on the robustness of lq regulators it is shown that in spite of its large gain and phase margins the linear quadratic state feedback regulator may suffer from poor robustness where small changes in the parameters of the system may lead to fast unstable closed loop modes

algebraic and proof theoretic characterizations of truth stressers for mtl and its extensions extensions of monoidal t norm logic mtl and related fuzzy logics with truth stresser modalities such as globalization and very true are presented here both algebraically in the framework of residuated lattices and proof theoretically as hypersequent calculi completeness with respect to standard algebras based on t norms embeddings between logics decidability and the finite embedding property are then investigated for these logics

variable domain fuzzy sets part ii apparatus abstract this is the second part of a two part paper on operations on fuzzy sets that may not share the same domain of definition part i described several ways of representing variable domain fuzzy sets and introduced a few basic variable domain fuzzy set operations the present largely self contained part ii further develops the theory of variable domain fuzzy sets based primarily on the representation of variable domains by a dummy membership degree that indicates the out of domain assignment error we first describe several families of algebraic operations on the extended set of membership degrees these are later employed in the investigation of basic notions of variable domain fuzzy set theory the fuzzy set operations studied in this paper include variable domain variants of unions and intersections kernels and supports heights and plinths equalities and inclusions cartesian products and fuzzy relational compositions besides the initial examination of these variable domain notions our aim is to demonstrate the viability of variable domain fuzzy set theory and highlight the similarity as well as differences between the fixed and variable domain treatments of fuzzy sets

the triple rotation method for constructing t norms given an involutive negator n and a left continuous t norm t that either has no zero divisors or is rotation invariant we build a rotation invariant t norm from a rescaled version of t and its left right and front rotation depending on the involutive negator n and the set of zero divisors of t some reshaping of the rescaled version of t may occur during the rotation process the rescaled version of t itself can be understood as the b zoom of the newly constructed rotation invariant t norm with b the unique fixpoint of n starting with a rotation invariant t norm t there is however one important restriction the triple rotation method based on the involutive negator n will yield a t norm if and only if the companion q of t is commutative on 0 1 2 when q is not commutative on 0 1 2 there even does not exist a rotation invariant t norm with b zoom equal to t

on fuzzy type theory this paper is a generalization of classical simple type theory we have developed a formal system of fuzzy type theory which differs from the classical one essentially in extension of truth values from two to infinitely many the structure of truth values is assumed to be an imtl algebra residuated lattice with prelinearity and double negation extended by the baaz delta operation various properties of fuzzy type theory are proved including its completeness

robustness bound for receding horizon finite memory control lyapunov krasovskii approach this article presents a robustness bound rb of receding horizon finite memory control rhfmc kwon w h and han s 2004 receding horizon finite memory controls of output feedback controls for state space systems ieee transactions on automatic control 49 1905 1915 for continuous time state space systems with norm bounded uncertainties the proposed rb is easily obtained by solving a convex problem in terms of a linear matrix inequality we show through a numerical example that the rhfmc can guarantee the robust stabilisation for a larger class of uncertain systems than linear quadratic gaussian controls when some poles of closed loop systems are close to an imaginary axis in the complex plane

on good eq algebras a special algebra called eq algebra has been recently introduced by vilem novak its original motivation comes from fuzzy type theory in which the main connective is fuzzy equality eq algebras have three binary operations meet multiplication and fuzzy equality and a unit element they open the door to an alternative development of fuzzy many valued logic with the basic connective being a fuzzy equality instead of an implication this direction is justified by the idea presented by g w leibniz that a fully satisfactory logical calculus must be an equational one in this paper we continue the study of eq algebras and their special cases we introduce and study the prefilters and filters of separated eq algebras we give great importance to the study of good eq algebras as we shall see in this paper the goodness property and thus separateness is necessary for reasonably behaving algebras we enrich good eq algebras with a unary operation d the so called baaz delta fulfilling some additional assumptions that are heavily used in fuzzy logic literature we show that the characterization theorem obtained until now for representable good eq algebras also hold for the enriched algebra

