the rubin stark conjecture for imaginary abelian fields of odd prime power conductor we build upon ideas developed in 9 as well as results of greither on a strong form of brumer s conjecture 2 4 and prove rubin s integral version of stark s conjecture for imaginary abelian extensions of q of odd prime power conductor

analysis of the fuzzy controllability property and stabilization for a class of t s fuzzy models in this paper the controllability property for a class of takagi sugeno t s fuzzy models is analyzed while a fully nonlinear stabilizer is designed in a practical way it is shown that global fuzzy stabilizers can be constructed in a nonconservative way by means of a relatively simple approach the existence of such controllers depends on the fuzzy controllability conditions which are derived in a straightforward way the main advantage of the proposed approach is that the convergence of the closed loop system can be imposed arbitrarily some examples are given in order to illustrate the validity of the method finally the proposed controller is applied on an underactuated system known as pendubot and the results are compared with an stabilizer designed on the basis of lmis

stability and stabilization analysis of positive polynomial fuzzy systems with time delay considering piecewise membership functions this paper is concerned with the stability analysis of positive polynomial fuzzy model based control systems with time delay each of the polynomial fuzzy model and the polynomial fuzzy controller is allowed to have its own set of premise membership functions which can improve the design and realization flexibility conditions showing the positivity of the system are first obtained then to deal with the difficulty resulting from the mismatched premise membership functions to the stability stabilization analysis approximated membership functions representing the original ones are constructed and the information of membership functions is brought into the stability conditions this way the stability conditions can be relaxed since the extra knowledge on the system is considered stability conditions are obtained by using the sum of squares approach based on the lyapunov stability theory a simulation example is provided to verify the effectiveness of this method

design of a polynomial fuzzy observer controller with sampled output measurements for nonlinear systems considering unmeasurable premise variables in this paper we propose a polynomial fuzzy observer controller for nonlinear systems where the design is achieved through the stability analysis of polynomial fuzzy model based pfmb observer control system the polynomial fuzzy observer estimates the system states using estimated premise variables the estimated states are then employed by the polynomial fuzzy controller for the feedback control of nonlinear systems represented by the polynomial fuzzy model the system stability of the pfmb observer control system is analyzed based on the lyapunov stability theory although using estimated premise variables in polynomial fuzzy observer can handle a wider class of nonlinear systems it leads to a significant drawback that the stability conditions obtained are nonconvex matrix decoupling technique is employed to achieve convex stability conditions in the form of sum of squares we further extend the design and analysis to polynomial fuzzy observer controller using a sampled data technique for nonlinear systems where only sampled output measurements are available simulation examples are presented to demonstrate the feasibility and validity of the design and analysis results

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

congruences between derivatives of abelian l functions at s 0 let k k be a finite abelian extension of global fields we prove that a natural equivariant leading term conjecture implies a family of explicit congruence relations between the values at s 0 of derivatives of the dirichlet l functions associated to k k we also show that these congruences provide a universal approach to the refined abelian stark conjectures formulated by inter alia stark gross rubin popescu and tate we thereby obtain the first proofs of amongst other things the rubin stark conjecture and the refined class number formulas of both gross and tate for all extensions k k in which k is either an abelian extension of or is a function field

model based control and stability analysis of discrete time polynomial fuzzy systems with time delay and positivity constraints this paper proposes a novel lyapunov stabilization analysis of discrete time polynomial fuzzy model based control systems with time delay under positivity constraint the polynomial fuzzy model is constructed to describe the dynamics of a nonlinear discrete time system with time delay a model based polynomial fuzzy controller is designed using nonparallel distributed compensation technique to stabilize the system while driving the system states to positive using the positivity constraints the lyapunov stability and positivity conditions are formulated as sum of squares to relax the conservativeness of the obtained stability results two main methods are proposed in this paper first the piecewise linear membership functions plmfs are used to introduce the approximate error between piecewise and the original membership functions into the stability analysis and second introduce the boundary information of the premise variables into the stability analysis since the premise variables hold rich nonlinearity information a numerical example is given to demonstrate the effectiveness of the proposed approach

