free lie algebras and formal power series let g be a group and k a field if v is a graded kg module of the form v v1plus sign in circlev2plus sign in circlecdots three dots centered where each vn is finite dimensional then the free lie algebra l v acquires the structure of a graded kg module l v l1 v plus sign in circlel2 v plus sign in circlecdots three dots centered the isomorphism types of v and l v may be described by the power series ngt or equal slanted1 vn tn and ngt or equal slanted1 ln v tn with coefficients from the green ring the main object of study is the function on power series which maps vn tn to ln v tn for every graded kg module v closed formulae are given in certain cases and these are closely related to character formulae of brandt and others

control design for fuzzy systems based on relaxed nonquadratic stability and h_ infty performance conditions in this paper new approaches to hinfin controller design for a class of discrete time nonlinear fuzzy systems are proposed based on a relaxed approach in which basis dependent lyapunov functions are used first two relaxed conditions of nonquadratic stability with h infin norm bound are presented for this class of systems the two relaxed conditions are shown to be useful in designing fuzzy control systems by introducing some additional instrumental matrix variables the two relaxed conditions are used to develop hinfin controllers in the control design the first relaxed condition has fewer inequality constraints but only admits a common additional matrix variable while the second one can admit multiple additional matrix variables finally two examples are given to demonstrate the applicability of the proposed approach

modular lie representations of groups of prime order let k be a field of prime characteristic p and let g be a group of order p for any finite dimensional kg module v and any positive integer n let ln v denote the nth homogeneous component of the free lie k algebra generated by a basis of v then ln v can be considered as a kg module called the nth lie power of v the main result of the paper is a formula which describes the module structure of ln v up to isomorphism

lie powers of modules for groups of prime order let l v be the free lie algebra on a finite dimensional vector space v over a field k with homogeneous components l n v for n geq 1 if g is a group and v is a kg module the action of g extends naturally to l v and the l n v become finite dimensional kg modules called the lie powers of v in the decomposition problem the aim is to identify the isomorphism types of indecomposable kg modules with their multiplicities in unrefinable direct decompositions of the lie powers this paper is concerned with the case where g has prime order p and k has characteristic p as is well known there are p indecomposables denoted here by j 1 dots j p where j r has dimension r a theory is developed which provides information about the overall module structure of l v and gives a recursive method for finding the multiplicities of j 1 dots j p in the lie powers l n v for example the theory yields decompositions of l v as a direct sum of modules isomorphic either to j 1 or to an infinite sum of the form j r oplus j p 1 oplus j p 1 oplus ldots with r geq 2 closed formulae are obtained for the multiplicities of j 1 dots j p in l n j p and l n j p 1 for r p 1 the indecomposables which occur with non zero multiplicity in l n j r are identified for all sufficiently large n

synchronization of chaotic systems from a fuzzy regulation approach in this paper some results on fuzzy regulation and fuzzy modeling are presented for synchronization of chaotic systems described by takagi sugeno ts fuzzy models using linear local controllers it is shown that the synchronization error is bounded if the local controller can be appropriately designed and that such an error is independent of initial conditions this feature allows synchronizing not only similar chaotic systems but under certain conditions different chaotic systems can be synchronized as well in other words this approach can be used to obtain either complete or generalized synchronization several simulations are carried out to illustrate how the problem can be solved in a practical way by using the linear matrix inequalities lmi technique

lmi based stability analysis for fuzzy model based control systems using artificial t s fuzzy model this paper investigates the stability of fuzzy model based fmb control systems an alternative stability analysis approach using an artificial fuzzy system based on the lyapunov stability theory is proposed to facilitate the stability analysis the continuous membership functions of the takagi sugeno t s fuzzy model are represented by the staircase ones with the nice property of the staircase membership functions it turns the set of infinite number of linear matrix inequality lmi based stability conditions into a finite one furthermore the staircase membership functions carrying system information can be brought to the stability conditions to relax the stability conditions the stability of the original fmb control systems is guaranteed by the satisfaction of the lmi based stability conditions the proposed stability analysis is applied to the fmb control systems of which the t s fuzzy model and fuzzy controller do not share the same premise membership functions and thus is able to enhance the design flexibility of the fuzzy controller a simulation example is given to illustrate the merits of the proposed approach

