of the form x u y u y x dy x r where the convolution root u l 2 r is complex valued with u x 0 for x 2 the result can be expressed equivalently as a factorization theorem for entire functions of finite exponential type this paper examines the boas kac representation under additional constraints if is real valued and even can the convolution root u be chosen as a real valued and or even function a complete answer in terms of the zeros of the fourier transform of is obtained furthermore the analogous problem for radially symmetric functions defined on r d is solved perhaps surprisingly there are compactly supported radial positive definite functions that do not admit a convolution root with half support however under the additional assumption of nonnegativity radially symmetric convolution roots with half support exist further results in this paper include a characterization of extreme points pointwise and integral bounds turan s problem and a unified solution to a minimization problem for compactly supported positive definite functions specifically if f is a probability density on r d whose characteristic function vanishes outside the unit ball then x 2 f x dx 0 4j 2 d 2 2 where j v denotes the first positive zero of the bessel function j v and the estimate is sharp applications to spatial moving average processes geostatistical simulation crystallography optics and phase retrieval are noted in particular a real valued half support convolution root of the spherical correlation function in r 2 does not exist

wiener s problem for positive definite functions we study the sharp constant w n d in wiener s inequality for positive definite functions begin aligned int mathbb t n f 2 dx le w n d d 1 int d f 2 dx quad d subset mathbb t n end aligned wiener proved that w 1 delta delta infty delta in 0 1 2 hlawka showed that w n d le 2 n where d is an origin symmetric convex body we sharpen hlawka s estimates for d being the ball delta b n and the cube delta i n in particular we prove that w n delta b n le 2 0 401 cdots o 1 n we also obtain a lower bound of w n d moreover for a cube d frac 1 q i n with q 3 4 ldots we obtain that w n d 2 n our proofs are based on the interrelation between wiener s problem and the problems of turan and delsarte

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

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

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

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

a systematic study of fuzzy pid controllers function based evaluation approach a function based evaluation approach is proposed for a systematic study of fuzzy proportional integral derivative pid like controllers this approach is applied for deriving process independent design guidelines from addressing two issues simplicity and nonlinearity to examine the simplicity of fuzzy pid controllers we conclude that direct action controllers exhibit simpler design properties than gain scheduling controllers then we evaluate the inference structures of direct action controllers in five criteria control action composition input coupling gain dependency gain role change and rule parameter growth three types of fuzzy pid controllers using one two and three input inference structures are analyzed the results according to the criteria demonstrate some shortcomings in mamdani s two input controllers for keeping the simplicity feature like a linear pid controller a one input fuzzy pid controller with one to three mapping inference engine is recommended we discuss three evaluation approaches in a nonlinear approximation study function estimation based generalization capability based and nonlinearity variation based approximations the study focuses on the last approach a nonlinearity evaluation is then performed for several one input fuzzy pid controllers based on two measures nonlinearity variation index and linearity approximation index using these quantitative indices one can make a reasonable selection of fuzzy reasoning mechanisms and membership functions without requiring any process information from the study we observed that the zadeh mamdani s max min gravity scheme produces the highest score in terms of nonlinearity variations which is superior to other schemes such as mizumoto s product sum gravity and takagi sugeno kang schemes

multivariable fuzzy control applied to the physical chemical treatment facility of a cellulose factory abstract feedback linearization is a well known technique in nonlinear control in which known system nonlinearities are canceled by the control input leaving a linear control problem feedback linearization requires an exact model for the system fundamental and advanced developments in neuro fuzzy synergy for modeling and control are used to apply the feedback linearization control law on second order plants in the models that are used the nonlinear plant is decomposed on six fuzzy systems necessary to apply the control signal to allow the following of a reference value a practical application is also presented using a waste water plant this method can be extended to multiple input multiple output mimo plants based on input output data pairs collected directly from the plant

theoretical analysis of crisp type fuzzy logic controllers using various t norm sum gravity inference methods the input output parametric relationship of a class of crisp type fuzzy logic controllers flcs using various t norm sum gravity inference methods is studied four most important t norms are used to calculate the matching level of each control rule and the explicit mathematical forms of reasoning surfaces obtained by using these four t norms are addressed reasoning surfaces of these crisp type flcs are proved to be composed of a two dimensional multilevel relay no matter which t norm is used and a local position dependent nonlinear compensator with output pattern influenced by the t norms is selected by analyzing the intrinsic operation of the four t norms the authors find that both standard intersection and algebraic product are suitable operators to perform the inference of the flc however bounded difference and drastic intersection are disqualified because they cannot satisfy some important criteria a measure of relative degree of nonlinearity is defined to examine the output figures of these crisp type flcs the ultimate behavior of these crisp type flcs as the number of linguistic terms approaches infinity is also explored the local stability criteria for the proportional integral pi type fuzzy control systems and the natural global stability characteristic for the proportional derivative pd type fuzzy control systems are also examined

theorectical analysis of a fuzzy logic controller with unequally spaced triangular membership functions abstract the mathematical analysis concerning the explicit input output relation of the fuzzy logic controller flc is addressed in this article under the assumption of simple rule mapping and the use of arbitrarily located triangular fuzzy partitions as the membership functions the inferred output of the flc can be decomposed into two terms the global multilevel relay and the local nonlinear compensator the ultimate control behavior of this flc and the equivalence to nonlinear proportional integral pi controller as the number of linguistic terms are approaching to infinity are also investigated the local stability criteria for pi type flcs are derived and the global stability properties for pd type flcs are established this analysis is a generalization to previous related research works where the flc is characterized by equally spaced triangular fuzzy partitions on the universal sets

new upper bounds on sphere packings i we continue the study of the linear programming bounds for sphere packing introduced by cohn and elkies we use theta series to give another proof of the principal theorem and present some related results and conjectures this article is in the arxiv as arxiv math mg 0110010

a new fuzzy logic learning control scheme for repetitive trajectory tracking problems in this paper we consider repeatable tracking control tasks using a new control approach pd type fuzzy logic learning control fllc fllc integrates two main control strategies fuzzy logic control as the basic control part and learning control as the refinement part the new fllc is constructed by simply adding an iterative learning mechanism to a fuzzy pd controller the incorporation of the learning function into fuzzy pd controllers ensures exact tracking because it completely nullifies the effects of reference signal and periodic disturbances on the tracking error through the rigorous proof based on energy function and functional analysis we show that the proposed fllc system achieves the following novel properties 1 the tracking error sequence converges uniformly to zero 2 learning control sequence converges to the desired control profile almost everywhere

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

two sample problem with bivariate exchangeable observations on utilise la propriete de symetrie de ce type de probleme et on propose une maniere alternative d analyser les donnees