linguistic characterization of time series the goal of this paper is to provide an overview of applications of special soft computing theories the fuzzy transform and fuzzy natural logic to analysis forecasting and mining information from time series the focus is especially placed on the ability of methods of fuzzy natural logic to provide information in sentences of natural language our approach is based on the decomposition of time series into three components the trend cycle seasonal component and noise the trend cycle is extracted using the f transform and its course is characterized by automatically generated linguistic description the latter is then used to forecast the trend cycle the trend cycle can be furthermore decomposed into trend general tendency and cycle the former is computed again using the f transform moreover the f1 transform makes it possible to estimate the direction of the trend which can then be characterized by expressions of natural language stagnating slightly increasing sharply decreasing etc finally we focus on selected problems of mining information from time series first we suggest an algorithm for finding intervals of monotonous behavior and then show how the theory of intermediate quantifiers a constituent of fuzzy natural logic and generalized aristotle s syllogisms can be applied to automatic summarization of information on time series

receding horizon unbiased fir filters for continuous time state space models without a priori initial state information a receding horizon unbiased finite impulse response filter rhuff is proposed for continuous time state space models linearity unbiasedness finite impulse response fir structure and independence of the initial state information will be required in advance in addition to a performance index of minimum variance the proposed rhuff is obtained by directly minimizing the performance index with the unbiasedness constraint the proposed rhuff is represented first in a standard fir form and then in an iterative form it is shown that the rhuff is equivalent to the existing receding horizon rh kalman fir filter the former is more systematic and logical while the latter is heuristic due to the handling of infinite covariance of the initial state information

monadic l fuzzy quantifiers of the type the aim of this paper is to introduce monadic l fuzzy quantifiers of the type and investigate some of their semantic properties that are well known in generalized quantifier theory relationships between the properties of generalized quantifiers are also studied for monadic l fuzzy quantifiers finally a property to be extensional with respect to a given l permutation equivalence is proposed and some properties are shown

fir filters and recursive forms for continuous time invariant state space models an fir finite impulse response filter and an fir smoother are introduced for continuous time invariant state space models it is shown in this note that finite impulse responses of the fir filter and smoother can be easily determined by solving a simple riccati type matrix differential equation on a finite interval especially for systems with stationary processes finite impulse responses of the fir filter and smoother become time invariant and can be computed from simpler equations for fast computational purposes recursive forms of the fir filter and smoother are derived by using adjoint variables in this case all gains for recursive forms are shown to be constant

logical structure of fuzzy if then rules abstract this paper provides a logical basis for manipulation with fuzzy if then rules our theory is wide enough and it encompasses not only finding a conclusion by means of the compositional rule of inference due to lotfi a zadeh but also other kinds of approximate reasoning methods e g perception based deduction provided that there exists a possibility to characterize them within a formal logical system in contrast with other approaches employing variants of multiple valued first order logic the approach presented here employs fuzzy type theory of v novak which has sufficient expressive power to present the essential concepts and results in a compact elegant and justifiable form within the effectively formalized representation developed here based on a complete logical system it is possible to reconstruct numerous well known properties of cri related fuzzy inference methods albeit not from the analytic point of view as usually presented but as formal derivations of the logical system employed the authors are confident that eventually all relevant knowledge about fuzzy inference methods based on fuzzy if then rule bases will be represented formalized and backed up by proof within the well founded logical representation presented here an immediate positive consequence of this approach is that suddenly all elements of a fuzzy inference method based on fuzzy if then rules are first class citizens of the representation there are clear logically founded definitions for fuzzy if then rule bases to be consistent complete or independent

eq algebras from the point of view of generalized algebras with fuzzy equalities eq algebras introduced by novak are algebras of truth values for a higher order fuzzy logic fuzzy type theory in this paper the compatibility of multiplication w r t the fuzzy equality in an arbitrary eq algebra is examined particularly an example indicates that the compatibility axiom does not always hold and then a class of eq algebras satisfying the compatibility axiom is characterized by introducing a residuated integral meet semilattice ordered monoid valued fuzzy algebra called a generalized algebra with a fuzzy equality

a formal theory of generalized intermediate syllogisms in this paper we continue developing the formal theory of intermediate quantifiers expressions such as most few almost all a lot of many a great deal of a large part of a small part of the theory is a fuzzy logic formalization of the concept introduced by peterson in his book we will syntactically prove that 105 generalized aristotle s syllogisms introduced in this book are valid in our theory at the same time we will also prove that syllogisms listed there as invalid are invalid also in our theory therefore we believe that our theory provides a reasonable mathematical model of the generalized syllogistics