output regulation of polynomial fuzzy model based control systems this paper investigates the output regulation problem of polynomial fuzzy model based pfmb control systems with the consideration of system stability and control synthesis using the sum of squares sos based approach a pfmb control system is formed by a polynomial fuzzy model and a polynomial fuzzy controller with integral action connected in a closed loop three cases of pfmb control systems regarding the premise membership functions and number of fuzzy rules which have their own properties and advantages are considered the stability of the pfmb control systems is investigated based on the lyapunov stability theory with consideration of the information of membership functions sos based stability conditions corresponding to each case are obtained to guarantee system stability and to realize output regulation simulation examples are given to compare the three cases and illustrate the effectiveness of the proposed approach

stability analysis and performance design for fuzzy model based control system under imperfect premise matching this paper investigates the stability analysis and performance design of nonlinear systems to facilitate the stability analysis the takagi sugeno t s fuzzy model is employed to represent the nonlinear plant under the imperfect premise matching in which t s fuzzy model and fuzzy controller do not share the same membership functions a fuzzy controller with enhanced design flexibility and robustness property is proposed to control the nonlinear plant however the nice characteristic given by the perfect premise matching leading to conservative stability conditions vanishes in this paper under the imperfect premise matching information of membership functions of the fuzzy model and controller are considered in stability analysis with the introduction of slack matrices relaxed linear matrix inequality lmi based stability conditions are derived using lyapunov based approach furthermore lmi based performance conditions are provided to guarantee system performance simulation examples are given to illustrate the effectiveness of the proposed approach

fuzzy model based quantized guaranteed cost control of nonlinear networked systems this paper is concerned with the problem of guaranteed cost control for a class of continuous time networked control systems based on the takagi sugeno fuzzy model approach in this study network induced transmission delays and packet dropouts are taken into simultaneous consideration by the improved lyapunov krasovskii functionals a less conservative delay dependent condition is proposed for the stability analysis of the resulting closed loop system furthermore the sufficient condition under which the guaranteed cost controller exists is formulated in terms of linear matrix inequalities and the suboptimum solution is obtained by a cone complementarity linearization algorithm finally numerical examples are provided to illustrate the effectiveness of the proposed method

quadratic stability analysis of fuzzy model based control systems using staircase membership functions this paper presents the stability analysis of fuzzy model based fmb control systems staircase membership functions are introduced to facilitate the stability analysis through the staircase membership functions approximating those of the fuzzy model and fuzzy controller the information of the membership functions can be brought into the stability analysis based on the lyapunov stability theory stability conditions in terms of linear matrix inequalities lmis are derived in a simple and easy to understand manner to guarantee the system stability the proposed stability analysis approach offers a nice property that includes the membership functions of both fuzzy model and fuzzy controller in the lmi based stability conditions for a dedicated fmb control system furthermore the proposed stability analysis approach can be applied to the fmb control systems of which the membership functions of both fuzzy model and fuzzy controller are not necessarily the same greater design flexibility is allowed to choose the membership functions during the design of fuzzy controllers by employing membership functions with simple structure it is possible to lower the structural complexity and the implementation cost simulation examples are given to illustrate the merits of the proposed approach

local stability analysis of continuous time takagi sugeno fuzzy systems a fuzzy lyapunov function approach this paper proposes a strategy to estimate invariant subsets of the domain of attraction da for asymptotically stable zero equilibrium points of continuous time takagi sugeno t s fuzzy systems specifically by using lyapunov s stability theory and the linear matrix inequality lmi technique sufficient conditions for proving the local stability are provided in terms of single parameter minimization problems subject to lmi constraints or eigenvalue problems which are solvable via convex optimizations the fuzzy lyapunov functions flfs expressed by the so called multi dimensional fuzzy summations are employed to characterize invariant subsets of the da as sublevel sets of the flfs to compute a larger inner estimate of the da an iterative lmi algorithm is also developed finally illustrative examples show the efficacy of the approach

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

stability analysis of polynomial fuzzy model based control systems with mismatched premise membership functions this paper investigates the stability of polynomial fuzzy model based pfmb control system which is formed by a polynomial fuzzy model and a polynomial fuzzy controller connected in a closed loop to enhance the design flexibility the number of rules and the shape of premise membership functions of the polynomial fuzzy controller are considered to be chosen freely and are different from those of the polynomial fuzzy model however which make the stability analysis more difficult and potentially lead to conservative stability analysis result a sum of squares sos based stability analysis approach using the lyapunov stability theory is proposed to investigate the stability of the pfmb control systems and synthesize the polynomial fuzzy controller to facilitate the stability analysis and relax the stability analysis result the property of the membership functions and the boundary information of the membership grades and premise variables are taken into account in the stability analysis and incorporated into the sos based stability conditions a simulation example is given to illustrate the effectiveness of the proposed approach