global solution for a non local eikonal equation modelling dislocation dynamics abstract in this paper we study a non local eikonal equation that arises from the theory of dislocation dynamics for this equation the global existence and uniqueness are available only for lipschitz continuous viscosity solutions in some particular cases based on a new gradient entropy estimate we prove the global existence of a continuous viscosity solution of this equation considering non signed velocity

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

fuzzy h_ infty filter design for a class of nonlinear discrete time systems with multiple time delays this paper studies the fuzzy hinfin filter design problem for signal estimation of nonlinear discrete time systems with multiple time delays and unknown bounded disturbances first the takagi sugeno t s fuzzy model is used to represent the state space model of nonlinear discrete time systems with time delays next we design a stable fuzzy hinfin filter based on the t s fuzzy model which guarantees asymptotic stability and a prescribed hinfin index for the filtering error system irrespective of the time delays and uncertain disturbances a sufficient condition for the existence of such a filter is established by using the linear matrix inequality lmi approach the proposed lmi problem can be efficiently solved with global convergence guarantee using convex optimization techniques such as the interior point algorithm simulation examples are provided to illustrate the design procedure of the present method

uniqueness results for nonlocal hamilton jacobi equations we are interested in nonlocal eikonal equations describing the evolution of interfaces moving with a nonlocal non monotone velocity for these equations only the existence of global in time weak solutions is available in some particular cases in this paper we propose a new approach for proving uniqueness of the solution when the front is expanding this approach simpli fies and extends existing results for dislocation dynamics it also provides the first uniqueness result for a fitzhugh nagumo system the key ingredients are some new perimeter estimates for the evolving fronts as well as some uniform interior cone property for these fronts

h filtering of discrete time fuzzy systems via basis dependent lyapunov function approach this paper is concerned with the h filtering problem for a class of discrete time fuzzy systems attention is focused on the design of a stable filter guaranteeing a prescribed noise attenuation level in the h sense by using basis dependent lyapunov functions sufficient conditions for the solvability of this problem are obtained it has been shown that the h filtering problem can be solved as a linear matrix inequality lmi optimization problem two examples are provided to demonstrate the applicability of the proposed approach

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

delayed impulsive control of takagi sugeno fuzzy delay systems in this paper the problem of exponential stability for takagi sugeno t s fuzzy delay systems with delayed impulses is addressed by means of the time dependent lyapunov function method combined with razumikhin technique a sufficient condition expressed in terms of linear matrix inequalities lmis is obtained for exponential stability of t s fuzzy delay systems with delayed impulses the derived stability condition depends both on the lower bound and the upper bound of impulsive intervals which is robust with respect to small impulse input delays by solving a set of lmis an impulsive state feedback controller can be easily constructed two numerical examples are discussed to illustrate the effectiveness of the theoretical findings

optifel a convergent heterogeneous particle swarm optimization algorithm for takagi sugeno fuzzy modeling data driven design of accurate and reliable takagi sugeno t s fuzzy systems has attracted a lot of attention where the model structures and parameters are important and often solved in an optimization framework the particle swarm optimization pso algorithm is widely applied in the field however the classical pso suffers from premature convergence and it is trapped easily into local optima which will significantly affect the model accuracy to overcome these drawbacks we have developed a new t s fuzzy system parameters searching strategy called optifel with a heterogeneous multiswarm pso mspso to enhance the searching performance mspso groups the whole population into multiple cooperative subswarms which perform different search behaviors for the potential solutions we have found that the multiple subswarms strategy proposed in this paper is greatly helpful for finding the optimal parameters suitable for the subspaces of the t s fuzzy model our theoretical proof has also demonstrated that the cooperation among the subswarms can maintain a balance between exploration and exploitation to ensure the particles converge to stable points experimental results show that mspso performs significantly better than traditional pso algorithms on six benchmark functions with the improved mspso optifel can generate a good fuzzy system model with high accuracy and strong generalization ability