fuzzy quantifiers defined over fuzzy domains abstract in this paper we introduce a generalization of the concept of fuzzy quantifier called fuzzy quantifiers over fuzzy domains the development of this generalization is motivated by the impossibility to satisfactorily define the important operation of relativization for standard fuzzy quantifiers we provide basic examples of these quantifiers and define and investigate also several semantic properties of them finally we prove a theorem saying among other things that all these two argument fuzzy quantifiers that possess properties of extension and conservativity are relativizations of one argument fuzzy quantifiers of this kind

poor modules with no proper poor direct summands abstract as a mean to provide intrinsic characterizations of poor modules the notion of a pauper module is introduced a module is a pauper if it is poor and has no proper poor direct summand we show that not all rings have pauper modules and explore conditions for their existence in addition we ponder the role of paupers in the characterization of poor modules over those rings that do have them by considering two possible types of ubiquity one according to which every poor module contains a pauper direct summand and a second one according to which every poor module contains a pauper as a pure submodule the second condition holds for the ring of integers and is just as significant as the first one for noetherian rings since in that context modules having poor pure submodules must themselves be poor it is shown that the existence of paupers is equivalent to the noetherian condition for rings with no middle class as indecomposable poor modules are pauper we study rings with no indecomposable right middle class i e the ring whose indecomposable right modules are pauper or injective we show that semiartinian v rings satisfy this property and also that a commutative noetherian ring r has no indecomposable middle class if and only if r is the direct product of finitely many fields and at most one ring of composition length 2 structure theorems are also provided for rings without indecomposable middle class when the rings are artinian serial or right artinian rings for which not having an indecomposable middle class suffices not to have a middle class include commutative noetherian and artinian serial rings the structure of poor modules is completely determined over commutative hereditary noetherian rings pauper abelian groups with torsion free rank one are fully characterized

triangular norm based predicate fuzzy logics the paper surveys the present state of knowledge on t norm based predicate fuzzy logics with their double semantics standard the set of truth values being the real interval 0 1 and general with abstract algebras of truth functions

non commutative first order eq logics eq algebras are meet semilattices endowed with two additional binary operations fuzzy equality and multiplication these algebras serve as a basic algebraic structure of truth values for many valued logics based on fuzzy equality instead of implication therefore they generalize residuated lattices in the sense that each residuated lattice is an eq algebra but not vice versa logics based on eq algebras are called eq logics and they can be considered as special kind of fuzzy logics after developing propositional and higher order ones we address in this paper the predicate first order eq logic first we overview some basic properties of eq algebras and the basic propositional eq logic analysis of necessary properties of the fuzzy equality that is in predicate eq logic considered not only between truth values the equivalence but also between objects revealed that we cannot consider the fuzzy equality in full generality without means enabling us to deal with the classical crisp equality this is possible using the connective therefore we pay a special attention to prelinear eq algebras and develop the corresponding propositional eq logic finally we in detail introduce syntax and semantics of the first order eq logics and prove various theorems characterizing its properties including completeness

maxima and minima in fuzzified linear orderings the paper studies the notions of fuzzy maxima minima suprema and infima in the setting of fuzzified linear orderings the main result shows that in fuzzified linear orders satisfying certain reasonable assumptions the fuzzy maxima and minima of all non empty fuzzy sets are non empty

logical foundations of rule based systems the evolution of a theory of fuzzy if then rules and its contribution to the establishment of fuzzy logic is a focus of this paper i am advocating in favour of hajek s fuzzy logic regarding it as a right methodology for special logical theories a theory of fuzzy if then rules as a special theory in the above considered sense is proposed this theory is for all practitioners who want to create a system of fuzzy if then rules free of conflicts logically consistent and rich enough to be able to make non trivial conclusions or answer inquires

brief a receding horizon unbiased fir filter for discrete time state space models this paper concerns with a new linear finite impulse response fir filter called the receding horizon unbiased fir rhuf filter for the state estimation in discrete time state space models to obtain the rhuf filter linearity unbiasedness and fir structure will be required beforehand in addition to a performance criteria of minimum variance the rhuf filter is obtained by directly solving an optimization problem with the unbiasedness constraint the rhuf filter has time invariance and deadbeat properties the rhuf filter is represented in both a batch form and an iterative form it is shown that the rhuf filter is equivalent to the existing receding horizon kalman fir rhkf filter whose optimality is not clear to understand the former is more systematic and logical while the latter is heuristic due to handling of infinite covariance of the initial state information

fuzzy sets and formal logics the paper discusses the relationship between fuzzy sets and formal logics as well as the influences fuzzy set theory had on the development of particular formal logics our focus is on the historical side of these developments