stability analysis of positive polynomial fuzzy model based control systems with time delay under imperfect premise matching this paper deals with the stability and positivity analysis of polynomial fuzzy model based pfmb control systems with time delay which is formed by a polynomial fuzzy model and a polynomial fuzzy controller connected in a closed loop under imperfect premise matching to improve the design and realization flexibility the polynomial fuzzy model and the polynomial fuzzy controller are allowed to have their own set of premise membership functions a sum of squares based stability analysis approach using the lyapunov stability theory is employed to investigate the positivity and stability of the pfmb control systems and synthesize the polynomial fuzzy controller in order to relax the stability results we propose two methods first membership functions are considered as symbolic variables in the stability analysis and second the property of the membership functions and the boundary information of the membership functions are considered in the stability analysis a simulation example is given to illustrate the effectiveness of the proposed approach

polynomial fuzzy observed state feedback stabilization via homogeneous lyapunov methods a homogeneously state dependent polynomial lyapunov function for an observed state feedback polynomial fuzzy control system is proposed from which an advanced sufficient sum of squares sos condition formulated in terms of state dependent matrix inequalities to achieve fuzzy stabilization is derived three main features compared with existing nonhomogeneously polynomial methods are as follows 1 homogeneous lyapunov function and its time derivative are nicely linked via the hessian matrix 2 removing the nonconvex obstacles on dot p x when deriving a nonhomogeneous lyapunov stabilization condition dot v x 3 separation principle holds since common p falls out as a special case the proposed homogeneous sos approach is placed ahead of the existing sos based methods found in polynomial fuzzy stabilization controls to verify the problem solving theories of the homogeneous polynomial method announced four examples are demonstrated to show the promising features of the approach adduced finally an appendix illustrating the solving procedure is added for clarification on how the examples are solved via sos

polynomial fuzzy models for nonlinear control a taylor series approach classical takagi sugeno t s fuzzy models are formed by convex combinations of linear consequent local models such fuzzy models can be obtained from nonlinear first principle equations by the well known sector nonlinearity modeling technique this paper extends the sector nonlinearity approach to the polynomial case this way generalized polynomial fuzzy models are obtained the new class of models is polynomial both in the membership functions and in the consequent models importantly t s models become a particular case of the proposed technique recent possibilities for stability analysis and controller synthesis are also discussed a set of examples shows that polynomial modeling is able to reduce conservativeness with respect to standard t s approaches as the degrees of the involved polynomials increase

polynomial fuzzy model based control systems stability analysis via piecewise linear membership functions this paper presents the stability analysis of polynomial fuzzy model based pfmb control systems using the sum of squares sos approach the pfmb control system under consideration requires that the polynomial fuzzy model and polynomial fuzzy controller share neither the same premise membership functions nor the same number of fuzzy rules this class of pfmb control systems offers a greater design flexibility to the polynomial fuzzy controller however due to the imperfectly matched membership functions it usually produces more conservative stability conditions by following the traditional stability analysis approach for the fmb control systems to facilitate the stability analysis piecewise linear membership functions plmfs are proposed which offer a nice property that the grades of membership are governed by a finite number of sample points thus it allows the plmfs to be brought to the sos based stability conditions which are applied to the pfmb control systems with the specified plmfs rather than any shapes the system stability can be examined by checking only the pfmb control system at the sample points it is worth mentioning that the plmfs which are not necessarily implemented physically are a mathematical tool to carry out the stability analysis to verify the stability analysis result a simulation example is given to demonstrate the effectiveness of the proposed approach

single hidden layer fuzzy recurrent wavelet neural network when a fuzzy wavelet neural network fwnn has a large number of neurons in its consequent part it may not be able to effectively follow fast variations in the process this paper aims to develop a single hidden layer fuzzy recurrent wavelet neural network slfrwnn for the function approximation and identification of dynamic systems in the proposed framework the consequent part of each fuzzy rule is developed by a single neuron with the capability of storing the past data of the network to guarantee the convergence and to speed up the process of the on line training algorithm the optimal learning rate olr is introduced based on lyapunov stability theory the modifications allow the slfrwnn to be much faster than fwnn and hence it is more appropriate in real time applications the efficiency of the modified model is investigated using three different types of wavelet families in approximating a benchmark piecewise function furthermore two nonlinear dynamic plants are considered to demonstrate the feasibility of the slfrwnn in the system identification the results indicate that slfrwnn achieves higher accuracy and needs a lower number of parameters than the other models