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

comparison principle for a generalized fast marching method in carlini falcone forcadel and monneau siam j numer anal 46 2008 pp 2920 2952 the authors have proposed a generalization of the classical fast marching method of sethian for the eikonal equation in the case where the normal velocity depends on space and time and can change sign the goal of this paper is to propose a modified version of the generalized fast marching method proposed in carlini falcone forcadel and monneau siam j numer anal 46 2008 pp 2920 2952 for which we state a general comparison principle we also prove the convergence of the new algorithm

fuzzy model based nonfragile control for nonlinear singularly perturbed systems with semi markov jump parameters this paper is concerned with the fuzzy model based nonfragile control problem for discrete time nonlinear singularly perturbed systems with stochastic jumping parameters the stochastic parameters are generated from the semi markov process the memory property of the transition probabilities among subsystems is fully considered in the investigated systems consequently the restriction that the transition probabilities are memoryless in widely used discrete time markov jump model can be removed based on the t s fuzzy model approach and semi markov kernel concept several criteria ensuring delta error mean square stability of the underlying closed loop system are established with the help of those criteria the designed procedures which could well deal with the fragility problem in the implementation of the proposed fuzzy model based controller are presented a technique is developed to estimate the permissible maximum value of singularly perturbed parameter for discrete time nonlinear semi markov jump singularly perturbed systems finally the validity of the established theoretical results is illustrated by a numerical example and a modified tunnel diode circuit model

output tracking control for fuzzy systems via output feedback design fuzzy observer based control design is proposed to deal with the output tracking problem for nonlinear systems for the purpose of tracking design the new concept of virtual desired variables and in turn the so called generalized kinematics are introduced to simplify the design procedure in light of this concept the design procedure is split into two steps i determine the virtual desired variables from the generalized kinematics and ii determine the control gains just like solving linear matrix inequalities for stabilization problem for immeasurable state variables output feedback design is proposed here we focus on a common feature held by many physical systems where their membership functions of fuzzy sets satisfy a lipschitz like property based on this setting control gains and observer gains can be designed separately moreover zero tracking error and estimation error are concluded three different types of systems including nonlinear mass spring systems dc dc converters and induction motors are considered to demonstrate the design procedure their satisfactory simulation results verify the proposed approach

dislocation dynamics short time existence and uniqueness of the solution we study a mathematical model describing dislocation dynamics in crystals we consider a single dislocation line moving in its slip plane the normal velocity is given by the peach koehler force created by the dislocation line itself the mathematical model is an eikonal equation with a velocity which is a non local quantity depending on the whole shape of the dislocation line we study the special case where the dislocation line is assumed to be a graph or a closed loop in the framework of discontinuous viscosity solutions for hamilton jacobi equations we prove the existence and uniqueness of a solution for small time we also give physical explanations and a formal derivation of the mathematical model finally we present numerical results based on a level sets formulation of the problem these results illustrate in particular the fact that there is no general inclusion principle for this model

fuzzy approximation based adaptive control of nonlinear delayed systems with unknown dead zone this paper focuses on the problem of adaptive fuzzy control for a class of nonlinear systems without a strict feedback form a new adaptive fuzzy control scheme is proposed for nonlinear delayed systems with unknown dead zone nonlinearity control design is achieved by introducing a variable separation approach and employing the unique structural property of fuzzy logic systems the proposed adaptive fuzzy controller guarantees that all the closed loop signals are semiglobally uniformly ultimately bounded and the tracking error converges to a small neighborhood of the origin two simulation examples illustrate the effectiveness of the proposed approach