a cut free calculus for second order godel logic abstract we prove that the extension of the known hypersequent calculus for standard first order godel logic with usual rules for second order quantifiers is sound and cut free complete for henkin style semantics for second order godel logic the proof is semantic and it is similar in nature to schutte and tait s proof of takeuti s conjecture

normal forms for fuzzy logic functions and their approximation ability abstract aiming at developing theoretical backgrounds of fuzzy logic we consider an mv algebra of fuzzy logic functions two kinds of formulas regarded as disjunctive and conjunctive normal forms are introduced as a formal translation of if then rules their ability for an approximate representation of uniformly continuous functions are investigated the problem of transformation of any formula of propositional fuzzy logic into either of the normal forms is discussed

fuzzy terms in this paper we will show how purely categorical constructions of terms are advantageous when investigating situations concerning uncertainty more specifically where uncertainty comes from and how uncertainty is integrated when dealing with terms over selected signatures there are basically two ways of invoking uncertainty for terms on one hand we may proceed by building composed monads where uncertainty is provided by some suitable monad composed with the traditional term monad on the other hand we can provide a strictly formal basis for term monads being created over categories themselves carrying uncertainty this is the distinction between computing with fuzzy and fuzzy computing and the fundamental question raised by these constructions is where uncertainty resides in language constructions for logic this paper also shows how the notion of signature often needs to be expanded to levels of signatures in particular when dealing with type constructors such levels allow us to strictly delineate e g primitive operations type terms and value level terms levels of signature will in this paper be exemplified by the construction of the signature of simply typed lambda calculus

elements of model theory in higher order fuzzy logic in this paper we turn our attention to model theory of higher order fuzzy logic fuzzy type theory this theory generalizes model theory of predicate logic but has some interesting specificities we will introduce few basic concepts related to homomorphism isomorphism submodel etc and show some properties of them

relations in fuzzy class theory this paper studies fuzzy relations in the graded framework of fuzzy class theory fct this includes i rephrasing existing work on graded properties of binary fuzzy relations in the framework of fuzzy class theory and ii generalizing existing crisp results on fuzzy relations to the graded framework our particular aim is to demonstrate that fuzzy class theory is a powerful and easy to use instrument for handling fuzzified properties of fuzzy relations this paper does not rephrase the whole theory of fuzzy relations instead it provides an illustrative introduction showing some representative results with a strong emphasis on fuzzy preorders and fuzzy equivalence relations

a receding horizon kalman fir filter for discrete time invariant systems a receding horizon kalman fir filter is presented that combines the kalman filter and the receding horizon strategy when the horizon initial state is assumed to be unknown the suggested filter is a fir filter form which has many good inherent properties it can always be defined irrespective of singularity problems caused by unknown information about the horizon initial state the suggested filter can be represented in either an iterative form or a standard fir form it is also shown that the suggested filter possesses the unbiasedness property and the remarkable deadbeat property irrespective of any horizon initial condition the validity of the suggested filter is illustrated by numerical examples

a graded approach to cardinal theory of finite fuzzy sets part ii fuzzy cardinality measures and their relationship to graded equipollence abstract in this article we propose an axiomatic system for fuzzy cardinality measures referred to as fuzzy c measures for short assigning to each finite fuzzy set a generalized cardinal that expresses the number of elements that the fuzzy set contains the system generalizes an axiomatic system introduced by j casasnovas and j torrens 2003 we show that each fuzzy c measure is determined by two appropriate homomorphisms between the reducts of residuated dually residuated rdr lattices for linearly ordered rdr lattices we prove that each fuzzy c measure is a product of a non decreasing and a non increasing fuzzy c measure which indicates that there is a close relation between fuzzy c measures and fgcount flcount and fecount provided by l a zadeh 1983 and generalized by m wygralak 2001 finally the relationship of fuzzy c measures to graded equipollence introduced in the first part of this contribution is analyzed

a receding horizon kalman fir filter for linear continuous time systems a receding horizon kalman finite impulse response fir filter is suggested for continuous time systems combining the kalman filter with the receding horizon strategy in the suggested filter the horizon initial state is assumed to be unknown it can always be obtained irrespective of unknown information on the horizon initial state the filter may be the first stochastic fir form for continuous time systems that may have many good inherent properties the suggested filter can be represented in an iterative form and also in a standard fir form the suggested filter turns out to be a remarkable deadbeat observer the validity of the suggested filter is illustrated by numerical examples