relaxed stability and performance lmi conditions for takagi sugeno fuzzy systems with polynomial constraints on membership function shapes most linear matrix inequality lmi fuzzy control results in literature are valid for any membership function i e independent of the actual membership shape hence they are conservative with respect to other nonlinear control approaches when specific knowledge of the shapes is available this paper presents relaxed lmi conditions for fuzzy control that incorporate such shape information in the form of polynomial constraints generalizing previous works by the authors interesting particular cases are overlap product bounds and ellipsoidal regions numerical examples illustrate the achieved improvements as well as the possibilities of solving some multiobjective problems the results also apply to polynomial in membership takagi sugeno fuzzy systems

stability analysis and estimation of domain of attraction for positive polynomial fuzzy systems with input saturation in this paper the stability and positivity of positive polynomial fuzzy model based ppfmb control system are investigated in which the positive polynomial fuzzy model and positive polynomial fuzzy controller are allowed to have different premise membership functions from each other these mismatched premise membership functions can increase the flexibility of controller design however it will lead to the conservative results when the stability is analyzed based on the lyapunov stability theory to relax the positivity stability conditions the improved taylor series membership functions dependent itsmfd method is introduced by introducing the sample points information of taylor series approximate membership functions local error information and boundary information of substate space of premise variables into the stability positivity conditions meanwhile the itsmfd method is extended to the ppfmb control system with input saturation to relax the estimation of domain of attraction finally simulation examples are presented to verify the feasibility of this method

more relaxed stability analysis and positivity analysis for positive polynomial fuzzy systems via membership functions dependent method abstract in this paper the conservatism source of the positivity and stability analysis results of positive polynomial fuzzy model based ppfmb control system are studied also in order to improve the flexibility of controller design a fuzzy controller that does not depend on the membership functions of the fuzzy model is designed in the existing literatures it is proved that the lclf can reduce the conservatism of stability results however the lclf generally results in non convex conditions which is still a conservatism source to handle the non convex conditions the sector nonlinear concept is applied to handle non convex terms in stability conditions and the obstacles caused by mismatched membership functions can be eliminated by plmf dependent method in addition to relax the conservatism caused by the lack of membership functions information the plmf dependent positivity analysis are performed for the first time meanwhile plmf dependent method is extended to stability conditions to obtained more relaxed conditions finally a simulation example is presented to verify the feasibility of this method

local stabilization of discrete time t s fuzzy systems with magnitude and energy bounded disturbances recently the local stability and local stabilization problems have been investigated by several authors for nonlinear systems described by takagi sugeno t s fuzzy systems in this paper we study the local stabilization problem for discrete time t s fuzzy systems with magnitude and energy bounded disturbances the control design problem is cast as a linear matrix inequality lmi optimization the state feedback controller is designed in such a way that if the initial state is within a certain domain then the state trajectories under bounded disturbances will not escape the fuzzy modelling region of the state space finally examples are given to demonstrate the validity of the proposed method

structure identification of fuzzy model abstract the problems of structure identification of a fuzzy model are formulated a criterion for the verification of a structure is discussed using the criterion an algorithm for identifying a structure is suggested further a successive identification algorithm of the parameters is suggested the proposed methods are applied to an example

on the rubin stark conjecture for a special class of cm extensions of totally real number fields we use greither s recent results on brumer s conjecture to prove rubin s integral version of stark s conjecture up to a power of 2 for an infinite class of cm extensions of totally real number fields called nice extensions under the assumption that a certain iwasawa invariant vanishes as a consequence and under the same assumption we show that the brumer stark conjecture is true for nice extensions up to a power of 2