approaches to quadratic stability conditions and h sub spl infin control designs for t s fuzzy systems in this paper the problems of quadratic stability conditions and h sub spl infin control designs for takagi sugeno t s fuzzy systems have been studied first a new quadratic stability condition which is more simple than that in a previous paper has been proposed second two new sufficient conditions in the terms of linear matrix inequalities lmis which guarantee the existence of the state feedback h sub spl infin control for the t s fuzzy systems have been proposed the conditions are not only simple but also consider the interactions among the fuzzy subsystems finally based on the lmis the h sub spl infin controller designing methods for the t s fuzzy systems have been given

on energy to peak filtering for nonuniformly sampled nonlinear systems a markovian jump system approach this paper focuses on the filter design for nonuniformly sampled nonlinear systems which can be approximated by takagi sugeno t s fuzzy systems the sampling periods of the measurements are time varying and the nonuniform observations of the outputs are modeled by a homogenous markov chain a mode dependent estimator with a fast sampling frequency is proposed such that the estimation can track the signal to be estimated with the nonuniformly sampled outputs the nonlinear systems are discretized with the fast sampling period by using an augmentation technique the corresponding stochastic estimation error system is obtained by studying the stochastic stability and the energy to peak performance of the estimation error system we derive the linear matrix inequality based sufficient conditions the parameters of the mode dependent estimator can be calculated by using the proposed iterative algorithm two examples are used to demonstrate the design procedure and the efficacy of the proposed design method

stability analysis and design of fuzzy control systems abstract the stability analysis and the design technique of fuzzy control systems using fuzzy block diagrams are discussed first we show the concept of fuzzy blocks and consider the connection problems of fuzzy blocks diagrams we derive some theorems and corollaries with respect to two basic types of connections of fuzzy blocks in order to preserve some properties in a connection of fuzzy blocks continuous piecewise polynomial membership functions are defined secondly a sufficient condition which guarantees the stability of fuzzy systems is obtained in terms of lyapunov s direct method we give an important fact based on this condition thirdly we propose a new design technique of a fuzzy controller the fuzzy block diagrams and the stability analysis are applied to the design problems of a model based fuzzy controller

modular lie powers and the solomon descent algebra let v be an r dimensional vector space over an infinite field f of prime characteristic p and let l n v denote the nth homogeneous component of the free lie algebra on v we study the structure of l n v as a module for the general linear group gl r f when n pk and k is not divisible by p and where r n our main result is an explicit 1 1 correspondence multiplicity preserving between the indecomposable direct summands of l k v and the indecomposable direct summands of l n v which are not isomorphic to direct summands of v n our approach uses idempotents of the solomon descent algebras and in addition a correspondence theorem for permutation modules of symmetric groups

generalized nonquadratic stability of continuous time takagi sugeno models this paper provides generalized nonquadratic stability conditions for continuous time nonlinear models in the takagi sugeno ts form obtained by sector nonlinearity approach should global quadratic stability fail for a given nonlinear model the proposed approach allows the obtaining of progressively better estimations of the stability domain via local asymptotic conditions in the form of linear matrix inequalities lmis which are efficiently solved by convex optimization techniques illustrative examples are presented to emphasize the broadening capabilities of the new stability analysis

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

further theoretical justification of the k samples variation approach for discrete time takagi sugeno fuzzy systems the recently developed k samples variation approach is known as a powerful way to reduce the conservativeness of existing stability and stabilization conditions for discrete time takagi sugeno t s fuzzy systems in this approach the lyapunov functions under consideration are not necessarily decreasing at every sample but are allowed to decrease every k samples which is evidently less restrictive than classical approaches consequently less conservative linear matrix inequality lmi conditions were derived in addition it was proved that for two positive integers k1 and k2 if the condition for k k1 is fulfilled then those corresponding to k k2 are also satisfied when k2 is the divisor of k1 in this letter we prove that if the condition for k k2 admits a solution then those corresponding to any k k2 are also solvable