equality relations as a basis for fuzzy control abstract the aim of this paper is to introduce a fuzzy control model with well founded semantics in order to explain the concepts applied in fuzzy control assuming that the domains of the input and output variables for the process are endowed with equality relations that reflect the indistinguishability of values lying closely together the use of triangular and trapezoidal membership functions can be justified and max inference where is a t norm turns out to be a consequence of our model distinguishing between a functional and a relational view of the control rules it is possible to explain when defuzzification strategies like mom or coa are appropriate or lead to undesired results

residuated eq algebras may not be residuated lattices abstract this paper aims to solve the question whether there is a residuated eq algebra which is not a residuated lattice in this paper we first introduce the notion of residuated meet semilattice monoids and research some properties of them we discuss the relations between residuated meet semilattice monoids and residuated lattices and prove that every complete or prelinear residuated meet semilattice monoid is a residuated lattice especially we construct a residuated meet semilattice monoid which has no lattice structure furthermore we discuss the relations between residuated meet semilattice monoids and residuated eq algebras finally by use of residuated meet semilattice monoids we give a theorem to show that there are some residuated eq algebras but they are not residuated lattices we also give an example which satisfies the conditions of the above theorem and hence it is a residuated eq algebra but it is not a residuated lattice

monoidal t norm based logic towards a logic for left continuous t norms hajek s bl logic is the fuzzy logic capturing the tautologies of continuous t norms and their residua in this paper we investigate a weaker logic mtl which is intended to cope with the tautologies of left continuous t norms and their residua the corresponding algebraic structures mtl algebras are defined and completeness of mtl with respect to linearly ordered mtl algebras is proved besides several schematic extensions of mtl are also considered as well as their corresponding predicate calculi

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

on the stability robustness of the continuous time lqg optimal control the robustness properties of the lqg optimal controller that is obtained by the asymptotic recovery method are discussed it is shown that in spite of the good recovered stability margins the resulting closed loop design is inherently unrobust in the sense that small parameter variations in the plant and or small realization errors in the controller may lead to an instantaneous instability of the design

the structure of generalized intermediate syllogisms abstract in this paper we continue development of the formal theory of intermediate quantifiers which are expressions of the natural language most many few etc in the previous paper we demonstrated that 105 generalized syllogisms are valid in our theory we introduce the generalization of all the figures and we show that for the proof of validity of all the generalized syllogisms we need to prove the validity of only few of them so that the validity of the other ones immediately follows

subtypes in fuzzy type theory abstract in this paper we introduce extension of the fuzzy type theory ftt by subtypes this demand arises in situations when for example it is useful to distinguish rational and real numbers where the set of rational numbers is a subset of the reals in the original ftt this causes a problem because we cannot assign rational numbers a special type the solution is to introduce special types for both of them and declare that the former is a subtype of the latter the resulting fuzzy type theory is denoted by sftt when constructing the frame for sftt we must realize that though all the considered functions are total if we take them as functions between subsets then they become partial

the lattice of prefilters of an eq algebra in this paper the notion of a prefilter generated by a nonempty subset of an eq algebra is introduced and a characterization of it is obtained it is proved that the set of all prefilters of an eq algebra is an algebraic lattice and it is a brouwerian lattice for an eq algebra furthermore it is shown that the set of all principal prefilters of an eq algebra is a sublattice of the lattice of prefilters then by defining an implication between two prefilters it is determined that the lattice of prefilters is a heyting algebra for an eq algebra finally the eq algebras for which the lattice of prefilters is a boolean algebra are given

results on equality algebras in this paper by considering the notion of equality algebra which is introduced by jenei in 18 as a possible algebraic semantic for fuzzy type theory we study and show that there are relations among equality algebras and some of other logical algebras such as residuated lattice mtl algebra bl algebra mv algebra hertz algebra heyting algebra boolean algebra eq algebra and hoop algebra the aim of this paper is to find that under which conditions equality algebras are equivalent to these logical algebras

a new solution to the induced l finite impulse response filtering problem based on two matrix inequalities in this paper we use an induced l approach to create a new filter with a finite impulse response fir structure for state space models with external disturbances this filter is called an induced l fir filter iliff the proposed iliff s gain matrix can be determined by solving a linear matrix inequality problem for a fixed positive scalar variable an illustrative example is given to demonstrate the effectiveness of the iliff