the exponential stability and asynchronous stabilization of a class of switched nonlinear system via the t s fuzzy model in this paper we investigate the problem of exponential stability and asynchronous stabilization for a class of switched nonlinear systems the takagi and sugeno t s fuzzy model is employed to approximate the subnonlinear dynamic systems with two level functions namely crisp switching functions and local fuzzy weighting functions we introduce continuous time switched fuzzy systems which inherently contain the features of the switched hybrid systems and t s fuzzy systems by the use of delicately constructed piecewise lyapunov like functions plfs and minimum dwell time method we obtain the exponential stability of the switched fuzzy systems which allows us to have stable and unstable nonlinear subsystems in practice for the control problem it inevitably takes some time to identify the system modes and apply the matched controller the asynchronous phenomena between the system modes switching and the controllers switching generally exists based on the result of stability the fuzzy state feedback controller under asynchronous switching is proposed for switched fuzzy systems in addition the lower bound of minimum dwell time can be obtained using convex optimization such that the switched fuzzy system can be exponentially stabilized if its minimum dwell time is larger than the bound the stability results and control laws of the switched fuzzy systems are formulated in the form of linear matrix inequalities that are numerically feasible finally two illustrated numerical examples are presented to show the effectiveness of the obtained theoretical results

interval type 2 regional switching t s fuzzy control for time delay systems via membership function dependent approach abstract this paper investigates the stabilization problem of the interval type 2 it2 takagi sugeno t s fuzzy model based control systems with time delay we extend the type 1 switching t s fuzzy model to deal with the it2 t s fuzzy model and the new constructed model is defined as the it2 regional switching t s fuzzy model in this paper furthermore a new type state feedback controller namely it2 regional switching fuzzy controller which switches according to system states is designed to perform the control process then based on the piecewise membership functions the membership function dependent approach is adopted specifically for time delay systems and hence several stability criteria are derived by utilizing the relaxed lyapunov krasovskii functional since the membership function dependent approach considers more information of lower and upper membership functions the conservatism of the stability conditions is reduced finally two numerical examples are provided to demonstrate that the theoretical results are effective and the membership function dependent approach plays an important role in relaxing the stability criteria

stabilization and separation principle of networked control systems using the t s fuzzy model approach this paper is concerned with the stabilization problem for a class of discrete time networked control systems ncss with bounded time delays and packet losses the controlled plant is represented by a takagi sugeno fuzzy model and both the state feedback control and output feedback control cases are considered by guaranteeing the decrement of lyapunov functional at each control signal updating step a less conservative stability condition for the state feedback ncss is derived and the corresponding stabilizing controller design method is also presented under an observer based framework the output feedback stabilization problem is further studied where the main contribution is the development of the separation principle for ncss illustrative examples are provided to show the advantage and effectiveness of the developed results

on the generalized local stability and local stabilization conditions for discrete time takagi sugeno fuzzy systems in this paper local stability and local stabilization problems are considered for discrete time nonlinear systems represented by takagi sugeno t s fuzzy systems improved methods to assess the local stability design locally stabilizing control laws and estimate the domain of attraction are developed in terms of single parameter minimization problems subject to linear matrix inequality lmi constraints the improvement is achieved by applying a convergent lmi relaxation technique to prove positivity of homogeneous polynomial parameter dependent matrices of arbitrary degree with variables in the simplex we also take advantage of the recently developed parameter variation modeling technique to deal with the case that the variation rate of the membership functions of the t s fuzzy systems is bounded and less than one finally several illustrative examples demonstrate the validity and efficiency of the proposed method

exact output regulation for nonlinear systems described by takagi sugeno fuzzy models the exact output regulation for takagi sugeno t s fuzzy models depends on two conditions 1 the local steady state zero error manifolds have to be the same for every local subsystem and 2 the local input matrices have to be the same for every local subsystem included in the t s fuzzy model these conditions are difficult to satisfy in general in this paper those conditions are relaxed by solving the fuzzy regulation problem directly on the overall t s fuzzy model instead of constructing the fuzzy regulator on the basis of linear local controllers by considering the fuzzy model as a special class of linear time varying systems existence conditions are rigorously derived these new conditions which can be solved by means of any mathematical software depend on the solution of a set of symbolic simultaneous linear equations depending on the membership values of the plant and or the exosystem two examples are given to illustrate the construction of the proposed regulator and to validate the improvement that is achieved with the proposed approach