robust h control for discrete time fuzzy systems via basis dependent lyapunov functions this paper deals with the robust h control problem for a class of discrete time fuzzy systems with uncertainty the uncertainty is assumed to be of structured linear fractional form by using basis dependent lyapunov function an h control design approach is developed the control design approach is facilitated by introducing some additional instrumental matrix variables these additional matrix variables decouple the lyapunov and the system matrices which makes the control design feasible the proposed approach leads to some sufficient results in the form of strict linear matrix inequalities lmis it is expected that the basis dependent results are less conservative than the basis independent ones due to the introduction of basis dependent lyapunov function finally numerical examples including the discrete chaotic lorenz system are also given to demonstrate the applicability of the proposed approach

a generalized fast marching method for dislocation dynamics in this paper we consider a generalized fast marching method gfmm as a numerical method to compute dislocation dynamics the dynamics of a dislocation hypersurface in mathbb r n with n 2 for physical applications is given by its normal velocity which is a nonlocal function of the whole shape of the hypersurface itself for this dynamics we show a convergence result of the gfmm as the mesh size goes to zero we also provide some numerical simulations in dimension n 2

a multiple lyapunov function approach to stabilization of fuzzy control systems this paper addresses stability analysis and stabilization for takagi sugeno fuzzy systems via a so called fuzzy lyapunov function which is a multiple lyapunov function the fuzzy lyapunov function is defined by fuzzily blending quadratic lyapunov functions based on the fuzzy lyapunov function approach we give stability conditions for open loop fuzzy systems and stabilization conditions for closed loop fuzzy systems to take full advantage of a fuzzy lyapunov function we propose a new parallel distributed compensation pdc scheme that feedbacks the time derivatives of premise membership functions the new pdc contains the ordinary pdc as a special case a design example illustrates the utility of the fuzzy lyapunov function approach and the new pdc stabilization method

exit time problems in optimal control and vanishing viscosity method the authors study the connections between deterministic exit time control problems and possibly discontinuous viscosity solutions of a first order hamilton jacobi hj equation up to the boundary this equation admits a maximum and a minimum solution that are the value functions associated to stopping time problems on the boundary when these solutions are equal they can be obtained through the vanishing viscosity method finally when the hj equation has a continuous solution it is proved to be the value function for the first exit time of the domain it is also the vanishing viscosity limit arising in particular in some large deviations problems

stability analysis of recurrent fuzzy systems a hybrid system and sos approach this paper presents a new approach to the stability analysis of recurrent fuzzy systems rfss rfss are rule based dynamic fuzzy systems that are usually obtained from heuristic or data driven modeling in the presented approach the stability of both continuous time and discrete time rfs can be analyzed in a common framework it is based on the representation of an rfs as a hybrid polynomial system due to the polynomial structure the recently developed method of sum of squares sos decomposition along with semidefinite programming can be employed to derive sufficient conditions for the stability of equilibrium points we consider stability analysis for known equilibrium points as well as unknown equilibrium points the latter results in a two step procedure in the first step a polynomial is constructed in the second verification step this polynomial is proven to be a lyapunov function for the rfs the applicability of the approach is shown by two systems formulated as a rule based rfs

adaptive fuzzy tracking control based barrier functions of uncertain nonlinear mimo systems with full state constraints and applications to chemical process an adaptive control approach based on the fuzzy systems for a class of uncertain nonlinear multi input multi output mimo systems is presented in this paper this class of systems is in the nested multiple coupling structure and their states are constrained in the corresponding compact sets the properties of the system structure are inevitable to bring about a complicated design and a difficult task the fuzzy logic systems are employed to approximate the unknown functions of systems and the decoupling backstepping way is proposed to design the stability controller and adaptation laws barrier lyapunov functions blfs are constructed in the backstepping design to guarantee that the constraint bounds are not violated based on lyapunov analysis in barrier form we can prove the stability of the closed loop system two simulation examples are viewed to verify the feasibility of the approach