stability analysis and region of attraction estimation using piecewise polynomial lyapunov functions polynomial fuzzy model approach this paper proposes sum of squares sos methodologies for stability analysis and region of attraction roa estimation for nonlinear systems represented by polynomial fuzzy models via piecewise polynomial lyapunov functions pplfs at first two sos based global stability criteria are proposed by applying maximum type and minimum type pplfs respectively it is known that less conservative results can be obtained by reducing global stability to local stability since it is usually the case for nonlinear systems that the stability cannot be reached globally therefore based on the two types of pplfs two local stability criteria are further proposed with the algorithms that enlarge the estimated roa as much as possible the constraints for checking global and local stability and enlarging the estimated roa are represented in terms of bilinear sos problems hence the path following method is applied to solve the bilinear sos problems in the proposed methodologies finally some examples are provided to illustrate the utility of the proposed methodologies

fuzzy numbers from raw discrete data using linear regression this paper focuses on modelling fuzzy numbers with meaningful membership functions more precisely it proposes a method to construct trapezoidal fuzzy number approximations from raw discrete data in many applications input information is numerical and therefore particular fuzzy sets such as fuzzy numbers hold great interest and relevance in managing data imprecision and vagueness the proposed technique provides an efficient way to obtain trapezoidal numbers using linear regression the technique is simple fast and effective preliminary tests are performed using different types of input data a gaussian function a sigmoidal function three datasets of synthetic discrete data and an histogram obtained from a colour satellite image

a novel approach to output feedback control of fuzzy stochastic systems this paper investigates the problem of hankel norm output feedback controller design for a class of t s fuzzy stochastic systems the full order output feedback controller design technique with the hankel norm performance is proposed by the fuzzy basis dependent lyapunov function approach and the conversion on the hankel norm controller parameters sufficient conditions are established to design the controllers such that the resulting closed loop system is stochastically stable and satisfies a prescribed performance the desired output feedback controller can be obtained by solving a convex optimization problem which can be efficiently solved by standard numerical algorithms finally a henon map system is used to illustrate the effectiveness of the proposed techniques

asymptotically necessary and sufficient conditions for stability and performance in fuzzy control applications of polya s theorem stability and performance requirements in fuzzy control of takagi sugeno systems are usually stated as fuzzy summations i e sums of terms related to lyapunov functions which are weighted by membership functions or products of them this paper presents an application to fuzzy control of polya s theorems on positive forms on the standard simplex the result provides a set of progressively less conservative sufficient conditions for proving positivity of fuzzy summations such conditions are less and less conservative as a complexity parameter n increases particular cases of such conditions are those in c h fang y s liu s w kau l hong c h lee a new lmi based approach to relaxed quadratic stabilization of t s fuzzy control systems ieee trans fuzzy systems 14 2006 286 397 x liu q zhang new approaches to h controller designs based on fuzzy observers for t s fuzzy systems via lmi automatica 39 9 2003 1571 1582 with n 2 and 3 respectively the proposed conditions are asymptotically exact i e necessary and sufficient when n tends to infinity or equivalently when a tolerance parameter tends to zero

membership function dependent stability analysis of fuzzy model based control systems using fuzzy lyapunov functions this paper investigates the stability of fuzzy model based fmb control system formed by a t s fuzzy model and a fuzzy controller connected in a close loop based on a fuzzy lyapunov function a general fmb control system that the t s fuzzy model and fuzzy controller not sharing the same premise membership functions and or the same number of fuzzy rules is considered a membership function dependent stability analysis approach is proposed to consider the membership functions of both the t s fuzzy model and fuzzy controller in the stability analysis and incorporate them in the stability conditions in the form of linear matrix inequalities as the stability conditions are membership function dependent they are dedicated to the fmb control system with the specified membership functions under consideration it is thus the membership function dependent stability conditions are more relaxed compared to the existing membership function independent stability conditions a fuzzy lyapunov function is a weighted sum of quadratic functions which are required to be positive definite in most of the existing work in this paper this criterion is not required and thus the stability conditions can be further relaxed some simulation examples are given to demonstrate the merits of the proposed approach

static output feedback stabilization of positive polynomial fuzzy systems this paper deals with the static output feedback stabilization of positive polynomial fuzzy model based control systems the positive polynomial fuzzy model does not need to share the same premise membership functions with the static output feedback polynomial fuzzy controller unlike the state feedback control case the static output feedback control usually leads to nonconvex stability conditions to circumvent the problem an approach is employed to transform the nonconvex stability conditions into convex ones by introducing a nonsingular transformation matrix initially the conditions guaranteeing the resultant closed loop systems to be positive and asymptotically stable are obtained moreover the divisional approximated membership functions that carry the local information of the membership functions are employed to facilitate the stability analysis and controller synthesis the relaxed stability conditions in terms of sum of squares are obtained based on lyapunov stability theory finally a simulation example is given to testify the validity of the analysis result