global existence for a system of non linear and non local transport equations describing the dynamics of dislocation densities in this paper we study the global in time existence problem for the groma balogh model describing the dynamics of dislocation densities this model is a two dimensional model where the dislocation densities satisfy a system of transport equations such that the velocity vector field is the shear stress in the material solving the equations of elasticity this shear stress can be expressed as some riesz transform of the dislocation densities the main tool in the proof of this result is the existence of an entropy for this system

t s fuzzy model identification based on a novel hyperplane shaped membership function the fuzzy membership function is a crucial factor that may affect the model structure and modeling accuracy although hyperplane shaped clustering hpsc has been widely used in takagi sugeno t s fuzzy model identification there is no well designed membership function in these approaches the commonly used bell shaped gaussian function is actually inappropriate to employ together with hpsc in fuzzy modeling in this paper a hyperplane shaped fuzzy membership function is designed to match hpsc for t s fuzzy model identification for the first time experimental results on several benchmark problems indicate that modeling accuracies have been greatly improved from the original modeling approach by replacing the proposed membership function

relaxed stability conditions for continuous time t s fuzzy control systems via augmented multi indexed matrix approach this paper is concerned with the problem of developing an advanced strategy to reduce the conservatism in stability analysis and control synthesis of continuous time takagi sugeno t s fuzzy systems a novel augmented multi indexed matrix approach is proposed to implement new right hand side slack variables technique for the homogenous polynomial setting combining with the finsler lemma with homogenous matrix lagrange multipliers convergent linear matrix inequality lmi relaxations for stability analysis are proposed by using the generalization of the polya theorem for the case of positive polynomials with matrix valued coefficients a new type of state feedback controller namely the homogeneous polynomially nonquadratic control law hpnqcl is developed to conceive less conservative stabilization conditions the obtained stability and stabilization conditions are further relaxed by using the proposed right hand side slack variables technique moreover the advantages over the existing control schemes are certificated in theory three numerical examples are also provided to illustrate the effectiveness of the proposed methods

comments on exact output regulation for nonlinear systems described by takagi sugeno fuzzy models this letter considers the work entitled exact output regulation for nonlinear systems described by takagi sugeno fuzzy models published previously in the ieee transactions on fuzzy s ystems where the authors try to provide a fuzzy approach to the well known problem of exact output regulation via a dynamic implementation of the nonlinear mapping between the state space and the steady state zero error manifold two problems invalidate this methodology 1 the results do not depend on the fuzzy structure as claimed by the authors and 2 the results have been obtained without properly taking into account the time derivative of the aforementioned nonlinear mapping example b in the above paper is revised and corrected to illustrate these issues

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

sos based stability analysis of polynomial fuzzy model based control systems via polynomial membership functions this paper presents stability analysis of polynomial fuzzy model based fmb control systems using the sum of squares sos approach recently stability analysis of the polynomial fuzzy control systems which is a generalized form of the well known takagi sugeno t s fmb control systems has been reported in the form of sos based stability conditions lack of information on the relations between membership functions and premise variables in the existing stability analysis approaches causes conservatism of their results in this paper to derive relaxed stability conditions for polynomial fmb control systems membership functions which are approximated with polynomials and carrying relations between membership functions and premise variables are brought into the stability analysis considering a polynomial fmb control system and based on the lyapunov stability theory stability conditions in the form of fuzzy summations are derived where each term contains product of membership functions of the polynomial fuzzy model and polynomial fuzzy controller each product term is approximated by a polynomial in order to obtain better approximation the operating domain of membership functions is partitioned to subregions then sos based stability conditions for all subregions are derived unlike some published stability analysis approaches the proposed one can be employed for stability analysis of polynomial fuzzy control systems under imperfect premise matching of which the fuzzy model and fuzzy controller do not share the same membership functions the solution of the sos based stability conditions can be found numerically using sostools which is a free third party matlab toolbox numerical examples are given to illustrate the effectiveness of the proposed stability conditions