further studies on relaxed stabilization conditions for discrete time two dimension takagi sugeno fuzzy systems this paper investigates the stabilization problem of discrete time two dimension 2 d takagi sugeno t s fuzzy systems based on a novel non parallel distributed compensation non pdc control scheme combined with a new non quadratic lyapunov function less conservative stabilization conditions are developed the proposed non quadratic lyapunov function is homogeneous polynomially parameter dependent on membership functions as the degree of the lyapunov function increases the conservatism of the obtained stabilization conditions is gradually reduced by exploiting the algebraic property of membership functions the stabilization conditions approach to exactness in the sense of convergence compared with the existing methods no slack variables are introduced in control synthesis and hence the same or less conservative results can be obtained with a lower computational cost a numerical example is given to illustrate the effectiveness of the proposed method

a new sum of squares design framework for robust control of polynomial fuzzy systems with uncertainties this paper presents a new sum of squares sos for brevity design framework for robust control of polynomial fuzzy systems with uncertainties two kinds of robust stabilization conditions are derived in terms of sos one is global sos robust stabilization conditions that guarantee the global and asymptotical stability of polynomial fuzzy control systems the other is semiglobal sos robust stabilization conditions the latter is available for very complicated systems that are difficult to guarantee the global and asymptotical stability of polynomial fuzzy control systems the main feature of all the sos robust stabilization conditions derived in this paper are to be expressed as nonconvex formulations with respect to polynomial lyapunov function parameters and polynomial feedback gains since a typical transformation from nonconvex sos design conditions to convex sos design conditions often results in some conservative issues the new design framework presented in this paper gives key ideas to avoid the conservative issues the first key idea is that we directly solve nonconvex sos design conditions without applying the typical transformation the second key idea is that we bring a so called copositivity concept these ideas provide some advantages in addition to relaxations to solve our sos robust stabilization conditions efficiently we introduce a gradient algorithm formulated as a minimizing optimization problem of the upper bound of the time derivative of an sos polynomial that can be regarded as a candidate of polynomial lyapunov functions three design examples are provided to illustrate the validity and applicability of the proposed design framework the examples demonstrate advantages of our new sos design framework for the existing linear matrix inequality approaches and the existing convex sos approach

design of polynomial fuzzy observer controller with membership functions using unmeasurable premise variables for nonlinear systems in this paper the stability of polynomial fuzzy model based pfmb observer control system is investigated via lyapunov stability theory the polynomial fuzzy observer with unmeasurable premise variables is designed to estimate the system states then the estimated system states are used for the state feedback control of nonlinear systems although the consideration of the polynomial fuzzy model and unmeasurable premise variables enhances the applicability of the fuzzy model based fmb control strategy it leads to non convex stability conditions therefore the refined completing square approach is proposed to derive convex stability conditions in the form of sum of squares sos with less manually designed parameters in addition the membership functions of the polynomial observer controller are optimized by the improved gradient descent method which outperforms the widely applied parallel distributed compensation pdc approach according to a general performance index simulation examples are provided to verify the proposed design and optimization scheme

polynomial fuzzy model based control systems stability analysis via approximated membership functions considering sector nonlinearity of control input this paper presents the stability analysis of polynomial fuzzy model based pfmb control systems in which both the polynomial fuzzy model and the polynomial fuzzy controller are allowed to have their own set of premise membership functions in order to address the input nonlinearity the control signal is considered to be bounded by a sector with nonlinear bounds these nonlinear lower and upper bounds of the sector are constructed by combining local bounds using fuzzy blending such that local information of input nonlinearity can be taken into account with the consideration of imperfectly matched membership functions and input nonlinearity the applicability of the pfmb control scheme can be further enhanced to facilitate the stability analysis a general form of approximated membership functions representing the original ones is introduced as a result approximated membership functions can be brought into the stability analysis leading to relaxed stability conditions the sum of squares approach is employed to obtain the stability conditions based on lyapunov stability theory simulation examples are presented to demonstrate the feasibility of the proposed method

membership dependent stability conditions for type 1 and interval type 2 t s fuzzy systems abstract this paper presents an idea to simplify and relax the stability conditions of takagi sugeno t s fuzzy systems based on the membership function extrema 1 by considering the distribution of membership functions in a unified membership space a graphical approach is provided to analyze the conservativeness of membership dependent stability conditions membership function extrema are used to construct a simple and tighter convex polyhedron that encloses the membership trajectory and produces less conservative linear matrix inequality lmi conditions the cases of both type 1 and interval type 2 t s fuzzy systems are considered and comparison with existing methods is made in the proposed membership vector framework

the expected value of a fuzzy number abstract the notions of expected interval and expected value of fuzzy number are introduced the expected interval is defined as the expected value of an interval random set generated by the fuzzy number and the expected value of this number is defined as the centre of the expected interval the linearity of the expected interval is studied

hankel norm model reduction of discrete time interval type 2 t s fuzzy systems with state delay this article focuses on the model reduction problem of discrete time time delay interval type 2 takagi sugeno t s fuzzy systems compared with the type 1 t s fuzzy system the interval type 2 t s fuzzy system has more advantages in expressing nonlinearity and capturing uncertainties in addition in order to simplify the analysis process complex high order systems can be approximated as low order systems which is called model reduction in previous studies there are few researches on model reduction of the interval type 2 t s fuzzy system with time delay hankel norm is adopted to limit the error after model reduction based on jensen s inequality a linear matrix inequality lmi condition for the hankel norm performance of the error system is obtained a membership function dependent method based on piecewise linear membership functions is utilized to deal with mismatched membership functions where information of membership functions will be used for relaxing analysis results next by a convex linearization design the model reduction problem is formulated as a convex lmi feasibility optimization condition numerical examples are given to verify the validity of the analysis

a motion compression reconstruction method based on max t norm composite fuzzy relational equations a motion compression reconstruction method based on max t norm composite fuzzy relational equations mcf is proposed where into intra pictures i pictures and predictive pictures p pictures of the original motion are compressed by uniform and non uniform coders respectively the non uniform coders of the proposed method can preserve edge information of p pictures on the compressed image to perform an effective compression reconstruction of the p pictures a design method of non uniform coders is proposed based on an overlap level of fuzzy sets and a fuzzy equalization an experiment using 10 p pictures confirms that the root mean square error of the reconstructed images obtained by the proposed non uniform coders is decreased to 89 4 of that one of the uniform coders under the condition that compression rate the ratio between the file size of compressed image and original one is 0 0057 two test motions tennis and woman 100 frames are compressed and reconstructed by the proposed mcf

a sum of squares approach to modeling and control of nonlinear dynamical systems with polynomial fuzzy systems this paper presents a sum of squares sos approach for modeling and control of nonlinear dynamical systems using polynomial fuzzy systems the proposed sos based framework provides a number of innovations and improvements over the existing linear matrix inequality lmi based approaches to takagi sugeno t s fuzzy modeling and control first we propose a polynomial fuzzy modeling and control framework that is more general and effective than the well known t s fuzzy modeling and control secondly we obtain stability and stabilizability conditions of the polynomial fuzzy systems based on polynomial lyapunov functions that contain quadratic lyapunov functions as a special case hence the stability and stabilizability conditions presented in this paper are more general and relaxed than those of the existing lmi based approaches to t s fuzzy modeling and control moreover the derived stability and stabilizability conditions are represented in terms of sos and can be numerically partially symbolically solved via the recently developed sostools to illustrate the validity and applicability of the proposed approach a number of analysis and design examples are provided the first example shows that the sos approach renders more relaxed stability results than those of both the lmi based approaches and a polynomial system approach the second example presents an extensive application of the sos approach in comparison with a piecewise lyapunov function approach the last example is a design exercise that demonstrates the viability of the sos based approach to synthesizing a stabilizing controller

a polynomial membership function approach for stability analysis of fuzzy systems for the stability analysis of a polynomial fuzzy system a new polynomial membership function approach is proposed to reduce the conservatism based on a state feedback closed loop system a polynomials fitting method is utilized and an improved membership functions transformation technique are proposed to approximate the membership functions of fuzzy system then the membership function dependent polynomials form stability conditions are derived the obtained polynomial membership functions and approximation errors will be involved in the stability analysis process based on the sum of squares optimization technique the polynomials conditions can be directly solved finally by several numerical and practical examples the conservatism reduction effects are shown by comparisons with existing methods